TPTP Problem File: SLH0027^1.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_00408_014339__19279122_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1316 ( 424 unt; 224 typ; 0 def)
% Number of atoms : 3004 (1031 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9774 ( 253 ~; 66 |; 105 &;7955 @)
% ( 0 <=>;1395 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 7 avg)
% Number of types : 28 ( 27 usr)
% Number of type conns : 475 ( 475 >; 0 *; 0 +; 0 <<)
% Number of symbols : 200 ( 197 usr; 18 con; 0-4 aty)
% Number of variables : 3160 ( 134 ^;2971 !; 55 ?;3160 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:35:11.130
%------------------------------------------------------------------------------
% Could-be-implicit typings (27)
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Complex__Ocomplex_J_J,type,
list_P6605091754902497125omplex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
set_set_mat_complex: $tType ).
thf(ty_n_t__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
list_mat_complex: $tType ).
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__Set__Oset_It__Matrix__Ovec_It__Complex__Ocomplex_J_J,type,
set_vec_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
set_mat_complex: $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__Matrix__Omat_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
mat_set_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
set_set_complex: $tType ).
thf(ty_n_t__List__Olist_It__Matrix__Omat_It__Nat__Onat_J_J,type,
list_mat_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__Matrix__Omat_It__Set__Oset_It__Nat__Onat_J_J,type,
mat_set_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__List__Olist_It__Matrix__Omat_Itf__a_J_J,type,
list_mat_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__Complex__Ocomplex_J,type,
mat_complex: $tType ).
thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
list_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Matrix__Omat_It__Nat__Onat_J,type,
mat_nat: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Matrix__Omat_Itf__a_J,type,
mat_a: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (197)
thf(sy_c_Column__Operations_Oadd__col__sub__row_001t__Complex__Ocomplex,type,
column6029646570091773654omplex: complex > nat > nat > mat_complex > mat_complex ).
thf(sy_c_Column__Operations_Omat__multcol_001t__Complex__Ocomplex,type,
column4410001698458707789omplex: nat > complex > mat_complex > mat_complex ).
thf(sy_c_Column__Operations_Omat__swapcols_001t__Complex__Ocomplex,type,
column4357519492343924999omplex: nat > nat > mat_complex > mat_complex ).
thf(sy_c_Column__Operations_Omat__swapcols_001tf__a,type,
column2528828918332591333cols_a: nat > nat > mat_a > mat_a ).
thf(sy_c_Column__Operations_Omult__col__div__row_001t__Complex__Ocomplex,type,
column217142795681433722omplex: complex > nat > mat_complex > mat_complex ).
thf(sy_c_Column__Operations_Oswap__cols__rows_001t__Complex__Ocomplex,type,
column7161609239796038556omplex: nat > nat > mat_complex > mat_complex ).
thf(sy_c_Column__Operations_Oswap__cols__rows_001t__Nat__Onat,type,
column141131285749525182ws_nat: nat > nat > mat_nat > mat_nat ).
thf(sy_c_Column__Operations_Oswap__cols__rows_001tf__a,type,
column5129559316938501008rows_a: nat > nat > mat_a > mat_a ).
thf(sy_c_Complex__Matrix_Odensity__operator,type,
comple5220265106149225959erator: mat_complex > $o ).
thf(sy_c_Complex__Matrix_Ohermitian_001t__Complex__Ocomplex,type,
comple8306762464034002205omplex: mat_complex > $o ).
thf(sy_c_Complex__Matrix_Olowner__le,type,
complex_lowner_le: mat_complex > mat_complex > $o ).
thf(sy_c_Complex__Matrix_Opartial__density__operator,type,
comple1169154605998056944erator: mat_complex > $o ).
thf(sy_c_Complex__Matrix_Opositive,type,
complex_positive: mat_complex > $o ).
thf(sy_c_Complex__Matrix_Otrace_001t__Complex__Ocomplex,type,
comple3184165445352484367omplex: mat_complex > complex ).
thf(sy_c_Complex__Matrix_Otrace_001tf__a,type,
complex_trace_a: mat_a > a ).
thf(sy_c_Complex__Matrix_Ounitary_001t__Complex__Ocomplex,type,
comple6660659447773130958omplex: mat_complex > $o ).
thf(sy_c_Factorial__Ring_Ocomm__semiring__1__class_Oirreducible_001t__Complex__Ocomplex,type,
factor4870819777162427853omplex: complex > $o ).
thf(sy_c_Factorial__Ring_Ocomm__semiring__1__class_Oirreducible_001t__Nat__Onat,type,
factor4388943552880185071le_nat: nat > $o ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
invers8013647133539491842omplex: complex > complex ).
thf(sy_c_Gates_Omat__incr,type,
mat_incr: nat > mat_complex ).
thf(sy_c_Gauss__Jordan__Elimination_Ogauss__jordan__single_001t__Complex__Ocomplex,type,
gauss_4244865067341541924omplex: mat_complex > mat_complex ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__multrow__gen_001t__Complex__Ocomplex,type,
gauss_2324787009747932227omplex: ( complex > complex > complex ) > nat > complex > mat_complex > mat_complex ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__multrow__gen_001t__Nat__Onat,type,
gauss_2409696420326117733en_nat: ( nat > nat > nat ) > nat > nat > mat_nat > mat_nat ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__multrow__gen_001tf__a,type,
gauss_5154200947219177641_gen_a: ( a > a > a ) > nat > a > mat_a > mat_a ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__swaprows_001t__Complex__Ocomplex,type,
gauss_1020679828357514249omplex: nat > nat > mat_complex > mat_complex ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__swaprows_001tf__a,type,
gauss_2482569599970757219rows_a: nat > nat > mat_a > mat_a ).
thf(sy_c_Gauss__Jordan__Elimination_Omultrow__mat_001t__Complex__Ocomplex,type,
gauss_6868829418328711927omplex: nat > nat > complex > mat_complex ).
thf(sy_c_Gauss__Jordan__Elimination_Omultrow__mat_001t__Nat__Onat,type,
gauss_3195076542185637913at_nat: nat > nat > nat > mat_nat ).
thf(sy_c_Gauss__Jordan__Elimination_Opivot__fun_001t__Complex__Ocomplex,type,
gauss_2609248829700396350omplex: mat_complex > ( nat > nat ) > nat > $o ).
thf(sy_c_Gauss__Jordan__Elimination_Opivot__fun_001t__Nat__Onat,type,
gauss_8416567519840421984un_nat: mat_nat > ( nat > nat ) > nat > $o ).
thf(sy_c_Gauss__Jordan__Elimination_Orow__echelon__form_001t__Complex__Ocomplex,type,
gauss_194721375535881179omplex: mat_complex > $o ).
thf(sy_c_Gauss__Jordan__Elimination_Oswaprows__mat_001t__Complex__Ocomplex,type,
gauss_8970452565587180529omplex: nat > nat > nat > mat_complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Set__Oset_It__Complex__Ocomplex_J,type,
one_one_set_complex: set_complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Set__Oset_It__Nat__Onat_J,type,
one_one_set_nat: set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
plus_plus_complex: complex > complex > complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
plus_p8323303612493835998omplex: mat_complex > mat_complex > mat_complex ).
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__Complex__Ocomplex_J,type,
plus_p7052360327008956141omplex: set_complex > set_complex > set_complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
plus_p4229080058245121342omplex: set_mat_complex > set_mat_complex > set_mat_complex ).
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__Complex__Ocomplex,type,
times_times_complex: complex > complex > complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
times_8009071140041733218omplex: mat_complex > mat_complex > mat_complex ).
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__Complex__Ocomplex_J,type,
times_6048082448287401577omplex: set_complex > set_complex > set_complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
times_6731331324747250370omplex: set_mat_complex > set_mat_complex > set_mat_complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Matrix__Omat_It__Nat__Onat_J_J,type,
times_5500231875258083300at_nat: set_mat_nat > set_mat_nat > set_mat_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_001t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
times_6103784797850505759omplex: set_set_complex > set_set_complex > set_set_complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
times_3957003352596167970omplex: set_set_mat_complex > set_set_mat_complex > set_set_mat_complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Set__Oset_It__Matrix__Omat_Itf__a_J_J_J,type,
times_5016826689369604684_mat_a: set_set_mat_a > set_set_mat_a > set_set_mat_a ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
times_4850922872519784769et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_Itf__a_J,type,
times_times_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Otimes__class_Otimes_001tf__a,type,
times_times_a: a > a > a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
zero_zero_a: a ).
thf(sy_c_If_001t__Complex__Ocomplex,type,
if_complex: $o > complex > complex > complex ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Jordan__Normal__Form_Ojordan__matrix_001t__Complex__Ocomplex,type,
jordan5739059635872469039omplex: list_P6605091754902497125omplex > mat_complex ).
thf(sy_c_Jordan__Normal__Form__Existence_Odiff__ev_001t__Complex__Ocomplex,type,
jordan8650160714669549932omplex: mat_complex > nat > nat > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Oev__block_001t__Complex__Ocomplex,type,
jordan8042990603089931364omplex: nat > mat_complex > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Oev__blocks_001t__Complex__Ocomplex,type,
jordan4650062548456832493omplex: nat > mat_complex > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Oev__blocks__part_001t__Complex__Ocomplex,type,
jordan4637981584770492064omplex: nat > mat_complex > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Oinv__all_001t__Complex__Ocomplex,type,
jordan5244935068081719878omplex: nat > ( mat_complex > nat > nat > $o ) > mat_complex > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Oinv__all_H_001t__Complex__Ocomplex,type,
jordan5032732407113867375omplex: ( mat_complex > nat > nat > $o ) > mat_complex > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Oinv__upto_001t__Complex__Ocomplex,type,
jordan5475473882837061487omplex: nat > ( mat_complex > nat > nat > $o ) > mat_complex > nat > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Ojb_001t__Complex__Ocomplex,type,
jordan4971026570492200526omplex: mat_complex > nat > nat > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Ojnf__vector_001t__Complex__Ocomplex,type,
jordan387279176131498413omplex: mat_complex > list_P6605091754902497125omplex ).
thf(sy_c_Jordan__Normal__Form__Existence_Ojnf__vector__main_001t__Complex__Ocomplex,type,
jordan4459423482773701094omplex: nat > mat_complex > list_P6605091754902497125omplex ).
thf(sy_c_Jordan__Normal__Form__Existence_Opartition__ev__blocks_001t__Complex__Ocomplex,type,
jordan5009815537632354121omplex: mat_complex > list_mat_complex > list_mat_complex ).
thf(sy_c_Jordan__Normal__Form__Existence_Osame__diag_001t__Complex__Ocomplex,type,
jordan2620430285385836103omplex: nat > mat_complex > mat_complex > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Ostep__1_001t__Complex__Ocomplex,type,
jordan2017415923357163885omplex: mat_complex > mat_complex ).
thf(sy_c_Jordan__Normal__Form__Existence_Ostep__1__main_001t__Complex__Ocomplex,type,
jordan9130142659770429862omplex: nat > nat > nat > mat_complex > mat_complex ).
thf(sy_c_Jordan__Normal__Form__Existence_Ostep__2_001t__Complex__Ocomplex,type,
jordan7871273693253786478omplex: mat_complex > mat_complex ).
thf(sy_c_Jordan__Normal__Form__Existence_Ostep__2__main_001t__Complex__Ocomplex,type,
jordan6916311984355858983omplex: nat > nat > mat_complex > mat_complex ).
thf(sy_c_Jordan__Normal__Form__Existence_Ostep__3_001t__Complex__Ocomplex,type,
jordan4501759426295633263omplex: mat_complex > mat_complex ).
thf(sy_c_Jordan__Normal__Form__Existence_Ostep__3__main_001t__Complex__Ocomplex,type,
jordan4702481308941288104omplex: nat > nat > mat_complex > mat_complex ).
thf(sy_c_Jordan__Normal__Form__Existence_Ouppert_001t__Complex__Ocomplex,type,
jordan3528196489273997576omplex: mat_complex > nat > nat > $o ).
thf(sy_c_Linear__Algebra__Complements_Ocpx__sq__mat__axioms,type,
linear2040860143340867312axioms: nat > nat > $o ).
thf(sy_c_Linear__Algebra__Complements_Oprojector_001t__Complex__Ocomplex,type,
linear5633924348262549461omplex: mat_complex > $o ).
thf(sy_c_List_Olist_ONil_001t__Complex__Ocomplex,type,
nil_complex: list_complex ).
thf(sy_c_List_Olist_ONil_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
nil_mat_complex: list_mat_complex ).
thf(sy_c_List_Olist_ONil_001t__Matrix__Omat_Itf__a_J,type,
nil_mat_a: list_mat_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
set_complex2: list_complex > set_complex ).
thf(sy_c_List_Olist_Oset_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
set_mat_complex2: list_mat_complex > set_mat_complex ).
thf(sy_c_List_Olist_Oset_001t__Matrix__Omat_It__Nat__Onat_J,type,
set_mat_nat2: list_mat_nat > set_mat_nat ).
thf(sy_c_List_Olist_Oset_001t__Matrix__Omat_Itf__a_J,type,
set_mat_a2: list_mat_a > set_mat_a ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Complex__Ocomplex,type,
carrier_mat_complex: nat > nat > set_mat_complex ).
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_Odiag__block__mat_001t__Complex__Ocomplex,type,
diag_b9145358668110806138omplex: list_mat_complex > mat_complex ).
thf(sy_c_Matrix_Odiagonal__mat_001t__Complex__Ocomplex,type,
diagonal_mat_complex: mat_complex > $o ).
thf(sy_c_Matrix_Odiagonal__mat_001t__Nat__Onat,type,
diagonal_mat_nat: mat_nat > $o ).
thf(sy_c_Matrix_Odiagonal__mat_001tf__a,type,
diagonal_mat_a: mat_a > $o ).
thf(sy_c_Matrix_Odim__col_001t__Complex__Ocomplex,type,
dim_col_complex: mat_complex > nat ).
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__Complex__Ocomplex,type,
dim_row_complex: mat_complex > 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_Oinvertible__mat_001t__Complex__Ocomplex,type,
invert2568027935824841882omplex: mat_complex > $o ).
thf(sy_c_Matrix_Oinverts__mat_001t__Complex__Ocomplex,type,
inverts_mat_complex: mat_complex > mat_complex > $o ).
thf(sy_c_Matrix_Oinverts__mat_001t__Nat__Onat,type,
inverts_mat_nat: mat_nat > mat_nat > $o ).
thf(sy_c_Matrix_Omk__diagonal_001t__Complex__Ocomplex,type,
mk_diagonal_complex: list_complex > mat_complex ).
thf(sy_c_Matrix_Omk__diagonal_001t__Nat__Onat,type,
mk_diagonal_nat: list_nat > mat_nat ).
thf(sy_c_Matrix_Omk__diagonal_001tf__a,type,
mk_diagonal_a: list_a > mat_a ).
thf(sy_c_Matrix_Oone__mat_001t__Complex__Ocomplex,type,
one_mat_complex: nat > mat_complex ).
thf(sy_c_Matrix_Oone__mat_001t__Nat__Onat,type,
one_mat_nat: nat > mat_nat ).
thf(sy_c_Matrix_Osimilar__mat_001t__Complex__Ocomplex,type,
similar_mat_complex: mat_complex > mat_complex > $o ).
thf(sy_c_Matrix_Osimilar__mat_001t__Nat__Onat,type,
similar_mat_nat: mat_nat > mat_nat > $o ).
thf(sy_c_Matrix_Osimilar__mat__wit_001t__Complex__Ocomplex,type,
simila5774310414453981135omplex: mat_complex > mat_complex > mat_complex > mat_complex > $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_Osmult__mat_001t__Complex__Ocomplex,type,
smult_mat_complex: complex > mat_complex > mat_complex ).
thf(sy_c_Matrix_Osmult__mat_001t__Nat__Onat,type,
smult_mat_nat: nat > mat_nat > mat_nat ).
thf(sy_c_Matrix_Osmult__mat_001t__Set__Oset_It__Complex__Ocomplex_J,type,
smult_4557042052056852367omplex: set_complex > mat_set_complex > mat_set_complex ).
thf(sy_c_Matrix_Osmult__mat_001t__Set__Oset_It__Nat__Onat_J,type,
smult_mat_set_nat: set_nat > mat_set_nat > mat_set_nat ).
thf(sy_c_Matrix_Osmult__mat_001tf__a,type,
smult_mat_a: a > mat_a > mat_a ).
thf(sy_c_Matrix_Osquare__mat_001t__Complex__Ocomplex,type,
square_mat_complex: mat_complex > $o ).
thf(sy_c_Matrix_Osquare__mat_001tf__a,type,
square_mat_a: mat_a > $o ).
thf(sy_c_Matrix_Otranspose__mat_001t__Complex__Ocomplex,type,
transp3074176993011536131omplex: mat_complex > mat_complex ).
thf(sy_c_Matrix_Otranspose__mat_001tf__a,type,
transpose_mat_a: mat_a > mat_a ).
thf(sy_c_Matrix_Oupdate__mat_001t__Complex__Ocomplex,type,
update_mat_complex: mat_complex > product_prod_nat_nat > complex > mat_complex ).
thf(sy_c_Matrix_Oupdate__mat_001t__Nat__Onat,type,
update_mat_nat: mat_nat > product_prod_nat_nat > nat > mat_nat ).
thf(sy_c_Matrix_Oupdate__mat_001tf__a,type,
update_mat_a: mat_a > product_prod_nat_nat > a > mat_a ).
thf(sy_c_Matrix_Oupper__triangular_001t__Complex__Ocomplex,type,
upper_4850907204721561915omplex: mat_complex > $o ).
thf(sy_c_Matrix_Oupper__triangular_001t__Nat__Onat,type,
upper_triangular_nat: mat_nat > $o ).
thf(sy_c_Matrix_Oupper__triangular_001tf__a,type,
upper_triangular_a: mat_a > $o ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
size_s3451745648224563538omplex: list_complex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
size_s5969786470865220249omplex: list_mat_complex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Matrix__Omat_It__Nat__Onat_J_J,type,
size_s66138613738048955at_nat: list_mat_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Matrix__Omat_Itf__a_J_J,type,
size_size_list_mat_a: list_mat_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Complex__Ocomplex,type,
ord_less_complex: complex > complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Complex__Ocomplex,type,
ord_less_eq_complex: complex > complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_le211207098394363844omplex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
ord_le3632134057777142183omplex: set_mat_complex > set_mat_complex > $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_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
power_power_complex: complex > nat > complex ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Projective__Measurements_Odensity__collapse,type,
projec3470689467825365843llapse: mat_complex > mat_complex > mat_complex ).
thf(sy_c_Projective__Measurements_Ohermitian__decomp_001t__Complex__Ocomplex,type,
projec5943904436471448624omplex: mat_complex > mat_complex > mat_complex > $o ).
thf(sy_c_Projective__Measurements_Omax__mix__density,type,
projec8360710381328234318ensity: nat > mat_complex ).
thf(sy_c_Quantum_Ocpx__mat__cnj,type,
cpx_mat_cnj: mat_complex > mat_complex ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
dvd_dvd_complex: complex > complex > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Schur__Decomposition_Ocorthogonal__mat_001t__Complex__Ocomplex,type,
schur_549222400177443379omplex: mat_complex > $o ).
thf(sy_c_Schur__Decomposition_Omat__adjoint_001t__Complex__Ocomplex,type,
schur_5982229384592763574omplex: mat_complex > mat_complex ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
collect_mat_complex: ( mat_complex > $o ) > set_mat_complex ).
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__Algebras_Oelt__set__times_001t__Complex__Ocomplex,type,
set_el8005228190238886239omplex: complex > set_complex > set_complex ).
thf(sy_c_Set__Algebras_Oelt__set__times_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
set_el176066062795894710omplex: mat_complex > set_mat_complex > set_mat_complex ).
thf(sy_c_Set__Algebras_Oelt__set__times_001t__Matrix__Omat_It__Nat__Onat_J,type,
set_el1310594772197002200at_nat: mat_nat > set_mat_nat > set_mat_nat ).
thf(sy_c_Set__Algebras_Oelt__set__times_001t__Matrix__Omat_Itf__a_J,type,
set_el1062546952344711308_mat_a: mat_a > set_mat_a > set_mat_a ).
thf(sy_c_Set__Algebras_Oelt__set__times_001t__Nat__Onat,type,
set_el2933305810450955905es_nat: nat > set_nat > set_nat ).
thf(sy_c_Set__Algebras_Oelt__set__times_001t__Set__Oset_It__Complex__Ocomplex_J,type,
set_el158709831744343061omplex: set_complex > set_set_complex > set_set_complex ).
thf(sy_c_Set__Algebras_Oelt__set__times_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
set_el2325834261644174870omplex: set_mat_complex > set_set_mat_complex > set_set_mat_complex ).
thf(sy_c_Set__Algebras_Oelt__set__times_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
set_el7210227519270355394_mat_a: set_mat_a > set_set_mat_a > set_set_mat_a ).
thf(sy_c_Set__Algebras_Oelt__set__times_001t__Set__Oset_It__Nat__Onat_J,type,
set_el3528970498207131191et_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set__Algebras_Oelt__set__times_001tf__a,type,
set_elt_set_times_a: a > set_a > set_a ).
thf(sy_c_Spectral__Theory__Complements_Omat__conj_001t__Complex__Ocomplex,type,
spectr5699176650994449695omplex: mat_complex > mat_complex > mat_complex ).
thf(sy_c_Tensor_Otensor__mat,type,
tensor_mat: mat_complex > mat_complex > mat_complex ).
thf(sy_c_VS__Connect_Ovec__space_Ocol__space_001t__Complex__Ocomplex,type,
vS_vec1879987866596122552omplex: nat > mat_complex > set_vec_complex ).
thf(sy_c_VS__Connect_Ovec__space_Orow__space_001t__Complex__Ocomplex,type,
vS_vec3284807721666986142omplex: nat > mat_complex > set_vec_complex ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
member_mat_complex: mat_complex > set_mat_complex > $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__Complex__Ocomplex_J,type,
member_set_complex: set_complex > set_set_complex > $o ).
thf(sy_c_member_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
member3612512168372279472omplex: set_mat_complex > set_set_mat_complex > $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_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: mat_a ).
thf(sy_v_B,type,
b: mat_a ).
thf(sy_v_i,type,
i: nat ).
thf(sy_v_j,type,
j: nat ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1086)
thf(fact_0_assms_I5_J,axiom,
ord_less_nat @ j @ n ).
% assms(5)
thf(fact_1_assms_I4_J,axiom,
ord_less_nat @ i @ n ).
% assms(4)
thf(fact_2_assms_I1_J,axiom,
diagonal_mat_a @ a2 ).
% assms(1)
thf(fact_3_assms_I2_J,axiom,
member_mat_a @ a2 @ ( carrier_mat_a @ n @ n ) ).
% assms(2)
thf(fact_4_assms_I3_J,axiom,
member_mat_a @ b @ ( carrier_mat_a @ n @ n ) ).
% assms(3)
thf(fact_5_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_6_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_7_index__mult__mat_I2_J,axiom,
! [A: mat_complex,B: mat_complex] :
( ( dim_row_complex @ ( times_8009071140041733218omplex @ A @ B ) )
= ( dim_row_complex @ A ) ) ).
% index_mult_mat(2)
thf(fact_8_Groups_Omult__ac_I3_J,axiom,
! [B2: set_complex,A2: set_complex,C: set_complex] :
( ( times_6048082448287401577omplex @ B2 @ ( times_6048082448287401577omplex @ A2 @ C ) )
= ( times_6048082448287401577omplex @ A2 @ ( times_6048082448287401577omplex @ B2 @ C ) ) ) ).
% Groups.mult_ac(3)
thf(fact_9_Groups_Omult__ac_I3_J,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( times_times_set_nat @ B2 @ ( times_times_set_nat @ A2 @ C ) )
= ( times_times_set_nat @ A2 @ ( times_times_set_nat @ B2 @ C ) ) ) ).
% Groups.mult_ac(3)
thf(fact_10_Groups_Omult__ac_I3_J,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( times_times_nat @ B2 @ ( times_times_nat @ A2 @ C ) )
= ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% Groups.mult_ac(3)
thf(fact_11_Groups_Omult__ac_I3_J,axiom,
! [B2: complex,A2: complex,C: complex] :
( ( times_times_complex @ B2 @ ( times_times_complex @ A2 @ C ) )
= ( times_times_complex @ A2 @ ( times_times_complex @ B2 @ C ) ) ) ).
% Groups.mult_ac(3)
thf(fact_12_Groups_Omult__ac_I2_J,axiom,
( times_6048082448287401577omplex
= ( ^ [A3: set_complex,B3: set_complex] : ( times_6048082448287401577omplex @ B3 @ A3 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_13_Groups_Omult__ac_I2_J,axiom,
( times_times_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( times_times_set_nat @ B3 @ A3 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_14_Groups_Omult__ac_I2_J,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_15_Groups_Omult__ac_I2_J,axiom,
( times_times_complex
= ( ^ [A3: complex,B3: complex] : ( times_times_complex @ B3 @ A3 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_16_Groups_Omult__ac_I1_J,axiom,
! [A2: set_complex,B2: set_complex,C: set_complex] :
( ( times_6048082448287401577omplex @ ( times_6048082448287401577omplex @ A2 @ B2 ) @ C )
= ( times_6048082448287401577omplex @ A2 @ ( times_6048082448287401577omplex @ B2 @ C ) ) ) ).
% Groups.mult_ac(1)
thf(fact_17_Groups_Omult__ac_I1_J,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( times_times_set_nat @ ( times_times_set_nat @ A2 @ B2 ) @ C )
= ( times_times_set_nat @ A2 @ ( times_times_set_nat @ B2 @ C ) ) ) ).
% Groups.mult_ac(1)
thf(fact_18_Groups_Omult__ac_I1_J,axiom,
! [A2: a,B2: a,C: a] :
( ( times_times_a @ ( times_times_a @ A2 @ B2 ) @ C )
= ( times_times_a @ A2 @ ( times_times_a @ B2 @ C ) ) ) ).
% Groups.mult_ac(1)
thf(fact_19_Groups_Omult__ac_I1_J,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% Groups.mult_ac(1)
thf(fact_20_Groups_Omult__ac_I1_J,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( times_times_complex @ ( times_times_complex @ A2 @ B2 ) @ C )
= ( times_times_complex @ A2 @ ( times_times_complex @ B2 @ C ) ) ) ).
% Groups.mult_ac(1)
thf(fact_21_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: set_complex,B2: set_complex,C: set_complex] :
( ( times_6048082448287401577omplex @ ( times_6048082448287401577omplex @ A2 @ B2 ) @ C )
= ( times_6048082448287401577omplex @ A2 @ ( times_6048082448287401577omplex @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_22_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( times_times_set_nat @ ( times_times_set_nat @ A2 @ B2 ) @ C )
= ( times_times_set_nat @ A2 @ ( times_times_set_nat @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_23_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_24_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( times_times_complex @ ( times_times_complex @ A2 @ B2 ) @ C )
= ( times_times_complex @ A2 @ ( times_times_complex @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_25_vector__space__over__itself_Ovector__space__assms_I3_J,axiom,
! [A2: complex,B2: complex,X: complex] :
( ( times_times_complex @ A2 @ ( times_times_complex @ B2 @ X ) )
= ( times_times_complex @ ( times_times_complex @ A2 @ B2 ) @ X ) ) ).
% vector_space_over_itself.vector_space_assms(3)
thf(fact_26_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_27_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_28_carrier__matD_I1_J,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( dim_row_complex @ A )
= Nr ) ) ).
% carrier_matD(1)
thf(fact_29_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_30_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_31_mult__carrier__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,B: mat_complex,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ Nc ) )
=> ( member_mat_complex @ ( times_8009071140041733218omplex @ A @ B ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_32_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_33_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_34_assoc__mult__mat,axiom,
! [A: mat_complex,N_1: nat,N_2: nat,B: mat_complex,N_3: nat,C2: mat_complex,N_4: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N_1 @ N_2 ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N_2 @ N_3 ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N_3 @ N_4 ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B ) @ C2 )
= ( times_8009071140041733218omplex @ A @ ( times_8009071140041733218omplex @ B @ C2 ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_35_vector__space__over__itself_Oscale__left__commute,axiom,
! [A2: complex,B2: complex,X: complex] :
( ( times_times_complex @ A2 @ ( times_times_complex @ B2 @ X ) )
= ( times_times_complex @ B2 @ ( times_times_complex @ A2 @ X ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_36_mk__diagonal__diagonal,axiom,
! [As: list_complex] : ( diagonal_mat_complex @ ( mk_diagonal_complex @ As ) ) ).
% mk_diagonal_diagonal
thf(fact_37_mk__diagonal__diagonal,axiom,
! [As: list_a] : ( diagonal_mat_a @ ( mk_diagonal_a @ As ) ) ).
% mk_diagonal_diagonal
thf(fact_38_inv__all_H__def,axiom,
( jordan5032732407113867375omplex
= ( ^ [P: mat_complex > nat > nat > $o,A4: mat_complex] :
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_complex @ A4 ) )
=> ( ( ord_less_nat @ J @ ( dim_row_complex @ A4 ) )
=> ( P @ A4 @ I @ J ) ) ) ) ) ).
% inv_all'_def
thf(fact_39_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_40_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_41_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_42_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_43_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_44_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P2 @ M2 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_45_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P2 @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P2 @ M2 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_46_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_47_pinf_I1_J,axiom,
! [P2: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ( P2 @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( ( ( P2 @ X3 )
& ( Q @ X3 ) )
= ( ( P3 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_48_pinf_I2_J,axiom,
! [P2: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ( P2 @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( ( ( P2 @ X3 )
| ( Q @ X3 ) )
= ( ( P3 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_49_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_50_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_51_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ~ ( ord_less_nat @ X3 @ T ) ) ).
% pinf(5)
thf(fact_52_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( ord_less_nat @ T @ X3 ) ) ).
