TPTP Problem File: SLH0323^1.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_00899_033708__19296310_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1393 ( 522 unt; 250 typ; 0 def)
% Number of atoms : 2949 (1495 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 12988 ( 218 ~; 41 |; 94 &;11304 @)
% ( 0 <=>;1331 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 6 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 514 ( 514 >; 0 *; 0 +; 0 <<)
% Number of symbols : 234 ( 231 usr; 16 con; 0-6 aty)
% Number of variables : 3232 ( 106 ^;3105 !; 21 ?;3232 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:35:36.322
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
thf(ty_n_t__Formal____Power____Series__Ofps_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
formal3495187508964346165omplex: $tType ).
thf(ty_n_t__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
formal670952693614245302omplex: $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__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
formal3361831859752904756s_real: $tType ).
thf(ty_n_t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
formal_Power_fps_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Real__Oreal_J_J,type,
set_mat_real: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
set_vec_a: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
set_mat_a: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
vec_complex: $tType ).
thf(ty_n_t__Matrix__Omat_It__Complex__Ocomplex_J,type,
mat_complex: $tType ).
thf(ty_n_t__Matrix__Omat_It__Real__Oreal_J,type,
mat_real: $tType ).
thf(ty_n_t__Matrix__Omat_It__Nat__Onat_J,type,
mat_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__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (231)
thf(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
gbinomial_complex: complex > nat > complex ).
thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
gbinomial_real: real > nat > real ).
thf(sy_c_Char__Poly_Ochar__matrix_001t__Complex__Ocomplex,type,
char_c872259621517735348omplex: mat_complex > complex > mat_complex ).
thf(sy_c_Char__Poly_Ochar__matrix_001t__Real__Oreal,type,
char_c4597223634827269298x_real: mat_real > real > mat_real ).
thf(sy_c_Char__Poly_Ochar__matrix_001tf__a,type,
char_char_matrix_a: mat_a > a > mat_a ).
thf(sy_c_Char__Poly_Oeigenvalue_001t__Complex__Ocomplex,type,
char_e7032225803028799586omplex: mat_complex > complex > $o ).
thf(sy_c_Char__Poly_Oeigenvalue_001t__Real__Oreal,type,
char_eigenvalue_real: mat_real > real > $o ).
thf(sy_c_Char__Poly_Oeigenvalue_001tf__a,type,
char_eigenvalue_a: mat_a > a > $o ).
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_Oadd__col__sub__row_001t__Real__Oreal,type,
column3494657893274022100w_real: real > nat > nat > mat_real > mat_real ).
thf(sy_c_Column__Operations_Oadd__col__sub__row_001tf__a,type,
column3081110322506813142_row_a: a > nat > nat > mat_a > mat_a ).
thf(sy_c_Column__Operations_Omat__addcol_001t__Complex__Ocomplex,type,
column896436094548437152omplex: complex > nat > nat > mat_complex > mat_complex ).
thf(sy_c_Column__Operations_Omat__addcol_001t__Real__Oreal,type,
column5677306341442300318l_real: real > nat > nat > mat_real > mat_real ).
thf(sy_c_Column__Operations_Omat__addcol_001tf__a,type,
column_mat_addcol_a: a > nat > nat > mat_a > mat_a ).
thf(sy_c_Column__Operations_Omat__multcol_001t__Complex__Ocomplex,type,
column4410001698458707789omplex: nat > complex > mat_complex > mat_complex ).
thf(sy_c_Column__Operations_Omat__multcol_001tf__a,type,
column_mat_multcol_a: nat > a > mat_a > mat_a ).
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_001t__Real__Oreal,type,
column2501654400089035909s_real: nat > nat > mat_real > mat_real ).
thf(sy_c_Column__Operations_Omat__swapcols_001tf__a,type,
column2528828918332591333cols_a: nat > nat > mat_a > mat_a ).
thf(sy_c_Column__Operations_Oswap__col__to__front_001t__Complex__Ocomplex,type,
column7264791363093833182omplex: mat_complex > nat > mat_complex ).
thf(sy_c_Column__Operations_Oswap__col__to__front_001t__Real__Oreal,type,
column6512727542960595932t_real: mat_real > nat > mat_real ).
thf(sy_c_Column__Operations_Oswap__col__to__front_001tf__a,type,
column2924081423933032910ront_a: mat_a > nat > mat_a ).
thf(sy_c_Column__Operations_Oswap__row__to__front_001t__Complex__Ocomplex,type,
column4342047067757093060omplex: mat_complex > nat > mat_complex ).
thf(sy_c_Column__Operations_Oswap__row__to__front_001t__Real__Oreal,type,
column3686191904915150786t_real: mat_real > nat > mat_real ).
thf(sy_c_Column__Operations_Oswap__row__to__front_001tf__a,type,
column973622294476583016ront_a: mat_a > nat > 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_Ohermitian_001tf__a,type,
complex_hermitian_a: mat_a > $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_001t__Real__Oreal,type,
complex_trace_real: mat_real > real ).
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_Complex__Matrix_Ounitary_001tf__a,type,
complex_unitary_a: mat_a > $o ).
thf(sy_c_Determinant_Oadj__mat_001t__Complex__Ocomplex,type,
adj_mat_complex: mat_complex > mat_complex ).
thf(sy_c_Determinant_Oadj__mat_001tf__a,type,
adj_mat_a: mat_a > mat_a ).
thf(sy_c_Determinant_Ocofactor_001t__Complex__Ocomplex,type,
cofactor_complex: mat_complex > nat > nat > complex ).
thf(sy_c_Determinant_Ocofactor_001t__Real__Oreal,type,
cofactor_real: mat_real > nat > nat > real ).
thf(sy_c_Determinant_Odelete__index,type,
delete_index: nat > nat > nat ).
thf(sy_c_Determinant_Odet_001t__Complex__Ocomplex,type,
det_complex: mat_complex > complex ).
thf(sy_c_Determinant_Odet_001t__Real__Oreal,type,
det_real: mat_real > real ).
thf(sy_c_Determinant_Odet_001tf__a,type,
det_a: mat_a > a ).
thf(sy_c_Determinant_Omat__delete_001t__Complex__Ocomplex,type,
mat_delete_complex: mat_complex > nat > nat > mat_complex ).
thf(sy_c_Determinant_Omat__delete_001t__Real__Oreal,type,
mat_delete_real: mat_real > nat > nat > mat_real ).
thf(sy_c_Determinant_Omat__delete_001tf__a,type,
mat_delete_a: mat_a > nat > nat > mat_a ).
thf(sy_c_Determinant_Opermutation__delete,type,
permutation_delete: ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Determinant_Opermutation__insert_001t__Complex__Ocomplex,type,
permut138581522262023397omplex: complex > nat > ( complex > nat ) > complex > nat ).
thf(sy_c_Determinant_Opermutation__insert_001t__Nat__Onat,type,
permut3695043542826343943rt_nat: nat > nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Determinant_Opermutation__insert_001t__Real__Oreal,type,
permut4060954620988167523t_real: real > nat > ( real > nat ) > real > nat ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Complex__Ocomplex,type,
comm_s2602460028002588243omplex: complex > nat > complex ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Nat__Onat,type,
comm_s4663373288045622133er_nat: nat > nat > nat ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
comm_s7457072308508201937r_real: real > nat > real ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
semiri5044797733671781792omplex: nat > complex ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
semiri2265585572941072030t_real: nat > real ).
thf(sy_c_Formal__Power__Series_Ofps_Ofps__nth_001t__Complex__Ocomplex,type,
formal3666518339620930912omplex: formal670952693614245302omplex > nat > complex ).
thf(sy_c_Formal__Power__Series_Ofps_Ofps__nth_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
formal6566246685975971957omplex: formal3495187508964346165omplex > nat > mat_complex ).
thf(sy_c_Formal__Power__Series_Ofps_Ofps__nth_001t__Nat__Onat,type,
formal3720337525774269570th_nat: formal_Power_fps_nat > nat > nat ).
thf(sy_c_Formal__Power__Series_Ofps_Ofps__nth_001t__Real__Oreal,type,
formal2580924720334399070h_real: formal3361831859752904756s_real > nat > real ).
thf(sy_c_Formal__Power__Series_Ofps__XD_001t__Complex__Ocomplex,type,
formal1655152611307539683omplex: formal670952693614245302omplex > formal670952693614245302omplex ).
thf(sy_c_Formal__Power__Series_Ofps__XDp_001t__Complex__Ocomplex,type,
formal5989188765539143467omplex: complex > formal670952693614245302omplex > formal670952693614245302omplex ).
thf(sy_c_Formal__Power__Series_Ofps__XDp_001t__Real__Oreal,type,
formal2839450981996073129p_real: real > formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__ln_001t__Complex__Ocomplex,type,
formal6928690614366948857omplex: complex > formal670952693614245302omplex ).
thf(sy_c_Formal__Power__Series_Ofps__ln_001t__Real__Oreal,type,
formal8688746759596762231n_real: real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__radical_001t__Complex__Ocomplex,type,
formal878984433073253857omplex: ( nat > complex > complex ) > nat > formal670952693614245302omplex > formal670952693614245302omplex ).
thf(sy_c_Formal__Power__Series_Ofps__radical_001t__Real__Oreal,type,
formal8604817403481219167l_real: ( nat > real > real ) > nat > formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Gauss__Jordan__Elimination_Oaddrow__mat_001t__Complex__Ocomplex,type,
gauss_947198734564870628omplex: nat > complex > nat > nat > mat_complex ).
thf(sy_c_Gauss__Jordan__Elimination_Oaddrow__mat_001t__Real__Oreal,type,
gauss_2378325378421436642t_real: nat > real > nat > nat > mat_real ).
thf(sy_c_Gauss__Jordan__Elimination_Oaddrow__mat_001tf__a,type,
gauss_8159914756388622152_mat_a: nat > a > nat > nat > mat_a ).
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_Ogauss__jordan__single_001tf__a,type,
gauss_4684855476144371464ngle_a: mat_a > mat_a ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__addrow__gen_001t__Complex__Ocomplex,type,
gauss_5252963565656066424omplex: ( complex > complex > complex ) > ( complex > complex > complex ) > complex > nat > nat > mat_complex > mat_complex ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__addrow__gen_001t__Real__Oreal,type,
gauss_4246877906280926838n_real: ( real > real > real ) > ( real > real > real ) > real > nat > nat > mat_real > mat_real ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__addrow__gen_001tf__a,type,
gauss_3441994962245461172_gen_a: ( a > a > a ) > ( a > a > a ) > a > nat > nat > mat_a > mat_a ).
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__Real__Oreal,type,
gauss_1037889766561479105n_real: ( real > real > real ) > nat > real > mat_real > mat_real ).
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_001t__Real__Oreal,type,
gauss_821192380332421767s_real: nat > nat > mat_real > mat_real ).
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_001tf__a,type,
gauss_5015385051186949877_mat_a: nat > nat > a > mat_a ).
thf(sy_c_Gauss__Jordan__Elimination_Orow__echelon__form_001t__Complex__Ocomplex,type,
gauss_194721375535881179omplex: mat_complex > $o ).
thf(sy_c_Gauss__Jordan__Elimination_Orow__echelon__form_001tf__a,type,
gauss_5855338539171749649form_a: mat_a > $o ).
thf(sy_c_Gauss__Jordan__Elimination_Oswaprows__mat_001t__Complex__Ocomplex,type,
gauss_8970452565587180529omplex: nat > nat > nat > mat_complex ).
thf(sy_c_Gauss__Jordan__Elimination_Oswaprows__mat_001t__Real__Oreal,type,
gauss_1271566072679876207t_real: nat > nat > nat > mat_real ).
thf(sy_c_Gauss__Jordan__Elimination_Oswaprows__mat_001tf__a,type,
gauss_110929411057020027_mat_a: nat > nat > nat > mat_a ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
abs_abs_real: real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
minus_minus_complex: complex > complex > complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
minus_2412168080157227406omplex: mat_complex > mat_complex > mat_complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Omat_It__Real__Oreal_J,type,
minus_minus_mat_real: mat_real > mat_real > mat_real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Omat_Itf__a_J,type,
minus_minus_mat_a: mat_a > mat_a > mat_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001tf__a,type,
minus_minus_a: a > a > a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
one_on1590755018477040891omplex: formal670952693614245302omplex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
one_on8598947968683843321s_real: formal3361831859752904756s_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oone__class_Oone_001tf__a,type,
one_one_a: a ).
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__Real__Oreal_J,type,
plus_plus_mat_real: mat_real > mat_real > mat_real ).
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__Real__Oreal,type,
plus_plus_real: real > real > real ).
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__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
times_1444617028055533883omplex: formal670952693614245302omplex > formal670952693614245302omplex > formal670952693614245302omplex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
times_7561426564079326009s_real: formal3361831859752904756s_real > formal3361831859752904756s_real > formal3361831859752904756s_real ).
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__Real__Oreal_J,type,
times_times_mat_real: mat_real > mat_real > mat_real ).
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__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001tf__a,type,
times_times_a: a > a > a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
uminus1482373934393186551omplex: complex > complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
uminus9178514011183859839omplex: formal670952693614245302omplex > formal670952693614245302omplex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Formal____Power____Series__Ofps_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
uminus8384360544929349932omplex: formal3495187508964346165omplex > formal3495187508964346165omplex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
uminus8389970968385878141s_real: formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
uminus467866341702955550omplex: mat_complex > mat_complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Matrix__Omat_It__Real__Oreal_J,type,
uminus18246009535971484t_real: mat_real > mat_real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Matrix__Omat_Itf__a_J,type,
uminus_uminus_mat_a: mat_a > mat_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001tf__a,type,
uminus_uminus_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_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
zero_zero_a: a ).
thf(sy_c_HOL_ONO__MATCH_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
nO_MAT8947977539597988553omplex: complex > complex > $o ).
thf(sy_c_HOL_ONO__MATCH_001t__Complex__Ocomplex_001t__Real__Oreal,type,
nO_MAT9165723751696580935x_real: complex > real > $o ).
thf(sy_c_HOL_ONO__MATCH_001t__Real__Oreal_001t__Real__Oreal,type,
nO_MATCH_real_real: real > real > $o ).
thf(sy_c_If_001t__Complex__Ocomplex,type,
if_complex: $o > complex > complex > complex ).
thf(sy_c_If_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
if_mat_complex: $o > mat_complex > mat_complex > mat_complex ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Linear__Algebra__Complements_Oprojector_001t__Complex__Ocomplex,type,
linear5633924348262549461omplex: mat_complex > $o ).
thf(sy_c_Linear__Algebra__Complements_Orank__1__proj_001t__Complex__Ocomplex,type,
linear1949544614684794075omplex: vec_complex > mat_complex ).
thf(sy_c_Matrix_Oappend__rows_001t__Complex__Ocomplex,type,
append_rows_complex: mat_complex > mat_complex > mat_complex ).
thf(sy_c_Matrix_Oappend__rows_001tf__a,type,
append_rows_a: mat_a > mat_a > mat_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__Real__Oreal,type,
carrier_mat_real: nat > nat > set_mat_real ).
thf(sy_c_Matrix_Ocarrier__mat_001tf__a,type,
carrier_mat_a: nat > nat > set_mat_a ).
thf(sy_c_Matrix_Ocomm__monoid__add__class_Osum__mat_001t__Complex__Ocomplex,type,
comm_m3586542329073673570omplex: mat_complex > complex ).
thf(sy_c_Matrix_Ocomm__monoid__add__class_Osum__mat_001t__Nat__Onat,type,
comm_m4056229327131402372at_nat: mat_nat > nat ).
thf(sy_c_Matrix_Ocomm__monoid__add__class_Osum__mat_001t__Real__Oreal,type,
comm_m8678487124704766304t_real: mat_real > real ).
thf(sy_c_Matrix_Ocomm__monoid__add__class_Osum__mat_001tf__a,type,
comm_m5291664705200495434_mat_a: mat_a > a ).
thf(sy_c_Matrix_Odiagonal__mat_001t__Complex__Ocomplex,type,
diagonal_mat_complex: mat_complex > $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__Real__Oreal,type,
dim_col_real: mat_real > 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__Real__Oreal,type,
dim_row_real: mat_real > nat ).
thf(sy_c_Matrix_Odim__row_001tf__a,type,
dim_row_a: mat_a > nat ).
thf(sy_c_Matrix_Ofour__block__mat_001t__Complex__Ocomplex,type,
four_b559179830521662709omplex: mat_complex > mat_complex > mat_complex > mat_complex > mat_complex ).
thf(sy_c_Matrix_Ofour__block__mat_001t__Real__Oreal,type,
four_block_mat_real: mat_real > mat_real > mat_real > mat_real > mat_real ).
thf(sy_c_Matrix_Ofour__block__mat_001tf__a,type,
four_block_mat_a: mat_a > mat_a > mat_a > mat_a > mat_a ).
thf(sy_c_Matrix_Oinvertible__mat_001t__Complex__Ocomplex,type,
invert2568027935824841882omplex: mat_complex > $o ).
thf(sy_c_Matrix_Oinvertible__mat_001tf__a,type,
invertible_mat_a: mat_a > $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_001tf__a,type,
inverts_mat_a: mat_a > mat_a > $o ).
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__Real__Oreal,type,
one_mat_real: nat > mat_real ).
thf(sy_c_Matrix_Oone__mat_001tf__a,type,
one_mat_a: nat > mat_a ).
thf(sy_c_Matrix_Opow__mat_001t__Complex__Ocomplex,type,
pow_mat_complex: mat_complex > nat > mat_complex ).
thf(sy_c_Matrix_Opow__mat_001tf__a,type,
pow_mat_a: mat_a > nat > mat_a ).
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_001tf__a,type,
similar_mat_wit_a: mat_a > mat_a > mat_a > mat_a > $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__Real__Oreal,type,
smult_mat_real: real > mat_real > mat_real ).
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_001t__Real__Oreal,type,
square_mat_real: mat_real > $o ).
thf(sy_c_Matrix_Oupper__triangular_001t__Complex__Ocomplex,type,
upper_4850907204721561915omplex: mat_complex > $o ).
thf(sy_c_Matrix_Oupper__triangular_001tf__a,type,
upper_triangular_a: mat_a > $o ).
thf(sy_c_Matrix_Ozero__mat_001t__Complex__Ocomplex,type,
zero_mat_complex: nat > nat > mat_complex ).
thf(sy_c_Matrix_Ozero__mat_001t__Nat__Onat,type,
zero_mat_nat: nat > nat > mat_nat ).
thf(sy_c_Matrix_Ozero__mat_001t__Real__Oreal,type,
zero_mat_real: nat > nat > mat_real ).
thf(sy_c_Matrix_Ozero__mat_001tf__a,type,
zero_mat_a: nat > nat > mat_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
semiri8010041392384452111omplex: nat > complex ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
semiri8948773824294531479omplex: nat > formal670952693614245302omplex ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
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_001t__Real__Oreal,type,
ord_less_real: real > real > $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__Real__Oreal,type,
ord_less_eq_real: real > real > $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__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
power_8487976900264310848omplex: formal670952693614245302omplex > nat > formal670952693614245302omplex ).
thf(sy_c_Power_Opower__class_Opower_001t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
power_568658719666546786ps_nat: formal_Power_fps_nat > nat > formal_Power_fps_nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
power_1846127563762588094s_real: formal3361831859752904756s_real > nat > formal3361831859752904756s_real ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Power_Opower__class_Opower_001tf__a,type,
power_power_a: a > nat > a ).
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_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
real_V4546457046886955230omplex: real > complex ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
divide1717551699836669952omplex: complex > complex > complex ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Rings_Odivide__class_Odivide_001tf__a,type,
divide_divide_a: a > a > a ).
thf(sy_c_Schur__Decomposition_Ocorthogonal__inv_001t__Complex__Ocomplex,type,
schur_4574106303853392228omplex: mat_complex > mat_complex ).
thf(sy_c_Schur__Decomposition_Ocorthogonal__mat_001t__Complex__Ocomplex,type,
schur_549222400177443379omplex: mat_complex > $o ).
thf(sy_c_Schur__Decomposition_Ocorthogonal__mat_001tf__a,type,
schur_4042290226164342457_mat_a: mat_a > $o ).
thf(sy_c_Schur__Decomposition_Omat__adjoint_001t__Complex__Ocomplex,type,
schur_5982229384592763574omplex: mat_complex > mat_complex ).
thf(sy_c_Schur__Decomposition_Omat__adjoint_001tf__a,type,
schur_mat_adjoint_a: mat_a > mat_a ).
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_Itf__a_J,type,
collect_mat_a: ( mat_a > $o ) > set_mat_a ).
thf(sy_c_Spectral__Theory__Complements_Omat__conj_001t__Complex__Ocomplex,type,
spectr5699176650994449695omplex: mat_complex > mat_complex > mat_complex ).
thf(sy_c_Spectral__Theory__Complements_Omat__conj_001tf__a,type,
spectr5828033140197310157conj_a: mat_a > mat_a > mat_a ).
thf(sy_c_Spectral__Theory__Complements_Ounitarily__equiv_001t__Complex__Ocomplex,type,
spectr6340060708231679580omplex: mat_complex > mat_complex > mat_complex > $o ).
thf(sy_c_Spectral__Theory__Complements_Ounitarily__equiv_001tf__a,type,
spectr4825054497075562704quiv_a: mat_a > mat_a > mat_a > $o ).
thf(sy_c_Spectral__Theory__Complements_Ounitary__diag_001t__Complex__Ocomplex,type,
spectr532731689276696518omplex: mat_complex > mat_complex > mat_complex > $o ).
thf(sy_c_Spectral__Theory__Complements_Ounitary__diag_001tf__a,type,
spectr4894841263502123494diag_a: mat_a > mat_a > mat_a > $o ).
thf(sy_c_VS__Connect_Ovec__space_Orow__space_001t__Complex__Ocomplex,type,
vS_vec3284807721666986142omplex: nat > mat_complex > set_vec_complex ).
thf(sy_c_VS__Connect_Ovec__space_Orow__space_001tf__a,type,
vS_vec_row_space_a: nat > mat_a > set_vec_a ).
thf(sy_c_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__Real__Oreal_J,type,
member_mat_real: mat_real > set_mat_real > $o ).
thf(sy_c_member_001t__Matrix__Omat_Itf__a_J,type,
member_mat_a: mat_a > set_mat_a > $o ).
thf(sy_v_A,type,
a2: mat_a ).
thf(sy_v_B,type,
b: mat_a ).
thf(sy_v_U,type,
u: mat_a ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1133)
thf(fact_0_assms_I4_J,axiom,
complex_unitary_a @ u ).
% assms(4)
thf(fact_1_assms_I3_J,axiom,
diagonal_mat_a @ b ).
% assms(3)
thf(fact_2_assms_I5_J,axiom,
( a2
= ( spectr5828033140197310157conj_a @ u @ b ) ) ).
% assms(5)
thf(fact_3_assms_I1_J,axiom,
member_mat_a @ a2 @ ( carrier_mat_a @ n @ n ) ).
% assms(1)
thf(fact_4_assms_I2_J,axiom,
member_mat_a @ b @ ( carrier_mat_a @ n @ n ) ).
% assms(2)
thf(fact_5_unitarily__equivD_I1_J,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( comple6660659447773130958omplex @ U ) ) ).
% unitarily_equivD(1)
thf(fact_6_unitarily__equivD_I1_J,axiom,
! [A: mat_a,B: mat_a,U: mat_a] :
( ( spectr4825054497075562704quiv_a @ A @ B @ U )
=> ( complex_unitary_a @ U ) ) ).
% unitarily_equivD(1)
thf(fact_7_unitarily__equiv__carrier_I2_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% unitarily_equiv_carrier(2)
thf(fact_8_unitarily__equiv__carrier_I2_J,axiom,
! [A: mat_a,N: nat,B: mat_a,U: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( spectr4825054497075562704quiv_a @ A @ B @ U )
=> ( member_mat_a @ U @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% unitarily_equiv_carrier(2)
thf(fact_9_unitarily__equiv__carrier_I1_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% unitarily_equiv_carrier(1)
thf(fact_10_unitarily__equiv__carrier_I1_J,axiom,
! [A: mat_a,N: nat,B: mat_a,U: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( spectr4825054497075562704quiv_a @ A @ B @ U )
=> ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% unitarily_equiv_carrier(1)
thf(fact_11_unitarily__equiv__adjoint,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( spectr6340060708231679580omplex @ B @ A @ ( schur_5982229384592763574omplex @ U ) ) ) ).
% unitarily_equiv_adjoint
thf(fact_12_unitarily__equiv__adjoint,axiom,
! [A: mat_a,B: mat_a,U: mat_a] :
( ( spectr4825054497075562704quiv_a @ A @ B @ U )
=> ( spectr4825054497075562704quiv_a @ B @ A @ ( schur_mat_adjoint_a @ U ) ) ) ).
% unitarily_equiv_adjoint
thf(fact_13_unitary__diag__def,axiom,
( spectr532731689276696518omplex
= ( ^ [A2: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( spectr6340060708231679580omplex @ A2 @ B2 @ U2 )
& ( diagonal_mat_complex @ B2 ) ) ) ) ).
% unitary_diag_def
thf(fact_14_unitary__diag__def,axiom,
( spectr4894841263502123494diag_a
= ( ^ [A2: mat_a,B2: mat_a,U2: mat_a] :
( ( spectr4825054497075562704quiv_a @ A2 @ B2 @ U2 )
& ( diagonal_mat_a @ B2 ) ) ) ) ).
% unitary_diag_def
thf(fact_15_unitarily__equivD_I2_J,axiom,
! [A: mat_a,B: mat_a,U: mat_a] :
( ( spectr4825054497075562704quiv_a @ A @ B @ U )
=> ( similar_mat_wit_a @ A @ B @ U @ ( schur_mat_adjoint_a @ U ) ) ) ).
% unitarily_equivD(2)
thf(fact_16_unitarily__equivD_I2_J,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( simila5774310414453981135omplex @ A @ B @ U @ ( schur_5982229384592763574omplex @ U ) ) ) ).
% unitarily_equivD(2)
thf(fact_17_unitarily__equiv__eq,axiom,
! [A: mat_a,B: mat_a,U: mat_a] :
( ( spectr4825054497075562704quiv_a @ A @ B @ U )
=> ( A
= ( times_times_mat_a @ ( times_times_mat_a @ U @ B ) @ ( schur_mat_adjoint_a @ U ) ) ) ) ).
% unitarily_equiv_eq
thf(fact_18_unitarily__equiv__eq,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ U @ B ) @ ( schur_5982229384592763574omplex @ U ) ) ) ) ).
% unitarily_equiv_eq
thf(fact_19_unitarily__equiv__commute,axiom,
! [A: mat_a,B: mat_a,U: mat_a,C: mat_a] :
( ( spectr4825054497075562704quiv_a @ A @ B @ U )
=> ( ( ( times_times_mat_a @ A @ C )
= ( times_times_mat_a @ C @ A ) )
=> ( ( times_times_mat_a @ B @ ( times_times_mat_a @ ( times_times_mat_a @ ( schur_mat_adjoint_a @ U ) @ C ) @ U ) )
= ( times_times_mat_a @ ( times_times_mat_a @ ( times_times_mat_a @ ( schur_mat_adjoint_a @ U ) @ C ) @ U ) @ B ) ) ) ) ).
% unitarily_equiv_commute
thf(fact_20_unitarily__equiv__commute,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex,C: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( ( ( times_8009071140041733218omplex @ A @ C )
= ( times_8009071140041733218omplex @ C @ A ) )
=> ( ( times_8009071140041733218omplex @ B @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ U ) @ C ) @ U ) )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ U ) @ C ) @ U ) @ B ) ) ) ) ).
% unitarily_equiv_commute
thf(fact_21_unitarily__equiv__uminus,axiom,
! [A: mat_a,N: nat,B: mat_a,U: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( spectr4825054497075562704quiv_a @ A @ B @ U )
=> ( spectr4825054497075562704quiv_a @ ( uminus_uminus_mat_a @ A ) @ ( uminus_uminus_mat_a @ B ) @ U ) ) ) ).
% unitarily_equiv_uminus
thf(fact_22_unitarily__equiv__uminus,axiom,
! [A: mat_complex,N: nat,B: mat_complex,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( spectr6340060708231679580omplex @ ( uminus467866341702955550omplex @ A ) @ ( uminus467866341702955550omplex @ B ) @ U ) ) ) ).
% unitarily_equiv_uminus
thf(fact_23_unitarily__equiv__smult,axiom,
! [A: mat_a,N: nat,B: mat_a,U: mat_a,X: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( spectr4825054497075562704quiv_a @ A @ B @ U )
=> ( spectr4825054497075562704quiv_a @ ( smult_mat_a @ X @ A ) @ ( smult_mat_a @ X @ B ) @ U ) ) ) ).
% unitarily_equiv_smult
thf(fact_24_unitarily__equiv__smult,axiom,
! [A: mat_complex,N: nat,B: mat_complex,U: mat_complex,X: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( spectr6340060708231679580omplex @ ( smult_mat_complex @ X @ A ) @ ( smult_mat_complex @ X @ B ) @ U ) ) ) ).
% unitarily_equiv_smult
thf(fact_25_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_26_adjoint__dim_H,axiom,
! [A: mat_a,N: nat,M: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( member_mat_a @ ( schur_mat_adjoint_a @ A ) @ ( carrier_mat_a @ M @ N ) ) ) ).
% adjoint_dim'
thf(fact_27_mat__conj__def,axiom,
( spectr5699176650994449695omplex
= ( ^ [U2: mat_complex,V: mat_complex] : ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ U2 @ V ) @ ( schur_5982229384592763574omplex @ U2 ) ) ) ) ).
% mat_conj_def
thf(fact_28_mat__conj__def,axiom,
( spectr5828033140197310157conj_a
= ( ^ [U2: mat_a,V: mat_a] : ( times_times_mat_a @ ( times_times_mat_a @ U2 @ V ) @ ( schur_mat_adjoint_a @ U2 ) ) ) ) ).
% mat_conj_def
thf(fact_29_unitary__diagI,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( simila5774310414453981135omplex @ A @ B @ U @ ( schur_5982229384592763574omplex @ U ) )
=> ( ( diagonal_mat_complex @ B )
=> ( ( comple6660659447773130958omplex @ U )
=> ( spectr532731689276696518omplex @ A @ B @ U ) ) ) ) ).
% unitary_diagI
thf(fact_30_unitary__diagI,axiom,
! [A: mat_a,B: mat_a,U: mat_a] :
( ( similar_mat_wit_a @ A @ B @ U @ ( schur_mat_adjoint_a @ U ) )
=> ( ( diagonal_mat_a @ B )
=> ( ( complex_unitary_a @ U )
=> ( spectr4894841263502123494diag_a @ A @ B @ U ) ) ) ) ).
% unitary_diagI
thf(fact_31_mat__conj__smult,axiom,
! [A: mat_complex,N: nat,U: mat_complex,B: mat_complex,X: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ U @ B ) @ ( schur_5982229384592763574omplex @ U ) ) )
=> ( ( smult_mat_complex @ X @ A )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ U @ ( smult_mat_complex @ X @ B ) ) @ ( schur_5982229384592763574omplex @ U ) ) ) ) ) ) ) ).
% mat_conj_smult
thf(fact_32_mat__conj__smult,axiom,
! [A: mat_a,N: nat,U: mat_a,B: mat_a,X: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ U @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( A
= ( times_times_mat_a @ ( times_times_mat_a @ U @ B ) @ ( schur_mat_adjoint_a @ U ) ) )
=> ( ( smult_mat_a @ X @ A )
= ( times_times_mat_a @ ( times_times_mat_a @ U @ ( smult_mat_a @ X @ B ) ) @ ( schur_mat_adjoint_a @ U ) ) ) ) ) ) ) ).
% mat_conj_smult
thf(fact_33_mat__conj__adjoint,axiom,
! [U: mat_complex,V2: mat_complex] :
( ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U ) @ V2 )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ U ) @ V2 ) @ U ) ) ).
% mat_conj_adjoint
thf(fact_34_mat__conj__adjoint,axiom,
! [U: mat_a,V2: mat_a] :
( ( spectr5828033140197310157conj_a @ ( schur_mat_adjoint_a @ U ) @ V2 )
= ( times_times_mat_a @ ( times_times_mat_a @ ( schur_mat_adjoint_a @ U ) @ V2 ) @ U ) ) ).
% mat_conj_adjoint
thf(fact_35_mat__conj__commute,axiom,
! [A: mat_complex,N: nat,B: mat_complex,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ U )
=> ( ( ( times_8009071140041733218omplex @ A @ B )
= ( times_8009071140041733218omplex @ B @ A ) )
=> ( ( times_8009071140041733218omplex @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U ) @ A ) @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U ) @ B ) )
= ( times_8009071140041733218omplex @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U ) @ B ) @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U ) @ A ) ) ) ) ) ) ) ) ).
% mat_conj_commute
thf(fact_36_mat__conj__commute,axiom,
! [A: mat_a,N: nat,B: mat_a,U: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ U @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_unitary_a @ U )
=> ( ( ( times_times_mat_a @ A @ B )
= ( times_times_mat_a @ B @ A ) )
=> ( ( times_times_mat_a @ ( spectr5828033140197310157conj_a @ ( schur_mat_adjoint_a @ U ) @ A ) @ ( spectr5828033140197310157conj_a @ ( schur_mat_adjoint_a @ U ) @ B ) )
= ( times_times_mat_a @ ( spectr5828033140197310157conj_a @ ( schur_mat_adjoint_a @ U ) @ B ) @ ( spectr5828033140197310157conj_a @ ( schur_mat_adjoint_a @ U ) @ A ) ) ) ) ) ) ) ) ).
% mat_conj_commute
thf(fact_37_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_38_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_39_mat__conj__uminus__eq,axiom,
! [A: mat_complex,N: nat,U: mat_complex,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( A
= ( spectr5699176650994449695omplex @ U @ B ) )
=> ( ( uminus467866341702955550omplex @ A )
= ( spectr5699176650994449695omplex @ U @ ( uminus467866341702955550omplex @ B ) ) ) ) ) ) ) ).
% mat_conj_uminus_eq
thf(fact_40_mat__conj__uminus__eq,axiom,
! [A: mat_a,N: nat,U: mat_a,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ U @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( A
= ( spectr5828033140197310157conj_a @ U @ B ) )
=> ( ( uminus_uminus_mat_a @ A )
= ( spectr5828033140197310157conj_a @ U @ ( uminus_uminus_mat_a @ B ) ) ) ) ) ) ) ).
% mat_conj_uminus_eq
thf(fact_41_diagonal__mat__uminus,axiom,
! [A: mat_a] :
( ( diagonal_mat_a @ A )
=> ( diagonal_mat_a @ ( uminus_uminus_mat_a @ A ) ) ) ).
% diagonal_mat_uminus
thf(fact_42_diagonal__mat__uminus,axiom,
! [A: mat_complex] :
( ( diagonal_mat_complex @ A )
=> ( diagonal_mat_complex @ ( uminus467866341702955550omplex @ A ) ) ) ).
% diagonal_mat_uminus
thf(fact_43_diagonal__mat__commute,axiom,
! [A: mat_a,N: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( diagonal_mat_a @ A )
=> ( ( diagonal_mat_a @ B )
=> ( ( times_times_mat_a @ A @ B )
= ( times_times_mat_a @ B @ A ) ) ) ) ) ) ).
% diagonal_mat_commute
thf(fact_44_diagonal__mat__commute,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 ) )
=> ( ( diagonal_mat_complex @ A )
=> ( ( diagonal_mat_complex @ B )
=> ( ( times_8009071140041733218omplex @ A @ B )
= ( times_8009071140041733218omplex @ B @ A ) ) ) ) ) ) ).
% diagonal_mat_commute
thf(fact_45_diagonal__mat__sq__diag,axiom,
! [B: mat_a,N: nat] :
( ( diagonal_mat_a @ B )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( diagonal_mat_a @ ( times_times_mat_a @ B @ B ) ) ) ) ).
% diagonal_mat_sq_diag
thf(fact_46_diagonal__mat__sq__diag,axiom,
! [B: mat_complex,N: nat] :
( ( diagonal_mat_complex @ B )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( diagonal_mat_complex @ ( times_8009071140041733218omplex @ B @ B ) ) ) ) ).
% diagonal_mat_sq_diag
thf(fact_47_mat__conj__unit__commute,axiom,
! [U: mat_complex,A: mat_complex,N: nat] :
( ( comple6660659447773130958omplex @ U )
=> ( ( ( times_8009071140041733218omplex @ U @ A )
= ( times_8009071140041733218omplex @ A @ U ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr5699176650994449695omplex @ U @ A )
= A ) ) ) ) ) ).
% mat_conj_unit_commute
thf(fact_48_mat__conj__unit__commute,axiom,
! [U: mat_a,A: mat_a,N: nat] :
( ( complex_unitary_a @ U )
=> ( ( ( times_times_mat_a @ U @ A )
= ( times_times_mat_a @ A @ U ) )
=> ( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ U @ ( carrier_mat_a @ N @ N ) )
=> ( ( spectr5828033140197310157conj_a @ U @ A )
= A ) ) ) ) ) ).
% mat_conj_unit_commute
thf(fact_49_unitary__mult__conjugate,axiom,
! [A: mat_complex,N: nat,V2: mat_complex,U: mat_complex,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ V2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ V2 )
=> ( ( ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ V2 ) @ A )
= ( spectr5699176650994449695omplex @ U @ B ) )
=> ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ V2 @ U ) @ B ) @ ( schur_5982229384592763574omplex @ ( times_8009071140041733218omplex @ V2 @ U ) ) ) ) ) ) ) ) ) ) ).
% unitary_mult_conjugate
thf(fact_50_unitary__mult__conjugate,axiom,
! [A: mat_a,N: nat,V2: mat_a,U: mat_a,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ V2 @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ U @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_unitary_a @ V2 )
=> ( ( ( spectr5828033140197310157conj_a @ ( schur_mat_adjoint_a @ V2 ) @ A )
= ( spectr5828033140197310157conj_a @ U @ B ) )
=> ( A
= ( times_times_mat_a @ ( times_times_mat_a @ ( times_times_mat_a @ V2 @ U ) @ B ) @ ( schur_mat_adjoint_a @ ( times_times_mat_a @ V2 @ U ) ) ) ) ) ) ) ) ) ) ).
% unitary_mult_conjugate
thf(fact_51_diagonal__mat__times__diag,axiom,
! [A: mat_a,N: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( diagonal_mat_a @ A )
=> ( ( diagonal_mat_a @ B )
=> ( diagonal_mat_a @ ( times_times_mat_a @ A @ B ) ) ) ) ) ) ).
% diagonal_mat_times_diag
thf(fact_52_diagonal__mat__times__diag,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 ) )
=> ( ( diagonal_mat_complex @ A )
=> ( ( diagonal_mat_complex @ B )
=> ( diagonal_mat_complex @ ( times_8009071140041733218omplex @ A @ B ) ) ) ) ) ) ).
% diagonal_mat_times_diag
thf(fact_53_unitarily__equiv__conjugate,axiom,
! [A: mat_a,N: nat,V2: mat_a,U: mat_a,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ V2 @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ U @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( spectr4825054497075562704quiv_a @ ( spectr5828033140197310157conj_a @ ( schur_mat_adjoint_a @ V2 ) @ A ) @ B @ U )
=> ( ( complex_unitary_a @ V2 )
=> ( spectr4825054497075562704quiv_a @ A @ B @ ( times_times_mat_a @ V2 @ U ) ) ) ) ) ) ) ) ).
% unitarily_equiv_conjugate
thf(fact_54_unitarily__equiv__conjugate,axiom,
! [A: mat_complex,N: nat,V2: mat_complex,U: mat_complex,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ V2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr6340060708231679580omplex @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ V2 ) @ A ) @ B @ U )
=> ( ( comple6660659447773130958omplex @ V2 )
=> ( spectr6340060708231679580omplex @ A @ B @ ( times_8009071140041733218omplex @ V2 @ U ) ) ) ) ) ) ) ) ).
% unitarily_equiv_conjugate
thf(fact_55_unitarily__equiv__def,axiom,
( spectr4825054497075562704quiv_a
= ( ^ [A2: mat_a,B2: mat_a,U2: mat_a] :
( ( complex_unitary_a @ U2 )
& ( similar_mat_wit_a @ A2 @ B2 @ U2 @ ( schur_mat_adjoint_a @ U2 ) ) ) ) ) ).
% unitarily_equiv_def
thf(fact_56_unitarily__equiv__def,axiom,
( spectr6340060708231679580omplex
= ( ^ [A2: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( comple6660659447773130958omplex @ U2 )
& ( simila5774310414453981135omplex @ A2 @ B2 @ U2 @ ( schur_5982229384592763574omplex @ U2 ) ) ) ) ) ).
% unitarily_equiv_def
thf(fact_57_unitarily__equivI,axiom,
! [A: mat_a,B: mat_a,U: mat_a] :
( ( similar_mat_wit_a @ A @ B @ U @ ( schur_mat_adjoint_a @ U ) )
=> ( ( complex_unitary_a @ U )
=> ( spectr4825054497075562704quiv_a @ A @ B @ U ) ) ) ).
% unitarily_equivI
thf(fact_58_unitarily__equivI,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( simila5774310414453981135omplex @ A @ B @ U @ ( schur_5982229384592763574omplex @ U ) )
=> ( ( comple6660659447773130958omplex @ U )
=> ( spectr6340060708231679580omplex @ A @ B @ U ) ) ) ).
% unitarily_equivI
thf(fact_59_unitarily__equivI_H,axiom,
! [A: mat_a,U: mat_a,B: mat_a,N: nat] :
( ( A
= ( spectr5828033140197310157conj_a @ U @ B ) )
=> ( ( complex_unitary_a @ U )
=> ( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( spectr4825054497075562704quiv_a @ A @ B @ U ) ) ) ) ) ).
% unitarily_equivI'
thf(fact_60_unitarily__equivI_H,axiom,
! [A: mat_complex,U: mat_complex,B: mat_complex,N: nat] :
( ( A
= ( spectr5699176650994449695omplex @ U @ B ) )
=> ( ( comple6660659447773130958omplex @ U )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( spectr6340060708231679580omplex @ A @ B @ U ) ) ) ) ) ).
% unitarily_equivI'
thf(fact_61_unitary__elim,axiom,
! [A: mat_complex,N: nat,B: mat_complex,P: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ P @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ P )
=> ( ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P @ A ) @ ( schur_5982229384592763574omplex @ P ) )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P @ B ) @ ( schur_5982229384592763574omplex @ P ) ) )
=> ( A = B ) ) ) ) ) ) ).
% unitary_elim
thf(fact_62_unitary__elim,axiom,
! [A: mat_a,N: nat,B: mat_a,P: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ P @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_unitary_a @ P )
=> ( ( ( times_times_mat_a @ ( times_times_mat_a @ P @ A ) @ ( schur_mat_adjoint_a @ P ) )
= ( times_times_mat_a @ ( times_times_mat_a @ P @ B ) @ ( schur_mat_adjoint_a @ P ) ) )
=> ( A = B ) ) ) ) ) ) ).
% unitary_elim
thf(fact_63_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_64_unitary__adjoint,axiom,
! [A: mat_a,N: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_unitary_a @ A )
=> ( complex_unitary_a @ ( schur_mat_adjoint_a @ A ) ) ) ) ).
% unitary_adjoint
thf(fact_65_similar__mat__witD2_I3_J,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a,P: mat_a,Q: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( A
= ( times_times_mat_a @ ( times_times_mat_a @ P @ B ) @ Q ) ) ) ) ).
% similar_mat_witD2(3)
thf(fact_66_similar__mat__witD2_I3_J,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,P: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q )
=> ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P @ B ) @ Q ) ) ) ) ).
% similar_mat_witD2(3)
thf(fact_67_unitary__times__unitary,axiom,
! [P: mat_a,N: nat,Q: mat_a] :
( ( member_mat_a @ P @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ Q @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_unitary_a @ P )
=> ( ( complex_unitary_a @ Q )
=> ( complex_unitary_a @ ( times_times_mat_a @ P @ Q ) ) ) ) ) ) ).
% unitary_times_unitary
thf(fact_68_unitary__times__unitary,axiom,
! [P: mat_complex,N: nat,Q: mat_complex] :
( ( member_mat_complex @ P @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ Q @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ P )
=> ( ( comple6660659447773130958omplex @ Q )
=> ( comple6660659447773130958omplex @ ( times_8009071140041733218omplex @ P @ Q ) ) ) ) ) ) ).
% unitary_times_unitary
thf(fact_69_mult__smult__distrib,axiom,
! [A: mat_a,Nr: nat,N: nat,B: mat_a,Nc: nat,K: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ A @ ( smult_mat_a @ K @ B ) )
= ( smult_mat_a @ K @ ( times_times_mat_a @ A @ B ) ) ) ) ) ).
% mult_smult_distrib
thf(fact_70_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_71_mult__smult__assoc__mat,axiom,
! [A: mat_a,Nr: nat,N: nat,B: mat_a,Nc: nat,K: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ ( smult_mat_a @ K @ A ) @ B )
= ( smult_mat_a @ K @ ( times_times_mat_a @ A @ B ) ) ) ) ) ).
% mult_smult_assoc_mat
thf(fact_72_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_73_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_74_adjoint__mult,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a,L: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ M @ L ) )
=> ( ( schur_mat_adjoint_a @ ( times_times_mat_a @ A @ B ) )
= ( times_times_mat_a @ ( schur_mat_adjoint_a @ B ) @ ( schur_mat_adjoint_a @ A ) ) ) ) ) ).
% adjoint_mult
thf(fact_75_similar__mat__wit__smult,axiom,
! [A: mat_complex,B: mat_complex,P: mat_complex,Q: mat_complex,K: complex] :
( ( simila5774310414453981135omplex @ A @ B @ P @ Q )
=> ( simila5774310414453981135omplex @ ( smult_mat_complex @ K @ A ) @ ( smult_mat_complex @ K @ B ) @ P @ Q ) ) ).
% similar_mat_wit_smult
thf(fact_76_similar__mat__wit__trans,axiom,
! [A: mat_complex,B: mat_complex,P: mat_complex,Q: mat_complex,C: mat_complex,P2: mat_complex,Q2: mat_complex] :
( ( simila5774310414453981135omplex @ A @ B @ P @ Q )
=> ( ( simila5774310414453981135omplex @ B @ C @ P2 @ Q2 )
=> ( simila5774310414453981135omplex @ A @ C @ ( times_8009071140041733218omplex @ P @ P2 ) @ ( times_8009071140041733218omplex @ Q2 @ Q ) ) ) ) ).
% similar_mat_wit_trans
thf(fact_77_uminus__carrier__mat,axiom,
! [A: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_mat_a @ ( uminus_uminus_mat_a @ A ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% uminus_carrier_mat
thf(fact_78_uminus__carrier__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( uminus467866341702955550omplex @ A ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% uminus_carrier_mat
thf(fact_79_uminus__carrier__iff__mat,axiom,
! [A: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ ( uminus_uminus_mat_a @ A ) @ ( carrier_mat_a @ Nr @ Nc ) )
= ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% uminus_carrier_iff_mat
thf(fact_80_uminus__carrier__iff__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ ( uminus467866341702955550omplex @ A ) @ ( carrier_mat_complex @ Nr @ Nc ) )
= ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% uminus_carrier_iff_mat
thf(fact_81_similar__mat__witD2_I7_J,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a,P: mat_a,Q: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( member_mat_a @ Q @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% similar_mat_witD2(7)
thf(fact_82_similar__mat__witD2_I7_J,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,P: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q )
=> ( member_mat_complex @ Q @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD2(7)
thf(fact_83_mem__Collect__eq,axiom,
! [A3: mat_a,P: mat_a > $o] :
( ( member_mat_a @ A3 @ ( collect_mat_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_84_mem__Collect__eq,axiom,
! [A3: mat_complex,P: mat_complex > $o] :
( ( member_mat_complex @ A3 @ ( collect_mat_complex @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_85_Collect__mem__eq,axiom,
! [A: set_mat_a] :
( ( collect_mat_a
@ ^ [X2: mat_a] : ( member_mat_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_86_Collect__mem__eq,axiom,
! [A: set_mat_complex] :
( ( collect_mat_complex
@ ^ [X2: mat_complex] : ( member_mat_complex @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_87_Complex__Matrix_Oadjoint__adjoint,axiom,
! [A: mat_complex] :
( ( schur_5982229384592763574omplex @ ( schur_5982229384592763574omplex @ A ) )
= A ) ).
% Complex_Matrix.adjoint_adjoint
thf(fact_88_Complex__Matrix_Oadjoint__adjoint,axiom,
! [A: mat_a] :
( ( schur_mat_adjoint_a @ ( schur_mat_adjoint_a @ A ) )
= A ) ).
% Complex_Matrix.adjoint_adjoint
thf(fact_89_uminus__uminus__mat,axiom,
! [A: mat_complex] :
( ( uminus467866341702955550omplex @ ( uminus467866341702955550omplex @ A ) )
= A ) ).
% uminus_uminus_mat
thf(fact_90_uminus__eq__mat,axiom,
! [A: mat_complex,B: mat_complex] :
( ( ( uminus467866341702955550omplex @ A )
= ( uminus467866341702955550omplex @ B ) )
= ( A = B ) ) ).
% uminus_eq_mat
thf(fact_91_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_92_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_93_assoc__mult__mat,axiom,
! [A: mat_a,N_1: nat,N_2: nat,B: mat_a,N_3: nat,C: 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 @ C @ ( carrier_mat_a @ N_3 @ N_4 ) )
=> ( ( times_times_mat_a @ ( times_times_mat_a @ A @ B ) @ C )
= ( times_times_mat_a @ A @ ( times_times_mat_a @ B @ C ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_94_assoc__mult__mat,axiom,
! [A: mat_complex,N_1: nat,N_2: nat,B: mat_complex,N_3: nat,C: 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 @ C @ ( carrier_mat_complex @ N_3 @ N_4 ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B ) @ C )
= ( times_8009071140041733218omplex @ A @ ( times_8009071140041733218omplex @ B @ C ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_95_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_96_adjoint__dim,axiom,
! [A: mat_a,N: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( member_mat_a @ ( schur_mat_adjoint_a @ A ) @ ( carrier_mat_a @ N @ N ) ) ) ).
% adjoint_dim
thf(fact_97_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_98_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_99_similar__mat__witD2_I4_J,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a,P: mat_a,Q: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% similar_mat_witD2(4)
thf(fact_100_similar__mat__witD2_I4_J,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,P: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q )
=> ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD2(4)
thf(fact_101_similar__mat__witD2_I5_J,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a,P: mat_a,Q: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% similar_mat_witD2(5)
thf(fact_102_similar__mat__witD2_I5_J,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,P: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q )
=> ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD2(5)
thf(fact_103_similar__mat__witD2_I6_J,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a,P: mat_a,Q: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( member_mat_a @ P @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% similar_mat_witD2(6)
thf(fact_104_similar__mat__witD2_I6_J,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,P: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q )
=> ( member_mat_complex @ P @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD2(6)
thf(fact_105_smult__smult__times,axiom,
! [A3: complex,K: complex,A: mat_complex] :
( ( smult_mat_complex @ A3 @ ( smult_mat_complex @ K @ A ) )
= ( smult_mat_complex @ ( times_times_complex @ A3 @ K ) @ A ) ) ).