% pinf(7)
thf(fact_53_mem__Collect__eq,axiom,
! [A2: complex,P2: complex > $o] :
( ( member_complex @ A2 @ ( collect_complex @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
! [A2: nat,P2: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_55_mem__Collect__eq,axiom,
! [A2: mat_nat,P2: mat_nat > $o] :
( ( member_mat_nat @ A2 @ ( collect_mat_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
! [A2: mat_a,P2: mat_a > $o] :
( ( member_mat_a @ A2 @ ( collect_mat_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_57_mem__Collect__eq,axiom,
! [A2: mat_complex,P2: mat_complex > $o] :
( ( member_mat_complex @ A2 @ ( collect_mat_complex @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A: set_complex] :
( ( collect_complex
@ ^ [X4: complex] : ( member_complex @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A: set_mat_nat] :
( ( collect_mat_nat
@ ^ [X4: mat_nat] : ( member_mat_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
! [A: set_mat_a] :
( ( collect_mat_a
@ ^ [X4: mat_a] : ( member_mat_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
! [A: set_mat_complex] :
( ( collect_mat_complex
@ ^ [X4: mat_complex] : ( member_mat_complex @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_63_Collect__cong,axiom,
! [P2: mat_complex > $o,Q: mat_complex > $o] :
( ! [X2: mat_complex] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_mat_complex @ P2 )
= ( collect_mat_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_64_Collect__cong,axiom,
! [P2: mat_a > $o,Q: mat_a > $o] :
( ! [X2: mat_a] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_mat_a @ P2 )
= ( collect_mat_a @ Q ) ) ) ).
% Collect_cong
thf(fact_65_minf_I1_J,axiom,
! [P2: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ( P2 @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( ( ( P2 @ X3 )
& ( Q @ X3 ) )
= ( ( P3 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_66_minf_I2_J,axiom,
! [P2: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ( P2 @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( ( ( P2 @ X3 )
| ( Q @ X3 ) )
= ( ( P3 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_67_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_68_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_69_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( ord_less_nat @ X3 @ T ) ) ).
% minf(5)
thf(fact_70_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ~ ( ord_less_nat @ T @ X3 ) ) ).
% minf(7)
thf(fact_71_order__less__imp__not__less,axiom,
! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ~ ( ord_less_complex @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_72_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_73_order__less__imp__not__eq2,axiom,
! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_74_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_75_order__less__imp__not__eq,axiom,
! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_76_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_77_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_78_order__less__imp__triv,axiom,
! [X: complex,Y: complex,P2: $o] :
( ( ord_less_complex @ X @ Y )
=> ( ( ord_less_complex @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_79_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P2: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_80_order__less__not__sym,axiom,
! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ~ ( ord_less_complex @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_81_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_82_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > complex,C: complex] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_complex @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_83_order__less__subst2,axiom,
! [A2: complex,B2: complex,F: complex > nat,C: nat] :
( ( ord_less_complex @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: complex,Y2: complex] :
( ( ord_less_complex @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_84_order__less__subst2,axiom,
! [A2: complex,B2: complex,F: complex > complex,C: complex] :
( ( ord_less_complex @ A2 @ B2 )
=> ( ( ord_less_complex @ ( F @ B2 ) @ C )
=> ( ! [X2: complex,Y2: complex] :
( ( ord_less_complex @ X2 @ Y2 )
=> ( ord_less_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_85_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_86_order__less__subst1,axiom,
! [A2: nat,F: complex > nat,B2: complex,C: complex] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_complex @ B2 @ C )
=> ( ! [X2: complex,Y2: complex] :
( ( ord_less_complex @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_87_order__less__subst1,axiom,
! [A2: complex,F: nat > complex,B2: nat,C: nat] :
( ( ord_less_complex @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_88_order__less__subst1,axiom,
! [A2: complex,F: complex > complex,B2: complex,C: complex] :
( ( ord_less_complex @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_complex @ B2 @ C )
=> ( ! [X2: complex,Y2: complex] :
( ( ord_less_complex @ X2 @ Y2 )
=> ( ord_less_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_89_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_90_order__less__irrefl,axiom,
! [X: complex] :
~ ( ord_less_complex @ X @ X ) ).
% order_less_irrefl
thf(fact_91_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_92_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > complex,C: complex] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_93_ord__less__eq__subst,axiom,
! [A2: complex,B2: complex,F: complex > nat,C: nat] :
( ( ord_less_complex @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: complex,Y2: complex] :
( ( ord_less_complex @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_94_ord__less__eq__subst,axiom,
! [A2: complex,B2: complex,F: complex > complex,C: complex] :
( ( ord_less_complex @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: complex,Y2: complex] :
( ( ord_less_complex @ X2 @ Y2 )
=> ( ord_less_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_95_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_96_ord__eq__less__subst,axiom,
! [A2: complex,F: nat > complex,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_97_ord__eq__less__subst,axiom,
! [A2: nat,F: complex > nat,B2: complex,C: complex] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_complex @ B2 @ C )
=> ( ! [X2: complex,Y2: complex] :
( ( ord_less_complex @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_98_ord__eq__less__subst,axiom,
! [A2: complex,F: complex > complex,B2: complex,C: complex] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_complex @ B2 @ C )
=> ( ! [X2: complex,Y2: complex] :
( ( ord_less_complex @ X2 @ Y2 )
=> ( ord_less_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_99_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_100_order__trans__rules_I28_J,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( A2 = B2 )
=> ( ( ord_less_complex @ B2 @ C )
=> ( ord_less_complex @ A2 @ C ) ) ) ).
% order_trans_rules(28)
thf(fact_101_order__trans__rules_I28_J,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order_trans_rules(28)
thf(fact_102_order__trans__rules_I27_J,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( ord_less_complex @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_complex @ A2 @ C ) ) ) ).
% order_trans_rules(27)
thf(fact_103_order__trans__rules_I27_J,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order_trans_rules(27)
thf(fact_104_order__trans__rules_I20_J,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_complex @ A2 @ B2 )
=> ~ ( ord_less_complex @ B2 @ A2 ) ) ).
% order_trans_rules(20)
thf(fact_105_order__trans__rules_I20_J,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_trans_rules(20)
thf(fact_106_order__trans__rules_I19_J,axiom,
! [X: complex,Y: complex,Z3: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ( ord_less_complex @ Y @ Z3 )
=> ( ord_less_complex @ X @ Z3 ) ) ) ).
% order_trans_rules(19)
thf(fact_107_order__trans__rules_I19_J,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_trans_rules(19)
thf(fact_108_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_109_neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% neqE
thf(fact_110_neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% neq_iff
thf(fact_111_less__imp__neq,axiom,
! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_112_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_113_less__asym,axiom,
! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ~ ( ord_less_complex @ Y @ X ) ) ).
% less_asym
thf(fact_114_less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% less_asym
thf(fact_115_order_Oasym,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_complex @ A2 @ B2 )
=> ~ ( ord_less_complex @ B2 @ A2 ) ) ).
% order.asym
thf(fact_116_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_117_order_Oirrefl,axiom,
! [A2: complex] :
~ ( ord_less_complex @ A2 @ A2 ) ).
% order.irrefl
thf(fact_118_order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% order.irrefl
thf(fact_119_less__induct,axiom,
! [P2: nat > $o,A2: nat] :
( ! [X2: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X2 )
=> ( P2 @ Y3 ) )
=> ( P2 @ X2 ) )
=> ( P2 @ A2 ) ) ).
% less_induct
thf(fact_120_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_121_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_122_dual__order_Oasym,axiom,
! [B2: complex,A2: complex] :
( ( ord_less_complex @ B2 @ A2 )
=> ~ ( ord_less_complex @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_123_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_124_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X5: nat] : ( P4 @ X5 ) )
= ( ^ [P5: nat > $o] :
? [N3: nat] :
( ( P5 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P5 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_125_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
=> ( P2 @ A5 @ B4 ) )
=> ( ! [A5: nat] : ( P2 @ A5 @ A5 )
=> ( ! [A5: nat,B4: nat] :
( ( P2 @ B4 @ A5 )
=> ( P2 @ A5 @ B4 ) )
=> ( P2 @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_126_order_Ostrict__trans,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( ord_less_complex @ A2 @ B2 )
=> ( ( ord_less_complex @ B2 @ C )
=> ( ord_less_complex @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_127_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_128_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_129_dual__order_Ostrict__trans,axiom,
! [B2: complex,A2: complex,C: complex] :
( ( ord_less_complex @ B2 @ A2 )
=> ( ( ord_less_complex @ C @ B2 )
=> ( ord_less_complex @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_130_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_131_order_Ostrict__implies__not__eq,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_complex @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_132_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_133_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: complex,A2: complex] :
( ( ord_less_complex @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_134_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_135_verit__comp__simplify_I1_J,axiom,
! [A2: complex] :
~ ( ord_less_complex @ A2 @ A2 ) ).
% verit_comp_simplify(1)
thf(fact_136_verit__comp__simplify_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify(1)
thf(fact_137_class__cring_Ofactors__equal,axiom,
! [A2: complex,B2: complex,C: complex,D: complex] :
( ( A2 = B2 )
=> ( ( C = D )
=> ( ( times_times_complex @ A2 @ C )
= ( times_times_complex @ B2 @ D ) ) ) ) ).
% class_cring.factors_equal
thf(fact_138_set__times__elim,axiom,
! [X: mat_nat,A: set_mat_nat,B: set_mat_nat] :
( ( member_mat_nat @ X @ ( times_5500231875258083300at_nat @ A @ B ) )
=> ~ ! [A5: mat_nat,B4: mat_nat] :
( ( X
= ( times_times_mat_nat @ A5 @ B4 ) )
=> ( ( member_mat_nat @ A5 @ A )
=> ~ ( member_mat_nat @ B4 @ B ) ) ) ) ).
% set_times_elim
thf(fact_139_set__times__elim,axiom,
! [X: set_complex,A: set_set_complex,B: set_set_complex] :
( ( member_set_complex @ X @ ( times_6103784797850505759omplex @ A @ B ) )
=> ~ ! [A5: set_complex,B4: set_complex] :
( ( X
= ( times_6048082448287401577omplex @ A5 @ B4 ) )
=> ( ( member_set_complex @ A5 @ A )
=> ~ ( member_set_complex @ B4 @ B ) ) ) ) ).
% set_times_elim
thf(fact_140_set__times__elim,axiom,
! [X: set_mat_complex,A: set_set_mat_complex,B: set_set_mat_complex] :
( ( member3612512168372279472omplex @ X @ ( times_3957003352596167970omplex @ A @ B ) )
=> ~ ! [A5: set_mat_complex,B4: set_mat_complex] :
( ( X
= ( times_6731331324747250370omplex @ A5 @ B4 ) )
=> ( ( member3612512168372279472omplex @ A5 @ A )
=> ~ ( member3612512168372279472omplex @ B4 @ B ) ) ) ) ).
% set_times_elim
thf(fact_141_set__times__elim,axiom,
! [X: set_mat_a,A: set_set_mat_a,B: set_set_mat_a] :
( ( member_set_mat_a @ X @ ( times_5016826689369604684_mat_a @ A @ B ) )
=> ~ ! [A5: set_mat_a,B4: set_mat_a] :
( ( X
= ( times_1230744552615602198_mat_a @ A5 @ B4 ) )
=> ( ( member_set_mat_a @ A5 @ A )
=> ~ ( member_set_mat_a @ B4 @ B ) ) ) ) ).
% set_times_elim
thf(fact_142_set__times__elim,axiom,
! [X: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ X @ ( times_4850922872519784769et_nat @ A @ B ) )
=> ~ ! [A5: set_nat,B4: set_nat] :
( ( X
= ( times_times_set_nat @ A5 @ B4 ) )
=> ( ( member_set_nat @ A5 @ A )
=> ~ ( member_set_nat @ B4 @ B ) ) ) ) ).
% set_times_elim
thf(fact_143_set__times__elim,axiom,
! [X: a,A: set_a,B: set_a] :
( ( member_a @ X @ ( times_times_set_a @ A @ B ) )
=> ~ ! [A5: a,B4: a] :
( ( X
= ( times_times_a @ A5 @ B4 ) )
=> ( ( member_a @ A5 @ A )
=> ~ ( member_a @ B4 @ B ) ) ) ) ).
% set_times_elim
thf(fact_144_set__times__elim,axiom,
! [X: mat_a,A: set_mat_a,B: set_mat_a] :
( ( member_mat_a @ X @ ( times_1230744552615602198_mat_a @ A @ B ) )
=> ~ ! [A5: mat_a,B4: mat_a] :
( ( X
= ( times_times_mat_a @ A5 @ B4 ) )
=> ( ( member_mat_a @ A5 @ A )
=> ~ ( member_mat_a @ B4 @ B ) ) ) ) ).
% set_times_elim
thf(fact_145_set__times__elim,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ( member_nat @ X @ ( times_times_set_nat @ A @ B ) )
=> ~ ! [A5: nat,B4: nat] :
( ( X
= ( times_times_nat @ A5 @ B4 ) )
=> ( ( member_nat @ A5 @ A )
=> ~ ( member_nat @ B4 @ B ) ) ) ) ).
% set_times_elim
thf(fact_146_set__times__elim,axiom,
! [X: mat_complex,A: set_mat_complex,B: set_mat_complex] :
( ( member_mat_complex @ X @ ( times_6731331324747250370omplex @ A @ B ) )
=> ~ ! [A5: mat_complex,B4: mat_complex] :
( ( X
= ( times_8009071140041733218omplex @ A5 @ B4 ) )
=> ( ( member_mat_complex @ A5 @ A )
=> ~ ( member_mat_complex @ B4 @ B ) ) ) ) ).
% set_times_elim
thf(fact_147_set__times__elim,axiom,
! [X: complex,A: set_complex,B: set_complex] :
( ( member_complex @ X @ ( times_6048082448287401577omplex @ A @ B ) )
=> ~ ! [A5: complex,B4: complex] :
( ( X
= ( times_times_complex @ A5 @ B4 ) )
=> ( ( member_complex @ A5 @ A )
=> ~ ( member_complex @ B4 @ B ) ) ) ) ).
% set_times_elim
thf(fact_148_set__times__intro,axiom,
! [A2: mat_nat,C2: set_mat_nat,B2: mat_nat,D2: set_mat_nat] :
( ( member_mat_nat @ A2 @ C2 )
=> ( ( member_mat_nat @ B2 @ D2 )
=> ( member_mat_nat @ ( times_times_mat_nat @ A2 @ B2 ) @ ( times_5500231875258083300at_nat @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_149_set__times__intro,axiom,
! [A2: set_complex,C2: set_set_complex,B2: set_complex,D2: set_set_complex] :
( ( member_set_complex @ A2 @ C2 )
=> ( ( member_set_complex @ B2 @ D2 )
=> ( member_set_complex @ ( times_6048082448287401577omplex @ A2 @ B2 ) @ ( times_6103784797850505759omplex @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_150_set__times__intro,axiom,
! [A2: set_mat_complex,C2: set_set_mat_complex,B2: set_mat_complex,D2: set_set_mat_complex] :
( ( member3612512168372279472omplex @ A2 @ C2 )
=> ( ( member3612512168372279472omplex @ B2 @ D2 )
=> ( member3612512168372279472omplex @ ( times_6731331324747250370omplex @ A2 @ B2 ) @ ( times_3957003352596167970omplex @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_151_set__times__intro,axiom,
! [A2: set_mat_a,C2: set_set_mat_a,B2: set_mat_a,D2: set_set_mat_a] :
( ( member_set_mat_a @ A2 @ C2 )
=> ( ( member_set_mat_a @ B2 @ D2 )
=> ( member_set_mat_a @ ( times_1230744552615602198_mat_a @ A2 @ B2 ) @ ( times_5016826689369604684_mat_a @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_152_set__times__intro,axiom,
! [A2: set_nat,C2: set_set_nat,B2: set_nat,D2: set_set_nat] :
( ( member_set_nat @ A2 @ C2 )
=> ( ( member_set_nat @ B2 @ D2 )
=> ( member_set_nat @ ( times_times_set_nat @ A2 @ B2 ) @ ( times_4850922872519784769et_nat @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_153_set__times__intro,axiom,
! [A2: a,C2: set_a,B2: a,D2: set_a] :
( ( member_a @ A2 @ C2 )
=> ( ( member_a @ B2 @ D2 )
=> ( member_a @ ( times_times_a @ A2 @ B2 ) @ ( times_times_set_a @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_154_set__times__intro,axiom,
! [A2: mat_a,C2: set_mat_a,B2: mat_a,D2: set_mat_a] :
( ( member_mat_a @ A2 @ C2 )
=> ( ( member_mat_a @ B2 @ D2 )
=> ( member_mat_a @ ( times_times_mat_a @ A2 @ B2 ) @ ( times_1230744552615602198_mat_a @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_155_set__times__intro,axiom,
! [A2: nat,C2: set_nat,B2: nat,D2: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( member_nat @ B2 @ D2 )
=> ( member_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_set_nat @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_156_set__times__intro,axiom,
! [A2: mat_complex,C2: set_mat_complex,B2: mat_complex,D2: set_mat_complex] :
( ( member_mat_complex @ A2 @ C2 )
=> ( ( member_mat_complex @ B2 @ D2 )
=> ( member_mat_complex @ ( times_8009071140041733218omplex @ A2 @ B2 ) @ ( times_6731331324747250370omplex @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_157_set__times__intro,axiom,
! [A2: complex,C2: set_complex,B2: complex,D2: set_complex] :
( ( member_complex @ A2 @ C2 )
=> ( ( member_complex @ B2 @ D2 )
=> ( member_complex @ ( times_times_complex @ A2 @ B2 ) @ ( times_6048082448287401577omplex @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_158_diagonal__imp__upper__triangular,axiom,
! [A: mat_nat,N: nat] :
( ( diagonal_mat_nat @ A )
=> ( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ N ) )
=> ( upper_triangular_nat @ A ) ) ) ).
% diagonal_imp_upper_triangular
thf(fact_159_diagonal__imp__upper__triangular,axiom,
! [A: mat_complex,N: nat] :
( ( diagonal_mat_complex @ A )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( upper_4850907204721561915omplex @ A ) ) ) ).
% diagonal_imp_upper_triangular
thf(fact_160_diagonal__imp__upper__triangular,axiom,
! [A: mat_a,N: nat] :
( ( diagonal_mat_a @ A )
=> ( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( upper_triangular_a @ A ) ) ) ).
% diagonal_imp_upper_triangular
thf(fact_161_normal__upper__triangular__matrix__is__diagonal,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( upper_4850907204721561915omplex @ A )
=> ( ( ( times_8009071140041733218omplex @ A @ ( schur_5982229384592763574omplex @ A ) )
= ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ A ) @ A ) )
=> ( diagonal_mat_complex @ A ) ) ) ) ).
% normal_upper_triangular_matrix_is_diagonal
thf(fact_162_dim__update__mat_I1_J,axiom,
! [A: mat_nat,Ij: product_prod_nat_nat,A2: nat] :
( ( dim_row_nat @ ( update_mat_nat @ A @ Ij @ A2 ) )
= ( dim_row_nat @ A ) ) ).
% dim_update_mat(1)
thf(fact_163_dim__update__mat_I1_J,axiom,
! [A: mat_a,Ij: product_prod_nat_nat,A2: a] :
( ( dim_row_a @ ( update_mat_a @ A @ Ij @ A2 ) )
= ( dim_row_a @ A ) ) ).
% dim_update_mat(1)
thf(fact_164_dim__update__mat_I1_J,axiom,
! [A: mat_complex,Ij: product_prod_nat_nat,A2: complex] :
( ( dim_row_complex @ ( update_mat_complex @ A @ Ij @ A2 ) )
= ( dim_row_complex @ A ) ) ).
% dim_update_mat(1)
thf(fact_165_triangular__to__jnf__steps__dims_I5_J,axiom,
! [A: mat_complex] :
( ( dim_row_complex @ ( jordan4501759426295633263omplex @ A ) )
= ( dim_row_complex @ A ) ) ).
% triangular_to_jnf_steps_dims(5)
thf(fact_166_mk__diagonal__dim_I1_J,axiom,
! [As: list_nat] :
( ( dim_row_nat @ ( mk_diagonal_nat @ As ) )
= ( size_size_list_nat @ As ) ) ).
% mk_diagonal_dim(1)
thf(fact_167_mk__diagonal__dim_I1_J,axiom,
! [As: list_a] :
( ( dim_row_a @ ( mk_diagonal_a @ As ) )
= ( size_size_list_a @ As ) ) ).
% mk_diagonal_dim(1)
thf(fact_168_mk__diagonal__dim_I1_J,axiom,
! [As: list_complex] :
( ( dim_row_complex @ ( mk_diagonal_complex @ As ) )
= ( size_s3451745648224563538omplex @ As ) ) ).
% mk_diagonal_dim(1)
thf(fact_169_vec__space_Orow__space__is__preserved,axiom,
! [P2: mat_complex,M: nat,A: mat_complex,N: nat] :
( ( invert2568027935824841882omplex @ P2 )
=> ( ( member_mat_complex @ P2 @ ( carrier_mat_complex @ M @ M ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ M @ N ) )
=> ( ( vS_vec3284807721666986142omplex @ N @ ( times_8009071140041733218omplex @ P2 @ A ) )
= ( vS_vec3284807721666986142omplex @ N @ A ) ) ) ) ) ).
% vec_space.row_space_is_preserved
thf(fact_170_similar__mat__witD_I3_J,axiom,
! [N: nat,A: mat_nat,B: mat_nat,P2: mat_nat,Q: mat_nat] :
( ( N
= ( dim_row_nat @ A ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P2 @ Q )
=> ( A
= ( times_times_mat_nat @ ( times_times_mat_nat @ P2 @ B ) @ Q ) ) ) ) ).
% similar_mat_witD(3)
thf(fact_171_similar__mat__witD_I3_J,axiom,
! [N: nat,A: mat_complex,B: mat_complex,P2: mat_complex,Q: mat_complex] :
( ( N
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P2 @ Q )
=> ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P2 @ B ) @ Q ) ) ) ) ).
% similar_mat_witD(3)
thf(fact_172_set__times__rearrange,axiom,
! [A2: set_complex,C2: set_set_complex,B2: set_complex,D2: set_set_complex] :
( ( times_6103784797850505759omplex @ ( set_el158709831744343061omplex @ A2 @ C2 ) @ ( set_el158709831744343061omplex @ B2 @ D2 ) )
= ( set_el158709831744343061omplex @ ( times_6048082448287401577omplex @ A2 @ B2 ) @ ( times_6103784797850505759omplex @ C2 @ D2 ) ) ) ).
% set_times_rearrange
thf(fact_173_set__times__rearrange,axiom,
! [A2: set_nat,C2: set_set_nat,B2: set_nat,D2: set_set_nat] :
( ( times_4850922872519784769et_nat @ ( set_el3528970498207131191et_nat @ A2 @ C2 ) @ ( set_el3528970498207131191et_nat @ B2 @ D2 ) )
= ( set_el3528970498207131191et_nat @ ( times_times_set_nat @ A2 @ B2 ) @ ( times_4850922872519784769et_nat @ C2 @ D2 ) ) ) ).
% set_times_rearrange
thf(fact_174_set__times__rearrange,axiom,
! [A2: nat,C2: set_nat,B2: nat,D2: set_nat] :
( ( times_times_set_nat @ ( set_el2933305810450955905es_nat @ A2 @ C2 ) @ ( set_el2933305810450955905es_nat @ B2 @ D2 ) )
= ( set_el2933305810450955905es_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_set_nat @ C2 @ D2 ) ) ) ).
% set_times_rearrange
thf(fact_175_set__times__rearrange,axiom,
! [A2: complex,C2: set_complex,B2: complex,D2: set_complex] :
( ( times_6048082448287401577omplex @ ( set_el8005228190238886239omplex @ A2 @ C2 ) @ ( set_el8005228190238886239omplex @ B2 @ D2 ) )
= ( set_el8005228190238886239omplex @ ( times_times_complex @ A2 @ B2 ) @ ( times_6048082448287401577omplex @ C2 @ D2 ) ) ) ).
% set_times_rearrange
thf(fact_176_similar__mat__witD2_I3_J,axiom,
! [A: mat_nat,N: nat,M: nat,B: mat_nat,P2: mat_nat,Q: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ M ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P2 @ Q )
=> ( A
= ( times_times_mat_nat @ ( times_times_mat_nat @ P2 @ B ) @ Q ) ) ) ) ).
% similar_mat_witD2(3)
thf(fact_177_similar__mat__witD2_I3_J,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,P2: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P2 @ Q )
=> ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P2 @ B ) @ Q ) ) ) ) ).
% similar_mat_witD2(3)
thf(fact_178_vec__space_Orow__space_Ocong,axiom,
vS_vec3284807721666986142omplex = vS_vec3284807721666986142omplex ).
% vec_space.row_space.cong
thf(fact_179_similar__mat__wit__sym,axiom,
! [A: mat_complex,B: mat_complex,P2: mat_complex,Q: mat_complex] :
( ( simila5774310414453981135omplex @ A @ B @ P2 @ Q )
=> ( simila5774310414453981135omplex @ B @ A @ Q @ P2 ) ) ).
% similar_mat_wit_sym
thf(fact_180_adjoint__dim,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( member_mat_complex @ ( schur_5982229384592763574omplex @ A ) @ ( carrier_mat_complex @ N @ N ) ) ) ).
% adjoint_dim
thf(fact_181_size__neq__size__imp__neq,axiom,
! [X: list_complex,Y: list_complex] :
( ( ( size_s3451745648224563538omplex @ X )
!= ( size_s3451745648224563538omplex @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_182_size__neq__size__imp__neq,axiom,
! [X: list_a,Y: list_a] :
( ( ( size_size_list_a @ X )
!= ( size_size_list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_183_size__neq__size__imp__neq,axiom,
! [X: list_mat_complex,Y: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ X )
!= ( size_s5969786470865220249omplex @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_184_size__neq__size__imp__neq,axiom,
! [X: list_mat_a,Y: list_mat_a] :
( ( ( size_size_list_mat_a @ X )
!= ( size_size_list_mat_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_185_adjoint__dim_H,axiom,
! [A: mat_complex,N: nat,M: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( member_mat_complex @ ( schur_5982229384592763574omplex @ A ) @ ( carrier_mat_complex @ M @ N ) ) ) ).
% adjoint_dim'
thf(fact_186_set__times__rearranges_I2_J,axiom,
! [A2: set_complex,B2: set_complex,C2: set_set_complex] :
( ( set_el158709831744343061omplex @ A2 @ ( set_el158709831744343061omplex @ B2 @ C2 ) )
= ( set_el158709831744343061omplex @ ( times_6048082448287401577omplex @ A2 @ B2 ) @ C2 ) ) ).
% set_times_rearranges(2)
thf(fact_187_set__times__rearranges_I2_J,axiom,
! [A2: set_nat,B2: set_nat,C2: set_set_nat] :
( ( set_el3528970498207131191et_nat @ A2 @ ( set_el3528970498207131191et_nat @ B2 @ C2 ) )
= ( set_el3528970498207131191et_nat @ ( times_times_set_nat @ A2 @ B2 ) @ C2 ) ) ).
% set_times_rearranges(2)
thf(fact_188_set__times__rearranges_I2_J,axiom,
! [A2: a,B2: a,C2: set_a] :
( ( set_elt_set_times_a @ A2 @ ( set_elt_set_times_a @ B2 @ C2 ) )
= ( set_elt_set_times_a @ ( times_times_a @ A2 @ B2 ) @ C2 ) ) ).
% set_times_rearranges(2)
thf(fact_189_set__times__rearranges_I2_J,axiom,
! [A2: nat,B2: nat,C2: set_nat] :
( ( set_el2933305810450955905es_nat @ A2 @ ( set_el2933305810450955905es_nat @ B2 @ C2 ) )
= ( set_el2933305810450955905es_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 ) ) ).
% set_times_rearranges(2)
thf(fact_190_set__times__rearranges_I2_J,axiom,
! [A2: complex,B2: complex,C2: set_complex] :
( ( set_el8005228190238886239omplex @ A2 @ ( set_el8005228190238886239omplex @ B2 @ C2 ) )
= ( set_el8005228190238886239omplex @ ( times_times_complex @ A2 @ B2 ) @ C2 ) ) ).
% set_times_rearranges(2)
thf(fact_191_set__times__intro2,axiom,
! [B2: mat_nat,C2: set_mat_nat,A2: mat_nat] :
( ( member_mat_nat @ B2 @ C2 )
=> ( member_mat_nat @ ( times_times_mat_nat @ A2 @ B2 ) @ ( set_el1310594772197002200at_nat @ A2 @ C2 ) ) ) ).
% set_times_intro2
thf(fact_192_set__times__intro2,axiom,
! [B2: set_complex,C2: set_set_complex,A2: set_complex] :
( ( member_set_complex @ B2 @ C2 )
=> ( member_set_complex @ ( times_6048082448287401577omplex @ A2 @ B2 ) @ ( set_el158709831744343061omplex @ A2 @ C2 ) ) ) ).
% set_times_intro2
thf(fact_193_set__times__intro2,axiom,
! [B2: set_mat_complex,C2: set_set_mat_complex,A2: set_mat_complex] :
( ( member3612512168372279472omplex @ B2 @ C2 )
=> ( member3612512168372279472omplex @ ( times_6731331324747250370omplex @ A2 @ B2 ) @ ( set_el2325834261644174870omplex @ A2 @ C2 ) ) ) ).
% set_times_intro2
thf(fact_194_set__times__intro2,axiom,
! [B2: set_mat_a,C2: set_set_mat_a,A2: set_mat_a] :
( ( member_set_mat_a @ B2 @ C2 )
=> ( member_set_mat_a @ ( times_1230744552615602198_mat_a @ A2 @ B2 ) @ ( set_el7210227519270355394_mat_a @ A2 @ C2 ) ) ) ).
% set_times_intro2
thf(fact_195_set__times__intro2,axiom,
! [B2: set_nat,C2: set_set_nat,A2: set_nat] :
( ( member_set_nat @ B2 @ C2 )
=> ( member_set_nat @ ( times_times_set_nat @ A2 @ B2 ) @ ( set_el3528970498207131191et_nat @ A2 @ C2 ) ) ) ).