% smult_smult_times
thf(fact_106_vector__space__over__itself_Oscale__minus__right,axiom,
! [A3: complex,X: complex] :
( ( times_times_complex @ A3 @ ( uminus1482373934393186551omplex @ X ) )
= ( uminus1482373934393186551omplex @ ( times_times_complex @ A3 @ X ) ) ) ).
% vector_space_over_itself.scale_minus_right
thf(fact_107_vector__space__over__itself_Oscale__minus__right,axiom,
! [A3: real,X: real] :
( ( times_times_real @ A3 @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( times_times_real @ A3 @ X ) ) ) ).
% vector_space_over_itself.scale_minus_right
thf(fact_108_vector__space__over__itself_Oscale__minus__left,axiom,
! [A3: complex,X: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ A3 ) @ X )
= ( uminus1482373934393186551omplex @ ( times_times_complex @ A3 @ X ) ) ) ).
% vector_space_over_itself.scale_minus_left
thf(fact_109_vector__space__over__itself_Oscale__minus__left,axiom,
! [A3: real,X: real] :
( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ X )
= ( uminus_uminus_real @ ( times_times_real @ A3 @ X ) ) ) ).
% vector_space_over_itself.scale_minus_left
thf(fact_110_minus__mult__commute,axiom,
! [A3: complex,B3: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ A3 ) @ B3 )
= ( times_times_complex @ A3 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ).
% minus_mult_commute
thf(fact_111_minus__mult__commute,axiom,
! [A3: real,B3: real] :
( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ B3 )
= ( times_times_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ).
% minus_mult_commute
thf(fact_112_mult__minus__right,axiom,
! [A3: complex,B3: complex] :
( ( times_times_complex @ A3 @ ( uminus1482373934393186551omplex @ B3 ) )
= ( uminus1482373934393186551omplex @ ( times_times_complex @ A3 @ B3 ) ) ) ).
% mult_minus_right
thf(fact_113_mult__minus__right,axiom,
! [A3: real,B3: real] :
( ( times_times_real @ A3 @ ( uminus_uminus_real @ B3 ) )
= ( uminus_uminus_real @ ( times_times_real @ A3 @ B3 ) ) ) ).
% mult_minus_right
thf(fact_114_minus__mult__minus,axiom,
! [A3: complex,B3: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ A3 ) @ ( uminus1482373934393186551omplex @ B3 ) )
= ( times_times_complex @ A3 @ B3 ) ) ).
% minus_mult_minus
thf(fact_115_minus__mult__minus,axiom,
! [A3: real,B3: real] :
( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) )
= ( times_times_real @ A3 @ B3 ) ) ).
% minus_mult_minus
thf(fact_116_mult__minus__left,axiom,
! [A3: complex,B3: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ A3 ) @ B3 )
= ( uminus1482373934393186551omplex @ ( times_times_complex @ A3 @ B3 ) ) ) ).
% mult_minus_left
thf(fact_117_mult__minus__left,axiom,
! [A3: real,B3: real] :
( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ B3 )
= ( uminus_uminus_real @ ( times_times_real @ A3 @ B3 ) ) ) ).
% mult_minus_left
thf(fact_118_square__eq__iff,axiom,
! [A3: complex,B3: complex] :
( ( ( times_times_complex @ A3 @ A3 )
= ( times_times_complex @ B3 @ B3 ) )
= ( ( A3 = B3 )
| ( A3
= ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).
% square_eq_iff
thf(fact_119_square__eq__iff,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ A3 )
= ( times_times_real @ B3 @ B3 ) )
= ( ( A3 = B3 )
| ( A3
= ( uminus_uminus_real @ B3 ) ) ) ) ).
% square_eq_iff
thf(fact_120_more__arith__simps_I10_J,axiom,
! [A3: complex] :
( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A3 ) )
= A3 ) ).
% more_arith_simps(10)
thf(fact_121_more__arith__simps_I10_J,axiom,
! [A3: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A3 ) )
= A3 ) ).
% more_arith_simps(10)
thf(fact_122_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_123_unitaryD2,axiom,
! [A: mat_a,N: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_unitary_a @ A )
=> ( inverts_mat_a @ ( schur_mat_adjoint_a @ A ) @ A ) ) ) ).
% unitaryD2
thf(fact_124_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_125_unitary__is__corthogonal,axiom,
! [U: mat_a,N: nat] :
( ( member_mat_a @ U @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_unitary_a @ U )
=> ( schur_4042290226164342457_mat_a @ U ) ) ) ).
% unitary_is_corthogonal
thf(fact_126_unitary__is__corthogonal,axiom,
! [U: mat_complex,N: nat] :
( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ U )
=> ( schur_549222400177443379omplex @ U ) ) ) ).
% unitary_is_corthogonal
thf(fact_127_hermitian__mat__conj_H,axiom,
! [A: mat_complex,N: nat,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple8306762464034002205omplex @ A )
=> ( comple8306762464034002205omplex @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U ) @ A ) ) ) ) ) ).
% hermitian_mat_conj'
thf(fact_128_hermitian__mat__conj_H,axiom,
! [A: mat_a,N: nat,U: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ U @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_hermitian_a @ A )
=> ( complex_hermitian_a @ ( spectr5828033140197310157conj_a @ ( schur_mat_adjoint_a @ U ) @ A ) ) ) ) ) ).
% hermitian_mat_conj'
thf(fact_129_trace__smult,axiom,
! [A: mat_a,N: nat,C2: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_trace_a @ ( smult_mat_a @ C2 @ A ) )
= ( times_times_a @ C2 @ ( complex_trace_a @ A ) ) ) ) ).
% trace_smult
thf(fact_130_trace__smult,axiom,
! [A: mat_complex,N: nat,C2: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple3184165445352484367omplex @ ( smult_mat_complex @ C2 @ A ) )
= ( times_times_complex @ C2 @ ( comple3184165445352484367omplex @ A ) ) ) ) ).
% trace_smult
thf(fact_131_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_132_mult__adjoint__hermitian,axiom,
! [A: mat_a,N: nat,M: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( complex_hermitian_a @ ( times_times_mat_a @ ( schur_mat_adjoint_a @ A ) @ A ) ) ) ).
% mult_adjoint_hermitian
thf(fact_133_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_134_unitary__simps_I2_J,axiom,
! [A: mat_a,N: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_unitary_a @ A )
=> ( ( times_times_mat_a @ A @ ( schur_mat_adjoint_a @ A ) )
= ( one_mat_a @ N ) ) ) ) ).
% unitary_simps(2)
thf(fact_135_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_136_unitary__simps_I1_J,axiom,
! [A: mat_a,N: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_unitary_a @ A )
=> ( ( times_times_mat_a @ ( schur_mat_adjoint_a @ A ) @ A )
= ( one_mat_a @ N ) ) ) ) ).
% unitary_simps(1)
thf(fact_137_mk__diagonal__diagonal,axiom,
! [As: list_a] : ( diagonal_mat_a @ ( mk_diagonal_a @ As ) ) ).
% mk_diagonal_diagonal
thf(fact_138_hermitian__mat__conj,axiom,
! [A: mat_complex,N: nat,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple8306762464034002205omplex @ A )
=> ( comple8306762464034002205omplex @ ( spectr5699176650994449695omplex @ U @ A ) ) ) ) ) ).
% hermitian_mat_conj
thf(fact_139_hermitian__mat__conj,axiom,
! [A: mat_a,N: nat,U: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ U @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_hermitian_a @ A )
=> ( complex_hermitian_a @ ( spectr5828033140197310157conj_a @ U @ A ) ) ) ) ) ).
% hermitian_mat_conj
thf(fact_140_upper__triangular__one,axiom,
! [N: nat] : ( upper_4850907204721561915omplex @ ( one_mat_complex @ N ) ) ).
% upper_triangular_one
thf(fact_141_hermitian__one,axiom,
! [N: nat] : ( comple8306762464034002205omplex @ ( one_mat_complex @ N ) ) ).
% hermitian_one
thf(fact_142_one__carrier__mat,axiom,
! [N: nat] : ( member_mat_a @ ( one_mat_a @ N ) @ ( carrier_mat_a @ N @ N ) ) ).
% one_carrier_mat
thf(fact_143_one__carrier__mat,axiom,
! [N: nat] : ( member_mat_complex @ ( one_mat_complex @ N ) @ ( carrier_mat_complex @ N @ N ) ) ).
% one_carrier_mat
thf(fact_144_unitary__one,axiom,
! [N: nat] : ( complex_unitary_a @ ( one_mat_a @ N ) ) ).
% unitary_one
thf(fact_145_unitary__one,axiom,
! [N: nat] : ( comple6660659447773130958omplex @ ( one_mat_complex @ N ) ) ).
% unitary_one
thf(fact_146_hermitian__square__hermitian,axiom,
! [A: mat_complex] :
( ( comple8306762464034002205omplex @ A )
=> ( comple8306762464034002205omplex @ ( times_8009071140041733218omplex @ A @ A ) ) ) ).
% hermitian_square_hermitian
thf(fact_147_hermitian__def,axiom,
( comple8306762464034002205omplex
= ( ^ [A2: mat_complex] :
( ( schur_5982229384592763574omplex @ A2 )
= A2 ) ) ) ).
% hermitian_def
thf(fact_148_hermitian__def,axiom,
( complex_hermitian_a
= ( ^ [A2: mat_a] :
( ( schur_mat_adjoint_a @ A2 )
= A2 ) ) ) ).
% hermitian_def
thf(fact_149_right__mult__one__mat,axiom,
! [A: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( times_times_mat_a @ A @ ( one_mat_a @ Nc ) )
= A ) ) ).
% right_mult_one_mat
thf(fact_150_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_151_left__mult__one__mat,axiom,
! [A: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( times_times_mat_a @ ( one_mat_a @ Nr ) @ A )
= A ) ) ).
% left_mult_one_mat
thf(fact_152_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_153_inverts__mat__unique,axiom,
! [A: mat_a,N: nat,B: mat_a,C: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ C @ ( carrier_mat_a @ N @ N ) )
=> ( ( inverts_mat_a @ A @ B )
=> ( ( inverts_mat_a @ A @ C )
=> ( B = C ) ) ) ) ) ) ).
% inverts_mat_unique
thf(fact_154_inverts__mat__unique,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( inverts_mat_complex @ A @ B )
=> ( ( inverts_mat_complex @ A @ C )
=> ( B = C ) ) ) ) ) ) ).
% inverts_mat_unique
thf(fact_155_inverts__mat__symm,axiom,
! [A: mat_a,N: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( inverts_mat_a @ A @ B )
=> ( inverts_mat_a @ B @ A ) ) ) ) ).
% inverts_mat_symm
thf(fact_156_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_157_similar__mat__wit__refl,axiom,
! [A: mat_a,N: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( similar_mat_wit_a @ A @ A @ ( one_mat_a @ N ) @ ( one_mat_a @ N ) ) ) ).
% similar_mat_wit_refl
thf(fact_158_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_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_similar__mat__witI,axiom,
! [P: mat_a,Q: mat_a,N: nat,A: mat_a,B: mat_a] :
( ( ( times_times_mat_a @ P @ Q )
= ( one_mat_a @ N ) )
=> ( ( ( times_times_mat_a @ Q @ P )
= ( one_mat_a @ N ) )
=> ( ( A
= ( times_times_mat_a @ ( times_times_mat_a @ P @ B ) @ Q ) )
=> ( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ P @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ Q @ ( carrier_mat_a @ N @ N ) )
=> ( similar_mat_wit_a @ A @ B @ P @ Q ) ) ) ) ) ) ) ) ).
% similar_mat_witI
thf(fact_162_similar__mat__witI,axiom,
! [P: mat_complex,Q: mat_complex,N: nat,A: mat_complex,B: mat_complex] :
( ( ( times_8009071140041733218omplex @ P @ Q )
= ( one_mat_complex @ N ) )
=> ( ( ( times_8009071140041733218omplex @ Q @ P )
= ( one_mat_complex @ N ) )
=> ( ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P @ B ) @ Q ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ P @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ Q @ ( carrier_mat_complex @ N @ N ) )
=> ( simila5774310414453981135omplex @ A @ B @ P @ Q ) ) ) ) ) ) ) ) ).
% similar_mat_witI
thf(fact_163_similar__mat__witD2_I1_J,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a,P: mat_a,Q: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( ( times_times_mat_a @ P @ Q )
= ( one_mat_a @ N ) ) ) ) ).
% similar_mat_witD2(1)
thf(fact_164_similar__mat__witD2_I1_J,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,P: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q )
=> ( ( times_8009071140041733218omplex @ P @ Q )
= ( one_mat_complex @ N ) ) ) ) ).
% similar_mat_witD2(1)
thf(fact_165_similar__mat__witD2_I2_J,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a,P: mat_a,Q: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( ( times_times_mat_a @ Q @ P )
= ( one_mat_a @ N ) ) ) ) ).
% similar_mat_witD2(2)
thf(fact_166_similar__mat__witD2_I2_J,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,P: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q )
=> ( ( times_8009071140041733218omplex @ Q @ P )
= ( one_mat_complex @ N ) ) ) ) ).
% similar_mat_witD2(2)
thf(fact_167_trace__comm,axiom,
! [A: mat_a,N: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_trace_a @ ( times_times_mat_a @ A @ B ) )
= ( complex_trace_a @ ( times_times_mat_a @ B @ A ) ) ) ) ) ).
% trace_comm
thf(fact_168_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_169_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_170_hermitian__is__normal,axiom,
! [A: mat_a] :
( ( complex_hermitian_a @ A )
=> ( ( times_times_mat_a @ A @ ( schur_mat_adjoint_a @ A ) )
= ( times_times_mat_a @ ( schur_mat_adjoint_a @ A ) @ A ) ) ) ).
% hermitian_is_normal
thf(fact_171_mat__mult__left__right__inverse,axiom,
! [A: mat_a,N: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( ( times_times_mat_a @ A @ B )
= ( one_mat_a @ N ) )
=> ( ( times_times_mat_a @ B @ A )
= ( one_mat_a @ N ) ) ) ) ) ).
% mat_mult_left_right_inverse
thf(fact_172_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_173_projector__def,axiom,
( linear5633924348262549461omplex
= ( ^ [M2: mat_complex] :
( ( comple8306762464034002205omplex @ M2 )
& ( ( times_8009071140041733218omplex @ M2 @ M2 )
= M2 ) ) ) ) ).
% projector_def
thf(fact_174_Complex__Matrix_Ounitary__def,axiom,
( comple6660659447773130958omplex
= ( ^ [A2: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ ( dim_row_complex @ A2 ) @ ( dim_row_complex @ A2 ) ) )
& ( inverts_mat_complex @ A2 @ ( schur_5982229384592763574omplex @ A2 ) ) ) ) ) ).
% Complex_Matrix.unitary_def
thf(fact_175_Complex__Matrix_Ounitary__def,axiom,
( complex_unitary_a
= ( ^ [A2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ ( dim_row_a @ A2 ) @ ( dim_row_a @ A2 ) ) )
& ( inverts_mat_a @ A2 @ ( schur_mat_adjoint_a @ A2 ) ) ) ) ) ).
% Complex_Matrix.unitary_def
thf(fact_176_gauss__jordan__single_I4_J,axiom,
! [A: mat_a,Nr: nat,Nc: nat,C: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( ( gauss_4684855476144371464ngle_a @ A )
= C )
=> ? [P3: mat_a,Q3: mat_a] :
( ( C
= ( times_times_mat_a @ P3 @ A ) )
& ( member_mat_a @ P3 @ ( carrier_mat_a @ Nr @ Nr ) )
& ( member_mat_a @ Q3 @ ( carrier_mat_a @ Nr @ Nr ) )
& ( ( times_times_mat_a @ P3 @ Q3 )
= ( one_mat_a @ Nr ) )
& ( ( times_times_mat_a @ Q3 @ P3 )
= ( one_mat_a @ Nr ) ) ) ) ) ).
% gauss_jordan_single(4)
thf(fact_177_gauss__jordan__single_I4_J,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,C: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( ( gauss_4244865067341541924omplex @ A )
= C )
=> ? [P3: mat_complex,Q3: mat_complex] :
( ( C
= ( times_8009071140041733218omplex @ P3 @ A ) )
& ( member_mat_complex @ P3 @ ( carrier_mat_complex @ Nr @ Nr ) )
& ( member_mat_complex @ Q3 @ ( carrier_mat_complex @ Nr @ Nr ) )
& ( ( times_8009071140041733218omplex @ P3 @ Q3 )
= ( one_mat_complex @ Nr ) )
& ( ( times_8009071140041733218omplex @ Q3 @ P3 )
= ( one_mat_complex @ Nr ) ) ) ) ) ).
% gauss_jordan_single(4)
thf(fact_178_projector__hermitian,axiom,
! [M3: mat_complex] :
( ( linear5633924348262549461omplex @ M3 )
=> ( comple8306762464034002205omplex @ M3 ) ) ).
% projector_hermitian
thf(fact_179_corthogonal__inv__result,axiom,
! [A: mat_complex] :
( ( schur_549222400177443379omplex @ A )
=> ( inverts_mat_complex @ ( schur_4574106303853392228omplex @ A ) @ A ) ) ).
% corthogonal_inv_result
thf(fact_180_inverts__mat__def,axiom,
( inverts_mat_complex
= ( ^ [A2: mat_complex,B2: mat_complex] :
( ( times_8009071140041733218omplex @ A2 @ B2 )
= ( one_mat_complex @ ( dim_row_complex @ A2 ) ) ) ) ) ).
% inverts_mat_def
thf(fact_181_projector__square__eq,axiom,
! [M3: mat_complex] :
( ( linear5633924348262549461omplex @ M3 )
=> ( ( times_8009071140041733218omplex @ M3 @ M3 )
= M3 ) ) ).
% projector_square_eq
thf(fact_182_similar__mat__witD_I1_J,axiom,
! [N: nat,A: mat_complex,B: mat_complex,P: mat_complex,Q: mat_complex] :
( ( N
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q )
=> ( ( times_8009071140041733218omplex @ P @ Q )
= ( one_mat_complex @ N ) ) ) ) ).
% similar_mat_witD(1)
thf(fact_183_similar__mat__witD_I2_J,axiom,
! [N: nat,A: mat_complex,B: mat_complex,P: mat_complex,Q: mat_complex] :
( ( N
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q )
=> ( ( times_8009071140041733218omplex @ Q @ P )
= ( one_mat_complex @ N ) ) ) ) ).
% similar_mat_witD(2)
thf(fact_184_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_185_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_186_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_187_gauss__jordan__single_I2_J,axiom,
! [A: mat_a,Nr: nat,Nc: nat,C: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( ( gauss_4684855476144371464ngle_a @ A )
= C )
=> ( member_mat_a @ C @ ( carrier_mat_a @ Nr @ Nc ) ) ) ) ).
% gauss_jordan_single(2)
thf(fact_188_gauss__jordan__single_I2_J,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,C: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( ( gauss_4244865067341541924omplex @ A )
= C )
=> ( member_mat_complex @ C @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% gauss_jordan_single(2)
thf(fact_189_index__one__mat_I2_J,axiom,
! [N: nat] :
( ( dim_row_complex @ ( one_mat_complex @ N ) )
= N ) ).
% index_one_mat(2)
thf(fact_190_index__smult__mat_I2_J,axiom,
! [A3: complex,A: mat_complex] :
( ( dim_row_complex @ ( smult_mat_complex @ A3 @ A ) )
= ( dim_row_complex @ A ) ) ).
% index_smult_mat(2)
thf(fact_191_index__uminus__mat_I2_J,axiom,
! [A: mat_complex] :
( ( dim_row_complex @ ( uminus467866341702955550omplex @ A ) )
= ( dim_row_complex @ A ) ) ).
% index_uminus_mat(2)
thf(fact_192_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_193_hermitian__square,axiom,
! [M3: mat_a] :
( ( complex_hermitian_a @ M3 )
=> ( member_mat_a @ M3 @ ( carrier_mat_a @ ( dim_row_a @ M3 ) @ ( dim_row_a @ M3 ) ) ) ) ).
% hermitian_square
thf(fact_194_hermitian__square,axiom,
! [M3: mat_complex] :
( ( comple8306762464034002205omplex @ M3 )
=> ( member_mat_complex @ M3 @ ( carrier_mat_complex @ ( dim_row_complex @ M3 ) @ ( dim_row_complex @ M3 ) ) ) ) ).
% hermitian_square
thf(fact_195_similar__mat__witD_I7_J,axiom,
! [N: nat,A: mat_a,B: mat_a,P: mat_a,Q: mat_a] :
( ( N
= ( dim_row_a @ A ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( member_mat_a @ Q @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% similar_mat_witD(7)
thf(fact_196_similar__mat__witD_I7_J,axiom,
! [N: nat,A: mat_complex,B: mat_complex,P: mat_complex,Q: mat_complex] :
( ( N
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q )
=> ( member_mat_complex @ Q @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD(7)
thf(fact_197_similar__mat__witD_I6_J,axiom,
! [N: nat,A: mat_a,B: mat_a,P: mat_a,Q: mat_a] :
( ( N
= ( dim_row_a @ A ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( member_mat_a @ P @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% similar_mat_witD(6)
thf(fact_198_similar__mat__witD_I6_J,axiom,
! [N: nat,A: mat_complex,B: mat_complex,P: mat_complex,Q: mat_complex] :
( ( N
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q )
=> ( member_mat_complex @ P @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD(6)
thf(fact_199_similar__mat__witD_I5_J,axiom,
! [N: nat,A: mat_a,B: mat_a,P: mat_a,Q: mat_a] :
( ( N
= ( dim_row_a @ A ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% similar_mat_witD(5)
thf(fact_200_similar__mat__witD_I5_J,axiom,
! [N: nat,A: mat_complex,B: mat_complex,P: mat_complex,Q: mat_complex] :
( ( N
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q )
=> ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD(5)
thf(fact_201_similar__mat__witD_I4_J,axiom,
! [N: nat,A: mat_a,B: mat_a,P: mat_a,Q: mat_a] :
( ( N
= ( dim_row_a @ A ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% similar_mat_witD(4)
thf(fact_202_similar__mat__witD_I4_J,axiom,
! [N: nat,A: mat_complex,B: mat_complex,P: mat_complex,Q: mat_complex] :
( ( N
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q )
=> ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD(4)
thf(fact_203_similar__mat__witD_I3_J,axiom,
! [N: nat,A: mat_complex,B: mat_complex,P: mat_complex,Q: mat_complex] :
( ( N
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q )
=> ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P @ B ) @ Q ) ) ) ) ).
% similar_mat_witD(3)
thf(fact_204_unitarily__equiv__carrier_H_I1_J,axiom,
! [A: mat_a,B: mat_a,U: mat_a] :
( ( spectr4825054497075562704quiv_a @ A @ B @ U )
=> ( member_mat_a @ A @ ( carrier_mat_a @ ( dim_row_a @ A ) @ ( dim_row_a @ A ) ) ) ) ).
% unitarily_equiv_carrier'(1)
thf(fact_205_unitarily__equiv__carrier_H_I1_J,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( member_mat_complex @ A @ ( carrier_mat_complex @ ( dim_row_complex @ A ) @ ( dim_row_complex @ A ) ) ) ) ).
% unitarily_equiv_carrier'(1)
thf(fact_206_unitarily__equiv__carrier_H_I2_J,axiom,
! [A: mat_a,B: mat_a,U: mat_a] :
( ( spectr4825054497075562704quiv_a @ A @ B @ U )
=> ( member_mat_a @ B @ ( carrier_mat_a @ ( dim_row_a @ A ) @ ( dim_row_a @ A ) ) ) ) ).
% unitarily_equiv_carrier'(2)
thf(fact_207_unitarily__equiv__carrier_H_I2_J,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( member_mat_complex @ B @ ( carrier_mat_complex @ ( dim_row_complex @ A ) @ ( dim_row_complex @ A ) ) ) ) ).
% unitarily_equiv_carrier'(2)
thf(fact_208_unitarily__equiv__carrier_H_I3_J,axiom,
! [A: mat_a,B: mat_a,U: mat_a] :
( ( spectr4825054497075562704quiv_a @ A @ B @ U )
=> ( member_mat_a @ U @ ( carrier_mat_a @ ( dim_row_a @ A ) @ ( dim_row_a @ A ) ) ) ) ).
% unitarily_equiv_carrier'(3)
thf(fact_209_unitarily__equiv__carrier_H_I3_J,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( member_mat_complex @ U @ ( carrier_mat_complex @ ( dim_row_complex @ A ) @ ( dim_row_complex @ A ) ) ) ) ).
% unitarily_equiv_carrier'(3)
thf(fact_210_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_211_gauss__jordan__single_I3_J,axiom,
! [A: mat_a,Nr: nat,Nc: nat,C: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( ( gauss_4684855476144371464ngle_a @ A )
= C )
=> ( gauss_5855338539171749649form_a @ C ) ) ) ).
% gauss_jordan_single(3)
thf(fact_212_gauss__jordan__single_I3_J,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,C: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( ( gauss_4244865067341541924omplex @ A )
= C )
=> ( gauss_194721375535881179omplex @ C ) ) ) ).
% gauss_jordan_single(3)
thf(fact_213_adj__mat_I1_J,axiom,
! [A: mat_a,N: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( member_mat_a @ ( adj_mat_a @ A ) @ ( carrier_mat_a @ N @ N ) ) ) ).
% adj_mat(1)
thf(fact_214_adj__mat_I1_J,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( member_mat_complex @ ( adj_mat_complex @ A ) @ ( carrier_mat_complex @ N @ N ) ) ) ).
% adj_mat(1)
thf(fact_215_uminus__mult__left__mat,axiom,
! [A: mat_complex,B: mat_complex] :
( ( ( dim_col_complex @ A )
= ( dim_row_complex @ B ) )
=> ( ( times_8009071140041733218omplex @ ( uminus467866341702955550omplex @ A ) @ B )
= ( uminus467866341702955550omplex @ ( times_8009071140041733218omplex @ A @ B ) ) ) ) ).
% uminus_mult_left_mat
thf(fact_216_uminus__mult__right__mat,axiom,
! [A: mat_complex,B: mat_complex] :
( ( ( dim_col_complex @ A )
= ( dim_row_complex @ B ) )
=> ( ( times_8009071140041733218omplex @ A @ ( uminus467866341702955550omplex @ B ) )
= ( uminus467866341702955550omplex @ ( times_8009071140041733218omplex @ A @ B ) ) ) ) ).
% uminus_mult_right_mat
thf(fact_217_verit__eq__simplify_I25_J,axiom,
! [A3: complex,B3: complex] :
( ( ( uminus1482373934393186551omplex @ A3 )
= ( uminus1482373934393186551omplex @ B3 ) )
= ( A3 = B3 ) ) ).
% verit_eq_simplify(25)
thf(fact_218_verit__eq__simplify_I25_J,axiom,
! [A3: real,B3: real] :
( ( ( uminus_uminus_real @ A3 )
= ( uminus_uminus_real @ B3 ) )
= ( A3 = B3 ) ) ).
% verit_eq_simplify(25)
thf(fact_219_verit__minus__simplify_I4_J,axiom,
! [B3: complex] :
( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B3 ) )
= B3 ) ).
% verit_minus_simplify(4)
thf(fact_220_verit__minus__simplify_I4_J,axiom,
! [B3: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B3 ) )
= B3 ) ).
% verit_minus_simplify(4)
thf(fact_221_verit__negate__coefficient_I3_J,axiom,
! [A3: complex,B3: complex] :
( ( A3 = B3 )
=> ( ( uminus1482373934393186551omplex @ A3 )
= ( uminus1482373934393186551omplex @ B3 ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_222_verit__negate__coefficient_I3_J,axiom,
! [A3: real,B3: real] :
( ( A3 = B3 )
=> ( ( uminus_uminus_real @ A3 )
= ( uminus_uminus_real @ B3 ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_223_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_224_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_225_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_226_index__one__mat_I3_J,axiom,
! [N: nat] :
( ( dim_col_complex @ ( one_mat_complex @ N ) )
= N ) ).
% index_one_mat(3)
thf(fact_227_index__smult__mat_I3_J,axiom,
! [A3: complex,A: mat_complex] :
( ( dim_col_complex @ ( smult_mat_complex @ A3 @ A ) )
= ( dim_col_complex @ A ) ) ).
% index_smult_mat(3)
thf(fact_228_index__uminus__mat_I3_J,axiom,
! [A: mat_complex] :
( ( dim_col_complex @ ( uminus467866341702955550omplex @ A ) )
= ( dim_col_complex @ A ) ) ).
% index_uminus_mat(3)
thf(fact_229_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_230_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_231_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_232_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_233_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_234_adjoint__dim__col,axiom,
! [A: mat_complex] :
( ( dim_col_complex @ ( schur_5982229384592763574omplex @ A ) )
= ( dim_row_complex @ A ) ) ).
% adjoint_dim_col
thf(fact_235_adjoint__dim__col,axiom,
! [A: mat_a] :
( ( dim_col_a @ ( schur_mat_adjoint_a @ A ) )
= ( dim_row_a @ A ) ) ).
% adjoint_dim_col
thf(fact_236_adjoint__dim__row,axiom,
! [A: mat_complex] :
( ( dim_row_complex @ ( schur_5982229384592763574omplex @ A ) )
= ( dim_col_complex @ A ) ) ).
% adjoint_dim_row
thf(fact_237_adjoint__dim__row,axiom,
! [A: mat_a] :
( ( dim_row_a @ ( schur_mat_adjoint_a @ A ) )
= ( dim_col_a @ A ) ) ).
% adjoint_dim_row
thf(fact_238_minus__equation__iff,axiom,
! [A3: complex,B3: complex] :
( ( ( uminus1482373934393186551omplex @ A3 )
= B3 )
= ( ( uminus1482373934393186551omplex @ B3 )
= A3 ) ) ).
% minus_equation_iff
thf(fact_239_minus__equation__iff,axiom,
! [A3: real,B3: real] :
( ( ( uminus_uminus_real @ A3 )
= B3 )
= ( ( uminus_uminus_real @ B3 )
= A3 ) ) ).
% minus_equation_iff
thf(fact_240_equation__minus__iff,axiom,
! [A3: complex,B3: complex] :
( ( A3
= ( uminus1482373934393186551omplex @ B3 ) )
= ( B3
= ( uminus1482373934393186551omplex @ A3 ) ) ) ).
% equation_minus_iff
thf(fact_241_equation__minus__iff,axiom,
! [A3: real,B3: real] :
( ( A3
= ( uminus_uminus_real @ B3 ) )
= ( B3
= ( uminus_uminus_real @ A3 ) ) ) ).
% equation_minus_iff
thf(fact_242_hermitian__decomp__dim__carrier,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( projec5943904436471448624omplex @ A @ B @ U )
=> ( member_mat_complex @ B @ ( carrier_mat_complex @ ( dim_row_complex @ A ) @ ( dim_col_complex @ A ) ) ) ) ).
% hermitian_decomp_dim_carrier
thf(fact_243_adj__mat_I2_J,axiom,
! [A: mat_a,N: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( times_times_mat_a @ A @ ( adj_mat_a @ A ) )
= ( smult_mat_a @ ( det_a @ A ) @ ( one_mat_a @ N ) ) ) ) ).
% adj_mat(2)
thf(fact_244_adj__mat_I2_J,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ A @ ( adj_mat_complex @ A ) )
= ( smult_mat_complex @ ( det_complex @ A ) @ ( one_mat_complex @ N ) ) ) ) ).
% adj_mat(2)
thf(fact_245_adj__mat_I3_J,axiom,
! [A: mat_a,N: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( times_times_mat_a @ ( adj_mat_a @ A ) @ A )
= ( smult_mat_a @ ( det_a @ A ) @ ( one_mat_a @ N ) ) ) ) ).
% adj_mat(3)
thf(fact_246_adj__mat_I3_J,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ ( adj_mat_complex @ A ) @ A )
= ( smult_mat_complex @ ( det_complex @ A ) @ ( one_mat_complex @ N ) ) ) ) ).
% adj_mat(3)
thf(fact_247_hermitian__decomp__sim,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( projec5943904436471448624omplex @ A @ B @ U )
=> ( simila5774310414453981135omplex @ A @ B @ U @ ( schur_5982229384592763574omplex @ U ) ) ) ).
% hermitian_decomp_sim
thf(fact_248_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_249_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_250_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_251_square__mat_Osimps,axiom,
( square_mat_complex
= ( ^ [A2: mat_complex] :
( ( dim_col_complex @ A2 )
= ( dim_row_complex @ A2 ) ) ) ) ).
% square_mat.simps
thf(fact_252_det__mult,axiom,
! [A: mat_a,N: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( det_a @ ( times_times_mat_a @ A @ B ) )
= ( times_times_a @ ( det_a @ A ) @ ( det_a @ B ) ) ) ) ) ).
% det_mult
thf(fact_253_det__mult,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 ) )
=> ( ( det_complex @ ( times_8009071140041733218omplex @ A @ B ) )
= ( times_times_complex @ ( det_complex @ A ) @ ( det_complex @ B ) ) ) ) ) ).
% det_mult
thf(fact_254_hermitian__decomp__unitary,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( projec5943904436471448624omplex @ A @ B @ U )
=> ( comple6660659447773130958omplex @ U ) ) ).
% hermitian_decomp_unitary
thf(fact_255_invertible__mat__def,axiom,
( invert2568027935824841882omplex
= ( ^ [A2: mat_complex] :
( ( square_mat_complex @ A2 )
& ? [B2: mat_complex] :
( ( inverts_mat_complex @ A2 @ B2 )
& ( inverts_mat_complex @ B2 @ A2 ) ) ) ) ) ).
% invertible_mat_def
thf(fact_256_det__smult,axiom,
! [A3: complex,A: mat_complex] :
( ( det_complex @ ( smult_mat_complex @ A3 @ A ) )
= ( times_times_complex @ ( power_power_complex @ A3 @ ( dim_col_complex @ A ) ) @ ( det_complex @ A ) ) ) ).
% det_smult
thf(fact_257_det__0__negate,axiom,
! [A: mat_a,N: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( ( det_a @ ( uminus_uminus_mat_a @ A ) )
= zero_zero_a )
= ( ( det_a @ A )
= zero_zero_a ) ) ) ).
% det_0_negate
thf(fact_258_det__0__negate,axiom,
! [A: mat_real,N: nat] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( ( det_real @ ( uminus18246009535971484t_real @ A ) )
= zero_zero_real )
= ( ( det_real @ A )
= zero_zero_real ) ) ) ).
% det_0_negate
thf(fact_259_det__0__negate,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ( det_complex @ ( uminus467866341702955550omplex @ A ) )
= zero_zero_complex )
= ( ( det_complex @ A )
= zero_zero_complex ) ) ) ).
% det_0_negate
thf(fact_260_rank__1__proj__square__mat,axiom,
! [V3: vec_complex] : ( square_mat_complex @ ( linear1949544614684794075omplex @ V3 ) ) ).
% rank_1_proj_square_mat
thf(fact_261_mult__right__cancel,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A3 @ C2 )
= ( times_times_nat @ B3 @ C2 ) )
= ( A3 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_262_mult__right__cancel,axiom,
! [C2: real,A3: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B3 @ C2 ) )
= ( A3 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_263_mult__right__cancel,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ C2 )
= ( times_times_complex @ B3 @ C2 ) )
= ( A3 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_264_mult__cancel__right,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ C2 )
= ( times_times_nat @ B3 @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_265_mult__cancel__right,axiom,
! [A3: real,C2: real,B3: real] :
( ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B3 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_266_mult__cancel__right,axiom,
! [A3: complex,C2: complex,B3: complex] :
( ( ( times_times_complex @ A3 @ C2 )
= ( times_times_complex @ B3 @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_267_mult__left__cancel,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A3 )
= ( times_times_nat @ C2 @ B3 ) )
= ( A3 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_268_mult__left__cancel,axiom,
! [C2: real,A3: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A3 )
= ( times_times_real @ C2 @ B3 ) )
= ( A3 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_269_mult__left__cancel,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ C2 @ A3 )
= ( times_times_complex @ C2 @ B3 ) )
= ( A3 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_270_mult__cancel__left,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ( times_times_nat @ C2 @ A3 )
= ( times_times_nat @ C2 @ B3 ) )
= ( ( C2 = zero_zero_nat )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_271_mult__cancel__left,axiom,
! [C2: real,A3: real,B3: real] :
( ( ( times_times_real @ C2 @ A3 )
= ( times_times_real @ C2 @ B3 ) )
= ( ( C2 = zero_zero_real )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_272_mult__cancel__left,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( ( times_times_complex @ C2 @ A3 )
= ( times_times_complex @ C2 @ B3 ) )
= ( ( C2 = zero_zero_complex )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_273_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: real,A3: real,B3: real] :
( ( X != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ X )
= ( times_times_real @ B3 @ X ) )
=> ( A3 = B3 ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_274_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: complex,A3: complex,B3: complex] :
( ( X != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ X )
= ( times_times_complex @ B3 @ X ) )
=> ( A3 = B3 ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_275_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A3: real,X: real,B3: real] :
( ( ( times_times_real @ A3 @ X )
= ( times_times_real @ B3 @ X ) )
= ( ( A3 = B3 )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_276_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A3: complex,X: complex,B3: complex] :
( ( ( times_times_complex @ A3 @ X )
= ( times_times_complex @ B3 @ X ) )
= ( ( A3 = B3 )
| ( X = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_277_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A3: real,X: real,Y: real] :
( ( A3 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ X )
= ( times_times_real @ A3 @ Y ) )
=> ( X = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_278_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A3: complex,X: complex,Y: complex] :
( ( A3 != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ X )
= ( times_times_complex @ A3 @ Y ) )
=> ( X = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_279_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A3: real,X: real,Y: real] :
( ( ( times_times_real @ A3 @ X )
= ( times_times_real @ A3 @ Y ) )
= ( ( X = Y )
| ( A3 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_280_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A3: complex,X: complex,Y: complex] :
( ( ( times_times_complex @ A3 @ X )
= ( times_times_complex @ A3 @ Y ) )
= ( ( X = Y )
| ( A3 = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_281_vector__space__over__itself_Oscale__zero__right,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_282_vector__space__over__itself_Oscale__zero__right,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ zero_zero_complex )
= zero_zero_complex ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_283_vector__space__over__itself_Oscale__zero__left,axiom,
! [X: real] :
( ( times_times_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_284_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_285_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A3: real,X: real] :
( ( ( times_times_real @ A3 @ X )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_286_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A3: complex,X: complex] :
( ( ( times_times_complex @ A3 @ X )
= zero_zero_complex )
= ( ( A3 = zero_zero_complex )
| ( X = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_287_no__zero__divisors,axiom,
! [A3: nat,B3: nat] :
( ( A3 != zero_zero_nat )
=> ( ( B3 != zero_zero_nat )
=> ( ( times_times_nat @ A3 @ B3 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_288_no__zero__divisors,axiom,
! [A3: real,B3: real] :
( ( A3 != zero_zero_real )
=> ( ( B3 != zero_zero_real )
=> ( ( times_times_real @ A3 @ B3 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_289_no__zero__divisors,axiom,
! [A3: complex,B3: complex] :
( ( A3 != zero_zero_complex )
=> ( ( B3 != zero_zero_complex )
=> ( ( times_times_complex @ A3 @ B3 )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_290_mult__eq__0__iff,axiom,
! [A3: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ B3 )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
| ( B3 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_291_mult__eq__0__iff,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ B3 )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( B3 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_292_mult__eq__0__iff,axiom,
! [A3: complex,B3: complex] :
( ( ( times_times_complex @ A3 @ B3 )
= zero_zero_complex )
= ( ( A3 = zero_zero_complex )
| ( B3 = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_293_divisors__zero,axiom,
! [A3: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ B3 )
= zero_zero_nat )
=> ( ( A3 = zero_zero_nat )
| ( B3 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_294_divisors__zero,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ B3 )
= zero_zero_real )
=> ( ( A3 = zero_zero_real )
| ( B3 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_295_divisors__zero,axiom,
! [A3: complex,B3: complex] :
( ( ( times_times_complex @ A3 @ B3 )
= zero_zero_complex )
=> ( ( A3 = zero_zero_complex )
| ( B3 = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_296_mult__zero__right,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_297_mult__zero__right,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_298_mult__zero__right,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_299_mult__zero__left,axiom,
! [A3: nat] :
( ( times_times_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_300_mult__zero__left,axiom,
! [A3: real] :
( ( times_times_real @ zero_zero_real @ A3 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_301_mult__zero__left,axiom,
! [A3: complex] :
( ( times_times_complex @ zero_zero_complex @ A3 )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_302_mult__not__zero,axiom,
! [A3: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ B3 )
!= zero_zero_nat )
=> ( ( A3 != zero_zero_nat )
& ( B3 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_303_mult__not__zero,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ B3 )
!= zero_zero_real )
=> ( ( A3 != zero_zero_real )
& ( B3 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_304_mult__not__zero,axiom,
! [A3: complex,B3: complex] :
( ( ( times_times_complex @ A3 @ B3 )
!= zero_zero_complex )
=> ( ( A3 != zero_zero_complex )
& ( B3 != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_305_neg__equal__zero,axiom,
! [A3: real] :
( ( ( uminus_uminus_real @ A3 )
= A3 )
= ( A3 = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_306_equal__neg__zero,axiom,
! [A3: real] :
( ( A3
= ( uminus_uminus_real @ A3 ) )
= ( A3 = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_307_neg__equal__0__iff__equal,axiom,
! [A3: complex] :
( ( ( uminus1482373934393186551omplex @ A3 )
= zero_zero_complex )
= ( A3 = zero_zero_complex ) ) ).
% neg_equal_0_iff_equal
thf(fact_308_neg__equal__0__iff__equal,axiom,
! [A3: real] :
( ( ( uminus_uminus_real @ A3 )
= zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_309_neg__0__equal__iff__equal,axiom,
! [A3: complex] :
( ( zero_zero_complex
= ( uminus1482373934393186551omplex @ A3 ) )
= ( zero_zero_complex = A3 ) ) ).
% neg_0_equal_iff_equal
thf(fact_310_neg__0__equal__iff__equal,axiom,
! [A3: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A3 ) )
= ( zero_zero_real = A3 ) ) ).
% neg_0_equal_iff_equal
thf(fact_311_add_Oinverse__neutral,axiom,
( ( uminus1482373934393186551omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% add.inverse_neutral
thf(fact_312_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_313_unitary__zero,axiom,
! [A: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ zero_zero_nat @ zero_zero_nat ) )
=> ( complex_unitary_a @ A ) ) ).
% unitary_zero
thf(fact_314_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_315_vec__space_Odet__nonzero__congruence,axiom,
! [A: mat_a,M3: mat_a,B: mat_a,N: nat] :
( ( ( times_times_mat_a @ A @ M3 )
= ( times_times_mat_a @ B @ M3 ) )
=> ( ( ( det_a @ M3 )
!= zero_zero_a )
=> ( ( member_mat_a @ M3 @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( A = B ) ) ) ) ) ) ).
% vec_space.det_nonzero_congruence
thf(fact_316_vec__space_Odet__nonzero__congruence,axiom,
! [A: mat_real,M3: mat_real,B: mat_real,N: nat] :
( ( ( times_times_mat_real @ A @ M3 )
= ( times_times_mat_real @ B @ M3 ) )
=> ( ( ( det_real @ M3 )
!= zero_zero_real )
=> ( ( member_mat_real @ M3 @ ( carrier_mat_real @ N @ N ) )
=> ( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( member_mat_real @ B @ ( carrier_mat_real @ N @ N ) )
=> ( A = B ) ) ) ) ) ) ).
% vec_space.det_nonzero_congruence
thf(fact_317_vec__space_Odet__nonzero__congruence,axiom,
! [A: mat_complex,M3: mat_complex,B: mat_complex,N: nat] :
( ( ( times_8009071140041733218omplex @ A @ M3 )
= ( times_8009071140041733218omplex @ B @ M3 ) )
=> ( ( ( det_complex @ M3 )
!= zero_zero_complex )
=> ( ( member_mat_complex @ M3 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( A = B ) ) ) ) ) ) ).
% vec_space.det_nonzero_congruence
thf(fact_318_class__ring_Oring__simprules_I21_J,axiom,
( ( uminus1482373934393186551omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% class_ring.ring_simprules(21)
thf(fact_319_class__ring_Oring__simprules_I21_J,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% class_ring.ring_simprules(21)
thf(fact_320_class__cring_Ocring__simprules_I22_J,axiom,
( ( uminus1482373934393186551omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% class_cring.cring_simprules(22)
thf(fact_321_class__cring_Ocring__simprules_I22_J,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% class_cring.cring_simprules(22)
thf(fact_322_vec__space_Orow__space__is__preserved,axiom,
! [P: mat_a,M: nat,A: mat_a,N: nat] :
( ( invertible_mat_a @ P )
=> ( ( member_mat_a @ P @ ( carrier_mat_a @ M @ M ) )
=> ( ( member_mat_a @ A @ ( carrier_mat_a @ M @ N ) )
=> ( ( vS_vec_row_space_a @ N @ ( times_times_mat_a @ P @ A ) )
= ( vS_vec_row_space_a @ N @ A ) ) ) ) ) ).
% vec_space.row_space_is_preserved
thf(fact_323_vec__space_Orow__space__is__preserved,axiom,
! [P: mat_complex,M: nat,A: mat_complex,N: nat] :
( ( invert2568027935824841882omplex @ P )
=> ( ( member_mat_complex @ P @ ( carrier_mat_complex @ M @ M ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ M @ N ) )
=> ( ( vS_vec3284807721666986142omplex @ N @ ( times_8009071140041733218omplex @ P @ A ) )
= ( vS_vec3284807721666986142omplex @ N @ A ) ) ) ) ) ).
% vec_space.row_space_is_preserved
thf(fact_324_mult__delta__right,axiom,
! [B3: $o,X: nat,Y: nat] :
( ( B3
=> ( ( times_times_nat @ X @ ( if_nat @ B3 @ Y @ zero_zero_nat ) )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B3
=> ( ( times_times_nat @ X @ ( if_nat @ B3 @ Y @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_325_mult__delta__right,axiom,
! [B3: $o,X: real,Y: real] :
( ( B3
=> ( ( times_times_real @ X @ ( if_real @ B3 @ Y @ zero_zero_real ) )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B3
=> ( ( times_times_real @ X @ ( if_real @ B3 @ Y @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_326_mult__delta__right,axiom,
! [B3: $o,X: complex,Y: complex] :
( ( B3
=> ( ( times_times_complex @ X @ ( if_complex @ B3 @ Y @ zero_zero_complex ) )
= ( times_times_complex @ X @ Y ) ) )
& ( ~ B3
=> ( ( times_times_complex @ X @ ( if_complex @ B3 @ Y @ zero_zero_complex ) )
= zero_zero_complex ) ) ) ).
% mult_delta_right
thf(fact_327_mult__delta__left,axiom,
! [B3: $o,X: nat,Y: nat] :
( ( B3
=> ( ( times_times_nat @ ( if_nat @ B3 @ X @ zero_zero_nat ) @ Y )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B3
=> ( ( times_times_nat @ ( if_nat @ B3 @ X @ zero_zero_nat ) @ Y )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_328_mult__delta__left,axiom,
! [B3: $o,X: real,Y: real] :
( ( B3
=> ( ( times_times_real @ ( if_real @ B3 @ X @ zero_zero_real ) @ Y )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B3
=> ( ( times_times_real @ ( if_real @ B3 @ X @ zero_zero_real ) @ Y )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_329_mult__delta__left,axiom,
! [B3: $o,X: complex,Y: complex] :
( ( B3
=> ( ( times_times_complex @ ( if_complex @ B3 @ X @ zero_zero_complex ) @ Y )
= ( times_times_complex @ X @ Y ) ) )
& ( ~ B3
=> ( ( times_times_complex @ ( if_complex @ B3 @ X @ zero_zero_complex ) @ Y )
= zero_zero_complex ) ) ) ).
% mult_delta_left
thf(fact_330_mult__hom_Ohom__zero,axiom,
! [C2: nat] :
( ( times_times_nat @ C2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_331_mult__hom_Ohom__zero,axiom,
! [C2: real] :
( ( times_times_real @ C2 @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_332_mult__hom_Ohom__zero,axiom,
! [C2: complex] :
( ( times_times_complex @ C2 @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_hom.hom_zero
thf(fact_333_smult__zero,axiom,
! [A: mat_real] :
( ( smult_mat_real @ zero_zero_real @ A )
= ( zero_mat_real @ ( dim_row_real @ A ) @ ( dim_col_real @ A ) ) ) ).
% smult_zero
thf(fact_334_smult__zero,axiom,
! [A: mat_complex] :
( ( smult_mat_complex @ zero_zero_complex @ A )
= ( zero_mat_complex @ ( dim_row_complex @ A ) @ ( dim_col_complex @ A ) ) ) ).
% smult_zero
thf(fact_335_det__dim__zero,axiom,
! [A: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ zero_zero_nat @ zero_zero_nat ) )
=> ( ( det_a @ A )
= one_one_a ) ) ).
% det_dim_zero
thf(fact_336_det__dim__zero,axiom,
! [A: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ zero_zero_nat @ zero_zero_nat ) )
=> ( ( det_complex @ A )
= one_one_complex ) ) ).
% det_dim_zero
thf(fact_337_det__dim__zero,axiom,
! [A: mat_real] :
( ( member_mat_real @ A @ ( carrier_mat_real @ zero_zero_nat @ zero_zero_nat ) )
=> ( ( det_real @ A )
= one_one_real ) ) ).
% det_dim_zero
thf(fact_338_pow__mat_Osimps_I1_J,axiom,
! [A: mat_complex] :
( ( pow_mat_complex @ A @ zero_zero_nat )
= ( one_mat_complex @ ( dim_row_complex @ A ) ) ) ).
% pow_mat.simps(1)
thf(fact_339_zero__carrier__mat,axiom,
! [Nr: nat,Nc: nat] : ( member_mat_a @ ( zero_mat_a @ Nr @ Nc ) @ ( carrier_mat_a @ Nr @ Nc ) ) ).
% zero_carrier_mat
thf(fact_340_zero__carrier__mat,axiom,
! [Nr: nat,Nc: nat] : ( member_mat_complex @ ( zero_mat_complex @ Nr @ Nc ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ).
% zero_carrier_mat
thf(fact_341_index__zero__mat_I2_J,axiom,
! [Nr: nat,Nc: nat] :
( ( dim_row_complex @ ( zero_mat_complex @ Nr @ Nc ) )
= Nr ) ).
% index_zero_mat(2)
thf(fact_342_index__zero__mat_I3_J,axiom,
! [Nr: nat,Nc: nat] :
( ( dim_col_complex @ ( zero_mat_complex @ Nr @ Nc ) )
= Nc ) ).
% index_zero_mat(3)
thf(fact_343_zero__adjoint,axiom,
! [N: nat,M: nat] :
( ( schur_5982229384592763574omplex @ ( zero_mat_complex @ N @ M ) )
= ( zero_mat_complex @ M @ N ) ) ).
% zero_adjoint
thf(fact_344_zero__adjoint,axiom,
! [N: nat,M: nat] :
( ( schur_mat_adjoint_a @ ( zero_mat_a @ N @ M ) )
= ( zero_mat_a @ M @ N ) ) ).
% zero_adjoint
thf(fact_345_smult__zero__mat,axiom,
! [K: complex,Nr: nat,Nc: nat] :
( ( smult_mat_complex @ K @ ( zero_mat_complex @ Nr @ Nc ) )
= ( zero_mat_complex @ Nr @ Nc ) ) ).