% set_times_intro2
thf(fact_196_set__times__intro2,axiom,
! [B2: a,C2: set_a,A2: a] :
( ( member_a @ B2 @ C2 )
=> ( member_a @ ( times_times_a @ A2 @ B2 ) @ ( set_elt_set_times_a @ A2 @ C2 ) ) ) ).
% set_times_intro2
thf(fact_197_set__times__intro2,axiom,
! [B2: mat_a,C2: set_mat_a,A2: mat_a] :
( ( member_mat_a @ B2 @ C2 )
=> ( member_mat_a @ ( times_times_mat_a @ A2 @ B2 ) @ ( set_el1062546952344711308_mat_a @ A2 @ C2 ) ) ) ).
% set_times_intro2
thf(fact_198_set__times__intro2,axiom,
! [B2: nat,C2: set_nat,A2: nat] :
( ( member_nat @ B2 @ C2 )
=> ( member_nat @ ( times_times_nat @ A2 @ B2 ) @ ( set_el2933305810450955905es_nat @ A2 @ C2 ) ) ) ).
% set_times_intro2
thf(fact_199_set__times__intro2,axiom,
! [B2: mat_complex,C2: set_mat_complex,A2: mat_complex] :
( ( member_mat_complex @ B2 @ C2 )
=> ( member_mat_complex @ ( times_8009071140041733218omplex @ A2 @ B2 ) @ ( set_el176066062795894710omplex @ A2 @ C2 ) ) ) ).
% set_times_intro2
thf(fact_200_set__times__intro2,axiom,
! [B2: complex,C2: set_complex,A2: complex] :
( ( member_complex @ B2 @ C2 )
=> ( member_complex @ ( times_times_complex @ A2 @ B2 ) @ ( set_el8005228190238886239omplex @ A2 @ C2 ) ) ) ).
% set_times_intro2
thf(fact_201_similar__mat__witD2_I4_J,axiom,
! [A: mat_nat,N: nat,M: nat,B: mat_nat,P2: mat_nat,Q: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ M ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P2 @ Q )
=> ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ N ) ) ) ) ).
% similar_mat_witD2(4)
thf(fact_202_similar__mat__witD2_I4_J,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,P2: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P2 @ Q )
=> ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD2(4)
thf(fact_203_similar__mat__witD2_I5_J,axiom,
! [A: mat_nat,N: nat,M: nat,B: mat_nat,P2: mat_nat,Q: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ M ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P2 @ Q )
=> ( member_mat_nat @ B @ ( carrier_mat_nat @ N @ N ) ) ) ) ).
% similar_mat_witD2(5)
thf(fact_204_similar__mat__witD2_I5_J,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,P2: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P2 @ Q )
=> ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD2(5)
thf(fact_205_similar__mat__witD2_I6_J,axiom,
! [A: mat_nat,N: nat,M: nat,B: mat_nat,P2: mat_nat,Q: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ M ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P2 @ Q )
=> ( member_mat_nat @ P2 @ ( carrier_mat_nat @ N @ N ) ) ) ) ).
% similar_mat_witD2(6)
thf(fact_206_similar__mat__witD2_I6_J,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,P2: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P2 @ Q )
=> ( member_mat_complex @ P2 @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD2(6)
thf(fact_207_similar__mat__witD2_I7_J,axiom,
! [A: mat_nat,N: nat,M: nat,B: mat_nat,P2: mat_nat,Q: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ M ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P2 @ Q )
=> ( member_mat_nat @ Q @ ( carrier_mat_nat @ N @ N ) ) ) ) ).
% similar_mat_witD2(7)
thf(fact_208_similar__mat__witD2_I7_J,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,P2: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P2 @ Q )
=> ( member_mat_complex @ Q @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD2(7)
thf(fact_209_similar__mat__wit__trans,axiom,
! [A: mat_complex,B: mat_complex,P2: mat_complex,Q: mat_complex,C2: mat_complex,P3: mat_complex,Q2: mat_complex] :
( ( simila5774310414453981135omplex @ A @ B @ P2 @ Q )
=> ( ( simila5774310414453981135omplex @ B @ C2 @ P3 @ Q2 )
=> ( simila5774310414453981135omplex @ A @ C2 @ ( times_8009071140041733218omplex @ P2 @ P3 ) @ ( times_8009071140041733218omplex @ Q2 @ Q ) ) ) ) ).
% similar_mat_wit_trans
thf(fact_210_adjoint__mult,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,L: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ M @ L ) )
=> ( ( schur_5982229384592763574omplex @ ( times_8009071140041733218omplex @ A @ B ) )
= ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ B ) @ ( schur_5982229384592763574omplex @ A ) ) ) ) ) ).
% adjoint_mult
thf(fact_211_set__times__rearranges_I4_J,axiom,
! [C2: set_complex,A2: complex,D2: set_complex] :
( ( times_6048082448287401577omplex @ C2 @ ( set_el8005228190238886239omplex @ A2 @ D2 ) )
= ( set_el8005228190238886239omplex @ A2 @ ( times_6048082448287401577omplex @ C2 @ D2 ) ) ) ).
% set_times_rearranges(4)
thf(fact_212_set__times__rearranges_I4_J,axiom,
! [C2: set_nat,A2: nat,D2: set_nat] :
( ( times_times_set_nat @ C2 @ ( set_el2933305810450955905es_nat @ A2 @ D2 ) )
= ( set_el2933305810450955905es_nat @ A2 @ ( times_times_set_nat @ C2 @ D2 ) ) ) ).
% set_times_rearranges(4)
thf(fact_213_set__times__rearranges_I3_J,axiom,
! [A2: complex,B: set_complex,C2: set_complex] :
( ( times_6048082448287401577omplex @ ( set_el8005228190238886239omplex @ A2 @ B ) @ C2 )
= ( set_el8005228190238886239omplex @ A2 @ ( times_6048082448287401577omplex @ B @ C2 ) ) ) ).
% set_times_rearranges(3)
thf(fact_214_set__times__rearranges_I3_J,axiom,
! [A2: nat,B: set_nat,C2: set_nat] :
( ( times_times_set_nat @ ( set_el2933305810450955905es_nat @ A2 @ B ) @ C2 )
= ( set_el2933305810450955905es_nat @ A2 @ ( times_times_set_nat @ B @ C2 ) ) ) ).
% set_times_rearranges(3)
thf(fact_215_similar__mat__witD_I7_J,axiom,
! [N: nat,A: mat_nat,B: mat_nat,P2: mat_nat,Q: mat_nat] :
( ( N
= ( dim_row_nat @ A ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P2 @ Q )
=> ( member_mat_nat @ Q @ ( carrier_mat_nat @ N @ N ) ) ) ) ).
% similar_mat_witD(7)
thf(fact_216_similar__mat__witD_I7_J,axiom,
! [N: nat,A: mat_complex,B: mat_complex,P2: mat_complex,Q: mat_complex] :
( ( N
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P2 @ Q )
=> ( member_mat_complex @ Q @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD(7)
thf(fact_217_similar__mat__witD_I6_J,axiom,
! [N: nat,A: mat_nat,B: mat_nat,P2: mat_nat,Q: mat_nat] :
( ( N
= ( dim_row_nat @ A ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P2 @ Q )
=> ( member_mat_nat @ P2 @ ( carrier_mat_nat @ N @ N ) ) ) ) ).
% similar_mat_witD(6)
thf(fact_218_similar__mat__witD_I6_J,axiom,
! [N: nat,A: mat_complex,B: mat_complex,P2: mat_complex,Q: mat_complex] :
( ( N
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P2 @ Q )
=> ( member_mat_complex @ P2 @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD(6)
thf(fact_219_similar__mat__witD_I5_J,axiom,
! [N: nat,A: mat_nat,B: mat_nat,P2: mat_nat,Q: mat_nat] :
( ( N
= ( dim_row_nat @ A ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P2 @ Q )
=> ( member_mat_nat @ B @ ( carrier_mat_nat @ N @ N ) ) ) ) ).
% similar_mat_witD(5)
thf(fact_220_similar__mat__witD_I5_J,axiom,
! [N: nat,A: mat_complex,B: mat_complex,P2: mat_complex,Q: mat_complex] :
( ( N
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P2 @ Q )
=> ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD(5)
thf(fact_221_similar__mat__witD_I4_J,axiom,
! [N: nat,A: mat_nat,B: mat_nat,P2: mat_nat,Q: mat_nat] :
( ( N
= ( dim_row_nat @ A ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P2 @ Q )
=> ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ N ) ) ) ) ).
% similar_mat_witD(4)
thf(fact_222_similar__mat__witD_I4_J,axiom,
! [N: nat,A: mat_complex,B: mat_complex,P2: mat_complex,Q: mat_complex] :
( ( N
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P2 @ Q )
=> ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD(4)
thf(fact_223_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_224_similar__mat__wit__dim__row,axiom,
! [A: mat_complex,B: mat_complex,Q: mat_complex,R: mat_complex] :
( ( simila5774310414453981135omplex @ A @ B @ Q @ R )
=> ( ( dim_row_complex @ B )
= ( dim_row_complex @ A ) ) ) ).
% similar_mat_wit_dim_row
thf(fact_225_length__induct,axiom,
! [P2: list_complex > $o,Xs: list_complex] :
( ! [Xs2: list_complex] :
( ! [Ys: list_complex] :
( ( ord_less_nat @ ( size_s3451745648224563538omplex @ Ys ) @ ( size_s3451745648224563538omplex @ Xs2 ) )
=> ( P2 @ Ys ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_226_length__induct,axiom,
! [P2: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys ) @ ( size_size_list_a @ Xs2 ) )
=> ( P2 @ Ys ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_227_length__induct,axiom,
! [P2: list_mat_complex > $o,Xs: list_mat_complex] :
( ! [Xs2: list_mat_complex] :
( ! [Ys: list_mat_complex] :
( ( ord_less_nat @ ( size_s5969786470865220249omplex @ Ys ) @ ( size_s5969786470865220249omplex @ Xs2 ) )
=> ( P2 @ Ys ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_228_length__induct,axiom,
! [P2: list_mat_a > $o,Xs: list_mat_a] :
( ! [Xs2: list_mat_a] :
( ! [Ys: list_mat_a] :
( ( ord_less_nat @ ( size_size_list_mat_a @ Ys ) @ ( size_size_list_mat_a @ Xs2 ) )
=> ( P2 @ Ys ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_229_unitary__elim,axiom,
! [A: mat_complex,N: nat,B: mat_complex,P2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ P2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ P2 )
=> ( ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P2 @ A ) @ ( schur_5982229384592763574omplex @ P2 ) )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P2 @ B ) @ ( schur_5982229384592763574omplex @ P2 ) ) )
=> ( A = B ) ) ) ) ) ) ).
% unitary_elim
thf(fact_230_mult__adjoint__hermitian,axiom,
! [A: mat_complex,N: nat,M: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( comple8306762464034002205omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ A ) @ A ) ) ) ).
% mult_adjoint_hermitian
thf(fact_231_mat__conj__def,axiom,
( spectr5699176650994449695omplex
= ( ^ [U: mat_complex,V: mat_complex] : ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ U @ V ) @ ( schur_5982229384592763574omplex @ U ) ) ) ) ).
% mat_conj_def
thf(fact_232_mat__conj__adjoint,axiom,
! [U2: mat_complex,V2: mat_complex] :
( ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U2 ) @ V2 )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ U2 ) @ V2 ) @ U2 ) ) ).
% mat_conj_adjoint
thf(fact_233_unitary__adjoint,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ A )
=> ( comple6660659447773130958omplex @ ( schur_5982229384592763574omplex @ A ) ) ) ) ).
% unitary_adjoint
thf(fact_234_mat__conj__smult,axiom,
! [A: mat_complex,N: nat,U2: mat_complex,B: mat_complex,X: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ U2 @ B ) @ ( schur_5982229384592763574omplex @ U2 ) ) )
=> ( ( smult_mat_complex @ X @ A )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ U2 @ ( smult_mat_complex @ X @ B ) ) @ ( schur_5982229384592763574omplex @ U2 ) ) ) ) ) ) ) ).
% mat_conj_smult
thf(fact_235_step__3__similar,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( similar_mat_complex @ ( jordan4501759426295633263omplex @ A ) @ A ) ) ).
% step_3_similar
thf(fact_236_similar__mat__trans,axiom,
! [A: mat_complex,B: mat_complex,C2: mat_complex] :
( ( similar_mat_complex @ A @ B )
=> ( ( similar_mat_complex @ B @ C2 )
=> ( similar_mat_complex @ A @ C2 ) ) ) ).
% similar_mat_trans
thf(fact_237_similar__mat__smult,axiom,
! [A: mat_complex,B: mat_complex,K: complex] :
( ( similar_mat_complex @ A @ B )
=> ( similar_mat_complex @ ( smult_mat_complex @ K @ A ) @ ( smult_mat_complex @ K @ B ) ) ) ).
% similar_mat_smult
thf(fact_238_similar__mat__sym,axiom,
! [A: mat_complex,B: mat_complex] :
( ( similar_mat_complex @ A @ B )
=> ( similar_mat_complex @ B @ A ) ) ).
% similar_mat_sym
thf(fact_239_hermitian__mat__conj,axiom,
! [A: mat_complex,N: nat,U2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple8306762464034002205omplex @ A )
=> ( comple8306762464034002205omplex @ ( spectr5699176650994449695omplex @ U2 @ A ) ) ) ) ) ).
% hermitian_mat_conj
thf(fact_240_smult__carrier__mat,axiom,
! [A: mat_nat,Nr: nat,Nc: nat,K: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( member_mat_nat @ ( smult_mat_nat @ K @ A ) @ ( carrier_mat_nat @ Nr @ Nc ) ) ) ).
% smult_carrier_mat
thf(fact_241_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_242_smult__carrier__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,K: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( smult_mat_complex @ K @ A ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% smult_carrier_mat
thf(fact_243_index__smult__mat_I2_J,axiom,
! [A2: nat,A: mat_nat] :
( ( dim_row_nat @ ( smult_mat_nat @ A2 @ A ) )
= ( dim_row_nat @ A ) ) ).
% index_smult_mat(2)
thf(fact_244_index__smult__mat_I2_J,axiom,
! [A2: a,A: mat_a] :
( ( dim_row_a @ ( smult_mat_a @ A2 @ A ) )
= ( dim_row_a @ A ) ) ).
% index_smult_mat(2)
thf(fact_245_index__smult__mat_I2_J,axiom,
! [A2: complex,A: mat_complex] :
( ( dim_row_complex @ ( smult_mat_complex @ A2 @ A ) )
= ( dim_row_complex @ A ) ) ).
% index_smult_mat(2)
thf(fact_246_similar__mat__wit__smult,axiom,
! [A: mat_complex,B: mat_complex,P2: mat_complex,Q: mat_complex,K: complex] :
( ( simila5774310414453981135omplex @ A @ B @ P2 @ Q )
=> ( simila5774310414453981135omplex @ ( smult_mat_complex @ K @ A ) @ ( smult_mat_complex @ K @ B ) @ P2 @ Q ) ) ).
% similar_mat_wit_smult
thf(fact_247_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_248_similar__mat__refl,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( similar_mat_complex @ A @ A ) ) ).
% similar_mat_refl
thf(fact_249_diagonal__mat__smult,axiom,
! [A: mat_complex,X: complex] :
( ( diagonal_mat_complex @ A )
=> ( diagonal_mat_complex @ ( smult_mat_complex @ X @ A ) ) ) ).
% diagonal_mat_smult
thf(fact_250_diagonal__mat__smult,axiom,
! [A: mat_a,X: a] :
( ( diagonal_mat_a @ A )
=> ( diagonal_mat_a @ ( smult_mat_a @ X @ A ) ) ) ).
% diagonal_mat_smult
thf(fact_251_similar__mat__def,axiom,
( similar_mat_complex
= ( ^ [A4: mat_complex,B5: mat_complex] :
? [P5: mat_complex,X6: mat_complex] : ( simila5774310414453981135omplex @ A4 @ B5 @ P5 @ X6 ) ) ) ).
% similar_mat_def
thf(fact_252_hermitian__square__hermitian,axiom,
! [A: mat_complex] :
( ( comple8306762464034002205omplex @ A )
=> ( comple8306762464034002205omplex @ ( times_8009071140041733218omplex @ A @ A ) ) ) ).
% hermitian_square_hermitian
thf(fact_253_hermitian__mat__conj_H,axiom,
! [A: mat_complex,N: nat,U2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple8306762464034002205omplex @ A )
=> ( comple8306762464034002205omplex @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U2 ) @ A ) ) ) ) ) ).
% hermitian_mat_conj'
thf(fact_254_mat__conj__unit__commute,axiom,
! [U2: mat_complex,A: mat_complex,N: nat] :
( ( comple6660659447773130958omplex @ U2 )
=> ( ( ( times_8009071140041733218omplex @ U2 @ A )
= ( times_8009071140041733218omplex @ A @ U2 ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr5699176650994449695omplex @ U2 @ A )
= A ) ) ) ) ) ).
% mat_conj_unit_commute
thf(fact_255_mult__smult__assoc__mat,axiom,
! [A: mat_nat,Nr: nat,N: nat,B: mat_nat,Nc: nat,K: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ N ) )
=> ( ( member_mat_nat @ B @ ( carrier_mat_nat @ N @ Nc ) )
=> ( ( times_times_mat_nat @ ( smult_mat_nat @ K @ A ) @ B )
= ( smult_mat_nat @ K @ ( times_times_mat_nat @ A @ B ) ) ) ) ) ).
% mult_smult_assoc_mat
thf(fact_256_mult__smult__assoc__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,B: mat_complex,Nc: nat,K: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( smult_mat_complex @ K @ A ) @ B )
= ( smult_mat_complex @ K @ ( times_8009071140041733218omplex @ A @ B ) ) ) ) ) ).
% mult_smult_assoc_mat
thf(fact_257_mult__smult__distrib,axiom,
! [A: mat_nat,Nr: nat,N: nat,B: mat_nat,Nc: nat,K: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ N ) )
=> ( ( member_mat_nat @ B @ ( carrier_mat_nat @ N @ Nc ) )
=> ( ( times_times_mat_nat @ A @ ( smult_mat_nat @ K @ B ) )
= ( smult_mat_nat @ K @ ( times_times_mat_nat @ A @ B ) ) ) ) ) ).
% mult_smult_distrib
thf(fact_258_mult__smult__distrib,axiom,
! [A: mat_complex,Nr: nat,N: nat,B: mat_complex,Nc: nat,K: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ A @ ( smult_mat_complex @ K @ B ) )
= ( smult_mat_complex @ K @ ( times_8009071140041733218omplex @ A @ B ) ) ) ) ) ).
% mult_smult_distrib
thf(fact_259_hermitian__is__normal,axiom,
! [A: mat_complex] :
( ( comple8306762464034002205omplex @ A )
=> ( ( times_8009071140041733218omplex @ A @ ( schur_5982229384592763574omplex @ A ) )
= ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ A ) @ A ) ) ) ).
% hermitian_is_normal
thf(fact_260_unitary__times__unitary,axiom,
! [P2: mat_complex,N: nat,Q: mat_complex] :
( ( member_mat_complex @ P2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ Q @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ P2 )
=> ( ( comple6660659447773130958omplex @ Q )
=> ( comple6660659447773130958omplex @ ( times_8009071140041733218omplex @ P2 @ Q ) ) ) ) ) ) ).
% unitary_times_unitary
thf(fact_261_hermitian__square,axiom,
! [M4: mat_complex] :
( ( comple8306762464034002205omplex @ M4 )
=> ( member_mat_complex @ M4 @ ( carrier_mat_complex @ ( dim_row_complex @ M4 ) @ ( dim_row_complex @ M4 ) ) ) ) ).
% hermitian_square
thf(fact_262_Complex__Matrix_Ounitary__def,axiom,
( comple6660659447773130958omplex
= ( ^ [A4: mat_complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ ( dim_row_complex @ A4 ) @ ( dim_row_complex @ A4 ) ) )
& ( inverts_mat_complex @ A4 @ ( schur_5982229384592763574omplex @ A4 ) ) ) ) ) ).
% Complex_Matrix.unitary_def
thf(fact_263_smult__smult__times,axiom,
! [A2: set_complex,K: set_complex,A: mat_set_complex] :
( ( smult_4557042052056852367omplex @ A2 @ ( smult_4557042052056852367omplex @ K @ A ) )
= ( smult_4557042052056852367omplex @ ( times_6048082448287401577omplex @ A2 @ K ) @ A ) ) ).
% smult_smult_times
thf(fact_264_smult__smult__times,axiom,
! [A2: set_nat,K: set_nat,A: mat_set_nat] :
( ( smult_mat_set_nat @ A2 @ ( smult_mat_set_nat @ K @ A ) )
= ( smult_mat_set_nat @ ( times_times_set_nat @ A2 @ K ) @ A ) ) ).
% smult_smult_times
thf(fact_265_smult__smult__times,axiom,
! [A2: a,K: a,A: mat_a] :
( ( smult_mat_a @ A2 @ ( smult_mat_a @ K @ A ) )
= ( smult_mat_a @ ( times_times_a @ A2 @ K ) @ A ) ) ).
% smult_smult_times
thf(fact_266_smult__smult__times,axiom,
! [A2: nat,K: nat,A: mat_nat] :
( ( smult_mat_nat @ A2 @ ( smult_mat_nat @ K @ A ) )
= ( smult_mat_nat @ ( times_times_nat @ A2 @ K ) @ A ) ) ).
% smult_smult_times
thf(fact_267_smult__smult__times,axiom,
! [A2: complex,K: complex,A: mat_complex] :
( ( smult_mat_complex @ A2 @ ( smult_mat_complex @ K @ A ) )
= ( smult_mat_complex @ ( times_times_complex @ A2 @ K ) @ A ) ) ).
% smult_smult_times
thf(fact_268_unitaryD2,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ A )
=> ( inverts_mat_complex @ ( schur_5982229384592763574omplex @ A ) @ A ) ) ) ).
% unitaryD2
thf(fact_269_unitary__is__corthogonal,axiom,
! [U2: mat_complex,N: nat] :
( ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ U2 )
=> ( schur_549222400177443379omplex @ U2 ) ) ) ).
% unitary_is_corthogonal
thf(fact_270_step__1__similar,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( similar_mat_complex @ ( jordan2017415923357163885omplex @ A ) @ A ) ) ).
% step_1_similar
thf(fact_271_step__2__similar,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( similar_mat_complex @ ( jordan7871273693253786478omplex @ A ) @ A ) ) ).
% step_2_similar
thf(fact_272_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_273_trace__smult,axiom,
! [A: mat_complex,N: nat,C: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple3184165445352484367omplex @ ( smult_mat_complex @ C @ A ) )
= ( times_times_complex @ C @ ( comple3184165445352484367omplex @ A ) ) ) ) ).
% trace_smult
thf(fact_274_inverts__mat__unique,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( inverts_mat_complex @ A @ B )
=> ( ( inverts_mat_complex @ A @ C2 )
=> ( B = C2 ) ) ) ) ) ) ).
% inverts_mat_unique
thf(fact_275_inverts__mat__symm,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( inverts_mat_complex @ A @ B )
=> ( inverts_mat_complex @ B @ A ) ) ) ) ).
% inverts_mat_symm
thf(fact_276_triangular__to__jnf__steps__dims_I3_J,axiom,
! [A: mat_complex] :
( ( dim_row_complex @ ( jordan7871273693253786478omplex @ A ) )
= ( dim_row_complex @ A ) ) ).
% triangular_to_jnf_steps_dims(3)
thf(fact_277_triangular__to__jnf__steps__dims_I1_J,axiom,
! [A: mat_complex] :
( ( dim_row_complex @ ( jordan2017415923357163885omplex @ A ) )
= ( dim_row_complex @ A ) ) ).
% triangular_to_jnf_steps_dims(1)
thf(fact_278_trace__comm,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ A @ B ) )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ B @ A ) ) ) ) ) ).
% trace_comm
thf(fact_279_step__1__2__inv_I3_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( upper_4850907204721561915omplex @ A )
=> ( ( B
= ( jordan7871273693253786478omplex @ ( jordan2017415923357163885omplex @ A ) ) )
=> ( jordan4650062548456832493omplex @ N @ B ) ) ) ) ).
% step_1_2_inv(3)
thf(fact_280_projector__def,axiom,
( linear5633924348262549461omplex
= ( ^ [M5: mat_complex] :
( ( comple8306762464034002205omplex @ M5 )
& ( ( times_8009071140041733218omplex @ M5 @ M5 )
= M5 ) ) ) ) ).
% projector_def
thf(fact_281_invertible__mat__def,axiom,
( invert2568027935824841882omplex
= ( ^ [A4: mat_complex] :
( ( square_mat_complex @ A4 )
& ? [B5: mat_complex] :
( ( inverts_mat_complex @ A4 @ B5 )
& ( inverts_mat_complex @ B5 @ A4 ) ) ) ) ) ).
% invertible_mat_def
thf(fact_282_step__1__2__inv_I2_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( upper_4850907204721561915omplex @ A )
=> ( ( B
= ( jordan7871273693253786478omplex @ ( jordan2017415923357163885omplex @ A ) ) )
=> ( jordan5244935068081719878omplex @ N @ jordan8650160714669549932omplex @ B ) ) ) ) ).
% step_1_2_inv(2)
thf(fact_283_step__1__2__inv_I1_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( upper_4850907204721561915omplex @ A )
=> ( ( B
= ( jordan7871273693253786478omplex @ ( jordan2017415923357163885omplex @ A ) ) )
=> ( jordan5244935068081719878omplex @ N @ jordan3528196489273997576omplex @ B ) ) ) ) ).
% step_1_2_inv(1)
thf(fact_284_unitary__simps_I1_J,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ A )
=> ( ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ A ) @ A )
= ( one_mat_complex @ N ) ) ) ) ).
% unitary_simps(1)
thf(fact_285_unitary__simps_I2_J,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ A )
=> ( ( times_8009071140041733218omplex @ A @ ( schur_5982229384592763574omplex @ A ) )
= ( one_mat_complex @ N ) ) ) ) ).
% unitary_simps(2)
thf(fact_286_projector__square__eq,axiom,
! [M4: mat_complex] :
( ( linear5633924348262549461omplex @ M4 )
=> ( ( times_8009071140041733218omplex @ M4 @ M4 )
= M4 ) ) ).
% projector_square_eq
thf(fact_287_one__carrier__mat,axiom,
! [N: nat] : ( member_mat_nat @ ( one_mat_nat @ N ) @ ( carrier_mat_nat @ N @ N ) ) ).
% one_carrier_mat
thf(fact_288_one__carrier__mat,axiom,
! [N: nat] : ( member_mat_complex @ ( one_mat_complex @ N ) @ ( carrier_mat_complex @ N @ N ) ) ).
% one_carrier_mat
thf(fact_289_index__one__mat_I2_J,axiom,
! [N: nat] :
( ( dim_row_nat @ ( one_mat_nat @ N ) )
= N ) ).
% index_one_mat(2)
thf(fact_290_index__one__mat_I2_J,axiom,
! [N: nat] :
( ( dim_row_complex @ ( one_mat_complex @ N ) )
= N ) ).
% index_one_mat(2)
thf(fact_291_upper__triangular__one,axiom,
! [N: nat] : ( upper_4850907204721561915omplex @ ( one_mat_complex @ N ) ) ).
% upper_triangular_one
thf(fact_292_inv__allD,axiom,
! [N: nat,P6: mat_complex > nat > nat > $o,A: mat_complex,I2: nat,J2: nat] :
( ( jordan5244935068081719878omplex @ N @ P6 @ A )
=> ( ( ord_less_nat @ I2 @ N )
=> ( ( ord_less_nat @ J2 @ N )
=> ( P6 @ A @ I2 @ J2 ) ) ) ) ).
% inv_allD
thf(fact_293_inv__all__def,axiom,
( jordan5244935068081719878omplex
= ( ^ [N3: nat,P: mat_complex > nat > nat > $o,A4: mat_complex] :
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ N3 )
=> ( ( ord_less_nat @ J @ N3 )
=> ( P @ A4 @ I @ J ) ) ) ) ) ).
% inv_all_def
thf(fact_294_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_295_right__mult__one__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( times_8009071140041733218omplex @ A @ ( one_mat_complex @ Nc ) )
= A ) ) ).
% right_mult_one_mat
thf(fact_296_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_297_left__mult__one__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( one_mat_complex @ Nr ) @ A )
= A ) ) ).
% left_mult_one_mat
thf(fact_298_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_299_left__mult__one__mat_H,axiom,
! [A: mat_complex,N: nat] :
( ( ( dim_row_complex @ A )
= N )
=> ( ( times_8009071140041733218omplex @ ( one_mat_complex @ N ) @ A )
= A ) ) ).
% left_mult_one_mat'
thf(fact_300_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_301_similar__mat__wit__refl,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( simila5774310414453981135omplex @ A @ A @ ( one_mat_complex @ N ) @ ( one_mat_complex @ N ) ) ) ).
% similar_mat_wit_refl
thf(fact_302_similar__mat__witD2_I2_J,axiom,
! [A: mat_nat,N: nat,M: nat,B: mat_nat,P2: mat_nat,Q: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ M ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P2 @ Q )
=> ( ( times_times_mat_nat @ Q @ P2 )
= ( one_mat_nat @ N ) ) ) ) ).
% similar_mat_witD2(2)
thf(fact_303_similar__mat__witD2_I2_J,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,P2: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P2 @ Q )
=> ( ( times_8009071140041733218omplex @ Q @ P2 )
= ( one_mat_complex @ N ) ) ) ) ).
% similar_mat_witD2(2)
thf(fact_304_similar__mat__witD2_I1_J,axiom,
! [A: mat_nat,N: nat,M: nat,B: mat_nat,P2: mat_nat,Q: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ M ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P2 @ Q )
=> ( ( times_times_mat_nat @ P2 @ Q )
= ( one_mat_nat @ N ) ) ) ) ).