% smult_zero_mat
thf(fact_346_zero__hermitian,axiom,
! [N: nat] : ( comple8306762464034002205omplex @ ( zero_mat_complex @ N @ N ) ) ).
% zero_hermitian
thf(fact_347_rel__simps_I93_J,axiom,
one_one_nat != zero_zero_nat ).
% rel_simps(93)
thf(fact_348_rel__simps_I93_J,axiom,
one_one_real != zero_zero_real ).
% rel_simps(93)
thf(fact_349_rel__simps_I93_J,axiom,
one_one_complex != zero_zero_complex ).
% rel_simps(93)
thf(fact_350_mult__1,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% mult_1
thf(fact_351_mult__1,axiom,
! [A3: complex] :
( ( times_times_complex @ one_one_complex @ A3 )
= A3 ) ).
% mult_1
thf(fact_352_mult__1,axiom,
! [A3: real] :
( ( times_times_real @ one_one_real @ A3 )
= A3 ) ).
% mult_1
thf(fact_353_mult_Ocomm__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.comm_neutral
thf(fact_354_mult_Ocomm__neutral,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ one_one_complex )
= A3 ) ).
% mult.comm_neutral
thf(fact_355_mult_Ocomm__neutral,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% mult.comm_neutral
thf(fact_356_mult_Oright__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.right_neutral
thf(fact_357_mult_Oright__neutral,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ one_one_complex )
= A3 ) ).
% mult.right_neutral
thf(fact_358_mult_Oright__neutral,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% mult.right_neutral
thf(fact_359_comm__monoid__mult__class_Omult__1,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_360_comm__monoid__mult__class_Omult__1,axiom,
! [A3: complex] :
( ( times_times_complex @ one_one_complex @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_361_comm__monoid__mult__class_Omult__1,axiom,
! [A3: real] :
( ( times_times_real @ one_one_real @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_362_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_363_vector__space__over__itself_Oscale__one,axiom,
! [X: real] :
( ( times_times_real @ one_one_real @ X )
= X ) ).
% vector_space_over_itself.scale_one
thf(fact_364_one__neq__neg__one,axiom,
( one_one_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% one_neq_neg_one
thf(fact_365_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_366_pow__carrier__mat,axiom,
! [A: mat_a,N: nat,K: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( member_mat_a @ ( pow_mat_a @ A @ K ) @ ( carrier_mat_a @ N @ N ) ) ) ).
% pow_carrier_mat
thf(fact_367_pow__carrier__mat,axiom,
! [A: mat_complex,N: nat,K: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( member_mat_complex @ ( pow_mat_complex @ A @ K ) @ ( carrier_mat_complex @ N @ N ) ) ) ).
% pow_carrier_mat
thf(fact_368_upper__triangular__zero,axiom,
! [N: nat] : ( upper_4850907204721561915omplex @ ( zero_mat_complex @ N @ N ) ) ).
% upper_triangular_zero
thf(fact_369_pow__mat__dim_I1_J,axiom,
! [A: mat_complex,K: nat] :
( ( dim_row_complex @ ( pow_mat_complex @ A @ K ) )
= ( dim_row_complex @ A ) ) ).
% pow_mat_dim(1)
thf(fact_370_zero__projector,axiom,
! [N: nat] : ( linear5633924348262549461omplex @ ( zero_mat_complex @ N @ N ) ) ).
% zero_projector
thf(fact_371_Linear__Algebra__Complements_Osmult__one,axiom,
! [A: mat_nat] :
( ( smult_mat_nat @ one_one_nat @ A )
= A ) ).
% Linear_Algebra_Complements.smult_one
thf(fact_372_Linear__Algebra__Complements_Osmult__one,axiom,
! [A: mat_complex] :
( ( smult_mat_complex @ one_one_complex @ A )
= A ) ).
% Linear_Algebra_Complements.smult_one
thf(fact_373_Linear__Algebra__Complements_Osmult__one,axiom,
! [A: mat_real] :
( ( smult_mat_real @ one_one_real @ A )
= A ) ).
% Linear_Algebra_Complements.smult_one
thf(fact_374_right__mult__zero__mat,axiom,
! [A: mat_a,Nr: nat,N: nat,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( times_times_mat_a @ A @ ( zero_mat_a @ N @ Nc ) )
= ( zero_mat_a @ Nr @ Nc ) ) ) ).
% right_mult_zero_mat
thf(fact_375_right__mult__zero__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( times_8009071140041733218omplex @ A @ ( zero_mat_complex @ N @ Nc ) )
= ( zero_mat_complex @ Nr @ Nc ) ) ) ).
% right_mult_zero_mat
thf(fact_376_left__mult__zero__mat,axiom,
! [A: mat_a,N: nat,Nc: nat,Nr: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ ( zero_mat_a @ Nr @ N ) @ A )
= ( zero_mat_a @ Nr @ Nc ) ) ) ).
% left_mult_zero_mat
thf(fact_377_left__mult__zero__mat,axiom,
! [A: mat_complex,N: nat,Nc: nat,Nr: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( zero_mat_complex @ Nr @ N ) @ A )
= ( zero_mat_complex @ Nr @ Nc ) ) ) ).
% left_mult_zero_mat
thf(fact_378_trace__zero,axiom,
! [N: nat] :
( ( complex_trace_real @ ( zero_mat_real @ N @ N ) )
= zero_zero_real ) ).
% trace_zero
thf(fact_379_trace__zero,axiom,
! [N: nat] :
( ( comple3184165445352484367omplex @ ( zero_mat_complex @ N @ N ) )
= zero_zero_complex ) ).
% trace_zero
thf(fact_380_mult__cancel__right2,axiom,
! [A3: real,C2: real] :
( ( ( times_times_real @ A3 @ C2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A3 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_381_mult__cancel__right2,axiom,
! [A3: complex,C2: complex] :
( ( ( times_times_complex @ A3 @ C2 )
= C2 )
= ( ( C2 = zero_zero_complex )
| ( A3 = one_one_complex ) ) ) ).
% mult_cancel_right2
thf(fact_382_mult__cancel__right1,axiom,
! [C2: real,B3: real] :
( ( C2
= ( times_times_real @ B3 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B3 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_383_mult__cancel__right1,axiom,
! [C2: complex,B3: complex] :
( ( C2
= ( times_times_complex @ B3 @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( B3 = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_384_mult__cancel__left2,axiom,
! [C2: real,A3: real] :
( ( ( times_times_real @ C2 @ A3 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A3 = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_385_mult__cancel__left2,axiom,
! [C2: complex,A3: complex] :
( ( ( times_times_complex @ C2 @ A3 )
= C2 )
= ( ( C2 = zero_zero_complex )
| ( A3 = one_one_complex ) ) ) ).
% mult_cancel_left2
thf(fact_386_mult__cancel__left1,axiom,
! [C2: real,B3: real] :
( ( C2
= ( times_times_real @ C2 @ B3 ) )
= ( ( C2 = zero_zero_real )
| ( B3 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_387_mult__cancel__left1,axiom,
! [C2: complex,B3: complex] :
( ( C2
= ( times_times_complex @ C2 @ B3 ) )
= ( ( C2 = zero_zero_complex )
| ( B3 = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_388_rel__simps_I89_J,axiom,
( zero_zero_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% rel_simps(89)
thf(fact_389_rel__simps_I89_J,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% rel_simps(89)
thf(fact_390_class__field_Oneg__1__not__0,axiom,
( ( uminus1482373934393186551omplex @ one_one_complex )
!= zero_zero_complex ) ).
% class_field.neg_1_not_0
thf(fact_391_class__field_Oneg__1__not__0,axiom,
( ( uminus_uminus_real @ one_one_real )
!= zero_zero_real ) ).
% class_field.neg_1_not_0
thf(fact_392_square__eq__1__iff,axiom,
! [X: complex] :
( ( ( times_times_complex @ X @ X )
= one_one_complex )
= ( ( X = one_one_complex )
| ( X
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% square_eq_1_iff
thf(fact_393_square__eq__1__iff,axiom,
! [X: real] :
( ( ( times_times_real @ X @ X )
= one_one_real )
= ( ( X = one_one_real )
| ( X
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% square_eq_1_iff
thf(fact_394_mult__minus1__right,axiom,
! [Z: complex] :
( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ Z ) ) ).
% mult_minus1_right
thf(fact_395_mult__minus1__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1_right
thf(fact_396_mult__minus1,axiom,
! [Z: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
= ( uminus1482373934393186551omplex @ Z ) ) ).
% mult_minus1
thf(fact_397_mult__minus1,axiom,
! [Z: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1
thf(fact_398_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_399_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_400_left__right__inverse__power,axiom,
! [X: real,Y: real,N: nat] :
( ( ( times_times_real @ X @ Y )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_401_pow__mat__dim__square_I1_J,axiom,
! [A: mat_a,N: nat,K: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( dim_row_a @ ( pow_mat_a @ A @ K ) )
= N ) ) ).
% pow_mat_dim_square(1)
thf(fact_402_pow__mat__dim__square_I1_J,axiom,
! [A: mat_complex,N: nat,K: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( dim_row_complex @ ( pow_mat_complex @ A @ K ) )
= N ) ) ).
% pow_mat_dim_square(1)
thf(fact_403_pow__mat__dim__square_I2_J,axiom,
! [A: mat_a,N: nat,K: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( dim_col_a @ ( pow_mat_a @ A @ K ) )
= N ) ) ).
% pow_mat_dim_square(2)
thf(fact_404_pow__mat__dim__square_I2_J,axiom,
! [A: mat_complex,N: nat,K: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( dim_col_complex @ ( pow_mat_complex @ A @ K ) )
= N ) ) ).
% pow_mat_dim_square(2)
thf(fact_405_similar__mat__wit__pow__id,axiom,
! [A: mat_complex,B: mat_complex,P: mat_complex,Q: mat_complex,K: nat] :
( ( simila5774310414453981135omplex @ A @ B @ P @ Q )
=> ( ( pow_mat_complex @ A @ K )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P @ ( pow_mat_complex @ B @ K ) ) @ Q ) ) ) ).
% similar_mat_wit_pow_id
thf(fact_406_det__one,axiom,
! [N: nat] :
( ( det_complex @ ( one_mat_complex @ N ) )
= one_one_complex ) ).
% det_one
thf(fact_407_det__one,axiom,
! [N: nat] :
( ( det_real @ ( one_mat_real @ N ) )
= one_one_real ) ).
% det_one
thf(fact_408_left__mult__zero__mat_H,axiom,
! [A: mat_complex,N: nat,Nr: nat] :
( ( ( dim_row_complex @ A )
= N )
=> ( ( times_8009071140041733218omplex @ ( zero_mat_complex @ Nr @ N ) @ A )
= ( zero_mat_complex @ Nr @ ( dim_col_complex @ A ) ) ) ) ).
% left_mult_zero_mat'
thf(fact_409_right__mult__zero__mat_H,axiom,
! [A: mat_complex,N: nat,Nc: nat] :
( ( ( dim_col_complex @ A )
= N )
=> ( ( times_8009071140041733218omplex @ A @ ( zero_mat_complex @ N @ Nc ) )
= ( zero_mat_complex @ ( dim_row_complex @ A ) @ Nc ) ) ) ).
% right_mult_zero_mat'
thf(fact_410_left__minus__one__mult__self,axiom,
! [N: nat,A3: complex] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A3 ) )
= A3 ) ).
% left_minus_one_mult_self
thf(fact_411_left__minus__one__mult__self,axiom,
! [N: nat,A3: real] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A3 ) )
= A3 ) ).
% left_minus_one_mult_self
thf(fact_412_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
= one_one_complex ) ).
% minus_one_mult_self
thf(fact_413_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
= one_one_real ) ).
% minus_one_mult_self
thf(fact_414_power__minus,axiom,
! [A3: complex,N: nat] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ A3 ) @ N )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A3 @ N ) ) ) ).
% power_minus
thf(fact_415_power__minus,axiom,
! [A3: real,N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ A3 ) @ N )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A3 @ N ) ) ) ).
% power_minus
thf(fact_416_pow__mat__dim_I2_J,axiom,
! [K: nat,A: mat_complex] :
( ( ( K = zero_zero_nat )
=> ( ( dim_col_complex @ ( pow_mat_complex @ A @ K ) )
= ( dim_row_complex @ A ) ) )
& ( ( K != zero_zero_nat )
=> ( ( dim_col_complex @ ( pow_mat_complex @ A @ K ) )
= ( dim_col_complex @ A ) ) ) ) ).
% pow_mat_dim(2)
thf(fact_417_mult__if__delta,axiom,
! [P: $o,Q4: nat] :
( ( P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q4 )
= Q4 ) )
& ( ~ P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q4 )
= zero_zero_nat ) ) ) ).
% mult_if_delta
thf(fact_418_mult__if__delta,axiom,
! [P: $o,Q4: real] :
( ( P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q4 )
= Q4 ) )
& ( ~ P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q4 )
= zero_zero_real ) ) ) ).
% mult_if_delta
thf(fact_419_mult__if__delta,axiom,
! [P: $o,Q4: complex] :
( ( P
=> ( ( times_times_complex @ ( if_complex @ P @ one_one_complex @ zero_zero_complex ) @ Q4 )
= Q4 ) )
& ( ~ P
=> ( ( times_times_complex @ ( if_complex @ P @ one_one_complex @ zero_zero_complex ) @ Q4 )
= zero_zero_complex ) ) ) ).
% mult_if_delta
thf(fact_420_det__four__block__mat__lower__right__zero__col,axiom,
! [A1: mat_a,N: nat,A22: mat_a,A32: mat_a,A4: mat_a] :
( ( member_mat_a @ A1 @ ( carrier_mat_a @ one_one_nat @ N ) )
=> ( ( member_mat_a @ A22 @ ( carrier_mat_a @ one_one_nat @ one_one_nat ) )
=> ( ( member_mat_a @ A32 @ ( carrier_mat_a @ N @ N ) )
=> ( ( A4
= ( zero_mat_a @ N @ one_one_nat ) )
=> ( ( det_a @ ( four_block_mat_a @ A1 @ A22 @ A32 @ A4 ) )
= ( times_times_a @ ( times_times_a @ ( power_power_a @ ( uminus_uminus_a @ one_one_a ) @ N ) @ ( det_a @ A22 ) ) @ ( det_a @ A32 ) ) ) ) ) ) ) ).
% det_four_block_mat_lower_right_zero_col
thf(fact_421_det__four__block__mat__lower__right__zero__col,axiom,
! [A1: mat_complex,N: nat,A22: mat_complex,A32: mat_complex,A4: mat_complex] :
( ( member_mat_complex @ A1 @ ( carrier_mat_complex @ one_one_nat @ N ) )
=> ( ( member_mat_complex @ A22 @ ( carrier_mat_complex @ one_one_nat @ one_one_nat ) )
=> ( ( member_mat_complex @ A32 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( A4
= ( zero_mat_complex @ N @ one_one_nat ) )
=> ( ( det_complex @ ( four_b559179830521662709omplex @ A1 @ A22 @ A32 @ A4 ) )
= ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( det_complex @ A22 ) ) @ ( det_complex @ A32 ) ) ) ) ) ) ) ).
% det_four_block_mat_lower_right_zero_col
thf(fact_422_det__four__block__mat__lower__right__zero__col,axiom,
! [A1: mat_real,N: nat,A22: mat_real,A32: mat_real,A4: mat_real] :
( ( member_mat_real @ A1 @ ( carrier_mat_real @ one_one_nat @ N ) )
=> ( ( member_mat_real @ A22 @ ( carrier_mat_real @ one_one_nat @ one_one_nat ) )
=> ( ( member_mat_real @ A32 @ ( carrier_mat_real @ N @ N ) )
=> ( ( A4
= ( zero_mat_real @ N @ one_one_nat ) )
=> ( ( det_real @ ( four_block_mat_real @ A1 @ A22 @ A32 @ A4 ) )
= ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( det_real @ A22 ) ) @ ( det_real @ A32 ) ) ) ) ) ) ) ).
% det_four_block_mat_lower_right_zero_col
thf(fact_423_pow__mat_Oelims,axiom,
! [X: mat_complex,Xa: nat,Y: mat_complex] :
( ( ( pow_mat_complex @ X @ Xa )
= Y )
=> ( ( ( Xa = zero_zero_nat )
=> ( Y
!= ( one_mat_complex @ ( dim_row_complex @ X ) ) ) )
=> ~ ! [K2: nat] :
( ( Xa
= ( suc @ K2 ) )
=> ( Y
!= ( times_8009071140041733218omplex @ ( pow_mat_complex @ X @ K2 ) @ X ) ) ) ) ) ).
% pow_mat.elims
thf(fact_424_power__minus_H,axiom,
! [X: complex,N: nat] :
( ( nO_MAT8947977539597988553omplex @ one_one_complex @ X )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ N )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ X @ N ) ) ) ) ).
% power_minus'
thf(fact_425_power__minus_H,axiom,
! [X: real,N: nat] :
( ( nO_MATCH_real_real @ one_one_real @ X )
=> ( ( power_power_real @ ( uminus_uminus_real @ X ) @ N )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ X @ N ) ) ) ) ).
% power_minus'
thf(fact_426_det__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( det_real @ ( zero_mat_real @ N @ N ) )
= zero_zero_real ) ) ).
% det_zero
thf(fact_427_det__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( det_complex @ ( zero_mat_complex @ N @ N ) )
= zero_zero_complex ) ) ).
% det_zero
thf(fact_428_similar__mat__wit__four__block,axiom,
! [A1: mat_a,B1: mat_a,P1: mat_a,Q1: mat_a,A22: mat_a,B22: mat_a,P22: mat_a,Q22: mat_a,URA: mat_a,UR: mat_a,LLA: mat_a,LL: mat_a,N: nat,M: nat] :
( ( similar_mat_wit_a @ A1 @ B1 @ P1 @ Q1 )
=> ( ( similar_mat_wit_a @ A22 @ B22 @ P22 @ Q22 )
=> ( ( URA
= ( times_times_mat_a @ ( times_times_mat_a @ P1 @ UR ) @ Q22 ) )
=> ( ( LLA
= ( times_times_mat_a @ ( times_times_mat_a @ P22 @ LL ) @ Q1 ) )
=> ( ( member_mat_a @ A1 @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ A22 @ ( carrier_mat_a @ M @ M ) )
=> ( ( member_mat_a @ LL @ ( carrier_mat_a @ M @ N ) )
=> ( ( member_mat_a @ UR @ ( carrier_mat_a @ N @ M ) )
=> ( similar_mat_wit_a @ ( four_block_mat_a @ A1 @ URA @ LLA @ A22 ) @ ( four_block_mat_a @ B1 @ UR @ LL @ B22 ) @ ( four_block_mat_a @ P1 @ ( zero_mat_a @ N @ M ) @ ( zero_mat_a @ M @ N ) @ P22 ) @ ( four_block_mat_a @ Q1 @ ( zero_mat_a @ N @ M ) @ ( zero_mat_a @ M @ N ) @ Q22 ) ) ) ) ) ) ) ) ) ) ).
% similar_mat_wit_four_block
thf(fact_429_similar__mat__wit__four__block,axiom,
! [A1: mat_complex,B1: mat_complex,P1: mat_complex,Q1: mat_complex,A22: mat_complex,B22: mat_complex,P22: mat_complex,Q22: mat_complex,URA: mat_complex,UR: mat_complex,LLA: mat_complex,LL: mat_complex,N: nat,M: nat] :
( ( simila5774310414453981135omplex @ A1 @ B1 @ P1 @ Q1 )
=> ( ( simila5774310414453981135omplex @ A22 @ B22 @ P22 @ Q22 )
=> ( ( URA
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P1 @ UR ) @ Q22 ) )
=> ( ( LLA
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P22 @ LL ) @ Q1 ) )
=> ( ( member_mat_complex @ A1 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ A22 @ ( carrier_mat_complex @ M @ M ) )
=> ( ( member_mat_complex @ LL @ ( carrier_mat_complex @ M @ N ) )
=> ( ( member_mat_complex @ UR @ ( carrier_mat_complex @ N @ M ) )
=> ( simila5774310414453981135omplex @ ( four_b559179830521662709omplex @ A1 @ URA @ LLA @ A22 ) @ ( four_b559179830521662709omplex @ B1 @ UR @ LL @ B22 ) @ ( four_b559179830521662709omplex @ P1 @ ( zero_mat_complex @ N @ M ) @ ( zero_mat_complex @ M @ N ) @ P22 ) @ ( four_b559179830521662709omplex @ Q1 @ ( zero_mat_complex @ N @ M ) @ ( zero_mat_complex @ M @ N ) @ Q22 ) ) ) ) ) ) ) ) ) ) ).
% similar_mat_wit_four_block
thf(fact_430_power__eq__if,axiom,
( power_power_nat
= ( ^ [P4: nat,M4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P4 @ ( power_power_nat @ P4 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_431_power__eq__if,axiom,
( power_power_complex
= ( ^ [P4: complex,M4: nat] : ( if_complex @ ( M4 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P4 @ ( power_power_complex @ P4 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_432_power__eq__if,axiom,
( power_power_real
= ( ^ [P4: real,M4: nat] : ( if_real @ ( M4 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P4 @ ( power_power_real @ P4 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_433_trace__minus__linear,axiom,
! [A: mat_a,N: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_trace_a @ ( minus_minus_mat_a @ A @ B ) )
= ( minus_minus_a @ ( complex_trace_a @ A ) @ ( complex_trace_a @ B ) ) ) ) ) ).
% trace_minus_linear
thf(fact_434_trace__minus__linear,axiom,
! [A: mat_real,N: nat,B: mat_real] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( member_mat_real @ B @ ( carrier_mat_real @ N @ N ) )
=> ( ( complex_trace_real @ ( minus_minus_mat_real @ A @ B ) )
= ( minus_minus_real @ ( complex_trace_real @ A ) @ ( complex_trace_real @ B ) ) ) ) ) ).
% trace_minus_linear
thf(fact_435_trace__minus__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 @ ( minus_2412168080157227406omplex @ A @ B ) )
= ( minus_minus_complex @ ( comple3184165445352484367omplex @ A ) @ ( comple3184165445352484367omplex @ B ) ) ) ) ) ).
% trace_minus_linear
thf(fact_436_arith__extra__simps_I13_J,axiom,
! [A3: complex] :
( ( minus_minus_complex @ A3 @ zero_zero_complex )
= A3 ) ).
% arith_extra_simps(13)
thf(fact_437_arith__extra__simps_I13_J,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ zero_zero_real )
= A3 ) ).
% arith_extra_simps(13)
thf(fact_438_cross3__simps_I25_J,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( times_times_nat @ A3 @ ( minus_minus_nat @ B3 @ C2 ) )
= ( minus_minus_nat @ ( times_times_nat @ A3 @ B3 ) @ ( times_times_nat @ A3 @ C2 ) ) ) ).
% cross3_simps(25)
thf(fact_439_cross3__simps_I25_J,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ A3 @ ( minus_minus_real @ B3 @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A3 @ B3 ) @ ( times_times_real @ A3 @ C2 ) ) ) ).
% cross3_simps(25)
thf(fact_440_cross3__simps_I26_J,axiom,
! [B3: nat,C2: nat,A3: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B3 @ C2 ) @ A3 )
= ( minus_minus_nat @ ( times_times_nat @ B3 @ A3 ) @ ( times_times_nat @ C2 @ A3 ) ) ) ).
% cross3_simps(26)
thf(fact_441_cross3__simps_I26_J,axiom,
! [B3: real,C2: real,A3: real] :
( ( times_times_real @ ( minus_minus_real @ B3 @ C2 ) @ A3 )
= ( minus_minus_real @ ( times_times_real @ B3 @ A3 ) @ ( times_times_real @ C2 @ A3 ) ) ) ).
% cross3_simps(26)
thf(fact_442_cross3__simps_I27_J,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ A3 @ ( minus_minus_real @ B3 @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A3 @ B3 ) @ ( times_times_real @ A3 @ C2 ) ) ) ).
% cross3_simps(27)
thf(fact_443_cross3__simps_I28_J,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ ( minus_minus_real @ A3 @ B3 ) @ C2 )
= ( minus_minus_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ).
% cross3_simps(28)
thf(fact_444_cross3__simps_I50_J,axiom,
! [A3: real,B3: real,X: real] :
( ( times_times_real @ ( minus_minus_real @ A3 @ B3 ) @ X )
= ( minus_minus_real @ ( times_times_real @ A3 @ X ) @ ( times_times_real @ B3 @ X ) ) ) ).
% cross3_simps(50)
thf(fact_445_cross3__simps_I51_J,axiom,
! [A3: real,X: real,Y: real] :
( ( times_times_real @ A3 @ ( minus_minus_real @ X @ Y ) )
= ( minus_minus_real @ ( times_times_real @ A3 @ X ) @ ( times_times_real @ A3 @ Y ) ) ) ).
% cross3_simps(51)
thf(fact_446_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_447_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_448_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_449_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_450_minus__diff__eq,axiom,
! [A3: complex,B3: complex] :
( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A3 @ B3 ) )
= ( minus_minus_complex @ B3 @ A3 ) ) ).
% minus_diff_eq
thf(fact_451_minus__diff__eq,axiom,
! [A3: real,B3: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A3 @ B3 ) )
= ( minus_minus_real @ B3 @ A3 ) ) ).
% minus_diff_eq
thf(fact_452_minus__diff__commute,axiom,
! [B3: complex,A3: complex] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B3 ) @ A3 )
= ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A3 ) @ B3 ) ) ).
% minus_diff_commute
thf(fact_453_minus__diff__commute,axiom,
! [B3: real,A3: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B3 ) @ A3 )
= ( minus_minus_real @ ( uminus_uminus_real @ A3 ) @ B3 ) ) ).
% minus_diff_commute
thf(fact_454_verit__negate__coefficient_I2_J,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_complex @ A3 @ B3 )
=> ( ord_less_complex @ ( uminus1482373934393186551omplex @ B3 ) @ ( uminus1482373934393186551omplex @ A3 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_455_verit__negate__coefficient_I2_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_456_less__minus__iff,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_complex @ A3 @ ( uminus1482373934393186551omplex @ B3 ) )
= ( ord_less_complex @ B3 @ ( uminus1482373934393186551omplex @ A3 ) ) ) ).
% less_minus_iff
thf(fact_457_less__minus__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ ( uminus_uminus_real @ B3 ) )
= ( ord_less_real @ B3 @ ( uminus_uminus_real @ A3 ) ) ) ).
% less_minus_iff
thf(fact_458_minus__less__iff,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_complex @ ( uminus1482373934393186551omplex @ A3 ) @ B3 )
= ( ord_less_complex @ ( uminus1482373934393186551omplex @ B3 ) @ A3 ) ) ).
% minus_less_iff
thf(fact_459_minus__less__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A3 ) @ B3 )
= ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ A3 ) ) ).
% minus_less_iff
thf(fact_460_neg__less__iff__less,axiom,
! [B3: complex,A3: complex] :
( ( ord_less_complex @ ( uminus1482373934393186551omplex @ B3 ) @ ( uminus1482373934393186551omplex @ A3 ) )
= ( ord_less_complex @ A3 @ B3 ) ) ).
% neg_less_iff_less
thf(fact_461_neg__less__iff__less,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_real @ A3 @ B3 ) ) ).
% neg_less_iff_less
thf(fact_462_unit__vecs__last_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A12: nat] :
( ! [N2: nat] : ( P @ N2 @ zero_zero_nat )
=> ( ! [N2: nat,I: nat] :
( ( P @ N2 @ I )
=> ( P @ N2 @ ( suc @ I ) ) )
=> ( P @ A0 @ A12 ) ) ) ).
% unit_vecs_last.induct
thf(fact_463_numeral__nat_I7_J,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% numeral_nat(7)
thf(fact_464_diff__numeral__special_I9_J,axiom,
( ( minus_minus_complex @ one_one_complex @ one_one_complex )
= zero_zero_complex ) ).
% diff_numeral_special(9)
thf(fact_465_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_466_semiring__norm_I57_J,axiom,
! [A3: complex] :
( ( minus_minus_complex @ zero_zero_complex @ A3 )
= ( uminus1482373934393186551omplex @ A3 ) ) ).
% semiring_norm(57)
thf(fact_467_semiring__norm_I57_J,axiom,
! [A3: real] :
( ( minus_minus_real @ zero_zero_real @ A3 )
= ( uminus_uminus_real @ A3 ) ) ).
% semiring_norm(57)
thf(fact_468_verit__minus__simplify_I3_J,axiom,
! [B3: complex] :
( ( minus_minus_complex @ zero_zero_complex @ B3 )
= ( uminus1482373934393186551omplex @ B3 ) ) ).
% verit_minus_simplify(3)
thf(fact_469_verit__minus__simplify_I3_J,axiom,
! [B3: real] :
( ( minus_minus_real @ zero_zero_real @ B3 )
= ( uminus_uminus_real @ B3 ) ) ).
% verit_minus_simplify(3)
thf(fact_470_zero__compare__simps_I10_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ B3 @ zero_zero_real ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B3 ) ) ) ) ).
% zero_compare_simps(10)
thf(fact_471_zero__compare__simps_I6_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ zero_zero_real @ B3 ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ B3 @ zero_zero_real ) ) ) ) ).
% zero_compare_simps(6)
thf(fact_472_mult__sign__intros_I8_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ).
% mult_sign_intros(8)
thf(fact_473_mult__sign__intros_I7_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ord_less_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(7)
thf(fact_474_mult__sign__intros_I7_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(7)
thf(fact_475_mult__sign__intros_I6_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(6)
thf(fact_476_mult__sign__intros_I6_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(6)
thf(fact_477_mult__sign__intros_I5_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ).
% mult_sign_intros(5)
thf(fact_478_mult__sign__intros_I5_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B3 ) ) ) ) ).
% mult_sign_intros(5)
thf(fact_479_not__square__less__zero,axiom,
! [A3: real] :
~ ( ord_less_real @ ( times_times_real @ A3 @ A3 ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_480_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B3 @ A3 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_481_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B3 @ A3 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_482_zero__less__mult__pos,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ B3 ) ) ) ).
% zero_less_mult_pos
thf(fact_483_zero__less__mult__pos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% zero_less_mult_pos
thf(fact_484_zero__less__mult__pos2,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B3 @ A3 ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ B3 ) ) ) ).
% zero_less_mult_pos2
thf(fact_485_zero__less__mult__pos2,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B3 @ A3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% zero_less_mult_pos2
thf(fact_486_mult__less__cancel__left__neg,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) )
= ( ord_less_real @ B3 @ A3 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_487_mult__less__cancel__left__pos,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) )
= ( ord_less_real @ A3 @ B3 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_488_mult__strict__left__mono__neg,axiom,
! [B3: real,A3: real,C2: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_489_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_490_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B3 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_491_mult__less__cancel__left__disj,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A3 @ B3 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_492_mult__strict__right__mono__neg,axiom,
! [B3: real,A3: real,C2: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_493_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_494_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_495_mult__less__cancel__right__disj,axiom,
! [A3: real,C2: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A3 @ B3 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_496_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_497_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B3 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_498_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_499_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_500_zero__less__one__class_Ozero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_less_one
thf(fact_501_zero__less__one__class_Ozero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_less_one
thf(fact_502_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_503_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_504_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_505_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_506_less__neg__neg,axiom,
! [A3: real] :
( ( ord_less_real @ A3 @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_507_neg__less__pos,axiom,
! [A3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A3 ) @ A3 )
= ( ord_less_real @ zero_zero_real @ A3 ) ) ).
% neg_less_pos
thf(fact_508_neg__0__less__iff__less,axiom,
! [A3: complex] :
( ( ord_less_complex @ zero_zero_complex @ ( uminus1482373934393186551omplex @ A3 ) )
= ( ord_less_complex @ A3 @ zero_zero_complex ) ) ).
% neg_0_less_iff_less
thf(fact_509_neg__0__less__iff__less,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_510_neg__less__0__iff__less,axiom,
! [A3: complex] :
( ( ord_less_complex @ ( uminus1482373934393186551omplex @ A3 ) @ zero_zero_complex )
= ( ord_less_complex @ zero_zero_complex @ A3 ) ) ).
% neg_less_0_iff_less
thf(fact_511_neg__less__0__iff__less,axiom,
! [A3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A3 ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A3 ) ) ).
% neg_less_0_iff_less
thf(fact_512_less__minus__one__simps_I2_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% less_minus_one_simps(2)
thf(fact_513_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(4)
thf(fact_514_four__block__mat__adjoint,axiom,
! [A: mat_complex,Nr1: nat,Nc1: nat,B: mat_complex,Nc2: nat,C: mat_complex,Nr2: nat,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr1 @ Nc1 ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ Nr1 @ Nc2 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ Nr2 @ Nc1 ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ Nr2 @ Nc2 ) )
=> ( ( schur_5982229384592763574omplex @ ( four_b559179830521662709omplex @ A @ B @ C @ D ) )
= ( four_b559179830521662709omplex @ ( schur_5982229384592763574omplex @ A ) @ ( schur_5982229384592763574omplex @ C ) @ ( schur_5982229384592763574omplex @ B ) @ ( schur_5982229384592763574omplex @ D ) ) ) ) ) ) ) ).
% four_block_mat_adjoint
thf(fact_515_four__block__mat__adjoint,axiom,
! [A: mat_a,Nr1: nat,Nc1: nat,B: mat_a,Nc2: nat,C: mat_a,Nr2: nat,D: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ( schur_mat_adjoint_a @ ( four_block_mat_a @ A @ B @ C @ D ) )
= ( four_block_mat_a @ ( schur_mat_adjoint_a @ A ) @ ( schur_mat_adjoint_a @ C ) @ ( schur_mat_adjoint_a @ B ) @ ( schur_mat_adjoint_a @ D ) ) ) ) ) ) ) ).
% four_block_mat_adjoint
thf(fact_516_smult__four__block__mat,axiom,
! [A: mat_a,Nr1: nat,Nc1: nat,B: mat_a,Nc2: nat,C: mat_a,Nr2: nat,D: mat_a,A3: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ( smult_mat_a @ A3 @ ( four_block_mat_a @ A @ B @ C @ D ) )
= ( four_block_mat_a @ ( smult_mat_a @ A3 @ A ) @ ( smult_mat_a @ A3 @ B ) @ ( smult_mat_a @ A3 @ C ) @ ( smult_mat_a @ A3 @ D ) ) ) ) ) ) ) ).
% smult_four_block_mat
thf(fact_517_smult__four__block__mat,axiom,
! [A: mat_complex,Nr1: nat,Nc1: nat,B: mat_complex,Nc2: nat,C: mat_complex,Nr2: nat,D: mat_complex,A3: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr1 @ Nc1 ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ Nr1 @ Nc2 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ Nr2 @ Nc1 ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ Nr2 @ Nc2 ) )
=> ( ( smult_mat_complex @ A3 @ ( four_b559179830521662709omplex @ A @ B @ C @ D ) )
= ( four_b559179830521662709omplex @ ( smult_mat_complex @ A3 @ A ) @ ( smult_mat_complex @ A3 @ B ) @ ( smult_mat_complex @ A3 @ C ) @ ( smult_mat_complex @ A3 @ D ) ) ) ) ) ) ) ).
% smult_four_block_mat
thf(fact_518_pow__mat_Osimps_I2_J,axiom,
! [A: mat_complex,K: nat] :
( ( pow_mat_complex @ A @ ( suc @ K ) )
= ( times_8009071140041733218omplex @ ( pow_mat_complex @ A @ K ) @ A ) ) ).
% pow_mat.simps(2)
thf(fact_519_diff__numeral__special_I12_J,axiom,
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= zero_zero_complex ) ).
% diff_numeral_special(12)
thf(fact_520_diff__numeral__special_I12_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% diff_numeral_special(12)
thf(fact_521_semiring__norm_I133_J,axiom,
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% semiring_norm(133)
thf(fact_522_semiring__norm_I131_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% semiring_norm(131)
thf(fact_523_power__less__power__Suc,axiom,
! [A3: real,N: nat] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ord_less_real @ ( power_power_real @ A3 @ N ) @ ( times_times_real @ A3 @ ( power_power_real @ A3 @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_524_power__less__power__Suc,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A3 )
=> ( ord_less_nat @ ( power_power_nat @ A3 @ N ) @ ( times_times_nat @ A3 @ ( power_power_nat @ A3 @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_525_power__gt1__lemma,axiom,
! [A3: real,N: nat] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ A3 @ ( power_power_real @ A3 @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_526_power__gt1__lemma,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A3 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A3 @ ( power_power_nat @ A3 @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_527_assoc__four__block__mat,axiom,
! [A: mat_complex,B: mat_complex,C: mat_complex] :
( ( four_b559179830521662709omplex @ A @ ( zero_mat_complex @ ( dim_row_complex @ A ) @ ( dim_col_complex @ ( four_b559179830521662709omplex @ B @ ( zero_mat_complex @ ( dim_row_complex @ B ) @ ( dim_col_complex @ C ) ) @ ( zero_mat_complex @ ( dim_row_complex @ C ) @ ( dim_col_complex @ B ) ) @ C ) ) ) @ ( zero_mat_complex @ ( dim_row_complex @ ( four_b559179830521662709omplex @ B @ ( zero_mat_complex @ ( dim_row_complex @ B ) @ ( dim_col_complex @ C ) ) @ ( zero_mat_complex @ ( dim_row_complex @ C ) @ ( dim_col_complex @ B ) ) @ C ) ) @ ( dim_col_complex @ A ) ) @ ( four_b559179830521662709omplex @ B @ ( zero_mat_complex @ ( dim_row_complex @ B ) @ ( dim_col_complex @ C ) ) @ ( zero_mat_complex @ ( dim_row_complex @ C ) @ ( dim_col_complex @ B ) ) @ C ) )
= ( four_b559179830521662709omplex @ ( four_b559179830521662709omplex @ A @ ( zero_mat_complex @ ( dim_row_complex @ A ) @ ( dim_col_complex @ B ) ) @ ( zero_mat_complex @ ( dim_row_complex @ B ) @ ( dim_col_complex @ A ) ) @ B ) @ ( zero_mat_complex @ ( dim_row_complex @ ( four_b559179830521662709omplex @ A @ ( zero_mat_complex @ ( dim_row_complex @ A ) @ ( dim_col_complex @ B ) ) @ ( zero_mat_complex @ ( dim_row_complex @ B ) @ ( dim_col_complex @ A ) ) @ B ) ) @ ( dim_col_complex @ C ) ) @ ( zero_mat_complex @ ( dim_row_complex @ C ) @ ( dim_col_complex @ ( four_b559179830521662709omplex @ A @ ( zero_mat_complex @ ( dim_row_complex @ A ) @ ( dim_col_complex @ B ) ) @ ( zero_mat_complex @ ( dim_row_complex @ B ) @ ( dim_col_complex @ A ) ) @ B ) ) ) @ C ) ) ).
% assoc_four_block_mat
thf(fact_528_pow__four__block__mat,axiom,
! [A: mat_a,N: nat,B: mat_a,M: nat,K: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ M @ M ) )
=> ( ( pow_mat_a @ ( four_block_mat_a @ A @ ( zero_mat_a @ N @ M ) @ ( zero_mat_a @ M @ N ) @ B ) @ K )
= ( four_block_mat_a @ ( pow_mat_a @ A @ K ) @ ( zero_mat_a @ N @ M ) @ ( zero_mat_a @ M @ N ) @ ( pow_mat_a @ B @ K ) ) ) ) ) ).
% pow_four_block_mat
thf(fact_529_pow__four__block__mat,axiom,
! [A: mat_complex,N: nat,B: mat_complex,M: nat,K: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ M @ M ) )
=> ( ( pow_mat_complex @ ( four_b559179830521662709omplex @ A @ ( zero_mat_complex @ N @ M ) @ ( zero_mat_complex @ M @ N ) @ B ) @ K )
= ( four_b559179830521662709omplex @ ( pow_mat_complex @ A @ K ) @ ( zero_mat_complex @ N @ M ) @ ( zero_mat_complex @ M @ N ) @ ( pow_mat_complex @ B @ K ) ) ) ) ) ).
% pow_four_block_mat
thf(fact_530_upper__triangular__four__block,axiom,
! [A: mat_a,N: nat,D: mat_a,M: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ D @ ( carrier_mat_a @ M @ M ) )
=> ( ( upper_triangular_a @ A )
=> ( ( upper_triangular_a @ D )
=> ( upper_triangular_a @ ( four_block_mat_a @ A @ B @ ( zero_mat_a @ M @ N ) @ D ) ) ) ) ) ) ).
% upper_triangular_four_block
thf(fact_531_upper__triangular__four__block,axiom,
! [A: mat_complex,N: nat,D: mat_complex,M: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ M @ M ) )
=> ( ( upper_4850907204721561915omplex @ A )
=> ( ( upper_4850907204721561915omplex @ D )
=> ( upper_4850907204721561915omplex @ ( four_b559179830521662709omplex @ A @ B @ ( zero_mat_complex @ M @ N ) @ D ) ) ) ) ) ) ).
% upper_triangular_four_block
thf(fact_532_det__four__block__mat__upper__right__zero,axiom,
! [A1: mat_a,N: nat,A22: mat_a,M: nat,A32: mat_a,A4: mat_a] :
( ( member_mat_a @ A1 @ ( carrier_mat_a @ N @ N ) )
=> ( ( A22
= ( zero_mat_a @ N @ M ) )
=> ( ( member_mat_a @ A32 @ ( carrier_mat_a @ M @ N ) )
=> ( ( member_mat_a @ A4 @ ( carrier_mat_a @ M @ M ) )
=> ( ( det_a @ ( four_block_mat_a @ A1 @ A22 @ A32 @ A4 ) )
= ( times_times_a @ ( det_a @ A1 ) @ ( det_a @ A4 ) ) ) ) ) ) ) ).
% det_four_block_mat_upper_right_zero
thf(fact_533_det__four__block__mat__upper__right__zero,axiom,
! [A1: mat_complex,N: nat,A22: mat_complex,M: nat,A32: mat_complex,A4: mat_complex] :
( ( member_mat_complex @ A1 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( A22
= ( zero_mat_complex @ N @ M ) )
=> ( ( member_mat_complex @ A32 @ ( carrier_mat_complex @ M @ N ) )
=> ( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ M @ M ) )
=> ( ( det_complex @ ( four_b559179830521662709omplex @ A1 @ A22 @ A32 @ A4 ) )
= ( times_times_complex @ ( det_complex @ A1 ) @ ( det_complex @ A4 ) ) ) ) ) ) ) ).
% det_four_block_mat_upper_right_zero
thf(fact_534_det__four__block__mat__lower__left__zero,axiom,
! [A1: mat_a,N: nat,A22: mat_a,M: nat,A32: mat_a,A4: mat_a] :
( ( member_mat_a @ A1 @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ A22 @ ( carrier_mat_a @ N @ M ) )
=> ( ( A32
= ( zero_mat_a @ M @ N ) )
=> ( ( member_mat_a @ A4 @ ( carrier_mat_a @ M @ M ) )
=> ( ( det_a @ ( four_block_mat_a @ A1 @ A22 @ A32 @ A4 ) )
= ( times_times_a @ ( det_a @ A1 ) @ ( det_a @ A4 ) ) ) ) ) ) ) ).
% det_four_block_mat_lower_left_zero
thf(fact_535_det__four__block__mat__lower__left__zero,axiom,
! [A1: mat_complex,N: nat,A22: mat_complex,M: nat,A32: mat_complex,A4: mat_complex] :
( ( member_mat_complex @ A1 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ A22 @ ( carrier_mat_complex @ N @ M ) )
=> ( ( A32
= ( zero_mat_complex @ M @ N ) )
=> ( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ M @ M ) )
=> ( ( det_complex @ ( four_b559179830521662709omplex @ A1 @ A22 @ A32 @ A4 ) )
= ( times_times_complex @ ( det_complex @ A1 ) @ ( det_complex @ A4 ) ) ) ) ) ) ) ).
% det_four_block_mat_lower_left_zero
thf(fact_536_det__four__block__mat__upper__right__zero__col,axiom,
! [A1: mat_a,N: nat,A22: mat_a,A32: mat_a,A4: mat_a] :
( ( member_mat_a @ A1 @ ( carrier_mat_a @ N @ N ) )
=> ( ( A22
= ( zero_mat_a @ N @ one_one_nat ) )
=> ( ( member_mat_a @ A32 @ ( carrier_mat_a @ one_one_nat @ N ) )
=> ( ( member_mat_a @ A4 @ ( carrier_mat_a @ one_one_nat @ one_one_nat ) )
=> ( ( det_a @ ( four_block_mat_a @ A1 @ A22 @ A32 @ A4 ) )
= ( times_times_a @ ( det_a @ A1 ) @ ( det_a @ A4 ) ) ) ) ) ) ) ).
% det_four_block_mat_upper_right_zero_col
thf(fact_537_det__four__block__mat__upper__right__zero__col,axiom,
! [A1: mat_complex,N: nat,A22: mat_complex,A32: mat_complex,A4: mat_complex] :
( ( member_mat_complex @ A1 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( A22
= ( zero_mat_complex @ N @ one_one_nat ) )
=> ( ( member_mat_complex @ A32 @ ( carrier_mat_complex @ one_one_nat @ N ) )
=> ( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ one_one_nat @ one_one_nat ) )
=> ( ( det_complex @ ( four_b559179830521662709omplex @ A1 @ A22 @ A32 @ A4 ) )
= ( times_times_complex @ ( det_complex @ A1 ) @ ( det_complex @ A4 ) ) ) ) ) ) ) ).
% det_four_block_mat_upper_right_zero_col
thf(fact_538_det__four__block__mat__lower__left__zero__col,axiom,
! [A1: mat_a,A22: mat_a,N: nat,A32: mat_a,A4: mat_a] :
( ( member_mat_a @ A1 @ ( carrier_mat_a @ one_one_nat @ one_one_nat ) )
=> ( ( member_mat_a @ A22 @ ( carrier_mat_a @ one_one_nat @ N ) )
=> ( ( A32
= ( zero_mat_a @ N @ one_one_nat ) )
=> ( ( member_mat_a @ A4 @ ( carrier_mat_a @ N @ N ) )
=> ( ( det_a @ ( four_block_mat_a @ A1 @ A22 @ A32 @ A4 ) )
= ( times_times_a @ ( det_a @ A1 ) @ ( det_a @ A4 ) ) ) ) ) ) ) ).
% det_four_block_mat_lower_left_zero_col
thf(fact_539_det__four__block__mat__lower__left__zero__col,axiom,
! [A1: mat_complex,A22: mat_complex,N: nat,A32: mat_complex,A4: mat_complex] :
( ( member_mat_complex @ A1 @ ( carrier_mat_complex @ one_one_nat @ one_one_nat ) )
=> ( ( member_mat_complex @ A22 @ ( carrier_mat_complex @ one_one_nat @ N ) )
=> ( ( A32
= ( zero_mat_complex @ N @ one_one_nat ) )
=> ( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( det_complex @ ( four_b559179830521662709omplex @ A1 @ A22 @ A32 @ A4 ) )
= ( times_times_complex @ ( det_complex @ A1 ) @ ( det_complex @ A4 ) ) ) ) ) ) ) ).
% det_four_block_mat_lower_left_zero_col
thf(fact_540_power__Suc__less,axiom,
! [A3: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ A3 @ one_one_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ ( power_power_real @ A3 @ N ) ) @ ( power_power_real @ A3 @ N ) ) ) ) ).
% power_Suc_less
thf(fact_541_power__Suc__less,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ A3 @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ ( power_power_nat @ A3 @ N ) ) @ ( power_power_nat @ A3 @ N ) ) ) ) ).
% power_Suc_less
thf(fact_542_permutation__insert__expand,axiom,
( permut138581522262023397omplex
= ( ^ [I2: complex,J: nat,P4: complex > nat,I3: complex] : ( if_nat @ ( ord_less_complex @ I3 @ I2 ) @ ( if_nat @ ( ord_less_nat @ ( P4 @ I3 ) @ J ) @ ( P4 @ I3 ) @ ( suc @ ( P4 @ I3 ) ) ) @ ( if_nat @ ( I3 = I2 ) @ J @ ( if_nat @ ( ord_less_nat @ ( P4 @ ( minus_minus_complex @ I3 @ one_one_complex ) ) @ J ) @ ( P4 @ ( minus_minus_complex @ I3 @ one_one_complex ) ) @ ( suc @ ( P4 @ ( minus_minus_complex @ I3 @ one_one_complex ) ) ) ) ) ) ) ) ).
% permutation_insert_expand
thf(fact_543_permutation__insert__expand,axiom,
( permut4060954620988167523t_real
= ( ^ [I2: real,J: nat,P4: real > nat,I3: real] : ( if_nat @ ( ord_less_real @ I3 @ I2 ) @ ( if_nat @ ( ord_less_nat @ ( P4 @ I3 ) @ J ) @ ( P4 @ I3 ) @ ( suc @ ( P4 @ I3 ) ) ) @ ( if_nat @ ( I3 = I2 ) @ J @ ( if_nat @ ( ord_less_nat @ ( P4 @ ( minus_minus_real @ I3 @ one_one_real ) ) @ J ) @ ( P4 @ ( minus_minus_real @ I3 @ one_one_real ) ) @ ( suc @ ( P4 @ ( minus_minus_real @ I3 @ one_one_real ) ) ) ) ) ) ) ) ).
% permutation_insert_expand
thf(fact_544_permutation__insert__expand,axiom,
( permut3695043542826343943rt_nat
= ( ^ [I2: nat,J: nat,P4: nat > nat,I3: nat] : ( if_nat @ ( ord_less_nat @ I3 @ I2 ) @ ( if_nat @ ( ord_less_nat @ ( P4 @ I3 ) @ J ) @ ( P4 @ I3 ) @ ( suc @ ( P4 @ I3 ) ) ) @ ( if_nat @ ( I3 = I2 ) @ J @ ( if_nat @ ( ord_less_nat @ ( P4 @ ( minus_minus_nat @ I3 @ one_one_nat ) ) @ J ) @ ( P4 @ ( minus_minus_nat @ I3 @ one_one_nat ) ) @ ( suc @ ( P4 @ ( minus_minus_nat @ I3 @ one_one_nat ) ) ) ) ) ) ) ) ).
% permutation_insert_expand
thf(fact_545_swap__row__to__front__det,axiom,
! [A: mat_a,N: nat,I4: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( ord_less_nat @ I4 @ N )
=> ( ( det_a @ ( column973622294476583016ront_a @ A @ I4 ) )
= ( times_times_a @ ( power_power_a @ ( uminus_uminus_a @ one_one_a ) @ I4 ) @ ( det_a @ A ) ) ) ) ) ).
% swap_row_to_front_det
thf(fact_546_swap__row__to__front__det,axiom,
! [A: mat_complex,N: nat,I4: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ I4 @ N )
=> ( ( det_complex @ ( column4342047067757093060omplex @ A @ I4 ) )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I4 ) @ ( det_complex @ A ) ) ) ) ) ).
% swap_row_to_front_det
thf(fact_547_swap__row__to__front__det,axiom,
! [A: mat_real,N: nat,I4: nat] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( ord_less_nat @ I4 @ N )
=> ( ( det_real @ ( column3686191904915150786t_real @ A @ I4 ) )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( det_real @ A ) ) ) ) ) ).
% swap_row_to_front_det
thf(fact_548_swap__col__to__front__det,axiom,
! [A: mat_a,N: nat,I4: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( ord_less_nat @ I4 @ N )
=> ( ( det_a @ ( column2924081423933032910ront_a @ A @ I4 ) )
= ( times_times_a @ ( power_power_a @ ( uminus_uminus_a @ one_one_a ) @ I4 ) @ ( det_a @ A ) ) ) ) ) ).