% similar_mat_witD2(1)
thf(fact_305_similar__mat__witD2_I1_J,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,P2: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P2 @ Q )
=> ( ( times_8009071140041733218omplex @ P2 @ Q )
= ( one_mat_complex @ N ) ) ) ) ).
% similar_mat_witD2(1)
thf(fact_306_similar__mat__witI,axiom,
! [P2: mat_nat,Q: mat_nat,N: nat,A: mat_nat,B: mat_nat] :
( ( ( times_times_mat_nat @ P2 @ Q )
= ( one_mat_nat @ N ) )
=> ( ( ( times_times_mat_nat @ Q @ P2 )
= ( one_mat_nat @ N ) )
=> ( ( A
= ( times_times_mat_nat @ ( times_times_mat_nat @ P2 @ B ) @ Q ) )
=> ( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ N ) )
=> ( ( member_mat_nat @ B @ ( carrier_mat_nat @ N @ N ) )
=> ( ( member_mat_nat @ P2 @ ( carrier_mat_nat @ N @ N ) )
=> ( ( member_mat_nat @ Q @ ( carrier_mat_nat @ N @ N ) )
=> ( similar_mat_wit_nat @ A @ B @ P2 @ Q ) ) ) ) ) ) ) ) ).
% similar_mat_witI
thf(fact_307_similar__mat__witI,axiom,
! [P2: mat_complex,Q: mat_complex,N: nat,A: mat_complex,B: mat_complex] :
( ( ( times_8009071140041733218omplex @ P2 @ Q )
= ( one_mat_complex @ N ) )
=> ( ( ( times_8009071140041733218omplex @ Q @ P2 )
= ( one_mat_complex @ N ) )
=> ( ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P2 @ B ) @ Q ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ P2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ Q @ ( carrier_mat_complex @ N @ N ) )
=> ( simila5774310414453981135omplex @ A @ B @ P2 @ Q ) ) ) ) ) ) ) ) ).
% similar_mat_witI
thf(fact_308_similar__mat__witD_I1_J,axiom,
! [N: nat,A: mat_nat,B: mat_nat,P2: mat_nat,Q: mat_nat] :
( ( N
= ( dim_row_nat @ A ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P2 @ Q )
=> ( ( times_times_mat_nat @ P2 @ Q )
= ( one_mat_nat @ N ) ) ) ) ).
% similar_mat_witD(1)
thf(fact_309_similar__mat__witD_I1_J,axiom,
! [N: nat,A: mat_complex,B: mat_complex,P2: mat_complex,Q: mat_complex] :
( ( N
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P2 @ Q )
=> ( ( times_8009071140041733218omplex @ P2 @ Q )
= ( one_mat_complex @ N ) ) ) ) ).
% similar_mat_witD(1)
thf(fact_310_similar__mat__witD_I2_J,axiom,
! [N: nat,A: mat_nat,B: mat_nat,P2: mat_nat,Q: mat_nat] :
( ( N
= ( dim_row_nat @ A ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P2 @ Q )
=> ( ( times_times_mat_nat @ Q @ P2 )
= ( one_mat_nat @ N ) ) ) ) ).
% similar_mat_witD(2)
thf(fact_311_similar__mat__witD_I2_J,axiom,
! [N: nat,A: mat_complex,B: mat_complex,P2: mat_complex,Q: mat_complex] :
( ( N
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P2 @ Q )
=> ( ( times_8009071140041733218omplex @ Q @ P2 )
= ( one_mat_complex @ N ) ) ) ) ).
% similar_mat_witD(2)
thf(fact_312_inverts__mat__def,axiom,
( inverts_mat_nat
= ( ^ [A4: mat_nat,B5: mat_nat] :
( ( times_times_mat_nat @ A4 @ B5 )
= ( one_mat_nat @ ( dim_row_nat @ A4 ) ) ) ) ) ).
% inverts_mat_def
thf(fact_313_inverts__mat__def,axiom,
( inverts_mat_complex
= ( ^ [A4: mat_complex,B5: mat_complex] :
( ( times_8009071140041733218omplex @ A4 @ B5 )
= ( one_mat_complex @ ( dim_row_complex @ A4 ) ) ) ) ) ).
% inverts_mat_def
thf(fact_314_mat__mult__left__right__inverse,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ( times_8009071140041733218omplex @ A @ B )
= ( one_mat_complex @ N ) )
=> ( ( times_8009071140041733218omplex @ B @ A )
= ( one_mat_complex @ N ) ) ) ) ) ).
% mat_mult_left_right_inverse
thf(fact_315_ev__blocks__def,axiom,
jordan4650062548456832493omplex = jordan4637981584770492064omplex ).
% ev_blocks_def
thf(fact_316_gauss__jordan__single_I4_J,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,C2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( ( gauss_4244865067341541924omplex @ A )
= C2 )
=> ? [P7: mat_complex,Q3: mat_complex] :
( ( C2
= ( times_8009071140041733218omplex @ P7 @ A ) )
& ( member_mat_complex @ P7 @ ( carrier_mat_complex @ Nr @ Nr ) )
& ( member_mat_complex @ Q3 @ ( carrier_mat_complex @ Nr @ Nr ) )
& ( ( times_8009071140041733218omplex @ P7 @ Q3 )
= ( one_mat_complex @ Nr ) )
& ( ( times_8009071140041733218omplex @ Q3 @ P7 )
= ( one_mat_complex @ Nr ) ) ) ) ) ).
% gauss_jordan_single(4)
thf(fact_317_jb__imp__diff__ev,axiom,
! [A: mat_complex,I2: nat,J2: nat] :
( ( jordan4971026570492200526omplex @ A @ I2 @ J2 )
=> ( jordan8650160714669549932omplex @ A @ I2 @ J2 ) ) ).
% jb_imp_diff_ev
thf(fact_318_swap__cols__rows__similar,axiom,
! [A: mat_nat,N: nat,K: nat,L: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ N ) )
=> ( ( ord_less_nat @ K @ N )
=> ( ( ord_less_nat @ L @ N )
=> ( similar_mat_nat @ ( column141131285749525182ws_nat @ K @ L @ A ) @ A ) ) ) ) ).
% swap_cols_rows_similar
thf(fact_319_swap__cols__rows__similar,axiom,
! [A: mat_complex,N: nat,K: nat,L: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ K @ N )
=> ( ( ord_less_nat @ L @ N )
=> ( similar_mat_complex @ ( column7161609239796038556omplex @ K @ L @ A ) @ A ) ) ) ) ).
% swap_cols_rows_similar
thf(fact_320_jb__imp__uppert,axiom,
! [A: mat_complex,I2: nat,J2: nat] :
( ( jordan4971026570492200526omplex @ A @ I2 @ J2 )
=> ( jordan3528196489273997576omplex @ A @ I2 @ J2 ) ) ).
% jb_imp_uppert
thf(fact_321_partition__jb_I1_J,axiom,
! [A: mat_complex,N: nat,Bs: list_mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( jordan5244935068081719878omplex @ N @ jordan3528196489273997576omplex @ A )
=> ( ( jordan5244935068081719878omplex @ N @ jordan8650160714669549932omplex @ A )
=> ( ( jordan4650062548456832493omplex @ N @ A )
=> ( ( ( jordan5009815537632354121omplex @ A @ nil_mat_complex )
= Bs )
=> ( A
= ( diag_b9145358668110806138omplex @ Bs ) ) ) ) ) ) ) ).
% partition_jb(1)
thf(fact_322_hermitian__decomp__diag__mat,axiom,
! [A: mat_complex,B: mat_complex,U2: mat_complex] :
( ( projec5943904436471448624omplex @ A @ B @ U2 )
=> ( diagonal_mat_complex @ B ) ) ).
% hermitian_decomp_diag_mat
thf(fact_323_swap__cols__rows__carrier_I3_J,axiom,
! [A: mat_nat,N: nat,K: nat,L: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ N ) )
=> ( member_mat_nat @ ( column141131285749525182ws_nat @ K @ L @ A ) @ ( carrier_mat_nat @ N @ N ) ) ) ).
% swap_cols_rows_carrier(3)
thf(fact_324_swap__cols__rows__carrier_I3_J,axiom,
! [A: mat_a,N: nat,K: nat,L: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( member_mat_a @ ( column5129559316938501008rows_a @ K @ L @ A ) @ ( carrier_mat_a @ N @ N ) ) ) ).
% swap_cols_rows_carrier(3)
thf(fact_325_swap__cols__rows__carrier_I3_J,axiom,
! [A: mat_complex,N: nat,K: nat,L: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( member_mat_complex @ ( column7161609239796038556omplex @ K @ L @ A ) @ ( carrier_mat_complex @ N @ N ) ) ) ).
% swap_cols_rows_carrier(3)
thf(fact_326_swap__cols__rows__carrier_I1_J,axiom,
! [K: nat,L: nat,A: mat_nat] :
( ( dim_row_nat @ ( column141131285749525182ws_nat @ K @ L @ A ) )
= ( dim_row_nat @ A ) ) ).
% swap_cols_rows_carrier(1)
thf(fact_327_swap__cols__rows__carrier_I1_J,axiom,
! [K: nat,L: nat,A: mat_a] :
( ( dim_row_a @ ( column5129559316938501008rows_a @ K @ L @ A ) )
= ( dim_row_a @ A ) ) ).
% swap_cols_rows_carrier(1)
thf(fact_328_swap__cols__rows__carrier_I1_J,axiom,
! [K: nat,L: nat,A: mat_complex] :
( ( dim_row_complex @ ( column7161609239796038556omplex @ K @ L @ A ) )
= ( dim_row_complex @ A ) ) ).
% swap_cols_rows_carrier(1)
thf(fact_329_gauss__jordan__single_I2_J,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,C2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( ( gauss_4244865067341541924omplex @ A )
= C2 )
=> ( member_mat_complex @ C2 @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% gauss_jordan_single(2)
thf(fact_330_partition__jb_I2_J,axiom,
! [A: mat_complex,N: nat,Bs: list_mat_complex,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( jordan5244935068081719878omplex @ N @ jordan3528196489273997576omplex @ A )
=> ( ( jordan5244935068081719878omplex @ N @ jordan8650160714669549932omplex @ A )
=> ( ( jordan4650062548456832493omplex @ N @ A )
=> ( ( ( jordan5009815537632354121omplex @ A @ nil_mat_complex )
= Bs )
=> ( ( member_mat_complex @ B @ ( set_mat_complex2 @ Bs ) )
=> ( ( jordan5032732407113867375omplex @ jordan3528196489273997576omplex @ B )
& ( jordan8042990603089931364omplex @ ( dim_col_complex @ B ) @ B )
& ( ( dim_row_complex @ B )
= ( dim_col_complex @ B ) ) ) ) ) ) ) ) ) ).
% partition_jb(2)
thf(fact_331_add__col__sub__row__similar,axiom,
! [A: mat_complex,N: nat,K: nat,L: nat,A2: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ K @ N )
=> ( ( ord_less_nat @ L @ N )
=> ( ( K != L )
=> ( similar_mat_complex @ ( column6029646570091773654omplex @ A2 @ K @ L @ A ) @ A ) ) ) ) ) ).
% add_col_sub_row_similar
thf(fact_332_uppert__to__jb,axiom,
! [N: nat,A: mat_complex] :
( ( jordan5244935068081719878omplex @ N @ jordan3528196489273997576omplex @ A )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( jordan5475473882837061487omplex @ N @ jordan4971026570492200526omplex @ A @ one_one_nat ) ) ) ).
% uppert_to_jb
thf(fact_333_swaprows__mat__inv,axiom,
! [K: nat,N: nat,L: nat] :
( ( ord_less_nat @ K @ N )
=> ( ( ord_less_nat @ L @ N )
=> ( ( times_8009071140041733218omplex @ ( gauss_8970452565587180529omplex @ N @ K @ L ) @ ( gauss_8970452565587180529omplex @ N @ K @ L ) )
= ( one_mat_complex @ N ) ) ) ) ).
% swaprows_mat_inv
thf(fact_334_gauss__jordan__single_I3_J,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,C2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( ( gauss_4244865067341541924omplex @ A )
= C2 )
=> ( gauss_194721375535881179omplex @ C2 ) ) ) ).
% gauss_jordan_single(3)
thf(fact_335_length__greater__0__conv,axiom,
! [Xs: list_complex] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs ) )
= ( Xs != nil_complex ) ) ).
% length_greater_0_conv
thf(fact_336_length__greater__0__conv,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
= ( Xs != nil_a ) ) ).
% length_greater_0_conv
thf(fact_337_length__greater__0__conv,axiom,
! [Xs: list_mat_complex] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5969786470865220249omplex @ Xs ) )
= ( Xs != nil_mat_complex ) ) ).
% length_greater_0_conv
thf(fact_338_length__greater__0__conv,axiom,
! [Xs: list_mat_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_mat_a @ Xs ) )
= ( Xs != nil_mat_a ) ) ).
% length_greater_0_conv
thf(fact_339_hermitian__decomp__dim__carrier,axiom,
! [A: mat_complex,B: mat_complex,U2: mat_complex] :
( ( projec5943904436471448624omplex @ A @ B @ U2 )
=> ( member_mat_complex @ B @ ( carrier_mat_complex @ ( dim_row_complex @ A ) @ ( dim_col_complex @ A ) ) ) ) ).
% hermitian_decomp_dim_carrier
thf(fact_340_verit__eq__simplify_I7_J,axiom,
zero_zero_nat != one_one_nat ).
% verit_eq_simplify(7)
thf(fact_341_verit__eq__simplify_I7_J,axiom,
zero_zero_complex != one_one_complex ).
% verit_eq_simplify(7)
thf(fact_342_verit__comp__simplify_I28_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% verit_comp_simplify(28)
thf(fact_343_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_344_less__numeral__extra_I2_J,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% less_numeral_extra(2)
thf(fact_345_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_346_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_347_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
= ( X = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_348_one__reorient,axiom,
! [X: complex] :
( ( one_one_complex = X )
= ( X = one_one_complex ) ) ).
% one_reorient
thf(fact_349_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_350_mult__cancel__left1,axiom,
! [C: complex,B2: complex] :
( ( C
= ( times_times_complex @ C @ B2 ) )
= ( ( C = zero_zero_complex )
| ( B2 = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_351_mult__cancel__left2,axiom,
! [C: complex,A2: complex] :
( ( ( times_times_complex @ C @ A2 )
= C )
= ( ( C = zero_zero_complex )
| ( A2 = one_one_complex ) ) ) ).
% mult_cancel_left2
thf(fact_352_mult__cancel__right1,axiom,
! [C: complex,B2: complex] :
( ( C
= ( times_times_complex @ B2 @ C ) )
= ( ( C = zero_zero_complex )
| ( B2 = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_353_mult__cancel__right2,axiom,
! [A2: complex,C: complex] :
( ( ( times_times_complex @ A2 @ C )
= C )
= ( ( C = zero_zero_complex )
| ( A2 = one_one_complex ) ) ) ).
% mult_cancel_right2
thf(fact_354_class__field_Ozero__not__one,axiom,
zero_zero_complex != one_one_complex ).
% class_field.zero_not_one
thf(fact_355_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_356_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_357_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_358_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_359_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_360_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_361_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_362_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_363_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_364_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_365_length__pos__if__in__set,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_366_length__pos__if__in__set,axiom,
! [X: mat_nat,Xs: list_mat_nat] :
( ( member_mat_nat @ X @ ( set_mat_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s66138613738048955at_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_367_length__pos__if__in__set,axiom,
! [X: complex,Xs: list_complex] :
( ( member_complex @ X @ ( set_complex2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_368_length__pos__if__in__set,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_369_length__pos__if__in__set,axiom,
! [X: mat_a,Xs: list_mat_a] :
( ( member_mat_a @ X @ ( set_mat_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_mat_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_370_length__pos__if__in__set,axiom,
! [X: mat_complex,Xs: list_mat_complex] :
( ( member_mat_complex @ X @ ( set_mat_complex2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s5969786470865220249omplex @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_371_carrier__matD_I2_J,axiom,
! [A: mat_nat,Nr: nat,Nc: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( dim_col_nat @ A )
= Nc ) ) ).
% carrier_matD(2)
thf(fact_372_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_373_carrier__matD_I2_J,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( dim_col_complex @ A )
= Nc ) ) ).
% carrier_matD(2)
thf(fact_374_index__mult__mat_I3_J,axiom,
! [A: mat_nat,B: mat_nat] :
( ( dim_col_nat @ ( times_times_mat_nat @ A @ B ) )
= ( dim_col_nat @ B ) ) ).
% index_mult_mat(3)
thf(fact_375_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_376_index__mult__mat_I3_J,axiom,
! [A: mat_complex,B: mat_complex] :
( ( dim_col_complex @ ( times_8009071140041733218omplex @ A @ B ) )
= ( dim_col_complex @ B ) ) ).
% index_mult_mat(3)
thf(fact_377_index__one__mat_I3_J,axiom,
! [N: nat] :
( ( dim_col_nat @ ( one_mat_nat @ N ) )
= N ) ).
% index_one_mat(3)
thf(fact_378_index__one__mat_I3_J,axiom,
! [N: nat] :
( ( dim_col_complex @ ( one_mat_complex @ N ) )
= N ) ).
% index_one_mat(3)
thf(fact_379_index__smult__mat_I3_J,axiom,
! [A2: a,A: mat_a] :
( ( dim_col_a @ ( smult_mat_a @ A2 @ A ) )
= ( dim_col_a @ A ) ) ).
% index_smult_mat(3)
thf(fact_380_index__smult__mat_I3_J,axiom,
! [A2: nat,A: mat_nat] :
( ( dim_col_nat @ ( smult_mat_nat @ A2 @ A ) )
= ( dim_col_nat @ A ) ) ).
% index_smult_mat(3)
thf(fact_381_index__smult__mat_I3_J,axiom,
! [A2: complex,A: mat_complex] :
( ( dim_col_complex @ ( smult_mat_complex @ A2 @ A ) )
= ( dim_col_complex @ A ) ) ).
% index_smult_mat(3)
thf(fact_382_mult__1,axiom,
! [A2: set_complex] :
( ( times_6048082448287401577omplex @ one_one_set_complex @ A2 )
= A2 ) ).
% mult_1
thf(fact_383_mult__1,axiom,
! [A2: set_nat] :
( ( times_times_set_nat @ one_one_set_nat @ A2 )
= A2 ) ).
% mult_1
thf(fact_384_mult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% mult_1
thf(fact_385_mult__1,axiom,
! [A2: complex] :
( ( times_times_complex @ one_one_complex @ A2 )
= A2 ) ).
% mult_1
thf(fact_386_vector__space__over__itself_Oscale__one,axiom,
! [X: complex] :
( ( times_times_complex @ one_one_complex @ X )
= X ) ).
% vector_space_over_itself.scale_one
thf(fact_387_mult_Ocomm__neutral,axiom,
! [A2: set_complex] :
( ( times_6048082448287401577omplex @ A2 @ one_one_set_complex )
= A2 ) ).
% mult.comm_neutral
thf(fact_388_mult_Ocomm__neutral,axiom,
! [A2: set_nat] :
( ( times_times_set_nat @ A2 @ one_one_set_nat )
= A2 ) ).
% mult.comm_neutral
thf(fact_389_mult_Ocomm__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.comm_neutral
thf(fact_390_mult_Ocomm__neutral,axiom,
! [A2: complex] :
( ( times_times_complex @ A2 @ one_one_complex )
= A2 ) ).
% mult.comm_neutral
thf(fact_391_mult_Oright__neutral,axiom,
! [A2: set_complex] :
( ( times_6048082448287401577omplex @ A2 @ one_one_set_complex )
= A2 ) ).
% mult.right_neutral
thf(fact_392_mult_Oright__neutral,axiom,
! [A2: set_nat] :
( ( times_times_set_nat @ A2 @ one_one_set_nat )
= A2 ) ).
% mult.right_neutral
thf(fact_393_mult_Oright__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.right_neutral
thf(fact_394_mult_Oright__neutral,axiom,
! [A2: complex] :
( ( times_times_complex @ A2 @ one_one_complex )
= A2 ) ).
% mult.right_neutral
thf(fact_395_comm__monoid__mult__class_Omult__1,axiom,
! [A2: set_complex] :
( ( times_6048082448287401577omplex @ one_one_set_complex @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_396_comm__monoid__mult__class_Omult__1,axiom,
! [A2: set_nat] :
( ( times_times_set_nat @ one_one_set_nat @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_397_comm__monoid__mult__class_Omult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_398_comm__monoid__mult__class_Omult__1,axiom,
! [A2: complex] :
( ( times_times_complex @ one_one_complex @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_399_rel__simps_I71_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% rel_simps(71)
thf(fact_400_mult__right__cancel,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B2 @ C ) )
= ( A2 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_401_mult__right__cancel,axiom,
! [C: complex,A2: complex,B2: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ A2 @ C )
= ( times_times_complex @ B2 @ C ) )
= ( A2 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_402_mult__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B2 @ C ) )
= ( ( C = zero_zero_nat )
| ( A2 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_403_mult__cancel__right,axiom,
! [A2: complex,C: complex,B2: complex] :
( ( ( times_times_complex @ A2 @ C )
= ( times_times_complex @ B2 @ C ) )
= ( ( C = zero_zero_complex )
| ( A2 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_404_mult__left__cancel,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B2 ) )
= ( A2 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_405_mult__left__cancel,axiom,
! [C: complex,A2: complex,B2: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ C @ A2 )
= ( times_times_complex @ C @ B2 ) )
= ( A2 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_406_mult__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B2 ) )
= ( ( C = zero_zero_nat )
| ( A2 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_407_mult__cancel__left,axiom,
! [C: complex,A2: complex,B2: complex] :
( ( ( times_times_complex @ C @ A2 )
= ( times_times_complex @ C @ B2 ) )
= ( ( C = zero_zero_complex )
| ( A2 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_408_no__zero__divisors,axiom,
! [A2: nat,B2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( B2 != zero_zero_nat )
=> ( ( times_times_nat @ A2 @ B2 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_409_no__zero__divisors,axiom,
! [A2: complex,B2: complex] :
( ( A2 != zero_zero_complex )
=> ( ( B2 != zero_zero_complex )
=> ( ( times_times_complex @ A2 @ B2 )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_410_mult__eq__0__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_411_mult__eq__0__iff,axiom,
! [A2: complex,B2: complex] :
( ( ( times_times_complex @ A2 @ B2 )
= zero_zero_complex )
= ( ( A2 = zero_zero_complex )
| ( B2 = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_412_divisors__zero,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
= zero_zero_nat )
=> ( ( A2 = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_413_divisors__zero,axiom,
! [A2: complex,B2: complex] :
( ( ( times_times_complex @ A2 @ B2 )
= zero_zero_complex )
=> ( ( A2 = zero_zero_complex )
| ( B2 = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_414_mult__zero__right,axiom,
! [A2: a] :
( ( times_times_a @ A2 @ zero_zero_a )
= zero_zero_a ) ).
% mult_zero_right
thf(fact_415_mult__zero__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_416_mult__zero__right,axiom,
! [A2: complex] :
( ( times_times_complex @ A2 @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_417_mult__zero__left,axiom,
! [A2: a] :
( ( times_times_a @ zero_zero_a @ A2 )
= zero_zero_a ) ).
% mult_zero_left
thf(fact_418_mult__zero__left,axiom,
! [A2: nat] :
( ( times_times_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_419_mult__zero__left,axiom,
! [A2: complex] :
( ( times_times_complex @ zero_zero_complex @ A2 )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_420_mult__not__zero,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
!= zero_zero_nat )
=> ( ( A2 != zero_zero_nat )
& ( B2 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_421_mult__not__zero,axiom,
! [A2: complex,B2: complex] :
( ( ( times_times_complex @ A2 @ B2 )
!= zero_zero_complex )
=> ( ( A2 != zero_zero_complex )
& ( B2 != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_422_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: complex,A2: complex,B2: complex] :
( ( X != zero_zero_complex )
=> ( ( ( times_times_complex @ A2 @ X )
= ( times_times_complex @ B2 @ X ) )
=> ( A2 = B2 ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_423_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A2: complex,X: complex,B2: complex] :
( ( ( times_times_complex @ A2 @ X )
= ( times_times_complex @ B2 @ X ) )
= ( ( A2 = B2 )
| ( X = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_424_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A2: complex,X: complex,Y: complex] :
( ( A2 != zero_zero_complex )
=> ( ( ( times_times_complex @ A2 @ X )
= ( times_times_complex @ A2 @ Y ) )
=> ( X = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_425_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A2: complex,X: complex,Y: complex] :
( ( ( times_times_complex @ A2 @ X )
= ( times_times_complex @ A2 @ Y ) )
= ( ( X = Y )
| ( A2 = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_426_vector__space__over__itself_Oscale__zero__right,axiom,
! [A2: complex] :
( ( times_times_complex @ A2 @ zero_zero_complex )
= zero_zero_complex ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_427_vector__space__over__itself_Oscale__zero__left,axiom,
! [X: complex] :
( ( times_times_complex @ zero_zero_complex @ X )
= zero_zero_complex ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_428_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A2: complex,X: complex] :
( ( ( times_times_complex @ A2 @ X )
= zero_zero_complex )
= ( ( A2 = zero_zero_complex )
| ( X = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_429_rel__simps_I70_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% rel_simps(70)
thf(fact_430_zero__order_I5_J,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% zero_order(5)
thf(fact_431_zero__order_I4_J,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_order(4)
thf(fact_432_zero__order_I3_J,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% zero_order(3)
thf(fact_433_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_434_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_435_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_436_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_437_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_438_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_439_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_440_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_441_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_442_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_443_mult__less__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_444_mult__less__mono2,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_445_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P2 @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P2 @ M2 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_446_mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel1
thf(fact_447_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_448_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_449_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_450_set__one__times,axiom,
! [C2: set_nat] :
( ( set_el2933305810450955905es_nat @ one_one_nat @ C2 )
= C2 ) ).
% set_one_times
thf(fact_451_triangular__to__jnf__steps__dims_I2_J,axiom,
! [A: mat_complex] :
( ( dim_col_complex @ ( jordan2017415923357163885omplex @ A ) )
= ( dim_col_complex @ A ) ) ).
% triangular_to_jnf_steps_dims(2)
thf(fact_452_triangular__to__jnf__steps__dims_I4_J,axiom,
! [A: mat_complex] :
( ( dim_col_complex @ ( jordan7871273693253786478omplex @ A ) )
= ( dim_col_complex @ A ) ) ).
% triangular_to_jnf_steps_dims(4)
thf(fact_453_add__col__sub__row__carrier_I3_J,axiom,
! [A: mat_complex,N: nat,A2: complex,K: nat,L: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( member_mat_complex @ ( column6029646570091773654omplex @ A2 @ K @ L @ A ) @ ( carrier_mat_complex @ N @ N ) ) ) ).
% add_col_sub_row_carrier(3)
thf(fact_454_swaprows__mat__carrier,axiom,
! [N: nat,K: nat,L: nat] : ( member_mat_complex @ ( gauss_8970452565587180529omplex @ N @ K @ L ) @ ( carrier_mat_complex @ N @ N ) ) ).
% swaprows_mat_carrier
thf(fact_455_index__mat__swaprows__mat_I2_J,axiom,
! [N: nat,K: nat,L: nat] :
( ( dim_row_complex @ ( gauss_8970452565587180529omplex @ N @ K @ L ) )
= N ) ).
% index_mat_swaprows_mat(2)
thf(fact_456_add__col__sub__row__carrier_I1_J,axiom,
! [A2: complex,K: nat,L: nat,A: mat_complex] :
( ( dim_row_complex @ ( column6029646570091773654omplex @ A2 @ K @ L @ A ) )
= ( dim_row_complex @ A ) ) ).
% add_col_sub_row_carrier(1)
thf(fact_457_triangular__to__jnf__steps__dims_I6_J,axiom,
! [A: mat_complex] :
( ( dim_col_complex @ ( jordan4501759426295633263omplex @ A ) )
= ( dim_col_complex @ A ) ) ).
% triangular_to_jnf_steps_dims(6)
thf(fact_458_dim__update__mat_I2_J,axiom,
! [A: mat_complex,Ij: product_prod_nat_nat,A2: complex] :
( ( dim_col_complex @ ( update_mat_complex @ A @ Ij @ A2 ) )
= ( dim_col_complex @ A ) ) ).
% dim_update_mat(2)
thf(fact_459_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_460_carrier__mat__triv,axiom,
! [M: mat_complex] : ( member_mat_complex @ M @ ( carrier_mat_complex @ ( dim_row_complex @ M ) @ ( dim_col_complex @ M ) ) ) ).
% carrier_mat_triv
thf(fact_461_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_462_carrier__matI,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( ( dim_row_complex @ A )
= Nr )
=> ( ( ( dim_col_complex @ A )
= Nc )
=> ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% carrier_matI
thf(fact_463_right__mult__one__mat_H,axiom,
! [A: mat_complex,N: nat] :
( ( ( dim_col_complex @ A )
= N )
=> ( ( times_8009071140041733218omplex @ A @ ( one_mat_complex @ N ) )
= A ) ) ).
% right_mult_one_mat'
thf(fact_464_adjoint__dim__row,axiom,
! [A: mat_complex] :
( ( dim_row_complex @ ( schur_5982229384592763574omplex @ A ) )
= ( dim_col_complex @ A ) ) ).
% adjoint_dim_row
thf(fact_465_adjoint__dim__col,axiom,
! [A: mat_complex] :
( ( dim_col_complex @ ( schur_5982229384592763574omplex @ A ) )
= ( dim_row_complex @ A ) ) ).
% adjoint_dim_col
thf(fact_466_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_467_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_468_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_469_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_470_zero__less__mult__pos2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_471_zero__less__mult__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_472_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_473_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_474_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_475_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_476_square__mat_Osimps,axiom,
( square_mat_a
= ( ^ [A4: mat_a] :
( ( dim_col_a @ A4 )
= ( dim_row_a @ A4 ) ) ) ) ).
% square_mat.simps
thf(fact_477_square__mat_Osimps,axiom,
( square_mat_complex
= ( ^ [A4: mat_complex] :
( ( dim_col_complex @ A4 )
= ( dim_row_complex @ A4 ) ) ) ) ).
% square_mat.simps
thf(fact_478_square__mat_Oelims_I1_J,axiom,
! [X: mat_a,Y: $o] :
( ( ( square_mat_a @ X )
= Y )
=> ( Y
= ( ( dim_col_a @ X )
= ( dim_row_a @ X ) ) ) ) ).
% square_mat.elims(1)
thf(fact_479_square__mat_Oelims_I1_J,axiom,
! [X: mat_complex,Y: $o] :
( ( ( square_mat_complex @ X )
= Y )
=> ( Y
= ( ( dim_col_complex @ X )
= ( dim_row_complex @ X ) ) ) ) ).
% square_mat.elims(1)
thf(fact_480_square__mat_Oelims_I2_J,axiom,
! [X: mat_a] :
( ( square_mat_a @ X )
=> ( ( dim_col_a @ X )
= ( dim_row_a @ X ) ) ) ).
% square_mat.elims(2)
thf(fact_481_square__mat_Oelims_I2_J,axiom,
! [X: mat_complex] :
( ( square_mat_complex @ X )
=> ( ( dim_col_complex @ X )
= ( dim_row_complex @ X ) ) ) ).
% square_mat.elims(2)
thf(fact_482_square__mat_Oelims_I3_J,axiom,
! [X: mat_a] :
( ~ ( square_mat_a @ X )
=> ( ( dim_col_a @ X )
!= ( dim_row_a @ X ) ) ) ).
% square_mat.elims(3)
thf(fact_483_square__mat_Oelims_I3_J,axiom,
! [X: mat_complex] :
( ~ ( square_mat_complex @ X )
=> ( ( dim_col_complex @ X )
!= ( dim_row_complex @ X ) ) ) ).
% square_mat.elims(3)
thf(fact_484_unitary__zero,axiom,
! [A: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ zero_zero_nat @ zero_zero_nat ) )
=> ( comple6660659447773130958omplex @ A ) ) ).
% unitary_zero
thf(fact_485_mk__diagonal__dim_I2_J,axiom,
! [As: list_complex] :
( ( dim_col_complex @ ( mk_diagonal_complex @ As ) )
= ( size_s3451745648224563538omplex @ As ) ) ).
% mk_diagonal_dim(2)
thf(fact_486_step__3__main__inv,axiom,
! [A: mat_complex,N: nat,K: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( jordan5244935068081719878omplex @ N @ jordan3528196489273997576omplex @ A )
=> ( ( jordan8042990603089931364omplex @ N @ A )
=> ( ( jordan5475473882837061487omplex @ N @ jordan4971026570492200526omplex @ A @ K )
=> ( ( jordan5244935068081719878omplex @ N @ jordan4971026570492200526omplex @ ( jordan4702481308941288104omplex @ N @ K @ A ) )
& ( jordan2620430285385836103omplex @ N @ A @ ( jordan4702481308941288104omplex @ N @ K @ A ) ) ) ) ) ) ) ) ).
% step_3_main_inv
thf(fact_487_mult__if__delta,axiom,
! [P2: $o,Q4: nat] :
( ( P2
=> ( ( times_times_nat @ ( if_nat @ P2 @ one_one_nat @ zero_zero_nat ) @ Q4 )
= Q4 ) )
& ( ~ P2
=> ( ( times_times_nat @ ( if_nat @ P2 @ one_one_nat @ zero_zero_nat ) @ Q4 )
= zero_zero_nat ) ) ) ).
% mult_if_delta
thf(fact_488_mult__if__delta,axiom,
! [P2: $o,Q4: complex] :
( ( P2
=> ( ( times_times_complex @ ( if_complex @ P2 @ one_one_complex @ zero_zero_complex ) @ Q4 )
= Q4 ) )
& ( ~ P2
=> ( ( times_times_complex @ ( if_complex @ P2 @ one_one_complex @ zero_zero_complex ) @ Q4 )
= zero_zero_complex ) ) ) ).
% mult_if_delta
thf(fact_489_step__3__main__dims_I1_J,axiom,
! [N: nat,J2: nat,A: mat_complex] :
( ( dim_row_complex @ ( jordan4702481308941288104omplex @ N @ J2 @ A ) )
= ( dim_row_complex @ A ) ) ).
% step_3_main_dims(1)
thf(fact_490_step__3__main__dims_I2_J,axiom,
! [N: nat,J2: nat,A: mat_complex] :
( ( dim_col_complex @ ( jordan4702481308941288104omplex @ N @ J2 @ A ) )
= ( dim_col_complex @ A ) ) ).
% step_3_main_dims(2)
thf(fact_491_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_492_step__3__main__dims__main,axiom,
! [N: nat,K: nat,A: mat_complex] :
( ( ( dim_row_complex @ ( jordan4702481308941288104omplex @ N @ K @ A ) )
= ( dim_row_complex @ A ) )
& ( ( dim_col_complex @ ( jordan4702481308941288104omplex @ N @ K @ A ) )
= ( dim_col_complex @ A ) ) ) ).
% step_3_main_dims_main
thf(fact_493_step__3__def,axiom,
( jordan4501759426295633263omplex
= ( ^ [A4: mat_complex] : ( jordan4702481308941288104omplex @ ( dim_row_complex @ A4 ) @ one_one_nat @ A4 ) ) ) ).
% step_3_def
thf(fact_494_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_495_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_496_swapcols__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,K: nat,L: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( ord_less_nat @ K @ N )
=> ( ( ord_less_nat @ L @ N )
=> ( ( column4357519492343924999omplex @ K @ L @ A )
= ( times_8009071140041733218omplex @ A @ ( gauss_8970452565587180529omplex @ N @ K @ L ) ) ) ) ) ) ).
% swapcols_mat
thf(fact_497_mult__col__div__row__similar,axiom,
! [A: mat_complex,N: nat,K: nat,A2: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ K @ N )
=> ( ( A2 != zero_zero_complex )
=> ( similar_mat_complex @ ( column217142795681433722omplex @ A2 @ K @ A ) @ A ) ) ) ) ).
% mult_col_div_row_similar
thf(fact_498_swapcols__carrier,axiom,
! [L: nat,K: nat,A: mat_a,N: nat,M: nat] :
( ( member_mat_a @ ( column2528828918332591333cols_a @ L @ K @ A ) @ ( carrier_mat_a @ N @ M ) )
= ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) ) ) ).
% swapcols_carrier
thf(fact_499_swapcols__carrier,axiom,
! [L: nat,K: nat,A: mat_complex,N: nat,M: nat] :
( ( member_mat_complex @ ( column4357519492343924999omplex @ L @ K @ A ) @ ( carrier_mat_complex @ N @ M ) )
= ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) ) ) ).
% swapcols_carrier
thf(fact_500_index__mat__swapcols_I2_J,axiom,
! [K: nat,L: nat,A: mat_a] :
( ( dim_row_a @ ( column2528828918332591333cols_a @ K @ L @ A ) )
= ( dim_row_a @ A ) ) ).
% index_mat_swapcols(2)
thf(fact_501_index__mat__swapcols_I2_J,axiom,
! [K: nat,L: nat,A: mat_complex] :
( ( dim_row_complex @ ( column4357519492343924999omplex @ K @ L @ A ) )
= ( dim_row_complex @ A ) ) ).
% index_mat_swapcols(2)
thf(fact_502_mult__col__div__row__carrier_I3_J,axiom,
! [A: mat_complex,N: nat,A2: complex,K: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( member_mat_complex @ ( column217142795681433722omplex @ A2 @ K @ A ) @ ( carrier_mat_complex @ N @ N ) ) ) ).
% mult_col_div_row_carrier(3)
thf(fact_503_mult__col__div__row__carrier_I1_J,axiom,
! [A2: complex,K: nat,A: mat_complex] :
( ( dim_row_complex @ ( column217142795681433722omplex @ A2 @ K @ A ) )
= ( dim_row_complex @ A ) ) ).
% mult_col_div_row_carrier(1)
thf(fact_504_density__collapse__carrier,axiom,
! [R: mat_complex,P2: mat_complex,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( dim_row_complex @ R ) )
=> ( ( member_mat_complex @ P2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ R @ ( carrier_mat_complex @ N @ N ) )
=> ( member_mat_complex @ ( projec3470689467825365843llapse @ R @ P2 ) @ ( carrier_mat_complex @ N @ N ) ) ) ) ) ).
% density_collapse_carrier
thf(fact_505_unitary__operator__keep__trace,axiom,
! [U2: mat_complex,N: nat,A: mat_complex] :
( ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ U2 )
=> ( ( comple3184165445352484367omplex @ A )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ U2 ) @ A ) @ U2 ) ) ) ) ) ) ).
% unitary_operator_keep_trace
thf(fact_506_mat__assoc__test_I10_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C2: mat_complex,D2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B ) @ C2 ) )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ B @ C2 ) @ A ) ) ) ) ) ) ) ).
% mat_assoc_test(10)
thf(fact_507_mat__assoc__test_I11_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C2: mat_complex,D2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B ) @ C2 ) @ D2 ) )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ C2 @ D2 ) @ A ) @ B ) ) ) ) ) ) ) ).
% mat_assoc_test(11)
thf(fact_508_mat__assoc__test_I2_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C2: mat_complex,D2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ ( times_8009071140041733218omplex @ A @ ( schur_5982229384592763574omplex @ B ) ) ) @ C2 )
= ( times_8009071140041733218omplex @ B @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ A ) @ C2 ) ) ) ) ) ) ) ).
% mat_assoc_test(2)
thf(fact_509_mat__assoc__test_I1_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C2: mat_complex,D2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B ) @ ( times_8009071140041733218omplex @ C2 @ D2 ) )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B ) @ C2 ) @ D2 ) ) ) ) ) ) ).
% mat_assoc_test(1)
thf(fact_510_smult__smult__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,K: complex,L: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( smult_mat_complex @ K @ ( smult_mat_complex @ L @ A ) )
= ( smult_mat_complex @ ( times_times_complex @ K @ L ) @ A ) ) ) ).
% smult_smult_mat
thf(fact_511_mat__assoc__test_I3_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C2: mat_complex,D2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ ( one_mat_complex @ N ) ) @ ( one_mat_complex @ N ) ) @ B ) @ ( one_mat_complex @ N ) )
= ( times_8009071140041733218omplex @ A @ B ) ) ) ) ) ) ).
% mat_assoc_test(3)
thf(fact_512_projector__collapse__trace,axiom,
! [P2: mat_complex,N: nat,R: mat_complex] :
( ( linear5633924348262549461omplex @ P2 )
=> ( ( member_mat_complex @ P2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ R @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P2 @ R ) @ P2 ) )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R @ P2 ) ) ) ) ) ) ).
% projector_collapse_trace
thf(fact_513_density__collapse__operator,axiom,
! [P2: mat_complex,R: mat_complex,N: nat] :
( ( linear5633924348262549461omplex @ P2 )
=> ( ( comple5220265106149225959erator @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_row_complex @ R ) )
=> ( ( member_mat_complex @ P2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ R @ ( carrier_mat_complex @ N @ N ) )
=> ( comple5220265106149225959erator @ ( projec3470689467825365843llapse @ R @ P2 ) ) ) ) ) ) ) ).
% density_collapse_operator
thf(fact_514_mult__delta__right,axiom,
! [B2: $o,X: nat,Y: nat] :
( ( B2
=> ( ( times_times_nat @ X @ ( if_nat @ B2 @ Y @ zero_zero_nat ) )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B2
=> ( ( times_times_nat @ X @ ( if_nat @ B2 @ Y @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_515_mult__delta__right,axiom,
! [B2: $o,X: complex,Y: complex] :
( ( B2
=> ( ( times_times_complex @ X @ ( if_complex @ B2 @ Y @ zero_zero_complex ) )
= ( times_times_complex @ X @ Y ) ) )
& ( ~ B2
=> ( ( times_times_complex @ X @ ( if_complex @ B2 @ Y @ zero_zero_complex ) )
= zero_zero_complex ) ) ) ).
% mult_delta_right
thf(fact_516_mult__delta__left,axiom,
! [B2: $o,X: nat,Y: nat] :
( ( B2
=> ( ( times_times_nat @ ( if_nat @ B2 @ X @ zero_zero_nat ) @ Y )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B2
=> ( ( times_times_nat @ ( if_nat @ B2 @ X @ zero_zero_nat ) @ Y )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_517_mult__delta__left,axiom,
! [B2: $o,X: complex,Y: complex] :
( ( B2
=> ( ( times_times_complex @ ( if_complex @ B2 @ X @ zero_zero_complex ) @ Y )
= ( times_times_complex @ X @ Y ) ) )
& ( ~ B2
=> ( ( times_times_complex @ ( if_complex @ B2 @ X @ zero_zero_complex ) @ Y )
= zero_zero_complex ) ) ) ).
% mult_delta_left
thf(fact_518_mult__hom_Ohom__zero,axiom,
! [C: nat] :
( ( times_times_nat @ C @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_519_mult__hom_Ohom__zero,axiom,
! [C: complex] :
( ( times_times_complex @ C @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_hom.hom_zero
thf(fact_520_step__2__def,axiom,
( jordan7871273693253786478omplex
= ( ^ [A4: mat_complex] : ( jordan6916311984355858983omplex @ ( dim_row_complex @ A4 ) @ zero_zero_nat @ A4 ) ) ) ).
% step_2_def
thf(fact_521_step__2__main__dims_I1_J,axiom,
! [N: nat,J2: nat,A: mat_complex] :
( ( dim_row_complex @ ( jordan6916311984355858983omplex @ N @ J2 @ A ) )
= ( dim_row_complex @ A ) ) ).
% step_2_main_dims(1)
thf(fact_522_step__2__main__dims_I2_J,axiom,
! [N: nat,J2: nat,A: mat_complex] :
( ( dim_col_complex @ ( jordan6916311984355858983omplex @ N @ J2 @ A ) )
= ( dim_col_complex @ A ) ) ).
% step_2_main_dims(2)
thf(fact_523_unitary__density,axiom,
! [R: mat_complex,U2: mat_complex,N: nat] :
( ( comple5220265106149225959erator @ R )
=> ( ( comple6660659447773130958omplex @ U2 )
=> ( ( member_mat_complex @ R @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N @ N ) )
=> ( comple5220265106149225959erator @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ U2 @ R ) @ ( schur_5982229384592763574omplex @ U2 ) ) ) ) ) ) ) ).
% unitary_density
thf(fact_524_step__2__main__dims__main,axiom,
! [N: nat,J2: nat,A: mat_complex] :
( ( ( dim_row_complex @ ( jordan6916311984355858983omplex @ N @ J2 @ A ) )
= ( dim_row_complex @ A ) )
& ( ( dim_col_complex @ ( jordan6916311984355858983omplex @ N @ J2 @ A ) )
= ( dim_col_complex @ A ) ) ) ).
% step_2_main_dims_main
thf(fact_525_max__mix__is__density,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( comple5220265106149225959erator @ ( projec8360710381328234318ensity @ N ) ) ) ).
% max_mix_is_density
thf(fact_526_trace__pdo__eq__imp__eq,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ! [Rho: mat_complex] :
( ( member_mat_complex @ Rho @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple1169154605998056944erator @ Rho )
=> ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ A @ Rho ) )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ B @ Rho ) ) ) ) )
=> ( A = B ) ) ) ) ).
% trace_pdo_eq_imp_eq
thf(fact_527_swaprows__mat,axiom,
! [A: mat_complex,N: nat,Nc: nat,K: nat,L: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( ord_less_nat @ K @ N )
=> ( ( ord_less_nat @ L @ N )
=> ( ( gauss_1020679828357514249omplex @ K @ L @ A )
= ( times_8009071140041733218omplex @ ( gauss_8970452565587180529omplex @ N @ K @ L ) @ A ) ) ) ) ) ).
% swaprows_mat
thf(fact_528_step__1__def,axiom,
( jordan2017415923357163885omplex
= ( ^ [A4: mat_complex] : ( jordan9130142659770429862omplex @ ( dim_row_complex @ A4 ) @ zero_zero_nat @ zero_zero_nat @ A4 ) ) ) ).
% step_1_def
thf(fact_529_cpx__sq__mat__axioms__def,axiom,
( linear2040860143340867312axioms
= ( ^ [DimR: nat,DimC: nat] :
( ( DimR = DimC )
& ( ord_less_nat @ zero_zero_nat @ DimR ) ) ) ) ).
% cpx_sq_mat_axioms_def
thf(fact_530_swaprows__carrier,axiom,
! [K: nat,L: nat,A: mat_a,N: nat,Nc: nat] :
( ( member_mat_a @ ( gauss_2482569599970757219rows_a @ K @ L @ A ) @ ( carrier_mat_a @ N @ Nc ) )
= ( member_mat_a @ A @ ( carrier_mat_a @ N @ Nc ) ) ) ).
% swaprows_carrier
thf(fact_531_swaprows__carrier,axiom,
! [K: nat,L: nat,A: mat_complex,N: nat,Nc: nat] :
( ( member_mat_complex @ ( gauss_1020679828357514249omplex @ K @ L @ A ) @ ( carrier_mat_complex @ N @ Nc ) )
= ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ Nc ) ) ) ).
% swaprows_carrier
thf(fact_532_index__mat__swaprows_I2_J,axiom,
! [K: nat,L: nat,A: mat_a] :
( ( dim_row_a @ ( gauss_2482569599970757219rows_a @ K @ L @ A ) )
= ( dim_row_a @ A ) ) ).
% index_mat_swaprows(2)
thf(fact_533_index__mat__swaprows_I2_J,axiom,
! [K: nat,L: nat,A: mat_complex] :
( ( dim_row_complex @ ( gauss_1020679828357514249omplex @ K @ L @ A ) )
= ( dim_row_complex @ A ) ) ).
% index_mat_swaprows(2)
thf(fact_534_step__1__main__dims_I1_J,axiom,
! [N: nat,I2: nat,J2: nat,A: mat_complex] :
( ( dim_row_complex @ ( jordan9130142659770429862omplex @ N @ I2 @ J2 @ A ) )
= ( dim_row_complex @ A ) ) ).
% step_1_main_dims(1)
thf(fact_535_step__1__main__dims_I2_J,axiom,
! [N: nat,I2: nat,J2: nat,A: mat_complex] :
( ( dim_col_complex @ ( jordan9130142659770429862omplex @ N @ I2 @ J2 @ A ) )
= ( dim_col_complex @ A ) ) ).
% step_1_main_dims(2)
thf(fact_536_step__1__main__dims__main,axiom,
! [N: nat,I2: nat,J2: nat,A: mat_complex] :
( ( ( dim_row_complex @ ( jordan9130142659770429862omplex @ N @ I2 @ J2 @ A ) )
= ( dim_row_complex @ A ) )
& ( ( dim_col_complex @ ( jordan9130142659770429862omplex @ N @ I2 @ J2 @ A ) )
= ( dim_col_complex @ A ) ) ) ).
% step_1_main_dims_main
thf(fact_537_cpx__sq__mat__axioms_Ointro,axiom,
! [DimR2: nat,DimC2: nat] :
( ( DimR2 = DimC2 )
=> ( ( ord_less_nat @ zero_zero_nat @ DimR2 )
=> ( linear2040860143340867312axioms @ DimR2 @ DimC2 ) ) ) ).
% cpx_sq_mat_axioms.intro
thf(fact_538_tensor__mat__unitary,axiom,
! [U2: mat_complex,V2: mat_complex] :
( ( comple6660659447773130958omplex @ U2 )
=> ( ( comple6660659447773130958omplex @ V2 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_row_complex @ U2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_row_complex @ V2 ) )
=> ( comple6660659447773130958omplex @ ( tensor_mat @ U2 @ V2 ) ) ) ) ) ) ).
% tensor_mat_unitary
thf(fact_539_swapcols__is__transp__swap__rows,axiom,
! [A: mat_a,N: nat,M: nat,K: nat,L: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( ord_less_nat @ K @ M )
=> ( ( ord_less_nat @ L @ M )
=> ( ( column2528828918332591333cols_a @ K @ L @ A )
= ( transpose_mat_a @ ( gauss_2482569599970757219rows_a @ K @ L @ ( transpose_mat_a @ A ) ) ) ) ) ) ) ).
% swapcols_is_transp_swap_rows
thf(fact_540_swapcols__is__transp__swap__rows,axiom,
! [A: mat_complex,N: nat,M: nat,K: nat,L: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( ord_less_nat @ K @ M )
=> ( ( ord_less_nat @ L @ M )
=> ( ( column4357519492343924999omplex @ K @ L @ A )
= ( transp3074176993011536131omplex @ ( gauss_1020679828357514249omplex @ K @ L @ ( transp3074176993011536131omplex @ A ) ) ) ) ) ) ) ).
% swapcols_is_transp_swap_rows
thf(fact_541_tensor__mat__adjoint,axiom,
! [M1: mat_complex,R1: nat,C1: nat,M22: mat_complex,R2: nat,C22: nat] :
( ( member_mat_complex @ M1 @ ( carrier_mat_complex @ R1 @ C1 ) )
=> ( ( member_mat_complex @ M22 @ ( carrier_mat_complex @ R2 @ C22 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C1 )
=> ( ( ord_less_nat @ zero_zero_nat @ C22 )
=> ( ( ord_less_nat @ zero_zero_nat @ R1 )
=> ( ( ord_less_nat @ zero_zero_nat @ R2 )
=> ( ( schur_5982229384592763574omplex @ ( tensor_mat @ M1 @ M22 ) )
= ( tensor_mat @ ( schur_5982229384592763574omplex @ M1 ) @ ( schur_5982229384592763574omplex @ M22 ) ) ) ) ) ) ) ) ) ).
% tensor_mat_adjoint
thf(fact_542_tensor__mat__id,axiom,
! [D1: nat,D22: nat] :
( ( ord_less_nat @ zero_zero_nat @ D1 )
=> ( ( ord_less_nat @ zero_zero_nat @ D22 )
=> ( ( tensor_mat @ ( one_mat_complex @ D1 ) @ ( one_mat_complex @ D22 ) )
= ( one_mat_complex @ ( times_times_nat @ D1 @ D22 ) ) ) ) ) ).
% tensor_mat_id
thf(fact_543_jnf__vector_I1_J,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ( ord_less_nat @ J3 @ N )
=> ( jordan4971026570492200526omplex @ A @ I3 @ J3 ) ) )
=> ( ( jordan8042990603089931364omplex @ N @ A )
=> ( ( jordan5739059635872469039omplex @ ( jordan387279176131498413omplex @ A ) )
= A ) ) ) ) ).
% jnf_vector(1)
thf(fact_544_Matrix_Otranspose__transpose,axiom,
! [A: mat_complex] :
( ( transp3074176993011536131omplex @ ( transp3074176993011536131omplex @ A ) )
= A ) ).
% Matrix.transpose_transpose
thf(fact_545_transpose__mat__eq,axiom,
! [A: mat_complex,B: mat_complex] :
( ( ( transp3074176993011536131omplex @ A )
= ( transp3074176993011536131omplex @ B ) )
= ( A = B ) ) ).
% transpose_mat_eq
thf(fact_546_transpose__carrier__mat,axiom,
! [A: mat_a,Nc: nat,Nr: nat] :
( ( member_mat_a @ ( transpose_mat_a @ A ) @ ( carrier_mat_a @ Nc @ Nr ) )
= ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% transpose_carrier_mat
thf(fact_547_transpose__carrier__mat,axiom,
! [A: mat_complex,Nc: nat,Nr: nat] :
( ( member_mat_complex @ ( transp3074176993011536131omplex @ A ) @ ( carrier_mat_complex @ Nc @ Nr ) )
= ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% transpose_carrier_mat
thf(fact_548_transpose__one,axiom,
! [N: nat] :
( ( transp3074176993011536131omplex @ ( one_mat_complex @ N ) )
= ( one_mat_complex @ N ) ) ).
% transpose_one
thf(fact_549_transpose__mult,axiom,
! [A: mat_complex,Nr: nat,N: nat,B: mat_complex,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( transp3074176993011536131omplex @ ( times_8009071140041733218omplex @ A @ B ) )
= ( times_8009071140041733218omplex @ ( transp3074176993011536131omplex @ B ) @ ( transp3074176993011536131omplex @ A ) ) ) ) ) ).
% transpose_mult
thf(fact_550_index__transpose__mat_I2_J,axiom,
! [A: mat_a] :
( ( dim_row_a @ ( transpose_mat_a @ A ) )
= ( dim_col_a @ A ) ) ).
% index_transpose_mat(2)
thf(fact_551_index__transpose__mat_I2_J,axiom,
! [A: mat_complex] :
( ( dim_row_complex @ ( transp3074176993011536131omplex @ A ) )
= ( dim_col_complex @ A ) ) ).
% index_transpose_mat(2)
thf(fact_552_index__transpose__mat_I3_J,axiom,
! [A: mat_a] :
( ( dim_col_a @ ( transpose_mat_a @ A ) )
= ( dim_row_a @ A ) ) ).
% index_transpose_mat(3)
thf(fact_553_index__transpose__mat_I3_J,axiom,
! [A: mat_complex] :
( ( dim_col_complex @ ( transp3074176993011536131omplex @ A ) )
= ( dim_row_complex @ A ) ) ).
% index_transpose_mat(3)
thf(fact_554_tensor__mat__hermitian,axiom,
! [A: mat_complex,N: nat,B: mat_complex,N4: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N4 @ N4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( comple8306762464034002205omplex @ A )
=> ( ( comple8306762464034002205omplex @ B )
=> ( comple8306762464034002205omplex @ ( tensor_mat @ A @ B ) ) ) ) ) ) ) ) ).
% tensor_mat_hermitian
thf(fact_555_tensor__mat__carrier,axiom,
! [U2: mat_complex,V2: mat_complex] : ( member_mat_complex @ ( tensor_mat @ U2 @ V2 ) @ ( carrier_mat_complex @ ( times_times_nat @ ( dim_row_complex @ U2 ) @ ( dim_row_complex @ V2 ) ) @ ( times_times_nat @ ( dim_col_complex @ U2 ) @ ( dim_col_complex @ V2 ) ) ) ) ).
% tensor_mat_carrier
thf(fact_556_mult__distr__tensor,axiom,
! [A: mat_complex,B: mat_complex,C2: mat_complex,D2: mat_complex] :
( ( ( dim_col_complex @ A )
= ( dim_row_complex @ B ) )
=> ( ( ( dim_col_complex @ C2 )
= ( dim_row_complex @ D2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_col_complex @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_col_complex @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_col_complex @ C2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_col_complex @ D2 ) )
=> ( ( tensor_mat @ ( times_8009071140041733218omplex @ A @ B ) @ ( times_8009071140041733218omplex @ C2 @ D2 ) )
= ( times_8009071140041733218omplex @ ( tensor_mat @ A @ C2 ) @ ( tensor_mat @ B @ D2 ) ) ) ) ) ) ) ) ) ).
% mult_distr_tensor
thf(fact_557_dim__row__tensor__mat,axiom,
! [A: mat_complex,B: mat_complex] :
( ( dim_row_complex @ ( tensor_mat @ A @ B ) )
= ( times_times_nat @ ( dim_row_complex @ A ) @ ( dim_row_complex @ B ) ) ) ).
% dim_row_tensor_mat
thf(fact_558_dim__col__tensor__mat,axiom,
! [A: mat_complex,B: mat_complex] :
( ( dim_col_complex @ ( tensor_mat @ A @ B ) )
= ( times_times_nat @ ( dim_col_complex @ A ) @ ( dim_col_complex @ B ) ) ) ).
% dim_col_tensor_mat
thf(fact_559_transpose__of__prod,axiom,
! [M4: mat_complex,N5: mat_complex] :
( ( ( dim_col_complex @ M4 )
= ( dim_row_complex @ N5 ) )
=> ( ( transp3074176993011536131omplex @ ( times_8009071140041733218omplex @ M4 @ N5 ) )
= ( times_8009071140041733218omplex @ ( transp3074176993011536131omplex @ N5 ) @ ( transp3074176993011536131omplex @ M4 ) ) ) ) ).
% transpose_of_prod
thf(fact_560_vec__space_Orow__space__eq__col__space__transpose,axiom,
( vS_vec3284807721666986142omplex
= ( ^ [N3: nat,A4: mat_complex] : ( vS_vec1879987866596122552omplex @ N3 @ ( transp3074176993011536131omplex @ A4 ) ) ) ) ).
% vec_space.row_space_eq_col_space_transpose
thf(fact_561_vec__space_Ocol__space__eq__row__space__transpose,axiom,
( vS_vec1879987866596122552omplex
= ( ^ [N3: nat,A4: mat_complex] : ( vS_vec3284807721666986142omplex @ N3 @ ( transp3074176993011536131omplex @ A4 ) ) ) ) ).
% vec_space.col_space_eq_row_space_transpose
thf(fact_562_cpx__mat__cnj__prod,axiom,
! [M4: mat_complex,N5: mat_complex] :
( ( ( dim_col_complex @ M4 )
= ( dim_row_complex @ N5 ) )
=> ( ( cpx_mat_cnj @ ( times_8009071140041733218omplex @ M4 @ N5 ) )
= ( times_8009071140041733218omplex @ ( cpx_mat_cnj @ M4 ) @ ( cpx_mat_cnj @ N5 ) ) ) ) ).
% cpx_mat_cnj_prod
thf(fact_563_jnf__vector__def,axiom,
( jordan387279176131498413omplex
= ( ^ [A4: mat_complex] : ( jordan4459423482773701094omplex @ ( dim_row_complex @ A4 ) @ A4 ) ) ) ).
% jnf_vector_def
thf(fact_564_dim__col__of__cjn__prod,axiom,
! [M4: mat_complex,N5: mat_complex] :
( ( dim_col_complex @ ( times_8009071140041733218omplex @ ( cpx_mat_cnj @ M4 ) @ ( cpx_mat_cnj @ N5 ) ) )
= ( dim_col_complex @ N5 ) ) ).