% swap_col_to_front_det
thf(fact_549_swap__col__to__front__det,axiom,
! [A: mat_complex,N: nat,I4: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ I4 @ N )
=> ( ( det_complex @ ( column7264791363093833182omplex @ A @ I4 ) )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I4 ) @ ( det_complex @ A ) ) ) ) ) ).
% swap_col_to_front_det
thf(fact_550_swap__col__to__front__det,axiom,
! [A: mat_real,N: nat,I4: nat] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( ord_less_nat @ I4 @ N )
=> ( ( det_real @ ( column6512727542960595932t_real @ A @ I4 ) )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( det_real @ A ) ) ) ) ) ).
% swap_col_to_front_det
thf(fact_551_append__rows__def,axiom,
( append_rows_complex
= ( ^ [A2: mat_complex,B2: mat_complex] : ( four_b559179830521662709omplex @ A2 @ ( zero_mat_complex @ ( dim_row_complex @ A2 ) @ zero_zero_nat ) @ B2 @ ( zero_mat_complex @ ( dim_row_complex @ B2 ) @ zero_zero_nat ) ) ) ) ).
% append_rows_def
thf(fact_552_poly__cancel__eq__conv,axiom,
! [X: complex,A3: complex,Y: complex,B3: complex] :
( ( X = zero_zero_complex )
=> ( ( A3 != zero_zero_complex )
=> ( ( Y = zero_zero_complex )
= ( ( minus_minus_complex @ ( times_times_complex @ A3 @ Y ) @ ( times_times_complex @ B3 @ X ) )
= zero_zero_complex ) ) ) ) ).
% poly_cancel_eq_conv
thf(fact_553_poly__cancel__eq__conv,axiom,
! [X: real,A3: real,Y: real,B3: real] :
( ( X = zero_zero_real )
=> ( ( A3 != zero_zero_real )
=> ( ( Y = zero_zero_real )
= ( ( minus_minus_real @ ( times_times_real @ A3 @ Y ) @ ( times_times_real @ B3 @ X ) )
= zero_zero_real ) ) ) ) ).
% poly_cancel_eq_conv
thf(fact_554_minus__carrier__mat_H,axiom,
! [A: mat_a,Nr: nat,Nc: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_mat_a @ ( minus_minus_mat_a @ A @ B ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ) ).
% minus_carrier_mat'
thf(fact_555_minus__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 @ ( minus_2412168080157227406omplex @ A @ B ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% minus_carrier_mat'
thf(fact_556_minus__carrier__mat,axiom,
! [B: mat_a,Nr: nat,Nc: nat,A: mat_a] :
( ( member_mat_a @ B @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_mat_a @ ( minus_minus_mat_a @ A @ B ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% minus_carrier_mat
thf(fact_557_minus__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 @ ( minus_2412168080157227406omplex @ A @ B ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% minus_carrier_mat
thf(fact_558_index__minus__mat_I2_J,axiom,
! [A: mat_complex,B: mat_complex] :
( ( dim_row_complex @ ( minus_2412168080157227406omplex @ A @ B ) )
= ( dim_row_complex @ B ) ) ).
% index_minus_mat(2)
thf(fact_559_index__minus__mat_I3_J,axiom,
! [A: mat_complex,B: mat_complex] :
( ( dim_col_complex @ ( minus_2412168080157227406omplex @ A @ B ) )
= ( dim_col_complex @ B ) ) ).
% index_minus_mat(3)
thf(fact_560_mult__minus__distrib__mat,axiom,
! [A: mat_a,Nr: nat,N: nat,B: mat_a,Nc: nat,C: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( member_mat_a @ C @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ A @ ( minus_minus_mat_a @ B @ C ) )
= ( minus_minus_mat_a @ ( times_times_mat_a @ A @ B ) @ ( times_times_mat_a @ A @ C ) ) ) ) ) ) ).
% mult_minus_distrib_mat
thf(fact_561_mult__minus__distrib__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,B: mat_complex,Nc: nat,C: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ A @ ( minus_2412168080157227406omplex @ B @ C ) )
= ( minus_2412168080157227406omplex @ ( times_8009071140041733218omplex @ A @ B ) @ ( times_8009071140041733218omplex @ A @ C ) ) ) ) ) ) ).
% mult_minus_distrib_mat
thf(fact_562_minus__mult__distrib__mat,axiom,
! [A: mat_a,Nr: nat,N: nat,B: mat_a,C: 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 @ C @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ ( minus_minus_mat_a @ A @ B ) @ C )
= ( minus_minus_mat_a @ ( times_times_mat_a @ A @ C ) @ ( times_times_mat_a @ B @ C ) ) ) ) ) ) ).
% minus_mult_distrib_mat
thf(fact_563_minus__mult__distrib__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,B: mat_complex,C: 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 @ C @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( minus_2412168080157227406omplex @ A @ B ) @ C )
= ( minus_2412168080157227406omplex @ ( times_8009071140041733218omplex @ A @ C ) @ ( times_8009071140041733218omplex @ B @ C ) ) ) ) ) ) ).
% minus_mult_distrib_mat
thf(fact_564_minus__r__inv__mat,axiom,
! [A: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( minus_minus_mat_a @ A @ A )
= ( zero_mat_a @ Nr @ Nc ) ) ) ).
% minus_r_inv_mat
thf(fact_565_minus__r__inv__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( minus_2412168080157227406omplex @ A @ A )
= ( zero_mat_complex @ Nr @ Nc ) ) ) ).
% minus_r_inv_mat
thf(fact_566_adjoint__minus,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 @ ( minus_2412168080157227406omplex @ A @ B ) )
= ( minus_2412168080157227406omplex @ ( schur_5982229384592763574omplex @ A ) @ ( schur_5982229384592763574omplex @ B ) ) ) ) ) ).
% adjoint_minus
thf(fact_567_adjoint__minus,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ M ) )
=> ( ( schur_mat_adjoint_a @ ( minus_minus_mat_a @ A @ B ) )
= ( minus_minus_mat_a @ ( schur_mat_adjoint_a @ A ) @ ( schur_mat_adjoint_a @ B ) ) ) ) ) ).
% adjoint_minus
thf(fact_568_smult__distrib__left__minus__mat,axiom,
! [A: mat_a,N: nat,B: mat_a,C2: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( smult_mat_a @ C2 @ ( minus_minus_mat_a @ B @ A ) )
= ( minus_minus_mat_a @ ( smult_mat_a @ C2 @ B ) @ ( smult_mat_a @ C2 @ A ) ) ) ) ) ).
% smult_distrib_left_minus_mat
thf(fact_569_smult__distrib__left__minus__mat,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C2: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( smult_mat_complex @ C2 @ ( minus_2412168080157227406omplex @ B @ A ) )
= ( minus_2412168080157227406omplex @ ( smult_mat_complex @ C2 @ B ) @ ( smult_mat_complex @ C2 @ A ) ) ) ) ) ).
% smult_distrib_left_minus_mat
thf(fact_570_hermitian__minus,axiom,
! [A: mat_a,N: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_hermitian_a @ A )
=> ( ( complex_hermitian_a @ B )
=> ( complex_hermitian_a @ ( minus_minus_mat_a @ A @ B ) ) ) ) ) ) ).
% hermitian_minus
thf(fact_571_hermitian__minus,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 @ ( minus_2412168080157227406omplex @ A @ B ) ) ) ) ) ) ).
% hermitian_minus
thf(fact_572_right__minus__zero__mat,axiom,
! [A: mat_complex] :
( ( minus_2412168080157227406omplex @ A @ ( zero_mat_complex @ ( dim_row_complex @ A ) @ ( dim_col_complex @ A ) ) )
= A ) ).
% right_minus_zero_mat
thf(fact_573_det__four__block__mat,axiom,
! [A: mat_a,N: nat,B: mat_a,C: mat_a,D: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ C @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ D @ ( carrier_mat_a @ N @ N ) )
=> ( ( ( times_times_mat_a @ C @ D )
= ( times_times_mat_a @ D @ C ) )
=> ( ( det_a @ ( four_block_mat_a @ A @ B @ C @ D ) )
= ( det_a @ ( minus_minus_mat_a @ ( times_times_mat_a @ A @ D ) @ ( times_times_mat_a @ B @ C ) ) ) ) ) ) ) ) ) ).
% det_four_block_mat
thf(fact_574_det__four__block__mat,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ( times_8009071140041733218omplex @ C @ D )
= ( times_8009071140041733218omplex @ D @ C ) )
=> ( ( det_complex @ ( four_b559179830521662709omplex @ A @ B @ C @ D ) )
= ( det_complex @ ( minus_2412168080157227406omplex @ ( times_8009071140041733218omplex @ A @ D ) @ ( times_8009071140041733218omplex @ B @ C ) ) ) ) ) ) ) ) ) ).
% det_four_block_mat
thf(fact_575_swap__col__to__front__four__block,axiom,
! [A1: mat_a,N1: nat,M: nat,A22: mat_a,A32: mat_a,N22: nat,A4: mat_a] :
( ( member_mat_a @ A1 @ ( carrier_mat_a @ N1 @ M ) )
=> ( ( member_mat_a @ A22 @ ( carrier_mat_a @ N1 @ one_one_nat ) )
=> ( ( member_mat_a @ A32 @ ( carrier_mat_a @ N22 @ M ) )
=> ( ( member_mat_a @ A4 @ ( carrier_mat_a @ N22 @ one_one_nat ) )
=> ( ( column2924081423933032910ront_a @ ( four_block_mat_a @ A1 @ A22 @ A32 @ A4 ) @ M )
= ( four_block_mat_a @ A22 @ A1 @ A4 @ A32 ) ) ) ) ) ) ).
% swap_col_to_front_four_block
thf(fact_576_swap__col__to__front__four__block,axiom,
! [A1: mat_complex,N1: nat,M: nat,A22: mat_complex,A32: mat_complex,N22: nat,A4: mat_complex] :
( ( member_mat_complex @ A1 @ ( carrier_mat_complex @ N1 @ M ) )
=> ( ( member_mat_complex @ A22 @ ( carrier_mat_complex @ N1 @ one_one_nat ) )
=> ( ( member_mat_complex @ A32 @ ( carrier_mat_complex @ N22 @ M ) )
=> ( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N22 @ one_one_nat ) )
=> ( ( column7264791363093833182omplex @ ( four_b559179830521662709omplex @ A1 @ A22 @ A32 @ A4 ) @ M )
= ( four_b559179830521662709omplex @ A22 @ A1 @ A4 @ A32 ) ) ) ) ) ) ).
% swap_col_to_front_four_block
thf(fact_577_swap__row__to__front__four__block,axiom,
! [A1: mat_a,N: nat,M1: nat,A22: mat_a,M22: nat,A32: mat_a,A4: mat_a] :
( ( member_mat_a @ A1 @ ( carrier_mat_a @ N @ M1 ) )
=> ( ( member_mat_a @ A22 @ ( carrier_mat_a @ N @ M22 ) )
=> ( ( member_mat_a @ A32 @ ( carrier_mat_a @ one_one_nat @ M1 ) )
=> ( ( member_mat_a @ A4 @ ( carrier_mat_a @ one_one_nat @ M22 ) )
=> ( ( column973622294476583016ront_a @ ( four_block_mat_a @ A1 @ A22 @ A32 @ A4 ) @ N )
= ( four_block_mat_a @ A32 @ A4 @ A1 @ A22 ) ) ) ) ) ) ).
% swap_row_to_front_four_block
thf(fact_578_swap__row__to__front__four__block,axiom,
! [A1: mat_complex,N: nat,M1: nat,A22: mat_complex,M22: nat,A32: mat_complex,A4: mat_complex] :
( ( member_mat_complex @ A1 @ ( carrier_mat_complex @ N @ M1 ) )
=> ( ( member_mat_complex @ A22 @ ( carrier_mat_complex @ N @ M22 ) )
=> ( ( member_mat_complex @ A32 @ ( carrier_mat_complex @ one_one_nat @ M1 ) )
=> ( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ one_one_nat @ M22 ) )
=> ( ( column4342047067757093060omplex @ ( four_b559179830521662709omplex @ A1 @ A22 @ A32 @ A4 ) @ N )
= ( four_b559179830521662709omplex @ A32 @ A4 @ A1 @ A22 ) ) ) ) ) ) ).
% swap_row_to_front_four_block
thf(fact_579_mult__less__iff1,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_580_delete__index__def,axiom,
( delete_index
= ( ^ [I2: nat,I3: nat] : ( if_nat @ ( ord_less_nat @ I3 @ I2 ) @ I3 @ ( minus_minus_nat @ I3 @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% delete_index_def
thf(fact_581_permutation__delete__expand,axiom,
( permutation_delete
= ( ^ [P4: nat > nat,I2: nat,J: nat] : ( if_nat @ ( ord_less_nat @ ( P4 @ ( if_nat @ ( ord_less_nat @ J @ I2 ) @ J @ ( suc @ J ) ) ) @ ( P4 @ I2 ) ) @ ( P4 @ ( if_nat @ ( ord_less_nat @ J @ I2 ) @ J @ ( suc @ J ) ) ) @ ( minus_minus_nat @ ( P4 @ ( if_nat @ ( ord_less_nat @ J @ I2 ) @ J @ ( suc @ J ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% permutation_delete_expand
thf(fact_582_density__collapse__carrier,axiom,
! [R: mat_complex,P: mat_complex,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( dim_row_complex @ R ) )
=> ( ( member_mat_complex @ P @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ R @ ( carrier_mat_complex @ N @ N ) )
=> ( member_mat_complex @ ( projec3470689467825365843llapse @ R @ P ) @ ( carrier_mat_complex @ N @ N ) ) ) ) ) ).
% density_collapse_carrier
thf(fact_583_det__swapcols,axiom,
! [K: nat,N: nat,L: nat,A: mat_a] :
( ( ord_less_nat @ K @ N )
=> ( ( ord_less_nat @ L @ N )
=> ( ( K != L )
=> ( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( det_a @ ( column2528828918332591333cols_a @ K @ L @ A ) )
= ( uminus_uminus_a @ ( det_a @ A ) ) ) ) ) ) ) ).
% det_swapcols
thf(fact_584_det__swapcols,axiom,
! [K: nat,N: nat,L: nat,A: mat_complex] :
( ( ord_less_nat @ K @ N )
=> ( ( ord_less_nat @ L @ N )
=> ( ( K != L )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( det_complex @ ( column4357519492343924999omplex @ K @ L @ A ) )
= ( uminus1482373934393186551omplex @ ( det_complex @ A ) ) ) ) ) ) ) ).
% det_swapcols
thf(fact_585_det__swapcols,axiom,
! [K: nat,N: nat,L: nat,A: mat_real] :
( ( ord_less_nat @ K @ N )
=> ( ( ord_less_nat @ L @ N )
=> ( ( K != L )
=> ( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( det_real @ ( column2501654400089035909s_real @ K @ L @ A ) )
= ( uminus_uminus_real @ ( det_real @ A ) ) ) ) ) ) ) ).
% det_swapcols
thf(fact_586_uminus__mat,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( uminus467866341702955550omplex @ A )
= ( smult_mat_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ A ) ) ) ).
% uminus_mat
thf(fact_587_projector__collapse__trace,axiom,
! [P: mat_complex,N: nat,R: mat_complex] :
( ( linear5633924348262549461omplex @ P )
=> ( ( member_mat_complex @ P @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ R @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P @ R ) @ P ) )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R @ P ) ) ) ) ) ) ).
% projector_collapse_trace
thf(fact_588_mat__assoc__test_I9_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ ( minus_2412168080157227406omplex @ B @ C ) ) @ D )
= ( minus_2412168080157227406omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B ) @ D ) @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ C ) @ D ) ) ) ) ) ) ) ).
% mat_assoc_test(9)
thf(fact_589_density__collapse__operator,axiom,
! [P: mat_complex,R: mat_complex,N: nat] :
( ( linear5633924348262549461omplex @ P )
=> ( ( comple5220265106149225959erator @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_row_complex @ R ) )
=> ( ( member_mat_complex @ P @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ R @ ( carrier_mat_complex @ N @ N ) )
=> ( comple5220265106149225959erator @ ( projec3470689467825365843llapse @ R @ P ) ) ) ) ) ) ) ).
% density_collapse_operator
thf(fact_590_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_591_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_592_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_593_unitary__density,axiom,
! [R: mat_complex,U: mat_complex,N: nat] :
( ( comple5220265106149225959erator @ R )
=> ( ( comple6660659447773130958omplex @ U )
=> ( ( member_mat_complex @ R @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( comple5220265106149225959erator @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ U @ R ) @ ( schur_5982229384592763574omplex @ U ) ) ) ) ) ) ) ).
% unitary_density
thf(fact_594_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_595_minus__diff__minus,axiom,
! [A3: complex,B3: complex] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A3 ) @ ( uminus1482373934393186551omplex @ B3 ) )
= ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A3 @ B3 ) ) ) ).
% minus_diff_minus
thf(fact_596_minus__diff__minus,axiom,
! [A3: real,B3: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) )
= ( uminus_uminus_real @ ( minus_minus_real @ A3 @ B3 ) ) ) ).
% minus_diff_minus
thf(fact_597_inf__period_I1_J,axiom,
! [P: real > $o,D: real,Q: real > $o] :
( ! [X3: real,K2: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D ) ) ) )
=> ( ! [X3: real,K2: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D ) ) ) )
=> ! [X4: real,K3: real] :
( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K3 @ D ) ) )
& ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K3 @ D ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_598_inf__period_I2_J,axiom,
! [P: real > $o,D: real,Q: real > $o] :
( ! [X3: real,K2: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D ) ) ) )
=> ( ! [X3: real,K2: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D ) ) ) )
=> ! [X4: real,K3: real] :
( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K3 @ D ) ) )
| ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K3 @ D ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_599_rank__1__proj__adjoint,axiom,
! [V3: vec_complex] :
( ( schur_5982229384592763574omplex @ ( linear1949544614684794075omplex @ V3 ) )
= ( linear1949544614684794075omplex @ V3 ) ) ).
% rank_1_proj_adjoint
thf(fact_600_addrow__mat__inv,axiom,
! [K: nat,N: nat,L: nat,A3: complex] :
( ( ord_less_nat @ K @ N )
=> ( ( ord_less_nat @ L @ N )
=> ( ( K != L )
=> ( ( times_8009071140041733218omplex @ ( gauss_947198734564870628omplex @ N @ A3 @ K @ L ) @ ( gauss_947198734564870628omplex @ N @ ( uminus1482373934393186551omplex @ A3 ) @ K @ L ) )
= ( one_mat_complex @ N ) ) ) ) ) ).
% addrow_mat_inv
thf(fact_601_addrow__mat__inv,axiom,
! [K: nat,N: nat,L: nat,A3: real] :
( ( ord_less_nat @ K @ N )
=> ( ( ord_less_nat @ L @ N )
=> ( ( K != L )
=> ( ( times_times_mat_real @ ( gauss_2378325378421436642t_real @ N @ A3 @ K @ L ) @ ( gauss_2378325378421436642t_real @ N @ ( uminus_uminus_real @ A3 ) @ K @ L ) )
= ( one_mat_real @ N ) ) ) ) ) ).
% addrow_mat_inv
thf(fact_602_swapcols__mat,axiom,
! [A: mat_a,Nr: nat,N: nat,K: nat,L: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( ord_less_nat @ K @ N )
=> ( ( ord_less_nat @ L @ N )
=> ( ( column2528828918332591333cols_a @ K @ L @ A )
= ( times_times_mat_a @ A @ ( gauss_110929411057020027_mat_a @ N @ K @ L ) ) ) ) ) ) ).
% swapcols_mat
thf(fact_603_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_604_mat__delete__dim_I2_J,axiom,
! [A: mat_complex,I5: nat,J2: nat] :
( ( dim_col_complex @ ( mat_delete_complex @ A @ I5 @ J2 ) )
= ( minus_minus_nat @ ( dim_col_complex @ A ) @ one_one_nat ) ) ).
% mat_delete_dim(2)
thf(fact_605_mat__delete__dim_I1_J,axiom,
! [A: mat_complex,I5: nat,J2: nat] :
( ( dim_row_complex @ ( mat_delete_complex @ A @ I5 @ J2 ) )
= ( minus_minus_nat @ ( dim_row_complex @ A ) @ one_one_nat ) ) ).
% mat_delete_dim(1)
thf(fact_606_rank__1__proj__hermitian,axiom,
! [V3: vec_complex] : ( comple8306762464034002205omplex @ ( linear1949544614684794075omplex @ V3 ) ) ).
% rank_1_proj_hermitian
thf(fact_607_swaprows__mat__carrier,axiom,
! [N: nat,K: nat,L: nat] : ( member_mat_a @ ( gauss_110929411057020027_mat_a @ N @ K @ L ) @ ( carrier_mat_a @ N @ N ) ) ).
% swaprows_mat_carrier
thf(fact_608_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_609_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_610_index__mat__swaprows__mat_I3_J,axiom,
! [N: nat,K: nat,L: nat] :
( ( dim_col_complex @ ( gauss_8970452565587180529omplex @ N @ K @ L ) )
= N ) ).
% index_mat_swaprows_mat(3)
thf(fact_611_addrow__mat__carrier,axiom,
! [N: nat,A3: a,K: nat,L: nat] : ( member_mat_a @ ( gauss_8159914756388622152_mat_a @ N @ A3 @ K @ L ) @ ( carrier_mat_a @ N @ N ) ) ).
% addrow_mat_carrier
thf(fact_612_addrow__mat__carrier,axiom,
! [N: nat,A3: complex,K: nat,L: nat] : ( member_mat_complex @ ( gauss_947198734564870628omplex @ N @ A3 @ K @ L ) @ ( carrier_mat_complex @ N @ N ) ) ).
% addrow_mat_carrier
thf(fact_613_index__mat__addrow__mat_I2_J,axiom,
! [N: nat,A3: complex,K: nat,L: nat] :
( ( dim_row_complex @ ( gauss_947198734564870628omplex @ N @ A3 @ K @ L ) )
= N ) ).
% index_mat_addrow_mat(2)
thf(fact_614_index__mat__addrow__mat_I3_J,axiom,
! [N: nat,A3: complex,K: nat,L: nat] :
( ( dim_col_complex @ ( gauss_947198734564870628omplex @ N @ A3 @ K @ L ) )
= N ) ).
% index_mat_addrow_mat(3)
thf(fact_615_det__addrow__mat,axiom,
! [K: nat,L: nat,N: nat,A3: complex] :
( ( K != L )
=> ( ( det_complex @ ( gauss_947198734564870628omplex @ N @ A3 @ K @ L ) )
= one_one_complex ) ) ).
% det_addrow_mat
thf(fact_616_det__addrow__mat,axiom,
! [K: nat,L: nat,N: nat,A3: real] :
( ( K != L )
=> ( ( det_real @ ( gauss_2378325378421436642t_real @ N @ A3 @ K @ L ) )
= one_one_real ) ) ).
% det_addrow_mat
thf(fact_617_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_618_det__swaprows__mat,axiom,
! [K: nat,N: nat,L: nat] :
( ( ord_less_nat @ K @ N )
=> ( ( ord_less_nat @ L @ N )
=> ( ( K != L )
=> ( ( det_complex @ ( gauss_8970452565587180529omplex @ N @ K @ L ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ) ).
% det_swaprows_mat
thf(fact_619_det__swaprows__mat,axiom,
! [K: nat,N: nat,L: nat] :
( ( ord_less_nat @ K @ N )
=> ( ( ord_less_nat @ L @ N )
=> ( ( K != L )
=> ( ( det_real @ ( gauss_1271566072679876207t_real @ N @ K @ L ) )
= ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% det_swaprows_mat
thf(fact_620_mat__delete__carrier,axiom,
! [A: mat_a,M: nat,N: nat,I5: nat,J2: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ M @ N ) )
=> ( member_mat_a @ ( mat_delete_a @ A @ I5 @ J2 ) @ ( carrier_mat_a @ ( minus_minus_nat @ M @ one_one_nat ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% mat_delete_carrier
thf(fact_621_mat__delete__carrier,axiom,
! [A: mat_complex,M: nat,N: nat,I5: nat,J2: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ M @ N ) )
=> ( member_mat_complex @ ( mat_delete_complex @ A @ I5 @ J2 ) @ ( carrier_mat_complex @ ( minus_minus_nat @ M @ one_one_nat ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% mat_delete_carrier
thf(fact_622_addcol__mat,axiom,
! [A: mat_a,Nr: nat,N: nat,K: nat,A3: a,L: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( ord_less_nat @ K @ N )
=> ( ( column_mat_addcol_a @ A3 @ L @ K @ A )
= ( times_times_mat_a @ A @ ( gauss_8159914756388622152_mat_a @ N @ A3 @ K @ L ) ) ) ) ) ).
% addcol_mat
thf(fact_623_addcol__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,K: nat,A3: complex,L: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( ord_less_nat @ K @ N )
=> ( ( column896436094548437152omplex @ A3 @ L @ K @ A )
= ( times_8009071140041733218omplex @ A @ ( gauss_947198734564870628omplex @ N @ A3 @ K @ L ) ) ) ) ) ).
% addcol_mat
thf(fact_624_mat__assoc__test_I8_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( minus_2412168080157227406omplex @ A @ B )
= ( plus_p8323303612493835998omplex @ A @ ( smult_mat_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ B ) ) ) ) ) ) ) ).
% mat_assoc_test(8)
thf(fact_625_det__addcol,axiom,
! [L: nat,N: nat,K: nat,A: mat_a,A3: a] :
( ( ord_less_nat @ L @ N )
=> ( ( K != L )
=> ( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( det_a @ ( column_mat_addcol_a @ A3 @ K @ L @ A ) )
= ( det_a @ A ) ) ) ) ) ).