% dim_col_of_cjn_prod
thf(fact_565_dim__row__of__cjn__prod,axiom,
! [M4: mat_complex,N5: mat_complex] :
( ( dim_row_complex @ ( times_8009071140041733218omplex @ ( cpx_mat_cnj @ M4 ) @ ( cpx_mat_cnj @ N5 ) ) )
= ( dim_row_complex @ M4 ) ) ).
% dim_row_of_cjn_prod
thf(fact_566_mat__incr__mult__adjoint__mat__incr,axiom,
! [N: nat] :
( ( times_8009071140041733218omplex @ ( mat_incr @ N ) @ ( schur_5982229384592763574omplex @ ( mat_incr @ N ) ) )
= ( one_mat_complex @ N ) ) ).
% mat_incr_mult_adjoint_mat_incr
thf(fact_567_lowner__le__keep__under__measurement,axiom,
! [M4: mat_complex,N: nat,A: mat_complex,B: mat_complex] :
( ( member_mat_complex @ M4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_lowner_le @ A @ B )
=> ( complex_lowner_le @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ M4 ) @ A ) @ M4 ) @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ M4 ) @ B ) @ M4 ) ) ) ) ) ) ).
% lowner_le_keep_under_measurement
thf(fact_568_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_569_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_570_dvd__triv__right,axiom,
! [A2: nat,B2: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ A2 ) ) ).
% dvd_triv_right
thf(fact_571_dvd__triv__right,axiom,
! [A2: complex,B2: complex] : ( dvd_dvd_complex @ A2 @ ( times_times_complex @ B2 @ A2 ) ) ).
% dvd_triv_right
thf(fact_572_dvd__mult__right,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
=> ( dvd_dvd_nat @ B2 @ C ) ) ).
% dvd_mult_right
thf(fact_573_dvd__mult__right,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A2 @ B2 ) @ C )
=> ( dvd_dvd_complex @ B2 @ C ) ) ).
% dvd_mult_right
thf(fact_574_mult__dvd__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_575_mult__dvd__mono,axiom,
! [A2: complex,B2: complex,C: complex,D: complex] :
( ( dvd_dvd_complex @ A2 @ B2 )
=> ( ( dvd_dvd_complex @ C @ D )
=> ( dvd_dvd_complex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B2 @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_576_dvd__triv__left,axiom,
! [A2: nat,B2: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ A2 @ B2 ) ) ).
% dvd_triv_left
thf(fact_577_dvd__triv__left,axiom,
! [A2: complex,B2: complex] : ( dvd_dvd_complex @ A2 @ ( times_times_complex @ A2 @ B2 ) ) ).
% dvd_triv_left
thf(fact_578_dvd__mult__left,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
=> ( dvd_dvd_nat @ A2 @ C ) ) ).
% dvd_mult_left
thf(fact_579_dvd__mult__left,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A2 @ B2 ) @ C )
=> ( dvd_dvd_complex @ A2 @ C ) ) ).
% dvd_mult_left
thf(fact_580_dvd__mult2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% dvd_mult2
thf(fact_581_dvd__mult2,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( dvd_dvd_complex @ A2 @ B2 )
=> ( dvd_dvd_complex @ A2 @ ( times_times_complex @ B2 @ C ) ) ) ).
% dvd_mult2
thf(fact_582_dvd__mult,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ C )
=> ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% dvd_mult
thf(fact_583_dvd__mult,axiom,
! [A2: complex,C: complex,B2: complex] :
( ( dvd_dvd_complex @ A2 @ C )
=> ( dvd_dvd_complex @ A2 @ ( times_times_complex @ B2 @ C ) ) ) ).
% dvd_mult
thf(fact_584_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B3: nat,A3: nat] :
? [K2: nat] :
( A3
= ( times_times_nat @ B3 @ K2 ) ) ) ) ).
% dvd_def
thf(fact_585_dvd__def,axiom,
( dvd_dvd_complex
= ( ^ [B3: complex,A3: complex] :
? [K2: complex] :
( A3
= ( times_times_complex @ B3 @ K2 ) ) ) ) ).
% dvd_def
thf(fact_586_dvdI,axiom,
! [A2: nat,B2: nat,K: nat] :
( ( A2
= ( times_times_nat @ B2 @ K ) )
=> ( dvd_dvd_nat @ B2 @ A2 ) ) ).
% dvdI
thf(fact_587_dvdI,axiom,
! [A2: complex,B2: complex,K: complex] :
( ( A2
= ( times_times_complex @ B2 @ K ) )
=> ( dvd_dvd_complex @ B2 @ A2 ) ) ).
% dvdI
thf(fact_588_dvdE,axiom,
! [B2: nat,A2: nat] :
( ( dvd_dvd_nat @ B2 @ A2 )
=> ~ ! [K3: nat] :
( A2
!= ( times_times_nat @ B2 @ K3 ) ) ) ).
% dvdE
thf(fact_589_dvdE,axiom,
! [B2: complex,A2: complex] :
( ( dvd_dvd_complex @ B2 @ A2 )
=> ~ ! [K3: complex] :
( A2
!= ( times_times_complex @ B2 @ K3 ) ) ) ).
% dvdE
thf(fact_590_dvd__field__iff,axiom,
( dvd_dvd_complex
= ( ^ [A3: complex,B3: complex] :
( ( A3 = zero_zero_complex )
=> ( B3 = zero_zero_complex ) ) ) ) ).
% dvd_field_iff
thf(fact_591_dvd__0__left,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
=> ( A2 = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_592_dvd__0__left,axiom,
! [A2: complex] :
( ( dvd_dvd_complex @ zero_zero_complex @ A2 )
=> ( A2 = zero_zero_complex ) ) ).
% dvd_0_left
thf(fact_593_dvd__0__right,axiom,
! [A2: nat] : ( dvd_dvd_nat @ A2 @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_594_dvd__0__right,axiom,
! [A2: complex] : ( dvd_dvd_complex @ A2 @ zero_zero_complex ) ).
% dvd_0_right
thf(fact_595_dvd__0__left__iff,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
= ( A2 = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_596_dvd__0__left__iff,axiom,
! [A2: complex] :
( ( dvd_dvd_complex @ zero_zero_complex @ A2 )
= ( A2 = zero_zero_complex ) ) ).
% dvd_0_left_iff
thf(fact_597_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_598_dvd__refl,axiom,
! [A2: nat] : ( dvd_dvd_nat @ A2 @ A2 ) ).
% dvd_refl
thf(fact_599_dvd__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ C )
=> ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% dvd_trans
thf(fact_600_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_601_dvd__unit__imp__unit,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ A2 @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_602_algebraic__semidom__class_Ounit__imp__dvd,axiom,
! [B2: nat,A2: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ B2 @ A2 ) ) ).
% algebraic_semidom_class.unit_imp_dvd
thf(fact_603_one__dvd,axiom,
! [A2: nat] : ( dvd_dvd_nat @ one_one_nat @ A2 ) ).
% one_dvd
thf(fact_604_algebraic__semidom__class_Odvd__times__right__cancel__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ A2 ) @ ( times_times_nat @ C @ A2 ) )
= ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_right_cancel_iff
thf(fact_605_algebraic__semidom__class_Odvd__times__left__cancel__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C ) )
= ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_left_cancel_iff
thf(fact_606_dvd__mult__cancel__right,axiom,
! [A2: complex,C: complex,B2: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B2 @ C ) )
= ( ( C = zero_zero_complex )
| ( dvd_dvd_complex @ A2 @ B2 ) ) ) ).
% dvd_mult_cancel_right
thf(fact_607_dvd__mult__cancel__left,axiom,
! [C: complex,A2: complex,B2: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ C @ A2 ) @ ( times_times_complex @ C @ B2 ) )
= ( ( C = zero_zero_complex )
| ( dvd_dvd_complex @ A2 @ B2 ) ) ) ).
% dvd_mult_cancel_left
thf(fact_608_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_609_algebraic__semidom__class_Ounit__mult__right__cancel,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( ( times_times_nat @ B2 @ A2 )
= ( times_times_nat @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% algebraic_semidom_class.unit_mult_right_cancel
thf(fact_610_algebraic__semidom__class_Ounit__mult__left__cancel,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( ( times_times_nat @ A2 @ B2 )
= ( times_times_nat @ A2 @ C ) )
= ( B2 = C ) ) ) ).
% algebraic_semidom_class.unit_mult_left_cancel
thf(fact_611_algebraic__semidom__class_Omult__unit__dvd__iff_H,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
= ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff'
thf(fact_612_algebraic__semidom__class_Odvd__mult__unit__iff_H,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C ) )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff'
thf(fact_613_algebraic__semidom__class_Omult__unit__dvd__iff,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff
thf(fact_614_algebraic__semidom__class_Odvd__mult__unit__iff,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ C @ B2 ) )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff
thf(fact_615_algebraic__semidom__class_Ois__unit__mult__iff,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A2 @ one_one_nat )
& ( dvd_dvd_nat @ B2 @ one_one_nat ) ) ) ).
% algebraic_semidom_class.is_unit_mult_iff
thf(fact_616_algebraic__semidom__class_Ounit__prod,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ).
% algebraic_semidom_class.unit_prod
thf(fact_617_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_618_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_619_unit__dvdE,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ~ ( ( A2 != zero_zero_nat )
=> ! [C3: nat] :
( B2
!= ( times_times_nat @ A2 @ C3 ) ) ) ) ).
% unit_dvdE
thf(fact_620_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_621_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_622_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_623_idom__class_Odvd__mult__unit__iff,axiom,
! [B2: complex,A2: complex,C: complex] :
( ( dvd_dvd_complex @ B2 @ one_one_complex )
=> ( ( dvd_dvd_complex @ A2 @ ( times_times_complex @ C @ B2 ) )
= ( dvd_dvd_complex @ A2 @ C ) ) ) ).
% idom_class.dvd_mult_unit_iff
thf(fact_624_idom__class_Odvd__mult__unit__iff_H,axiom,
! [B2: complex,A2: complex,C: complex] :
( ( dvd_dvd_complex @ B2 @ one_one_complex )
=> ( ( dvd_dvd_complex @ A2 @ ( times_times_complex @ B2 @ C ) )
= ( dvd_dvd_complex @ A2 @ C ) ) ) ).
% idom_class.dvd_mult_unit_iff'
thf(fact_625_comm__monoid__mult__class_Ounit__prod,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_626_comm__monoid__mult__class_Ounit__prod,axiom,
! [A2: complex,B2: complex] :
( ( dvd_dvd_complex @ A2 @ one_one_complex )
=> ( ( dvd_dvd_complex @ B2 @ one_one_complex )
=> ( dvd_dvd_complex @ ( times_times_complex @ A2 @ B2 ) @ one_one_complex ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_627_division__decomp,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C ) )
=> ? [B6: nat,C4: nat] :
( ( A2
= ( times_times_nat @ B6 @ C4 ) )
& ( dvd_dvd_nat @ B6 @ B2 )
& ( dvd_dvd_nat @ C4 @ C ) ) ) ).
% division_decomp
thf(fact_628_dvd__productE,axiom,
! [P6: nat,A2: nat,B2: nat] :
( ( dvd_dvd_nat @ P6 @ ( times_times_nat @ A2 @ B2 ) )
=> ~ ! [X2: nat,Y2: nat] :
( ( P6
= ( times_times_nat @ X2 @ Y2 ) )
=> ( ( dvd_dvd_nat @ X2 @ A2 )
=> ~ ( dvd_dvd_nat @ Y2 @ B2 ) ) ) ) ).
% dvd_productE
thf(fact_629_idom__class_Odvd__times__right__cancel__iff,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( A2 != zero_zero_complex )
=> ( ( dvd_dvd_complex @ ( times_times_complex @ B2 @ A2 ) @ ( times_times_complex @ C @ A2 ) )
= ( dvd_dvd_complex @ B2 @ C ) ) ) ).
% idom_class.dvd_times_right_cancel_iff
thf(fact_630_idom__class_Odvd__times__left__cancel__iff,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( A2 != zero_zero_complex )
=> ( ( dvd_dvd_complex @ ( times_times_complex @ A2 @ B2 ) @ ( times_times_complex @ A2 @ C ) )
= ( dvd_dvd_complex @ B2 @ C ) ) ) ).
% idom_class.dvd_times_left_cancel_iff
thf(fact_631_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
= ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_632_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( dvd_dvd_complex @ A2 @ one_one_complex )
=> ( ( dvd_dvd_complex @ ( times_times_complex @ A2 @ B2 ) @ C )
= ( dvd_dvd_complex @ B2 @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_633_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_634_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B2: complex,A2: complex,C: complex] :
( ( dvd_dvd_complex @ B2 @ one_one_complex )
=> ( ( dvd_dvd_complex @ ( times_times_complex @ A2 @ B2 ) @ C )
= ( dvd_dvd_complex @ A2 @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_635_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A2 @ one_one_nat )
& ( dvd_dvd_nat @ B2 @ one_one_nat ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_636_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A2: complex,B2: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A2 @ B2 ) @ one_one_complex )
= ( ( dvd_dvd_complex @ A2 @ one_one_complex )
& ( dvd_dvd_complex @ B2 @ one_one_complex ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_637_idom__class_Ounit__mult__right__cancel,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( dvd_dvd_complex @ A2 @ one_one_complex )
=> ( ( ( times_times_complex @ B2 @ A2 )
= ( times_times_complex @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% idom_class.unit_mult_right_cancel
thf(fact_638_idom__class_Ounit__mult__left__cancel,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( dvd_dvd_complex @ A2 @ one_one_complex )
=> ( ( ( times_times_complex @ A2 @ B2 )
= ( times_times_complex @ A2 @ C ) )
= ( B2 = C ) ) ) ).
% idom_class.unit_mult_left_cancel
thf(fact_639_lowner__le__traceI,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ! [Rho: mat_complex] :
( ( member_mat_complex @ Rho @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple1169154605998056944erator @ Rho )
=> ( ord_less_eq_complex @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ A @ Rho ) ) @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ B @ Rho ) ) ) ) )
=> ( complex_lowner_le @ A @ B ) ) ) ) ).
% lowner_le_traceI
thf(fact_640_lowner__le__traceD,axiom,
! [A: mat_complex,N: nat,B: mat_complex,Rho2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ Rho2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_lowner_le @ A @ B )
=> ( ( comple1169154605998056944erator @ Rho2 )
=> ( ord_less_eq_complex @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ A @ Rho2 ) ) @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ B @ Rho2 ) ) ) ) ) ) ) ) ).
% lowner_le_traceD
thf(fact_641_lowner__le__trace,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_lowner_le @ A @ B )
= ( ! [X4: mat_complex] :
( ( member_mat_complex @ X4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple1169154605998056944erator @ X4 )
=> ( ord_less_eq_complex @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ A @ X4 ) ) @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ B @ X4 ) ) ) ) ) ) ) ) ) ).
% lowner_le_trace
thf(fact_642_not__irreducibleE,axiom,
! [X: nat] :
( ~ ( factor4388943552880185071le_nat @ X )
=> ( ( X != zero_zero_nat )
=> ( ~ ( dvd_dvd_nat @ X @ one_one_nat )
=> ~ ! [A5: nat,B4: nat] :
( ( X
= ( times_times_nat @ A5 @ B4 ) )
=> ( ~ ( dvd_dvd_nat @ A5 @ one_one_nat )
=> ( dvd_dvd_nat @ B4 @ one_one_nat ) ) ) ) ) ) ).
% not_irreducibleE
thf(fact_643_not__irreducibleE,axiom,
! [X: complex] :
( ~ ( factor4870819777162427853omplex @ X )
=> ( ( X != zero_zero_complex )
=> ( ~ ( dvd_dvd_complex @ X @ one_one_complex )
=> ~ ! [A5: complex,B4: complex] :
( ( X
= ( times_times_complex @ A5 @ B4 ) )
=> ( ~ ( dvd_dvd_complex @ A5 @ one_one_complex )
=> ( dvd_dvd_complex @ B4 @ one_one_complex ) ) ) ) ) ) ).
% not_irreducibleE
thf(fact_644_rel__simps_I46_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% rel_simps(46)
thf(fact_645_zero__order_I2_J,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% zero_order(2)
thf(fact_646_zero__order_I1_J,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_order(1)
thf(fact_647_complete__interval,axiom,
! [A2: nat,B2: nat,P2: nat > $o] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( P2 @ A2 )
=> ( ~ ( P2 @ B2 )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A2 @ C3 )
& ( ord_less_eq_nat @ C3 @ B2 )
& ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ C3 ) )
=> ( P2 @ X3 ) )
& ! [D3: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A2 @ X2 )
& ( ord_less_nat @ X2 @ D3 ) )
=> ( P2 @ X2 ) )
=> ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_648_order__trans__rules_I22_J,axiom,
! [X: complex,Y: complex,Z3: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ( ord_less_eq_complex @ Y @ Z3 )
=> ( ord_less_complex @ X @ Z3 ) ) ) ).
% order_trans_rules(22)
thf(fact_649_order__trans__rules_I22_J,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_trans_rules(22)
thf(fact_650_order__trans__rules_I21_J,axiom,
! [X: complex,Y: complex,Z3: complex] :
( ( ord_less_eq_complex @ X @ Y )
=> ( ( ord_less_complex @ Y @ Z3 )
=> ( ord_less_complex @ X @ Z3 ) ) ) ).
% order_trans_rules(21)
thf(fact_651_order__trans__rules_I21_J,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_trans_rules(21)
thf(fact_652_order__trans__rules_I18_J,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_complex @ A2 @ B2 ) ) ) ).
% order_trans_rules(18)
thf(fact_653_order__trans__rules_I18_J,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_trans_rules(18)
thf(fact_654_order__trans__rules_I17_J,axiom,
! [A2: complex,B2: complex] :
( ( A2 != B2 )
=> ( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ord_less_complex @ A2 @ B2 ) ) ) ).
% order_trans_rules(17)
thf(fact_655_order__trans__rules_I17_J,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_trans_rules(17)
thf(fact_656_order__trans__rules_I6_J,axiom,
! [A2: complex,F: complex > complex,B2: complex,C: complex] :
( ( ord_less_complex @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_complex @ B2 @ C )
=> ( ! [X2: complex,Y2: complex] :
( ( ord_less_eq_complex @ X2 @ Y2 )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_657_order__trans__rules_I6_J,axiom,
! [A2: nat,F: complex > nat,B2: complex,C: complex] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_complex @ B2 @ C )
=> ( ! [X2: complex,Y2: complex] :
( ( ord_less_eq_complex @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_658_order__trans__rules_I6_J,axiom,
! [A2: complex,F: nat > complex,B2: nat,C: nat] :
( ( ord_less_complex @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_659_order__trans__rules_I6_J,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_660_order__trans__rules_I5_J,axiom,
! [A2: nat,B2: nat,F: nat > complex,C: complex] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_complex @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(5)
thf(fact_661_order__trans__rules_I5_J,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(5)
thf(fact_662_order__trans__rules_I4_J,axiom,
! [A2: complex,F: nat > complex,B2: nat,C: nat] :
( ( ord_less_eq_complex @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_663_order__trans__rules_I4_J,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_664_order__trans__rules_I3_J,axiom,
! [A2: complex,B2: complex,F: complex > complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ( ord_less_complex @ ( F @ B2 ) @ C )
=> ( ! [X2: complex,Y2: complex] :
( ( ord_less_eq_complex @ X2 @ Y2 )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(3)
thf(fact_665_order__trans__rules_I3_J,axiom,
! [A2: complex,B2: complex,F: complex > nat,C: nat] :
( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: complex,Y2: complex] :
( ( ord_less_eq_complex @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(3)
thf(fact_666_order__trans__rules_I3_J,axiom,
! [A2: nat,B2: nat,F: nat > complex,C: complex] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_complex @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(3)
thf(fact_667_order__trans__rules_I3_J,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(3)
thf(fact_668_leD,axiom,
! [Y: complex,X: complex] :
( ( ord_less_eq_complex @ Y @ X )
=> ~ ( ord_less_complex @ X @ Y ) ) ).
% leD
thf(fact_669_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_670_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_671_le__less,axiom,
( ord_less_eq_complex
= ( ^ [X4: complex,Y4: complex] :
( ( ord_less_complex @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% le_less
thf(fact_672_le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% le_less
thf(fact_673_less__le,axiom,
( ord_less_complex
= ( ^ [X4: complex,Y4: complex] :
( ( ord_less_eq_complex @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% less_le
thf(fact_674_less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% less_le
thf(fact_675_nless__le,axiom,
! [A2: complex,B2: complex] :
( ( ~ ( ord_less_complex @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_complex @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_676_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_677_not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% not_le
thf(fact_678_not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% not_less
thf(fact_679_antisym__conv1,axiom,
! [X: complex,Y: complex] :
( ~ ( ord_less_complex @ X @ Y )
=> ( ( ord_less_eq_complex @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_680_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_681_antisym__conv2,axiom,
! [X: complex,Y: complex] :
( ( ord_less_eq_complex @ X @ Y )
=> ( ( ~ ( ord_less_complex @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_682_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_683_less__imp__le,axiom,
! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ord_less_eq_complex @ X @ Y ) ) ).
% less_imp_le
thf(fact_684_less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% less_imp_le
thf(fact_685_le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% le_less_linear
thf(fact_686_le__imp__less__or__eq,axiom,
! [X: complex,Y: complex] :
( ( ord_less_eq_complex @ X @ Y )
=> ( ( ord_less_complex @ X @ Y )
| ( X = Y ) ) ) ).
% le_imp_less_or_eq
thf(fact_687_le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% le_imp_less_or_eq
thf(fact_688_less__le__not__le,axiom,
( ord_less_complex
= ( ^ [X4: complex,Y4: complex] :
( ( ord_less_eq_complex @ X4 @ Y4 )
& ~ ( ord_less_eq_complex @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_689_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_690_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_691_order_Oorder__iff__strict,axiom,
( ord_less_eq_complex
= ( ^ [A3: complex,B3: complex] :
( ( ord_less_complex @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_692_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_693_order_Ostrict__iff__order,axiom,
( ord_less_complex
= ( ^ [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_694_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_695_order_Ostrict__trans1,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ( ord_less_complex @ B2 @ C )
=> ( ord_less_complex @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_696_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_697_order_Ostrict__trans2,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( ord_less_complex @ A2 @ B2 )
=> ( ( ord_less_eq_complex @ B2 @ C )
=> ( ord_less_complex @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_698_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_699_order_Ostrict__iff__not,axiom,
( ord_less_complex
= ( ^ [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
& ~ ( ord_less_eq_complex @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_700_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_701_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_complex
= ( ^ [B3: complex,A3: complex] :
( ( ord_less_complex @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_702_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_703_dual__order_Ostrict__iff__order,axiom,
( ord_less_complex
= ( ^ [B3: complex,A3: complex] :
( ( ord_less_eq_complex @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_704_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_705_dual__order_Ostrict__trans1,axiom,
! [B2: complex,A2: complex,C: complex] :
( ( ord_less_eq_complex @ B2 @ A2 )
=> ( ( ord_less_complex @ C @ B2 )
=> ( ord_less_complex @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_706_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_707_dual__order_Ostrict__trans2,axiom,
! [B2: complex,A2: complex,C: complex] :
( ( ord_less_complex @ B2 @ A2 )
=> ( ( ord_less_eq_complex @ C @ B2 )
=> ( ord_less_complex @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_708_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_709_dual__order_Ostrict__iff__not,axiom,
( ord_less_complex
= ( ^ [B3: complex,A3: complex] :
( ( ord_less_eq_complex @ B3 @ A3 )
& ~ ( ord_less_eq_complex @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_710_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_711_order_Ostrict__implies__order,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_complex @ A2 @ B2 )
=> ( ord_less_eq_complex @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_712_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_713_dual__order_Ostrict__implies__order,axiom,
! [B2: complex,A2: complex] :
( ( ord_less_complex @ B2 @ A2 )
=> ( ord_less_eq_complex @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_714_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_715_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X3 ) ) ).
% minf(8)
thf(fact_716_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ T ) ) ).
% minf(6)
thf(fact_717_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( ord_less_eq_nat @ T @ X3 ) ) ).
% pinf(8)
thf(fact_718_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ T ) ) ).
% pinf(6)
thf(fact_719_verit__comp__simplify_I3_J,axiom,
! [B7: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A6 ) )
= ( ord_less_nat @ A6 @ B7 ) ) ).
% verit_comp_simplify(3)
thf(fact_720_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_721_order__trans__rules_I26_J,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( A2 = B2 )
=> ( ( ord_less_eq_complex @ B2 @ C )
=> ( ord_less_eq_complex @ A2 @ C ) ) ) ).
% order_trans_rules(26)
thf(fact_722_order__trans__rules_I26_J,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order_trans_rules(26)
thf(fact_723_order__trans__rules_I25_J,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_complex @ A2 @ C ) ) ) ).
% order_trans_rules(25)
thf(fact_724_order__trans__rules_I25_J,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order_trans_rules(25)
thf(fact_725_order__trans__rules_I24_J,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ( ord_less_eq_complex @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_trans_rules(24)
thf(fact_726_order__trans__rules_I24_J,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_trans_rules(24)
thf(fact_727_order__trans__rules_I23_J,axiom,
! [X: complex,Y: complex,Z3: complex] :
( ( ord_less_eq_complex @ X @ Y )
=> ( ( ord_less_eq_complex @ Y @ Z3 )
=> ( ord_less_eq_complex @ X @ Z3 ) ) ) ).
% order_trans_rules(23)
thf(fact_728_order__trans__rules_I23_J,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans_rules(23)
thf(fact_729_order__trans__rules_I10_J,axiom,
! [A2: complex,F: complex > complex,B2: complex,C: complex] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_complex @ B2 @ C )
=> ( ! [X2: complex,Y2: complex] :
( ( ord_less_eq_complex @ X2 @ Y2 )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_730_order__trans__rules_I10_J,axiom,
! [A2: nat,F: complex > nat,B2: complex,C: complex] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_complex @ B2 @ C )
=> ( ! [X2: complex,Y2: complex] :
( ( ord_less_eq_complex @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_731_order__trans__rules_I10_J,axiom,
! [A2: complex,F: nat > complex,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_732_order__trans__rules_I10_J,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_733_order__trans__rules_I9_J,axiom,
! [A2: complex,B2: complex,F: complex > complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: complex,Y2: complex] :
( ( ord_less_eq_complex @ X2 @ Y2 )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_734_order__trans__rules_I9_J,axiom,
! [A2: complex,B2: complex,F: complex > nat,C: nat] :
( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: complex,Y2: complex] :
( ( ord_less_eq_complex @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_735_order__trans__rules_I9_J,axiom,
! [A2: nat,B2: nat,F: nat > complex,C: complex] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_736_order__trans__rules_I9_J,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_737_order__trans__rules_I8_J,axiom,
! [A2: complex,F: complex > complex,B2: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_complex @ B2 @ C )
=> ( ! [X2: complex,Y2: complex] :
( ( ord_less_eq_complex @ X2 @ Y2 )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_738_order__trans__rules_I8_J,axiom,
! [A2: complex,F: nat > complex,B2: nat,C: nat] :
( ( ord_less_eq_complex @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_739_order__trans__rules_I8_J,axiom,
! [A2: nat,F: complex > nat,B2: complex,C: complex] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_complex @ B2 @ C )
=> ( ! [X2: complex,Y2: complex] :
( ( ord_less_eq_complex @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_740_order__trans__rules_I8_J,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_741_order__trans__rules_I7_J,axiom,
! [A2: complex,B2: complex,F: complex > complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ( ord_less_eq_complex @ ( F @ B2 ) @ C )
=> ( ! [X2: complex,Y2: complex] :
( ( ord_less_eq_complex @ X2 @ Y2 )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_742_order__trans__rules_I7_J,axiom,
! [A2: complex,B2: complex,F: complex > nat,C: nat] :
( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: complex,Y2: complex] :
( ( ord_less_eq_complex @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_743_order__trans__rules_I7_J,axiom,
! [A2: nat,B2: nat,F: nat > complex,C: complex] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_complex @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_744_order__trans__rules_I7_J,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_745_linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linear
thf(fact_746_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_747_eq__refl,axiom,
! [X: complex,Y: complex] :
( ( X = Y )
=> ( ord_less_eq_complex @ X @ Y ) ) ).
% eq_refl
thf(fact_748_eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% eq_refl
thf(fact_749_le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% le_cases
thf(fact_750_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_751_antisym__conv,axiom,
! [Y: complex,X: complex] :
( ( ord_less_eq_complex @ Y @ X )
=> ( ( ord_less_eq_complex @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_752_antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_753_order_Oeq__iff,axiom,
( ( ^ [Y5: complex,Z4: complex] : ( Y5 = Z4 ) )
= ( ^ [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
& ( ord_less_eq_complex @ B3 @ A3 ) ) ) ) ).
% order.eq_iff
thf(fact_754_order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.eq_iff
thf(fact_755_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: complex,Z4: complex] : ( Y5 = Z4 ) )
= ( ^ [X4: complex,Y4: complex] :
( ( ord_less_eq_complex @ X4 @ Y4 )
& ( ord_less_eq_complex @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_756_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_757_order__antisym,axiom,
! [X: complex,Y: complex] :
( ( ord_less_eq_complex @ X @ Y )
=> ( ( ord_less_eq_complex @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_758_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_759_order_Orefl,axiom,
! [A2: complex] : ( ord_less_eq_complex @ A2 @ A2 ) ).
% order.refl
thf(fact_760_order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% order.refl
thf(fact_761_order__refl,axiom,
! [X: complex] : ( ord_less_eq_complex @ X @ X ) ).
% order_refl
thf(fact_762_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_763_order_Otrans,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ( ord_less_eq_complex @ B2 @ C )
=> ( ord_less_eq_complex @ A2 @ C ) ) ) ).
% order.trans
thf(fact_764_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_765_linorder__wlog,axiom,
! [P2: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
=> ( P2 @ A5 @ B4 ) )
=> ( ! [A5: nat,B4: nat] :
( ( P2 @ B4 @ A5 )
=> ( P2 @ A5 @ B4 ) )
=> ( P2 @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_766_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: complex,Z4: complex] : ( Y5 = Z4 ) )
= ( ^ [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ B3 @ A3 )
& ( ord_less_eq_complex @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_767_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_768_dual__order_Oantisym,axiom,
! [B2: complex,A2: complex] :
( ( ord_less_eq_complex @ B2 @ A2 )
=> ( ( ord_less_eq_complex @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_769_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_770_dual__order_Otrans,axiom,
! [B2: complex,A2: complex,C: complex] :
( ( ord_less_eq_complex @ B2 @ A2 )
=> ( ( ord_less_eq_complex @ C @ B2 )
=> ( ord_less_eq_complex @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_771_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_772_verit__comp__simplify_I2_J,axiom,
! [A2: complex] : ( ord_less_eq_complex @ A2 @ A2 ) ).
% verit_comp_simplify(2)
thf(fact_773_verit__comp__simplify_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify(2)
thf(fact_774_verit__la__disequality,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_nat @ A2 @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_775_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ord_less_eq_complex @ ( times_times_complex @ C @ A2 ) @ ( times_times_complex @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_776_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_777_mult__nonneg__nonpos2,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A2 )
=> ( ( ord_less_eq_complex @ B2 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ B2 @ A2 ) @ zero_zero_complex ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_778_mult__nonneg__nonpos2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_779_mult__nonpos__nonneg,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq_complex @ A2 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B2 )
=> ( ord_less_eq_complex @ ( times_times_complex @ A2 @ B2 ) @ zero_zero_complex ) ) ) ).
% mult_nonpos_nonneg
thf(fact_780_mult__nonpos__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_781_mult__nonneg__nonpos,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A2 )
=> ( ( ord_less_eq_complex @ B2 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ A2 @ B2 ) @ zero_zero_complex ) ) ) ).
% mult_nonneg_nonpos
thf(fact_782_mult__nonneg__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_783_mult__nonneg__nonneg,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B2 )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A2 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_784_mult__nonneg__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_785_split__mult__neg__le,axiom,
! [A2: complex,B2: complex] :
( ( ( ( ord_less_eq_complex @ zero_zero_complex @ A2 )
& ( ord_less_eq_complex @ B2 @ zero_zero_complex ) )
| ( ( ord_less_eq_complex @ A2 @ zero_zero_complex )
& ( ord_less_eq_complex @ zero_zero_complex @ B2 ) ) )
=> ( ord_less_eq_complex @ ( times_times_complex @ A2 @ B2 ) @ zero_zero_complex ) ) ).
% split_mult_neg_le
thf(fact_786_split__mult__neg__le,axiom,
! [A2: nat,B2: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
& ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_787_mult__right__mono,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ord_less_eq_complex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_788_mult__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_789_mult__right__mono__neg,axiom,
! [B2: complex,A2: complex,C: complex] :
( ( ord_less_eq_complex @ B2 @ A2 )
=> ( ( ord_less_eq_complex @ C @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B2 @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_790_mult__left__mono,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ord_less_eq_complex @ ( times_times_complex @ C @ A2 ) @ ( times_times_complex @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_791_mult__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_792_mult__nonpos__nonpos,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq_complex @ A2 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ B2 @ zero_zero_complex )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A2 @ B2 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_793_mult__left__mono__neg,axiom,
! [B2: complex,A2: complex,C: complex] :
( ( ord_less_eq_complex @ B2 @ A2 )
=> ( ( ord_less_eq_complex @ C @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ C @ A2 ) @ ( times_times_complex @ C @ B2 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_794_split__mult__pos__le,axiom,
! [A2: complex,B2: complex] :
( ( ( ( ord_less_eq_complex @ zero_zero_complex @ A2 )
& ( ord_less_eq_complex @ zero_zero_complex @ B2 ) )
| ( ( ord_less_eq_complex @ A2 @ zero_zero_complex )
& ( ord_less_eq_complex @ B2 @ zero_zero_complex ) ) )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A2 @ B2 ) ) ) ).
% split_mult_pos_le
thf(fact_795_mult__mono_H,axiom,
! [A2: complex,B2: complex,C: complex,D: complex] :
( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ( ord_less_eq_complex @ C @ D )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ A2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ord_less_eq_complex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B2 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_796_mult__mono_H,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_797_mult__mono,axiom,
! [A2: complex,B2: complex,C: complex,D: complex] :
( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ( ord_less_eq_complex @ C @ D )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ord_less_eq_complex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B2 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_798_mult__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_799_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_800_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_801_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_802_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_803_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_804_mult__right__le__imp__le,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_805_mult__left__le__imp__le,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_806_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_807_mult__right__less__imp__less,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_808_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_809_mult__left__less__imp__less,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_810_mult__left__le,axiom,
! [C: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_811_mult__le__one,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_812_irreducible__mult,axiom,
! [A2: complex,B2: complex] :
( ( factor4870819777162427853omplex @ ( times_times_complex @ A2 @ B2 ) )
= ( ( ( dvd_dvd_complex @ A2 @ one_one_complex )
& ( factor4870819777162427853omplex @ B2 ) )
| ( ( dvd_dvd_complex @ B2 @ one_one_complex )
& ( factor4870819777162427853omplex @ A2 ) ) ) ) ).
% irreducible_mult
thf(fact_813_irreducible__multD,axiom,
! [A2: complex,B2: complex] :
( ( factor4870819777162427853omplex @ ( times_times_complex @ A2 @ B2 ) )
=> ( ( ( dvd_dvd_complex @ A2 @ one_one_complex )
& ( factor4870819777162427853omplex @ B2 ) )
| ( ( dvd_dvd_complex @ B2 @ one_one_complex )
& ( factor4870819777162427853omplex @ A2 ) ) ) ) ).
% irreducible_multD
thf(fact_814_idom__class_Oirreducible__mult__unit__left,axiom,
! [A2: complex,P6: complex] :
( ( dvd_dvd_complex @ A2 @ one_one_complex )
=> ( ( factor4870819777162427853omplex @ ( times_times_complex @ A2 @ P6 ) )
= ( factor4870819777162427853omplex @ P6 ) ) ) ).
% idom_class.irreducible_mult_unit_left
thf(fact_815_irreducible__mult__unit__right,axiom,
! [A2: complex,P6: complex] :
( ( dvd_dvd_complex @ A2 @ one_one_complex )
=> ( ( factor4870819777162427853omplex @ ( times_times_complex @ P6 @ A2 ) )
= ( factor4870819777162427853omplex @ P6 ) ) ) ).
% irreducible_mult_unit_right
thf(fact_816_trace__adjoint__positive,axiom,
! [A: mat_complex] : ( ord_less_eq_complex @ zero_zero_complex @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ A @ ( schur_5982229384592763574omplex @ A ) ) ) ) ).
% trace_adjoint_positive
thf(fact_817_irreducibleE,axiom,
! [P6: nat] :
( ( factor4388943552880185071le_nat @ P6 )
=> ~ ( ( P6 != zero_zero_nat )
=> ( ~ ( dvd_dvd_nat @ P6 @ one_one_nat )
=> ~ ! [A7: nat,B8: nat] :
( ( P6
= ( times_times_nat @ A7 @ B8 ) )
=> ( ( dvd_dvd_nat @ A7 @ one_one_nat )
| ( dvd_dvd_nat @ B8 @ one_one_nat ) ) ) ) ) ) ).
% irreducibleE
thf(fact_818_irreducibleE,axiom,
! [P6: complex] :
( ( factor4870819777162427853omplex @ P6 )
=> ~ ( ( P6 != zero_zero_complex )
=> ( ~ ( dvd_dvd_complex @ P6 @ one_one_complex )
=> ~ ! [A7: complex,B8: complex] :
( ( P6
= ( times_times_complex @ A7 @ B8 ) )
=> ( ( dvd_dvd_complex @ A7 @ one_one_complex )
| ( dvd_dvd_complex @ B8 @ one_one_complex ) ) ) ) ) ) ).
% irreducibleE
thf(fact_819_irreducibleI,axiom,
! [P6: nat] :
( ( P6 != zero_zero_nat )
=> ( ~ ( dvd_dvd_nat @ P6 @ one_one_nat )
=> ( ! [A5: nat,B4: nat] :
( ( P6
= ( times_times_nat @ A5 @ B4 ) )
=> ( ( dvd_dvd_nat @ A5 @ one_one_nat )
| ( dvd_dvd_nat @ B4 @ one_one_nat ) ) )
=> ( factor4388943552880185071le_nat @ P6 ) ) ) ) ).
% irreducibleI
thf(fact_820_irreducibleI,axiom,
! [P6: complex] :
( ( P6 != zero_zero_complex )
=> ( ~ ( dvd_dvd_complex @ P6 @ one_one_complex )
=> ( ! [A5: complex,B4: complex] :
( ( P6
= ( times_times_complex @ A5 @ B4 ) )
=> ( ( dvd_dvd_complex @ A5 @ one_one_complex )
| ( dvd_dvd_complex @ B4 @ one_one_complex ) ) )
=> ( factor4870819777162427853omplex @ P6 ) ) ) ) ).
% irreducibleI
thf(fact_821_irreducible__def,axiom,
( factor4388943552880185071le_nat
= ( ^ [P: nat] :
( ( P != zero_zero_nat )
& ~ ( dvd_dvd_nat @ P @ one_one_nat )
& ! [A3: nat,B3: nat] :
( ( P
= ( times_times_nat @ A3 @ B3 ) )
=> ( ( dvd_dvd_nat @ A3 @ one_one_nat )
| ( dvd_dvd_nat @ B3 @ one_one_nat ) ) ) ) ) ) ).
% irreducible_def
thf(fact_822_irreducible__def,axiom,
( factor4870819777162427853omplex
= ( ^ [P: complex] :
( ( P != zero_zero_complex )
& ~ ( dvd_dvd_complex @ P @ one_one_complex )
& ! [A3: complex,B3: complex] :
( ( P
= ( times_times_complex @ A3 @ B3 ) )
=> ( ( dvd_dvd_complex @ A3 @ one_one_complex )
| ( dvd_dvd_complex @ B3 @ one_one_complex ) ) ) ) ) ) ).
% irreducible_def
thf(fact_823_irreducibleD,axiom,
! [P6: nat,A2: nat,B2: nat] :
( ( factor4388943552880185071le_nat @ P6 )
=> ( ( P6
= ( times_times_nat @ A2 @ B2 ) )
=> ( ( dvd_dvd_nat @ A2 @ one_one_nat )
| ( dvd_dvd_nat @ B2 @ one_one_nat ) ) ) ) ).
% irreducibleD
thf(fact_824_irreducibleD,axiom,
! [P6: complex,A2: complex,B2: complex] :
( ( factor4870819777162427853omplex @ P6 )
=> ( ( P6
= ( times_times_complex @ A2 @ B2 ) )
=> ( ( dvd_dvd_complex @ A2 @ one_one_complex )
| ( dvd_dvd_complex @ B2 @ one_one_complex ) ) ) ) ).
% irreducibleD
thf(fact_825_algebraic__semidom__class_Oirreducible__mult__unit__left,axiom,
! [A2: nat,P6: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( factor4388943552880185071le_nat @ ( times_times_nat @ A2 @ P6 ) )
= ( factor4388943552880185071le_nat @ P6 ) ) ) ).
% algebraic_semidom_class.irreducible_mult_unit_left
thf(fact_826_mult__le__mono2,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_827_mult__le__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).
% mult_le_mono1
thf(fact_828_mult__le__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_829_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_830_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_831_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_832_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_833_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_834_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_835_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_836_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_837_le__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% le_simps(1)
thf(fact_838_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_839_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_840_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_841_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_842_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J2: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_843_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_844_set__times__mono5,axiom,
! [A2: mat_a,C2: set_mat_a,B: set_mat_a,D2: set_mat_a] :
( ( member_mat_a @ A2 @ C2 )
=> ( ( ord_le3318621148231462513_mat_a @ B @ D2 )
=> ( ord_le3318621148231462513_mat_a @ ( set_el1062546952344711308_mat_a @ A2 @ B ) @ ( times_1230744552615602198_mat_a @ C2 @ D2 ) ) ) ) ).
% set_times_mono5
thf(fact_845_set__times__mono5,axiom,
! [A2: mat_complex,C2: set_mat_complex,B: set_mat_complex,D2: set_mat_complex] :
( ( member_mat_complex @ A2 @ C2 )
=> ( ( ord_le3632134057777142183omplex @ B @ D2 )
=> ( ord_le3632134057777142183omplex @ ( set_el176066062795894710omplex @ A2 @ B ) @ ( times_6731331324747250370omplex @ C2 @ D2 ) ) ) ) ).
% set_times_mono5
thf(fact_846_set__times__mono3,axiom,
! [A2: mat_a,C2: set_mat_a,D2: set_mat_a] :
( ( member_mat_a @ A2 @ C2 )
=> ( ord_le3318621148231462513_mat_a @ ( set_el1062546952344711308_mat_a @ A2 @ D2 ) @ ( times_1230744552615602198_mat_a @ C2 @ D2 ) ) ) ).
% set_times_mono3
thf(fact_847_set__times__mono3,axiom,
! [A2: mat_complex,C2: set_mat_complex,D2: set_mat_complex] :
( ( member_mat_complex @ A2 @ C2 )
=> ( ord_le3632134057777142183omplex @ ( set_el176066062795894710omplex @ A2 @ D2 ) @ ( times_6731331324747250370omplex @ C2 @ D2 ) ) ) ).
% set_times_mono3
thf(fact_848_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_849_mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel1
thf(fact_850_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_851_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ) ).
% dvd_imp_le
thf(fact_852_mult__eq__1,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A2 )
=> ( ( ord_less_eq_complex @ A2 @ one_one_complex )
=> ( ( ord_less_eq_complex @ B2 @ one_one_complex )
=> ( ( ( times_times_complex @ A2 @ B2 )
= one_one_complex )
= ( ( A2 = one_one_complex )
& ( B2 = one_one_complex ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_853_mult__eq__1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ B2 @ one_one_nat )
=> ( ( ( times_times_nat @ A2 @ B2 )
= one_one_nat )
= ( ( A2 = one_one_nat )
& ( B2 = one_one_nat ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_854_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_855_le__trans,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_856_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_857_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_858_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_859_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B2: nat] :
( ( P2 @ K )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ? [X2: nat] :
( ( P2 @ X2 )
& ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_860_Set_Obasic__monos_I7_J,axiom,
! [A: set_mat_a,B: set_mat_a,X: mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ B )
=> ( ( member_mat_a @ X @ A )
=> ( member_mat_a @ X @ B ) ) ) ).
% Set.basic_monos(7)
thf(fact_861_Set_Obasic__monos_I7_J,axiom,
! [A: set_mat_complex,B: set_mat_complex,X: mat_complex] :
( ( ord_le3632134057777142183omplex @ A @ B )
=> ( ( member_mat_complex @ X @ A )
=> ( member_mat_complex @ X @ B ) ) ) ).
% Set.basic_monos(7)
thf(fact_862_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_863_basic__trans__rules_I31_J,axiom,
! [A: set_mat_complex,B: set_mat_complex,C: mat_complex] :
( ( ord_le3632134057777142183omplex @ A @ B )
=> ( ( member_mat_complex @ C @ A )
=> ( member_mat_complex @ C @ B ) ) ) ).
% basic_trans_rules(31)
thf(fact_864_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_865_subsetI,axiom,
! [A: set_mat_complex,B: set_mat_complex] :
( ! [X2: mat_complex] :
( ( member_mat_complex @ X2 @ A )
=> ( member_mat_complex @ X2 @ B ) )
=> ( ord_le3632134057777142183omplex @ A @ B ) ) ).
% subsetI
thf(fact_866_subset__eq,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [A4: set_mat_a,B5: set_mat_a] :
! [X4: mat_a] :
( ( member_mat_a @ X4 @ A4 )
=> ( member_mat_a @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_867_subset__eq,axiom,
( ord_le3632134057777142183omplex
= ( ^ [A4: set_mat_complex,B5: set_mat_complex] :
! [X4: mat_complex] :
( ( member_mat_complex @ X4 @ A4 )
=> ( member_mat_complex @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_868_subset__iff,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [A4: set_mat_a,B5: set_mat_a] :
! [T2: mat_a] :
( ( member_mat_a @ T2 @ A4 )
=> ( member_mat_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_869_subset__iff,axiom,
( ord_le3632134057777142183omplex
= ( ^ [A4: set_mat_complex,B5: set_mat_complex] :
! [T2: mat_complex] :
( ( member_mat_complex @ T2 @ A4 )
=> ( member_mat_complex @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_870_positive__proj__trace,axiom,
! [P2: mat_complex,R: mat_complex,N: nat] :
( ( linear5633924348262549461omplex @ P2 )
=> ( ( complex_positive @ R )
=> ( ( member_mat_complex @ P2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ R @ ( carrier_mat_complex @ N @ N ) )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R @ P2 ) ) ) ) ) ) ) ).
% positive_proj_trace
thf(fact_871_positive__close__under__left__right__mult__adjoint,axiom,
! [M4: mat_complex,N: nat,A: mat_complex] :
( ( member_mat_complex @ M4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_positive @ A )
=> ( complex_positive @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ M4 @ A ) @ ( schur_5982229384592763574omplex @ M4 ) ) ) ) ) ) ).
% positive_close_under_left_right_mult_adjoint
thf(fact_872_positive__only__if__decomp,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_positive @ A )
=> ? [X2: mat_complex] :
( ( member_mat_complex @ X2 @ ( carrier_mat_complex @ N @ N ) )
& ( ( times_8009071140041733218omplex @ X2 @ ( schur_5982229384592763574omplex @ X2 ) )
= A ) ) ) ) ).
% positive_only_if_decomp
thf(fact_873_positive__iff__decomp,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_positive @ A )
= ( ? [X4: mat_complex] :
( ( member_mat_complex @ X4 @ ( carrier_mat_complex @ N @ N ) )
& ( ( times_8009071140041733218omplex @ X4 @ ( schur_5982229384592763574omplex @ X4 ) )
= A ) ) ) ) ) ).
% positive_iff_decomp
thf(fact_874_positive__if__decomp,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ? [M6: mat_complex] :
( ( times_8009071140041733218omplex @ M6 @ ( schur_5982229384592763574omplex @ M6 ) )
= A )
=> ( complex_positive @ A ) ) ) ).
% positive_if_decomp
thf(fact_875_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K3: nat] :
? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( F @ K3 @ I4 ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ! [K4: nat] :
? [K5: nat] :
( ( ord_less_eq_nat @ K4 @ K5 )
& ( F @ K5 @ I3 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_876_nat__descend__induct,axiom,
! [N: nat,P2: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P2 @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K3 @ I4 )
=> ( P2 @ I4 ) )
=> ( P2 @ K3 ) ) )
=> ( P2 @ M ) ) ) ).
% nat_descend_induct
thf(fact_877_pivot__fun__swaprows,axiom,
! [A: mat_complex,F: nat > nat,Jj: nat,Nr: nat,Nc: nat,L: nat,K: nat] :
( ( gauss_2609248829700396350omplex @ A @ F @ Jj )
=> ( ( ( dim_row_complex @ A )
= Nr )
=> ( ( ( dim_col_complex @ A )
= Nc )
=> ( ( ( F @ L )
= Jj )
=> ( ( ( F @ K )
= Jj )
=> ( ( ord_less_nat @ L @ Nr )
=> ( ( ord_less_nat @ K @ Nr )
=> ( ( ord_less_eq_nat @ Jj @ Nc )
=> ( gauss_2609248829700396350omplex @ ( gauss_1020679828357514249omplex @ L @ K @ A ) @ F @ Jj ) ) ) ) ) ) ) ) ) ).
% pivot_fun_swaprows
thf(fact_878_pivot__funD_I1_J,axiom,
! [A: mat_complex,Nr: nat,F: nat > nat,Nc: nat,I2: nat] :
( ( ( dim_row_complex @ A )
= Nr )
=> ( ( gauss_2609248829700396350omplex @ A @ F @ Nc )
=> ( ( ord_less_nat @ I2 @ Nr )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ Nc ) ) ) ) ).
% pivot_funD(1)
thf(fact_879_pivot__fun__multrow,axiom,
! [A: mat_nat,F: nat > nat,Jj: nat,Nr: nat,Nc: nat,I0: nat,A2: nat] :
( ( gauss_8416567519840421984un_nat @ A @ F @ Jj )
=> ( ( ( dim_row_nat @ A )
= Nr )
=> ( ( ( dim_col_nat @ A )
= Nc )
=> ( ( ( F @ I0 )
= Jj )
=> ( ( ord_less_eq_nat @ Jj @ Nc )
=> ( gauss_8416567519840421984un_nat @ ( gauss_2409696420326117733en_nat @ times_times_nat @ I0 @ A2 @ A ) @ F @ Jj ) ) ) ) ) ) ).
% pivot_fun_multrow
thf(fact_880_pivot__fun__multrow,axiom,
! [A: mat_complex,F: nat > nat,Jj: nat,Nr: nat,Nc: nat,I0: nat,A2: complex] :
( ( gauss_2609248829700396350omplex @ A @ F @ Jj )
=> ( ( ( dim_row_complex @ A )
= Nr )
=> ( ( ( dim_col_complex @ A )
= Nc )
=> ( ( ( F @ I0 )
= Jj )
=> ( ( ord_less_eq_nat @ Jj @ Nc )
=> ( gauss_2609248829700396350omplex @ ( gauss_2324787009747932227omplex @ times_times_complex @ I0 @ A2 @ A ) @ F @ Jj ) ) ) ) ) ) ).
% pivot_fun_multrow
thf(fact_881_index__mat__multrow_I4_J,axiom,
! [Mul: a > a > a,K: nat,A2: a,A: mat_a] :
( ( dim_row_a @ ( gauss_5154200947219177641_gen_a @ Mul @ K @ A2 @ A ) )
= ( dim_row_a @ A ) ) ).
% index_mat_multrow(4)
thf(fact_882_index__mat__multrow_I4_J,axiom,
! [Mul: complex > complex > complex,K: nat,A2: complex,A: mat_complex] :
( ( dim_row_complex @ ( gauss_2324787009747932227omplex @ Mul @ K @ A2 @ A ) )
= ( dim_row_complex @ A ) ) ).
% index_mat_multrow(4)
thf(fact_883_multrow__carrier,axiom,
! [Mul: a > a > a,K: nat,A2: a,A: mat_a,N: nat,Nc: nat] :
( ( member_mat_a @ ( gauss_5154200947219177641_gen_a @ Mul @ K @ A2 @ A ) @ ( carrier_mat_a @ N @ Nc ) )
= ( member_mat_a @ A @ ( carrier_mat_a @ N @ Nc ) ) ) ).
% multrow_carrier
thf(fact_884_multrow__carrier,axiom,
! [Mul: complex > complex > complex,K: nat,A2: complex,A: mat_complex,N: nat,Nc: nat] :
( ( member_mat_complex @ ( gauss_2324787009747932227omplex @ Mul @ K @ A2 @ A ) @ ( carrier_mat_complex @ N @ Nc ) )
= ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ Nc ) ) ) ).
% multrow_carrier
thf(fact_885_multrow__mat,axiom,
! [A: mat_nat,N: nat,Nc: nat,K: nat,A2: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ Nc ) )
=> ( ( gauss_2409696420326117733en_nat @ times_times_nat @ K @ A2 @ A )
= ( times_times_mat_nat @ ( gauss_3195076542185637913at_nat @ N @ K @ A2 ) @ A ) ) ) ).
% multrow_mat
thf(fact_886_multrow__mat,axiom,
! [A: mat_complex,N: nat,Nc: nat,K: nat,A2: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( gauss_2324787009747932227omplex @ times_times_complex @ K @ A2 @ A )
= ( times_8009071140041733218omplex @ ( gauss_6868829418328711927omplex @ N @ K @ A2 ) @ A ) ) ) ).
% multrow_mat
thf(fact_887_multrow__mat__carrier,axiom,
! [N: nat,K: nat,A2: complex] : ( member_mat_complex @ ( gauss_6868829418328711927omplex @ N @ K @ A2 ) @ ( carrier_mat_complex @ N @ N ) ) ).
% multrow_mat_carrier
thf(fact_888_index__mat__multrow__mat_I2_J,axiom,
! [N: nat,K: nat,A2: complex] :
( ( dim_row_complex @ ( gauss_6868829418328711927omplex @ N @ K @ A2 ) )
= N ) ).
% index_mat_multrow_mat(2)
thf(fact_889_multcol__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,K: nat,A2: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( column4410001698458707789omplex @ K @ A2 @ A )
= ( times_8009071140041733218omplex @ A @ ( gauss_6868829418328711927omplex @ N @ K @ A2 ) ) ) ) ).
% multcol_mat
thf(fact_890_index__mat__multcol_I4_J,axiom,
! [K: nat,A2: complex,A: mat_complex] :
( ( dim_row_complex @ ( column4410001698458707789omplex @ K @ A2 @ A ) )
= ( dim_row_complex @ A ) ) ).
% index_mat_multcol(4)
thf(fact_891_multrow__mat__inv,axiom,
! [K: nat,N: nat,A2: complex] :
( ( ord_less_nat @ K @ N )
=> ( ( A2 != zero_zero_complex )
=> ( ( times_8009071140041733218omplex @ ( gauss_6868829418328711927omplex @ N @ K @ A2 ) @ ( gauss_6868829418328711927omplex @ N @ K @ ( invers8013647133539491842omplex @ A2 ) ) )
= ( one_mat_complex @ N ) ) ) ) ).
% multrow_mat_inv
thf(fact_892_nonzero__imp__inverse__nonzero,axiom,
! [A2: complex] :
( ( A2 != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ A2 )
!= zero_zero_complex ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_893_inverse__nonzero__iff__nonzero,axiom,
! [A2: complex] :
( ( ( invers8013647133539491842omplex @ A2 )
= zero_zero_complex )
= ( A2 = zero_zero_complex ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_894_nonzero__inverse__inverse__eq,axiom,
! [A2: complex] :
( ( A2 != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A2 ) )
= A2 ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_895_nonzero__inverse__eq__imp__eq,axiom,
! [A2: complex,B2: complex] :
( ( ( invers8013647133539491842omplex @ A2 )
= ( invers8013647133539491842omplex @ B2 ) )
=> ( ( A2 != zero_zero_complex )
=> ( ( B2 != zero_zero_complex )
=> ( A2 = B2 ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_896_inverse__zero__imp__zero,axiom,
! [A2: complex] :
( ( ( invers8013647133539491842omplex @ A2 )
= zero_zero_complex )
=> ( A2 = zero_zero_complex ) ) ).
% inverse_zero_imp_zero
thf(fact_897_inverse__zero,axiom,
( ( invers8013647133539491842omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% inverse_zero
thf(fact_898_field__class_Ofield__inverse__zero,axiom,
( ( invers8013647133539491842omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% field_class.field_inverse_zero
thf(fact_899_nonzero__inverse__mult__distrib,axiom,
! [A2: complex,B2: complex] :
( ( A2 != zero_zero_complex )
=> ( ( B2 != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ ( times_times_complex @ A2 @ B2 ) )
= ( times_times_complex @ ( invers8013647133539491842omplex @ B2 ) @ ( invers8013647133539491842omplex @ A2 ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_900_inverse__unique,axiom,
! [A2: complex,B2: complex] :
( ( ( times_times_complex @ A2 @ B2 )
= one_one_complex )
=> ( ( invers8013647133539491842omplex @ A2 )
= B2 ) ) ).
% inverse_unique
thf(fact_901_mult__commute__imp__mult__inverse__commute,axiom,
! [Y: complex,X: complex] :
( ( ( times_times_complex @ Y @ X )
= ( times_times_complex @ X @ Y ) )
=> ( ( times_times_complex @ ( invers8013647133539491842omplex @ Y ) @ X )
= ( times_times_complex @ X @ ( invers8013647133539491842omplex @ Y ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_902_inverse__mult__distrib,axiom,
! [A2: complex,B2: complex] :
( ( invers8013647133539491842omplex @ ( times_times_complex @ A2 @ B2 ) )
= ( times_times_complex @ ( invers8013647133539491842omplex @ A2 ) @ ( invers8013647133539491842omplex @ B2 ) ) ) ).
% inverse_mult_distrib
thf(fact_903_right__inverse,axiom,
! [A2: complex] :
( ( A2 != zero_zero_complex )
=> ( ( times_times_complex @ A2 @ ( invers8013647133539491842omplex @ A2 ) )
= one_one_complex ) ) ).
% right_inverse
thf(fact_904_left__inverse,axiom,
! [A2: complex] :
( ( A2 != zero_zero_complex )
=> ( ( times_times_complex @ ( invers8013647133539491842omplex @ A2 ) @ A2 )
= one_one_complex ) ) ).
% left_inverse
thf(fact_905_field__class_Ofield__inverse,axiom,
! [A2: complex] :
( ( A2 != zero_zero_complex )
=> ( ( times_times_complex @ ( invers8013647133539491842omplex @ A2 ) @ A2 )
= one_one_complex ) ) ).
% field_class.field_inverse
thf(fact_906_mult__col__div__row__def,axiom,
( column217142795681433722omplex
= ( ^ [A3: complex,K2: nat,A4: mat_complex] : ( gauss_2324787009747932227omplex @ times_times_complex @ K2 @ ( invers8013647133539491842omplex @ A3 ) @ ( column4410001698458707789omplex @ K2 @ A3 @ A4 ) ) ) ) ).
% mult_col_div_row_def
thf(fact_907_power__strict__mono,axiom,
! [A2: nat,B2: nat,N: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_908_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_909_set__zero__plus2,axiom,
! [A: set_complex,B: set_complex] :
( ( member_complex @ zero_zero_complex @ A )
=> ( ord_le211207098394363844omplex @ B @ ( plus_p7052360327008956141omplex @ A @ B ) ) ) ).
% set_zero_plus2
thf(fact_910_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: complex,J2: complex,K: complex,L: complex] :
( ( ( ord_less_eq_complex @ I2 @ J2 )
& ( K = L ) )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ I2 @ K ) @ ( plus_plus_complex @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_911_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_912_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: complex,J2: complex,K: complex,L: complex] :
( ( ( I2 = J2 )
& ( ord_less_eq_complex @ K @ L ) )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ I2 @ K ) @ ( plus_plus_complex @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_913_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( I2 = J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_914_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: complex,J2: complex,K: complex,L: complex] :
( ( ( ord_less_eq_complex @ I2 @ J2 )
& ( ord_less_eq_complex @ K @ L ) )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ I2 @ K ) @ ( plus_plus_complex @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_915_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_916_add__mono,axiom,
! [A2: complex,B2: complex,C: complex,D: complex] :
( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ( ord_less_eq_complex @ C @ D )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A2 @ C ) @ ( plus_plus_complex @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_917_add__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_918_add__left__mono,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ C @ A2 ) @ ( plus_plus_complex @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_919_add__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_920_less__eqE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ! [C3: nat] :
( B2
!= ( plus_plus_nat @ A2 @ C3 ) ) ) ).
% less_eqE
thf(fact_921_add__right__mono,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A2 @ C ) @ ( plus_plus_complex @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_922_add__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_923_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C5: nat] :
( B3
= ( plus_plus_nat @ A3 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_924_add__le__cancel__left,axiom,
! [C: complex,A2: complex,B2: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ C @ A2 ) @ ( plus_plus_complex @ C @ B2 ) )
= ( ord_less_eq_complex @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_925_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_926_add__le__imp__le__left,axiom,
! [C: complex,A2: complex,B2: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ C @ A2 ) @ ( plus_plus_complex @ C @ B2 ) )
=> ( ord_less_eq_complex @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_927_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_928_add__le__cancel__right,axiom,
! [A2: complex,C: complex,B2: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ A2 @ C ) @ ( plus_plus_complex @ B2 @ C ) )
= ( ord_less_eq_complex @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_929_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_930_add__le__imp__le__right,axiom,
! [A2: complex,C: complex,B2: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ A2 @ C ) @ ( plus_plus_complex @ B2 @ C ) )
=> ( ord_less_eq_complex @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_931_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_932_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_933_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_934_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_935_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_936_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_937_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_938_add__le__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_939_add__le__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_940_trans__le__add1,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_le_add1
thf(fact_941_trans__le__add2,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_le_add2
thf(fact_942_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_943_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_944_add__sign__intros_I8_J,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq_complex @ A2 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ B2 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A2 @ B2 ) @ zero_zero_complex ) ) ) ).
% add_sign_intros(8)
thf(fact_945_add__sign__intros_I8_J,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_sign_intros(8)
thf(fact_946_add__sign__intros_I4_J,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B2 )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( plus_plus_complex @ A2 @ B2 ) ) ) ) ).
% add_sign_intros(4)
thf(fact_947_add__sign__intros_I4_J,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_sign_intros(4)
thf(fact_948_add__decreasing,axiom,
! [A2: complex,C: complex,B2: complex] :
( ( ord_less_eq_complex @ A2 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ C @ B2 )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_949_add__decreasing,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_950_add__increasing,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A2 )
=> ( ( ord_less_eq_complex @ B2 @ C )
=> ( ord_less_eq_complex @ B2 @ ( plus_plus_complex @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_951_add__increasing,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_952_add__decreasing2,axiom,
! [C: complex,A2: complex,B2: complex] :
( ( ord_less_eq_complex @ C @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ A2 @ B2 )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_953_add__decreasing2,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_954_add__increasing2,axiom,
! [C: complex,B2: complex,A2: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ( ord_less_eq_complex @ B2 @ A2 )
=> ( ord_less_eq_complex @ B2 @ ( plus_plus_complex @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_955_add__increasing2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_956_add__nonneg__eq__0__iff,axiom,
! [X: complex,Y: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ X )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ Y )
=> ( ( ( plus_plus_complex @ X @ Y )
= zero_zero_complex )
= ( ( X = zero_zero_complex )
& ( Y = zero_zero_complex ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_957_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_958_add__nonpos__eq__0__iff,axiom,
! [X: complex,Y: complex] :
( ( ord_less_eq_complex @ X @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ Y @ zero_zero_complex )
=> ( ( ( plus_plus_complex @ X @ Y )
= zero_zero_complex )
= ( ( X = zero_zero_complex )
& ( Y = zero_zero_complex ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_959_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_960_add__le__same__cancel1,axiom,
! [B2: complex,A2: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_complex @ A2 @ zero_zero_complex ) ) ).
% add_le_same_cancel1
thf(fact_961_add__le__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_962_add__le__same__cancel2,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_complex @ A2 @ zero_zero_complex ) ) ).
% add_le_same_cancel2
thf(fact_963_add__le__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_964_le__add__same__cancel1,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq_complex @ A2 @ ( plus_plus_complex @ A2 @ B2 ) )
= ( ord_less_eq_complex @ zero_zero_complex @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_965_le__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_966_le__add__same__cancel2,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq_complex @ A2 @ ( plus_plus_complex @ B2 @ A2 ) )
= ( ord_less_eq_complex @ zero_zero_complex @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_967_le__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_968_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_969_add__smult__distrib__left__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,B: mat_complex,K: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( smult_mat_complex @ K @ ( plus_p8323303612493835998omplex @ A @ B ) )
= ( plus_p8323303612493835998omplex @ ( smult_mat_complex @ K @ A ) @ ( smult_mat_complex @ K @ B ) ) ) ) ) ).
% add_smult_distrib_left_mat
thf(fact_970_adjoint__add,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ M ) )
=> ( ( schur_5982229384592763574omplex @ ( plus_p8323303612493835998omplex @ A @ B ) )
= ( plus_p8323303612493835998omplex @ ( schur_5982229384592763574omplex @ A ) @ ( schur_5982229384592763574omplex @ B ) ) ) ) ) ).
% adjoint_add
thf(fact_971_mat__assoc__test_I7_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C2: mat_complex,D2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ ( plus_p8323303612493835998omplex @ A @ B ) @ ( plus_p8323303612493835998omplex @ B @ C2 ) )
= ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ A @ B ) @ ( times_8009071140041733218omplex @ B @ B ) ) @ ( times_8009071140041733218omplex @ A @ C2 ) ) @ ( times_8009071140041733218omplex @ B @ C2 ) ) ) ) ) ) ) ).
% mat_assoc_test(7)
thf(fact_972_hermitian__add,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple8306762464034002205omplex @ A )
=> ( ( comple8306762464034002205omplex @ B )
=> ( comple8306762464034002205omplex @ ( plus_p8323303612493835998omplex @ A @ B ) ) ) ) ) ) ).
% hermitian_add
thf(fact_973_transpose__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 ) )
=> ( ( transpose_mat_a @ ( plus_plus_mat_a @ A @ B ) )
= ( plus_plus_mat_a @ ( transpose_mat_a @ A ) @ ( transpose_mat_a @ B ) ) ) ) ) ).
% transpose_add
thf(fact_974_transpose__add,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( transp3074176993011536131omplex @ ( plus_p8323303612493835998omplex @ A @ B ) )
= ( plus_p8323303612493835998omplex @ ( transp3074176993011536131omplex @ A ) @ ( transp3074176993011536131omplex @ B ) ) ) ) ) ).
% transpose_add
thf(fact_975_index__add__mat_I3_J,axiom,
! [A: mat_complex,B: mat_complex] :
( ( dim_col_complex @ ( plus_p8323303612493835998omplex @ A @ B ) )
= ( dim_col_complex @ B ) ) ).
% index_add_mat(3)
thf(fact_976_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_977_add__carrier__mat_H,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( plus_p8323303612493835998omplex @ A @ B ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% add_carrier_mat'
thf(fact_978_swap__plus__mat,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ B ) @ C2 )
= ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ C2 ) @ B ) ) ) ) ) ).
% swap_plus_mat
thf(fact_979_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_980_add__carrier__mat,axiom,
! [B: mat_complex,Nr: nat,Nc: nat,A: mat_complex] :
( ( member_mat_complex @ B @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( plus_p8323303612493835998omplex @ A @ B ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% add_carrier_mat
thf(fact_981_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_982_assoc__add__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,B: mat_complex,C2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ B ) @ C2 )
= ( plus_p8323303612493835998omplex @ A @ ( plus_p8323303612493835998omplex @ B @ C2 ) ) ) ) ) ) ).
% assoc_add_mat
thf(fact_983_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_984_comm__add__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( plus_p8323303612493835998omplex @ A @ B )
= ( plus_p8323303612493835998omplex @ B @ A ) ) ) ) ).
% comm_add_mat
thf(fact_985_class__semiring_Onat__pow__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% class_semiring.nat_pow_one
thf(fact_986_nat__arith_Oadd2,axiom,
! [B: nat,K: nat,B2: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% nat_arith.add2
thf(fact_987_nat__arith_Oadd2,axiom,
! [B: complex,K: complex,B2: complex,A2: complex] :
( ( B
= ( plus_plus_complex @ K @ B2 ) )
=> ( ( plus_plus_complex @ A2 @ B )
= ( plus_plus_complex @ K @ ( plus_plus_complex @ A2 @ B2 ) ) ) ) ).
% nat_arith.add2
thf(fact_988_nat__arith_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% nat_arith.add1
thf(fact_989_nat__arith_Oadd1,axiom,
! [A: complex,K: complex,A2: complex,B2: complex] :
( ( A
= ( plus_plus_complex @ K @ A2 ) )
=> ( ( plus_plus_complex @ A @ B2 )
= ( plus_plus_complex @ K @ ( plus_plus_complex @ A2 @ B2 ) ) ) ) ).
% nat_arith.add1
thf(fact_990_add__right__imp__eq,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_991_add__right__imp__eq,axiom,
! [B2: complex,A2: complex,C: complex] :
( ( ( plus_plus_complex @ B2 @ A2 )
= ( plus_plus_complex @ C @ A2 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_992_add__right__cancel,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_993_add__right__cancel,axiom,
! [B2: complex,A2: complex,C: complex] :
( ( ( plus_plus_complex @ B2 @ A2 )
= ( plus_plus_complex @ C @ A2 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_994_add__left__imp__eq,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_995_add__left__imp__eq,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( ( plus_plus_complex @ A2 @ B2 )
= ( plus_plus_complex @ A2 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_996_add__left__cancel,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_997_add__left__cancel,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( ( plus_plus_complex @ A2 @ B2 )
= ( plus_plus_complex @ A2 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_998_add_Oleft__commute,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_999_add_Oleft__commute,axiom,
! [B2: complex,A2: complex,C: complex] :
( ( plus_plus_complex @ B2 @ ( plus_plus_complex @ A2 @ C ) )
= ( plus_plus_complex @ A2 @ ( plus_plus_complex @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_1000_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_1001_add_Ocommute,axiom,
( plus_plus_complex
= ( ^ [A3: complex,B3: complex] : ( plus_plus_complex @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_1002_add_Oright__cancel,axiom,
! [B2: complex,A2: complex,C: complex] :
( ( ( plus_plus_complex @ B2 @ A2 )
= ( plus_plus_complex @ C @ A2 ) )
= ( B2 = C ) ) ).
% add.right_cancel
thf(fact_1003_add_Oleft__cancel,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( ( plus_plus_complex @ A2 @ B2 )
= ( plus_plus_complex @ A2 @ C ) )
= ( B2 = C ) ) ).
% add.left_cancel
thf(fact_1004_add_Oassoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_1005_add_Oassoc,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A2 @ B2 ) @ C )
= ( plus_plus_complex @ A2 @ ( plus_plus_complex @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_1006_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( I2 = J2 )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1007_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: complex,J2: complex,K: complex,L: complex] :
( ( ( I2 = J2 )
& ( K = L ) )
=> ( ( plus_plus_complex @ I2 @ K )
= ( plus_plus_complex @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1008_is__num__normalize_I1_J,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A2 @ B2 ) @ C )
= ( plus_plus_complex @ A2 @ ( plus_plus_complex @ B2 @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_1009_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1010_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A2 @ B2 ) @ C )
= ( plus_plus_complex @ A2 @ ( plus_plus_complex @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1011_set__plus__intro,axiom,
! [A2: mat_a,C2: set_mat_a,B2: mat_a,D2: set_mat_a] :
( ( member_mat_a @ A2 @ C2 )
=> ( ( member_mat_a @ B2 @ D2 )
=> ( member_mat_a @ ( plus_plus_mat_a @ A2 @ B2 ) @ ( plus_plus_set_mat_a @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_1012_set__plus__intro,axiom,
! [A2: nat,C2: set_nat,B2: nat,D2: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( member_nat @ B2 @ D2 )
=> ( member_nat @ ( plus_plus_nat @ A2 @ B2 ) @ ( plus_plus_set_nat @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_1013_set__plus__intro,axiom,
! [A2: mat_complex,C2: set_mat_complex,B2: mat_complex,D2: set_mat_complex] :
( ( member_mat_complex @ A2 @ C2 )
=> ( ( member_mat_complex @ B2 @ D2 )
=> ( member_mat_complex @ ( plus_p8323303612493835998omplex @ A2 @ B2 ) @ ( plus_p4229080058245121342omplex @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_1014_set__plus__intro,axiom,
! [A2: complex,C2: set_complex,B2: complex,D2: set_complex] :
( ( member_complex @ A2 @ C2 )
=> ( ( member_complex @ B2 @ D2 )
=> ( member_complex @ ( plus_plus_complex @ A2 @ B2 ) @ ( plus_p7052360327008956141omplex @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_1015_set__plus__elim,axiom,
! [X: mat_a,A: set_mat_a,B: set_mat_a] :
( ( member_mat_a @ X @ ( plus_plus_set_mat_a @ A @ B ) )
=> ~ ! [A5: mat_a,B4: mat_a] :
( ( X
= ( plus_plus_mat_a @ A5 @ B4 ) )
=> ( ( member_mat_a @ A5 @ A )
=> ~ ( member_mat_a @ B4 @ B ) ) ) ) ).
% set_plus_elim
thf(fact_1016_set__plus__elim,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ( member_nat @ X @ ( plus_plus_set_nat @ A @ B ) )
=> ~ ! [A5: nat,B4: nat] :
( ( X
= ( plus_plus_nat @ A5 @ B4 ) )
=> ( ( member_nat @ A5 @ A )
=> ~ ( member_nat @ B4 @ B ) ) ) ) ).
% set_plus_elim
thf(fact_1017_set__plus__elim,axiom,
! [X: mat_complex,A: set_mat_complex,B: set_mat_complex] :
( ( member_mat_complex @ X @ ( plus_p4229080058245121342omplex @ A @ B ) )
=> ~ ! [A5: mat_complex,B4: mat_complex] :
( ( X
= ( plus_p8323303612493835998omplex @ A5 @ B4 ) )
=> ( ( member_mat_complex @ A5 @ A )
=> ~ ( member_mat_complex @ B4 @ B ) ) ) ) ).
% set_plus_elim
thf(fact_1018_set__plus__elim,axiom,
! [X: complex,A: set_complex,B: set_complex] :
( ( member_complex @ X @ ( plus_p7052360327008956141omplex @ A @ B ) )
=> ~ ! [A5: complex,B4: complex] :
( ( X
= ( plus_plus_complex @ A5 @ B4 ) )
=> ( ( member_complex @ A5 @ A )
=> ~ ( member_complex @ B4 @ B ) ) ) ) ).
% set_plus_elim
thf(fact_1019_class__semiring_Oadd_Ofactors__equal,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( A2 = B2 )
=> ( ( C = D )
=> ( ( plus_plus_nat @ A2 @ C )
= ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% class_semiring.add.factors_equal
thf(fact_1020_class__semiring_Oadd_Ofactors__equal,axiom,
! [A2: complex,B2: complex,C: complex,D: complex] :
( ( A2 = B2 )
=> ( ( C = D )
=> ( ( plus_plus_complex @ A2 @ C )
= ( plus_plus_complex @ B2 @ D ) ) ) ) ).
% class_semiring.add.factors_equal
thf(fact_1021_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_1022_index__add__mat_I2_J,axiom,
! [A: mat_complex,B: mat_complex] :
( ( dim_row_complex @ ( plus_p8323303612493835998omplex @ A @ B ) )
= ( dim_row_complex @ B ) ) ).
% index_add_mat(2)
thf(fact_1023_power__commutes,axiom,
! [A2: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ A2 )
= ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ).
% power_commutes
thf(fact_1024_power__commutes,axiom,
! [A2: complex,N: nat] :
( ( times_times_complex @ ( power_power_complex @ A2 @ N ) @ A2 )
= ( times_times_complex @ A2 @ ( power_power_complex @ A2 @ N ) ) ) ).
% power_commutes
thf(fact_1025_power__mult__distrib,axiom,
! [A2: nat,B2: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A2 @ B2 ) @ N )
= ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ).
% power_mult_distrib
thf(fact_1026_power__mult__distrib,axiom,
! [A2: complex,B2: complex,N: nat] :
( ( power_power_complex @ ( times_times_complex @ A2 @ B2 ) @ N )
= ( times_times_complex @ ( power_power_complex @ A2 @ N ) @ ( power_power_complex @ B2 @ N ) ) ) ).
% power_mult_distrib
thf(fact_1027_power__commuting__commutes,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= ( times_times_nat @ Y @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_1028_power__commuting__commutes,axiom,
! [X: complex,Y: complex,N: nat] :
( ( ( times_times_complex @ X @ Y )
= ( times_times_complex @ Y @ X ) )
=> ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ Y )
= ( times_times_complex @ Y @ ( power_power_complex @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_1029_power__add,axiom,
! [A2: nat,M: nat,N: nat] :
( ( power_power_nat @ A2 @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) ) ) ).
% power_add
thf(fact_1030_power__add,axiom,
! [A2: complex,M: nat,N: nat] :
( ( power_power_complex @ A2 @ ( plus_plus_nat @ M @ N ) )
= ( times_times_complex @ ( power_power_complex @ A2 @ M ) @ ( power_power_complex @ A2 @ N ) ) ) ).
% power_add
thf(fact_1031_Rings_Oring__distribs_I2_J,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).
% Rings.ring_distribs(2)
thf(fact_1032_Rings_Oring__distribs_I2_J,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A2 @ B2 ) @ C )
= ( plus_plus_complex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B2 @ C ) ) ) ).
% Rings.ring_distribs(2)
thf(fact_1033_Rings_Oring__distribs_I1_J,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( times_times_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% Rings.ring_distribs(1)
thf(fact_1034_Rings_Oring__distribs_I1_J,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( times_times_complex @ A2 @ ( plus_plus_complex @ B2 @ C ) )
= ( plus_plus_complex @ ( times_times_complex @ A2 @ B2 ) @ ( times_times_complex @ A2 @ C ) ) ) ).
% Rings.ring_distribs(1)
thf(fact_1035_ring__class_Oring__distribs_I2_J,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A2 @ B2 ) @ C )
= ( plus_plus_complex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B2 @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1036_ring__class_Oring__distribs_I1_J,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( times_times_complex @ A2 @ ( plus_plus_complex @ B2 @ C ) )
= ( plus_plus_complex @ ( times_times_complex @ A2 @ B2 ) @ ( times_times_complex @ A2 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1037_comm__semiring__class_Odistrib,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1038_comm__semiring__class_Odistrib,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A2 @ B2 ) @ C )
= ( plus_plus_complex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B2 @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1039_combine__common__factor,axiom,
! [A2: nat,E: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A2 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B2 @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1040_combine__common__factor,axiom,
! [A2: complex,E: complex,B2: complex,C: complex] :
( ( plus_plus_complex @ ( times_times_complex @ A2 @ E ) @ ( plus_plus_complex @ ( times_times_complex @ B2 @ E ) @ C ) )
= ( plus_plus_complex @ ( times_times_complex @ ( plus_plus_complex @ A2 @ B2 ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1041_vector__space__over__itself_Oscale__right__distrib,axiom,
! [A2: complex,X: complex,Y: complex] :
( ( times_times_complex @ A2 @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_complex @ ( times_times_complex @ A2 @ X ) @ ( times_times_complex @ A2 @ Y ) ) ) ).
% vector_space_over_itself.scale_right_distrib
thf(fact_1042_vector__space__over__itself_Oscale__left__distrib,axiom,
! [A2: complex,B2: complex,X: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A2 @ B2 ) @ X )
= ( plus_plus_complex @ ( times_times_complex @ A2 @ X ) @ ( times_times_complex @ B2 @ X ) ) ) ).
% vector_space_over_itself.scale_left_distrib
thf(fact_1043_mult__hom_Ohom__add,axiom,
! [C: nat,X: nat,Y: nat] :
( ( times_times_nat @ C @ ( plus_plus_nat @ X @ Y ) )
= ( plus_plus_nat @ ( times_times_nat @ C @ X ) @ ( times_times_nat @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_1044_mult__hom_Ohom__add,axiom,
! [C: complex,X: complex,Y: complex] :
( ( times_times_complex @ C @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_complex @ ( times_times_complex @ C @ X ) @ ( times_times_complex @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_1045_left__add__mult__distrib,axiom,
! [I2: nat,U3: nat,J2: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I2 @ U3 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U3 ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J2 ) @ U3 ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1046_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1047_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1048_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_1049_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_1050_add__smult__distrib__right__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,K: complex,L: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( smult_mat_complex @ ( plus_plus_complex @ K @ L ) @ A )
= ( plus_p8323303612493835998omplex @ ( smult_mat_complex @ K @ A ) @ ( smult_mat_complex @ L @ A ) ) ) ) ).
% add_smult_distrib_right_mat
thf(fact_1051_trace__add__linear,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple3184165445352484367omplex @ ( plus_p8323303612493835998omplex @ A @ B ) )
= ( plus_plus_complex @ ( comple3184165445352484367omplex @ A ) @ ( comple3184165445352484367omplex @ B ) ) ) ) ) ).
% trace_add_linear
thf(fact_1052_dvd__add__triv__right__iff,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( dvd_dvd_nat @ A2 @ B2 ) ) ).
% dvd_add_triv_right_iff
thf(fact_1053_dvd__add__triv__right__iff,axiom,
! [A2: complex,B2: complex] :
( ( dvd_dvd_complex @ A2 @ ( plus_plus_complex @ B2 @ A2 ) )
= ( dvd_dvd_complex @ A2 @ B2 ) ) ).
% dvd_add_triv_right_iff
thf(fact_1054_dvd__add__triv__left__iff,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( dvd_dvd_nat @ A2 @ B2 ) ) ).
% dvd_add_triv_left_iff
thf(fact_1055_dvd__add__triv__left__iff,axiom,
! [A2: complex,B2: complex] :
( ( dvd_dvd_complex @ A2 @ ( plus_plus_complex @ A2 @ B2 ) )
= ( dvd_dvd_complex @ A2 @ B2 ) ) ).
% dvd_add_triv_left_iff
thf(fact_1056_dvd__add__right__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_1057_dvd__add__right__iff,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( dvd_dvd_complex @ A2 @ B2 )
=> ( ( dvd_dvd_complex @ A2 @ ( plus_plus_complex @ B2 @ C ) )
= ( dvd_dvd_complex @ A2 @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_1058_dvd__add__left__iff,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ C )
=> ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) )
= ( dvd_dvd_nat @ A2 @ B2 ) ) ) ).
% dvd_add_left_iff
thf(fact_1059_dvd__add__left__iff,axiom,
! [A2: complex,C: complex,B2: complex] :
( ( dvd_dvd_complex @ A2 @ C )
=> ( ( dvd_dvd_complex @ A2 @ ( plus_plus_complex @ B2 @ C ) )
= ( dvd_dvd_complex @ A2 @ B2 ) ) ) ).
% dvd_add_left_iff
thf(fact_1060_dvd__add,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ A2 @ C )
=> ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ) ).
% dvd_add
thf(fact_1061_dvd__add,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( dvd_dvd_complex @ A2 @ B2 )
=> ( ( dvd_dvd_complex @ A2 @ C )
=> ( dvd_dvd_complex @ A2 @ ( plus_plus_complex @ B2 @ C ) ) ) ) ).
% dvd_add
thf(fact_1062_bezout__lemma__nat,axiom,
! [D: nat,A2: nat,B2: nat,X: nat,Y: nat] :
( ( dvd_dvd_nat @ D @ A2 )
=> ( ( dvd_dvd_nat @ D @ B2 )
=> ( ( ( ( times_times_nat @ A2 @ X )
= ( plus_plus_nat @ ( times_times_nat @ B2 @ Y ) @ D ) )
| ( ( times_times_nat @ B2 @ X )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y ) @ D ) ) )
=> ? [X2: nat,Y2: nat] :
( ( dvd_dvd_nat @ D @ A2 )
& ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A2 @ B2 ) )
& ( ( ( times_times_nat @ A2 @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ Y2 ) @ D ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y2 ) @ D ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1063_bezout__add__nat,axiom,
! [A2: nat,B2: nat] :
? [D4: nat,X2: nat,Y2: nat] :
( ( dvd_dvd_nat @ D4 @ A2 )
& ( dvd_dvd_nat @ D4 @ B2 )
& ( ( ( times_times_nat @ A2 @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ B2 @ Y2 ) @ D4 ) )
| ( ( times_times_nat @ B2 @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y2 ) @ D4 ) ) ) ) ).
% bezout_add_nat
thf(fact_1064_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_1065_add__mult__distrib__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,B: mat_complex,C2: mat_complex,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( plus_p8323303612493835998omplex @ A @ B ) @ C2 )
= ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ A @ C2 ) @ ( times_8009071140041733218omplex @ B @ C2 ) ) ) ) ) ) ).
% add_mult_distrib_mat
thf(fact_1066_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_1067_mult__add__distrib__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,B: mat_complex,Nc: nat,C2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ A @ ( plus_p8323303612493835998omplex @ B @ C2 ) )
= ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ A @ B ) @ ( times_8009071140041733218omplex @ A @ C2 ) ) ) ) ) ) ).
% mult_add_distrib_mat
thf(fact_1068_dvd__add__times__triv__right__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ ( times_times_nat @ C @ A2 ) ) )
= ( dvd_dvd_nat @ A2 @ B2 ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_1069_dvd__add__times__triv__right__iff,axiom,
! [A2: complex,B2: complex,C: complex] :
( ( dvd_dvd_complex @ A2 @ ( plus_plus_complex @ B2 @ ( times_times_complex @ C @ A2 ) ) )
= ( dvd_dvd_complex @ A2 @ B2 ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_1070_dvd__add__times__triv__left__iff,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ ( times_times_nat @ C @ A2 ) @ B2 ) )
= ( dvd_dvd_nat @ A2 @ B2 ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_1071_dvd__add__times__triv__left__iff,axiom,
! [A2: complex,C: complex,B2: complex] :
( ( dvd_dvd_complex @ A2 @ ( plus_plus_complex @ ( times_times_complex @ C @ A2 ) @ B2 ) )
= ( dvd_dvd_complex @ A2 @ B2 ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_1072_mult__hom_Ohom__add__eq__zero,axiom,
! [X: nat,Y: nat,C: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
=> ( ( plus_plus_nat @ ( times_times_nat @ C @ X ) @ ( times_times_nat @ C @ Y ) )
= zero_zero_nat ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_1073_mult__hom_Ohom__add__eq__zero,axiom,
! [X: complex,Y: complex,C: complex] :
( ( ( plus_plus_complex @ X @ Y )
= zero_zero_complex )
=> ( ( plus_plus_complex @ ( times_times_complex @ C @ X ) @ ( times_times_complex @ C @ Y ) )
= zero_zero_complex ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_1074_class__semiring_Onat__pow__0,axiom,
! [X: nat] :
( ( power_power_nat @ X @ zero_zero_nat )
= one_one_nat ) ).
% class_semiring.nat_pow_0
thf(fact_1075_class__semiring_Onat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% class_semiring.nat_pow_zero
thf(fact_1076_class__semiring_Onat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ).
% class_semiring.nat_pow_zero
thf(fact_1077_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_1078_left__right__inverse__power,axiom,
! [X: complex,Y: complex,N: nat] :
( ( ( times_times_complex @ X @ Y )
= one_one_complex )
=> ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) )
= one_one_complex ) ) ).
% left_right_inverse_power
thf(fact_1079_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_1080_Linear__Algebra__Complements_Otrace__add,axiom,
! [A: mat_complex,B: mat_complex] :
( ( square_mat_complex @ A )
=> ( ( square_mat_complex @ B )
=> ( ( ( dim_row_complex @ A )
= ( dim_row_complex @ B ) )
=> ( ( comple3184165445352484367omplex @ ( plus_p8323303612493835998omplex @ A @ B ) )
= ( plus_plus_complex @ ( comple3184165445352484367omplex @ A ) @ ( comple3184165445352484367omplex @ B ) ) ) ) ) ) ).
% Linear_Algebra_Complements.trace_add
thf(fact_1081_mat__assoc__test_I12_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C2: mat_complex,D2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple3184165445352484367omplex @ ( plus_p8323303612493835998omplex @ A @ ( times_8009071140041733218omplex @ B @ C2 ) ) )
= ( plus_plus_complex @ ( comple3184165445352484367omplex @ A ) @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ C2 @ B ) ) ) ) ) ) ) ) ).
% mat_assoc_test(12)
thf(fact_1082_arith__simps_I50_J,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% arith_simps(50)
thf(fact_1083_arith__simps_I50_J,axiom,
! [A2: complex] :
( ( plus_plus_complex @ A2 @ zero_zero_complex )
= A2 ) ).
% arith_simps(50)
thf(fact_1084_arith__simps_I49_J,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% arith_simps(49)
thf(fact_1085_arith__simps_I49_J,axiom,
! [A2: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A2 )
= A2 ) ).
% arith_simps(49)
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Complex__Ocomplex_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( dim_row_a @ ( times_times_mat_a @ a2 @ b ) )
= n ) ).
%------------------------------------------------------------------------------