% det_addcol
thf(fact_626_det__addcol,axiom,
! [L: nat,N: nat,K: nat,A: mat_complex,A3: complex] :
( ( ord_less_nat @ L @ N )
=> ( ( K != L )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( det_complex @ ( column896436094548437152omplex @ A3 @ K @ L @ A ) )
= ( det_complex @ A ) ) ) ) ) ).
% det_addcol
thf(fact_627_trace__add__linear,axiom,
! [A: mat_a,N: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_trace_a @ ( plus_plus_mat_a @ A @ B ) )
= ( plus_plus_a @ ( complex_trace_a @ A ) @ ( complex_trace_a @ B ) ) ) ) ) ).
% trace_add_linear
thf(fact_628_trace__add__linear,axiom,
! [A: mat_real,N: nat,B: mat_real] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( member_mat_real @ B @ ( carrier_mat_real @ N @ N ) )
=> ( ( complex_trace_real @ ( plus_plus_mat_real @ A @ B ) )
= ( plus_plus_real @ ( complex_trace_real @ A ) @ ( complex_trace_real @ B ) ) ) ) ) ).
% trace_add_linear
thf(fact_629_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_630_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_631_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_632_add__smult__distrib__right__mat,axiom,
! [A: mat_real,Nr: nat,Nc: nat,K: real,L: real] :
( ( member_mat_real @ A @ ( carrier_mat_real @ Nr @ Nc ) )
=> ( ( smult_mat_real @ ( plus_plus_real @ K @ L ) @ A )
= ( plus_plus_mat_real @ ( smult_mat_real @ K @ A ) @ ( smult_mat_real @ L @ A ) ) ) ) ).
% add_smult_distrib_right_mat
thf(fact_633_mult__hom_Ohom__add,axiom,
! [C2: complex,X: complex,Y: complex] :
( ( times_times_complex @ C2 @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_complex @ ( times_times_complex @ C2 @ X ) @ ( times_times_complex @ C2 @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_634_mult__hom_Ohom__add,axiom,
! [C2: real,X: real,Y: real] :
( ( times_times_real @ C2 @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ C2 @ X ) @ ( times_times_real @ C2 @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_635_vector__space__over__itself_Oscale__right__distrib,axiom,
! [A3: complex,X: complex,Y: complex] :
( ( times_times_complex @ A3 @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ X ) @ ( times_times_complex @ A3 @ Y ) ) ) ).
% vector_space_over_itself.scale_right_distrib
thf(fact_636_vector__space__over__itself_Oscale__right__distrib,axiom,
! [A3: real,X: real,Y: real] :
( ( times_times_real @ A3 @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ A3 @ X ) @ ( times_times_real @ A3 @ Y ) ) ) ).
% vector_space_over_itself.scale_right_distrib
thf(fact_637_vector__space__over__itself_Oscale__left__distrib,axiom,
! [A3: complex,B3: complex,X: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A3 @ B3 ) @ X )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ X ) @ ( times_times_complex @ B3 @ X ) ) ) ).
% vector_space_over_itself.scale_left_distrib
thf(fact_638_vector__space__over__itself_Oscale__left__distrib,axiom,
! [A3: real,B3: real,X: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ X )
= ( plus_plus_real @ ( times_times_real @ A3 @ X ) @ ( times_times_real @ B3 @ X ) ) ) ).
% vector_space_over_itself.scale_left_distrib
thf(fact_639_combine__common__factor,axiom,
! [A3: complex,E: complex,B3: complex,C2: complex] :
( ( plus_plus_complex @ ( times_times_complex @ A3 @ E ) @ ( plus_plus_complex @ ( times_times_complex @ B3 @ E ) @ C2 ) )
= ( plus_plus_complex @ ( times_times_complex @ ( plus_plus_complex @ A3 @ B3 ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_640_combine__common__factor,axiom,
! [A3: real,E: real,B3: real,C2: real] :
( ( plus_plus_real @ ( times_times_real @ A3 @ E ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E ) @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_641_distrib__right,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A3 @ B3 ) @ C2 )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B3 @ C2 ) ) ) ).
% distrib_right
thf(fact_642_distrib__right,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ).
% distrib_right
thf(fact_643_distrib__left,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( times_times_complex @ A3 @ ( plus_plus_complex @ B3 @ C2 ) )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ B3 ) @ ( times_times_complex @ A3 @ C2 ) ) ) ).
% distrib_left
thf(fact_644_distrib__left,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ A3 @ ( plus_plus_real @ B3 @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A3 @ B3 ) @ ( times_times_real @ A3 @ C2 ) ) ) ).
% distrib_left
thf(fact_645_comm__semiring__class_Odistrib,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A3 @ B3 ) @ C2 )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B3 @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_646_comm__semiring__class_Odistrib,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_647_ring__class_Oring__distribs_I1_J,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( times_times_complex @ A3 @ ( plus_plus_complex @ B3 @ C2 ) )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ B3 ) @ ( times_times_complex @ A3 @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_648_ring__class_Oring__distribs_I1_J,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ A3 @ ( plus_plus_real @ B3 @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A3 @ B3 ) @ ( times_times_real @ A3 @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_649_ring__class_Oring__distribs_I2_J,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A3 @ B3 ) @ C2 )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B3 @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_650_ring__class_Oring__distribs_I2_J,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_651_add_Oinverse__distrib__swap,axiom,
! [A3: complex,B3: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A3 @ B3 ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B3 ) @ ( uminus1482373934393186551omplex @ A3 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_652_add_Oinverse__distrib__swap,axiom,
! [A3: real,B3: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A3 @ B3 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_653_minus__add__distrib,axiom,
! [A3: complex,B3: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A3 @ B3 ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A3 ) @ ( uminus1482373934393186551omplex @ B3 ) ) ) ).
% minus_add_distrib
thf(fact_654_minus__add__distrib,axiom,
! [A3: real,B3: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A3 @ B3 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) ) ) ).
% minus_add_distrib
thf(fact_655_minus__add__cancel,axiom,
! [A3: complex,B3: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A3 ) @ ( plus_plus_complex @ A3 @ B3 ) )
= B3 ) ).
% minus_add_cancel
thf(fact_656_minus__add__cancel,axiom,
! [A3: real,B3: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ ( plus_plus_real @ A3 @ B3 ) )
= B3 ) ).
% minus_add_cancel
thf(fact_657_add__minus__cancel,axiom,
! [A3: complex,B3: complex] :
( ( plus_plus_complex @ A3 @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A3 ) @ B3 ) )
= B3 ) ).
% add_minus_cancel
thf(fact_658_add__minus__cancel,axiom,
! [A3: real,B3: real] :
( ( plus_plus_real @ A3 @ ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ B3 ) )
= B3 ) ).
% add_minus_cancel
thf(fact_659_group__cancel_Oneg1,axiom,
! [A: complex,K: complex,A3: complex] :
( ( A
= ( plus_plus_complex @ K @ A3 ) )
=> ( ( uminus1482373934393186551omplex @ A )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A3 ) ) ) ) ).
% group_cancel.neg1
thf(fact_660_group__cancel_Oneg1,axiom,
! [A: real,K: real,A3: real] :
( ( A
= ( plus_plus_real @ K @ A3 ) )
=> ( ( uminus_uminus_real @ A )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A3 ) ) ) ) ).
% group_cancel.neg1
thf(fact_661_is__num__normalize_I8_J,axiom,
! [A3: complex,B3: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A3 @ B3 ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B3 ) @ ( uminus1482373934393186551omplex @ A3 ) ) ) ).
% is_num_normalize(8)
thf(fact_662_is__num__normalize_I8_J,axiom,
! [A3: real,B3: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A3 @ B3 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) ) ) ).
% is_num_normalize(8)
thf(fact_663_is__num__normalize_I1_J,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A3 @ B3 ) @ C2 )
= ( plus_plus_complex @ A3 @ ( plus_plus_complex @ B3 @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_664_is__num__normalize_I1_J,axiom,
! [A3: real,B3: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A3 @ B3 ) @ C2 )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B3 @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_665_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_666_arith__extra__simps_I5_J,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% arith_extra_simps(5)
thf(fact_667_arith__extra__simps_I5_J,axiom,
! [A3: real] :
( ( plus_plus_real @ zero_zero_real @ A3 )
= A3 ) ).
% arith_extra_simps(5)
thf(fact_668_arith__extra__simps_I5_J,axiom,
! [A3: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A3 )
= A3 ) ).
% arith_extra_simps(5)
thf(fact_669_arith__extra__simps_I6_J,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% arith_extra_simps(6)
thf(fact_670_arith__extra__simps_I6_J,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ zero_zero_real )
= A3 ) ).
% arith_extra_simps(6)
thf(fact_671_arith__extra__simps_I6_J,axiom,
! [A3: complex] :
( ( plus_plus_complex @ A3 @ zero_zero_complex )
= A3 ) ).
% arith_extra_simps(6)
thf(fact_672_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_673_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_674_swap__plus__mat,axiom,
! [A: mat_a,N: nat,B: mat_a,C: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ C @ ( carrier_mat_a @ N @ N ) )
=> ( ( plus_plus_mat_a @ ( plus_plus_mat_a @ A @ B ) @ C )
= ( plus_plus_mat_a @ ( plus_plus_mat_a @ A @ C ) @ B ) ) ) ) ) ).
% swap_plus_mat
thf(fact_675_swap__plus__mat,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ B ) @ C )
= ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ C ) @ B ) ) ) ) ) ).
% swap_plus_mat
thf(fact_676_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_677_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_678_assoc__add__mat,axiom,
! [A: mat_a,Nr: nat,Nc: nat,B: mat_a,C: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ C @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ ( plus_plus_mat_a @ A @ B ) @ C )
= ( plus_plus_mat_a @ A @ ( plus_plus_mat_a @ B @ C ) ) ) ) ) ) ).
% assoc_add_mat
thf(fact_679_assoc__add__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,B: mat_complex,C: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ B ) @ C )
= ( plus_p8323303612493835998omplex @ A @ ( plus_p8323303612493835998omplex @ B @ C ) ) ) ) ) ) ).
% assoc_add_mat
thf(fact_680_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_681_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_682_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_683_mat__assoc__test_I15_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ B ) @ ( plus_p8323303612493835998omplex @ C @ D ) )
= ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ C ) @ ( plus_p8323303612493835998omplex @ B @ D ) ) ) ) ) ) ) ).
% mat_assoc_test(15)
thf(fact_684_mat__assoc__test_I14_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ B ) @ C )
= ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ C @ B ) @ A ) ) ) ) ) ) ).
% mat_assoc_test(14)
thf(fact_685_mat__assoc__test_I13_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( plus_p8323303612493835998omplex @ A @ B )
= ( plus_p8323303612493835998omplex @ B @ A ) ) ) ) ) ) ).
% mat_assoc_test(13)
thf(fact_686_Linear__Algebra__Complements_Otrace__add,axiom,
! [A: mat_real,B: mat_real] :
( ( square_mat_real @ A )
=> ( ( square_mat_real @ B )
=> ( ( ( dim_row_real @ A )
= ( dim_row_real @ B ) )
=> ( ( complex_trace_real @ ( plus_plus_mat_real @ A @ B ) )
= ( plus_plus_real @ ( complex_trace_real @ A ) @ ( complex_trace_real @ B ) ) ) ) ) ) ).
% Linear_Algebra_Complements.trace_add
thf(fact_687_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_688_mult__hom_Ohom__add__eq__zero,axiom,
! [X: nat,Y: nat,C2: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
=> ( ( plus_plus_nat @ ( times_times_nat @ C2 @ X ) @ ( times_times_nat @ C2 @ Y ) )
= zero_zero_nat ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_689_mult__hom_Ohom__add__eq__zero,axiom,
! [X: real,Y: real,C2: real] :
( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
=> ( ( plus_plus_real @ ( times_times_real @ C2 @ X ) @ ( times_times_real @ C2 @ Y ) )
= zero_zero_real ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_690_mult__hom_Ohom__add__eq__zero,axiom,
! [X: complex,Y: complex,C2: complex] :
( ( ( plus_plus_complex @ X @ Y )
= zero_zero_complex )
=> ( ( plus_plus_complex @ ( times_times_complex @ C2 @ X ) @ ( times_times_complex @ C2 @ Y ) )
= zero_zero_complex ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_691_sum__squares__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_692_add__less__zeroD,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_693_add__mono1,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ one_one_real ) @ ( plus_plus_real @ B3 @ one_one_real ) ) ) ).
% add_mono1
thf(fact_694_add__mono1,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) @ ( plus_plus_nat @ B3 @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_695_less__add__one,axiom,
! [A3: real] : ( ord_less_real @ A3 @ ( plus_plus_real @ A3 @ one_one_real ) ) ).
% less_add_one
thf(fact_696_less__add__one,axiom,
! [A3: nat] : ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ).
% less_add_one
thf(fact_697_ab__left__minus,axiom,
! [A3: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A3 ) @ A3 )
= zero_zero_complex ) ).
% ab_left_minus
thf(fact_698_ab__left__minus,axiom,
! [A3: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ A3 )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_699_neg__eq__iff__add__eq__0,axiom,
! [A3: complex,B3: complex] :
( ( ( uminus1482373934393186551omplex @ A3 )
= B3 )
= ( ( plus_plus_complex @ A3 @ B3 )
= zero_zero_complex ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_700_neg__eq__iff__add__eq__0,axiom,
! [A3: real,B3: real] :
( ( ( uminus_uminus_real @ A3 )
= B3 )
= ( ( plus_plus_real @ A3 @ B3 )
= zero_zero_real ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_701_eq__neg__iff__add__eq__0,axiom,
! [A3: complex,B3: complex] :
( ( A3
= ( uminus1482373934393186551omplex @ B3 ) )
= ( ( plus_plus_complex @ A3 @ B3 )
= zero_zero_complex ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_702_eq__neg__iff__add__eq__0,axiom,
! [A3: real,B3: real] :
( ( A3
= ( uminus_uminus_real @ B3 ) )
= ( ( plus_plus_real @ A3 @ B3 )
= zero_zero_real ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_703_add_Oinverse__unique,axiom,
! [A3: complex,B3: complex] :
( ( ( plus_plus_complex @ A3 @ B3 )
= zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ A3 )
= B3 ) ) ).
% add.inverse_unique
thf(fact_704_add_Oinverse__unique,axiom,
! [A3: real,B3: real] :
( ( ( plus_plus_real @ A3 @ B3 )
= zero_zero_real )
=> ( ( uminus_uminus_real @ A3 )
= B3 ) ) ).
% add.inverse_unique
thf(fact_705_add_Oright__inverse,axiom,
! [A3: complex] :
( ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ A3 ) )
= zero_zero_complex ) ).
% add.right_inverse
thf(fact_706_add_Oright__inverse,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ ( uminus_uminus_real @ A3 ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_707_ab__group__add__class_Oab__left__minus,axiom,
! [A3: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A3 ) @ A3 )
= zero_zero_complex ) ).
% ab_group_add_class.ab_left_minus
thf(fact_708_ab__group__add__class_Oab__left__minus,axiom,
! [A3: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ A3 )
= zero_zero_real ) ).
% ab_group_add_class.ab_left_minus
thf(fact_709_add__eq__0__iff,axiom,
! [A3: complex,B3: complex] :
( ( ( plus_plus_complex @ A3 @ B3 )
= zero_zero_complex )
= ( B3
= ( uminus1482373934393186551omplex @ A3 ) ) ) ).
% add_eq_0_iff
thf(fact_710_add__eq__0__iff,axiom,
! [A3: real,B3: real] :
( ( ( plus_plus_real @ A3 @ B3 )
= zero_zero_real )
= ( B3
= ( uminus_uminus_real @ A3 ) ) ) ).
% add_eq_0_iff
thf(fact_711_mult__diff__mult,axiom,
! [X: complex,Y: complex,A3: complex,B3: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X @ Y ) @ ( times_times_complex @ A3 @ B3 ) )
= ( plus_plus_complex @ ( times_times_complex @ X @ ( minus_minus_complex @ Y @ B3 ) ) @ ( times_times_complex @ ( minus_minus_complex @ X @ A3 ) @ B3 ) ) ) ).
% mult_diff_mult
thf(fact_712_mult__diff__mult,axiom,
! [X: real,Y: real,A3: real,B3: real] :
( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A3 @ B3 ) )
= ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B3 ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A3 ) @ B3 ) ) ) ).
% mult_diff_mult
thf(fact_713_square__diff__square__factored,axiom,
! [X: complex,Y: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ ( times_times_complex @ Y @ Y ) )
= ( times_times_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_complex @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_714_square__diff__square__factored,axiom,
! [X: real,Y: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_715_eq__add__iff2,axiom,
! [A3: complex,E: complex,C2: complex,B3: complex,D2: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ A3 @ E ) @ C2 )
= ( plus_plus_complex @ ( times_times_complex @ B3 @ E ) @ D2 ) )
= ( C2
= ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B3 @ A3 ) @ E ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_716_eq__add__iff2,axiom,
! [A3: real,E: real,C2: real,B3: real,D2: real] :
( ( ( plus_plus_real @ ( times_times_real @ A3 @ E ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ B3 @ E ) @ D2 ) )
= ( C2
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B3 @ A3 ) @ E ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_717_eq__add__iff1,axiom,
! [A3: complex,E: complex,C2: complex,B3: complex,D2: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ A3 @ E ) @ C2 )
= ( plus_plus_complex @ ( times_times_complex @ B3 @ E ) @ D2 ) )
= ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A3 @ B3 ) @ E ) @ C2 )
= D2 ) ) ).
% eq_add_iff1
thf(fact_718_eq__add__iff1,axiom,
! [A3: real,E: real,C2: real,B3: real,D2: real] :
( ( ( plus_plus_real @ ( times_times_real @ A3 @ E ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ B3 @ E ) @ D2 ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A3 @ B3 ) @ E ) @ C2 )
= D2 ) ) ).
% eq_add_iff1
thf(fact_719_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A3: real,B3: real] :
( ~ ( ord_less_real @ A3 @ B3 )
=> ( ( plus_plus_real @ B3 @ ( minus_minus_real @ A3 @ B3 ) )
= A3 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_720_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A3: nat,B3: nat] :
( ~ ( ord_less_nat @ A3 @ B3 )
=> ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A3 @ B3 ) )
= A3 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_721_pth__2,axiom,
( minus_minus_complex
= ( ^ [X2: complex,Y2: complex] : ( plus_plus_complex @ X2 @ ( uminus1482373934393186551omplex @ Y2 ) ) ) ) ).
% pth_2
thf(fact_722_pth__2,axiom,
( minus_minus_real
= ( ^ [X2: real,Y2: real] : ( plus_plus_real @ X2 @ ( uminus_uminus_real @ Y2 ) ) ) ) ).
% pth_2
thf(fact_723_class__ring_Ominus__eq,axiom,
( minus_minus_complex
= ( ^ [X2: complex,Y2: complex] : ( plus_plus_complex @ X2 @ ( uminus1482373934393186551omplex @ Y2 ) ) ) ) ).
% class_ring.minus_eq
thf(fact_724_class__ring_Ominus__eq,axiom,
( minus_minus_real
= ( ^ [X2: real,Y2: real] : ( plus_plus_real @ X2 @ ( uminus_uminus_real @ Y2 ) ) ) ) ).
% class_ring.minus_eq
thf(fact_725_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_complex
= ( ^ [A5: complex,B4: complex] : ( plus_plus_complex @ A5 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_726_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A5: real,B4: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_727_uminus__add__conv__diff,axiom,
! [A3: complex,B3: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A3 ) @ B3 )
= ( minus_minus_complex @ B3 @ A3 ) ) ).
% uminus_add_conv_diff
thf(fact_728_uminus__add__conv__diff,axiom,
! [A3: real,B3: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ B3 )
= ( minus_minus_real @ B3 @ A3 ) ) ).
% uminus_add_conv_diff
thf(fact_729_diff__conv__add__uminus,axiom,
( minus_minus_complex
= ( ^ [A5: complex,B4: complex] : ( plus_plus_complex @ A5 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_730_diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A5: real,B4: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_731_diff__minus__eq__add,axiom,
! [A3: complex,B3: complex] :
( ( minus_minus_complex @ A3 @ ( uminus1482373934393186551omplex @ B3 ) )
= ( plus_plus_complex @ A3 @ B3 ) ) ).
% diff_minus_eq_add
thf(fact_732_diff__minus__eq__add,axiom,
! [A3: real,B3: real] :
( ( minus_minus_real @ A3 @ ( uminus_uminus_real @ B3 ) )
= ( plus_plus_real @ A3 @ B3 ) ) ).
% diff_minus_eq_add
thf(fact_733_group__cancel_Osub2,axiom,
! [B: complex,K: complex,B3: complex,A3: complex] :
( ( B
= ( plus_plus_complex @ K @ B3 ) )
=> ( ( minus_minus_complex @ A3 @ B )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A3 @ B3 ) ) ) ) ).
% group_cancel.sub2
thf(fact_734_group__cancel_Osub2,axiom,
! [B: real,K: real,B3: real,A3: real] :
( ( B
= ( plus_plus_real @ K @ B3 ) )
=> ( ( minus_minus_real @ A3 @ B )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A3 @ B3 ) ) ) ) ).
% group_cancel.sub2
thf(fact_735_index__mat__addcol_I4_J,axiom,
! [A3: complex,K: nat,L: nat,A: mat_complex] :
( ( dim_row_complex @ ( column896436094548437152omplex @ A3 @ K @ L @ A ) )
= ( dim_row_complex @ A ) ) ).
% index_mat_addcol(4)
thf(fact_736_mult__add__distrib__mat,axiom,
! [A: mat_a,Nr: nat,N: nat,B: mat_a,Nc: nat,C: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( member_mat_a @ C @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ A @ ( plus_plus_mat_a @ B @ C ) )
= ( plus_plus_mat_a @ ( times_times_mat_a @ A @ B ) @ ( times_times_mat_a @ A @ C ) ) ) ) ) ) ).
% mult_add_distrib_mat
thf(fact_737_mult__add__distrib__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,B: mat_complex,Nc: nat,C: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ A @ ( plus_p8323303612493835998omplex @ B @ C ) )
= ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ A @ B ) @ ( times_8009071140041733218omplex @ A @ C ) ) ) ) ) ) ).
% mult_add_distrib_mat
thf(fact_738_add__mult__distrib__mat,axiom,
! [A: mat_a,Nr: nat,N: nat,B: mat_a,C: 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 @ C @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ ( plus_plus_mat_a @ A @ B ) @ C )
= ( plus_plus_mat_a @ ( times_times_mat_a @ A @ C ) @ ( times_times_mat_a @ B @ C ) ) ) ) ) ) ).
% add_mult_distrib_mat
thf(fact_739_add__mult__distrib__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,B: mat_complex,C: 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 @ C @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( plus_p8323303612493835998omplex @ A @ B ) @ C )
= ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ A @ C ) @ ( times_8009071140041733218omplex @ B @ C ) ) ) ) ) ) ).
% add_mult_distrib_mat
thf(fact_740_Complex__Matrix_Oright__add__zero__mat,axiom,
! [A: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ A @ ( zero_mat_a @ Nr @ Nc ) )
= A ) ) ).
% Complex_Matrix.right_add_zero_mat
thf(fact_741_Complex__Matrix_Oright__add__zero__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( plus_p8323303612493835998omplex @ A @ ( zero_mat_complex @ Nr @ Nc ) )
= A ) ) ).
% Complex_Matrix.right_add_zero_mat
thf(fact_742_add__inv__exists__mat,axiom,
! [A: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ? [X3: mat_a] :
( ( member_mat_a @ X3 @ ( carrier_mat_a @ Nr @ Nc ) )
& ( ( plus_plus_mat_a @ X3 @ A )
= ( zero_mat_a @ Nr @ Nc ) )
& ( ( plus_plus_mat_a @ A @ X3 )
= ( zero_mat_a @ Nr @ Nc ) ) ) ) ).
% add_inv_exists_mat
thf(fact_743_add__inv__exists__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ? [X3: mat_complex] :
( ( member_mat_complex @ X3 @ ( carrier_mat_complex @ Nr @ Nc ) )
& ( ( plus_p8323303612493835998omplex @ X3 @ A )
= ( zero_mat_complex @ Nr @ Nc ) )
& ( ( plus_p8323303612493835998omplex @ A @ X3 )
= ( zero_mat_complex @ Nr @ Nc ) ) ) ) ).
% add_inv_exists_mat
thf(fact_744_left__add__zero__mat,axiom,
! [A: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ ( zero_mat_a @ Nr @ Nc ) @ A )
= A ) ) ).
% left_add_zero_mat
thf(fact_745_left__add__zero__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( plus_p8323303612493835998omplex @ ( zero_mat_complex @ Nr @ Nc ) @ A )
= A ) ) ).
% left_add_zero_mat
thf(fact_746_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_747_adjoint__add,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ M ) )
=> ( ( schur_mat_adjoint_a @ ( plus_plus_mat_a @ A @ B ) )
= ( plus_plus_mat_a @ ( schur_mat_adjoint_a @ A ) @ ( schur_mat_adjoint_a @ B ) ) ) ) ) ).
% adjoint_add
thf(fact_748_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_749_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_750_add__four__block__mat,axiom,
! [A1: mat_a,Nr1: nat,Nc1: nat,B1: mat_a,Nc2: nat,C1: mat_a,Nr2: nat,D1: mat_a,A22: mat_a,B22: mat_a,C22: mat_a,D22: mat_a] :
( ( member_mat_a @ A1 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B1 @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C1 @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D1 @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ( member_mat_a @ A22 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B22 @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C22 @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D22 @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ( plus_plus_mat_a @ ( four_block_mat_a @ A1 @ B1 @ C1 @ D1 ) @ ( four_block_mat_a @ A22 @ B22 @ C22 @ D22 ) )
= ( four_block_mat_a @ ( plus_plus_mat_a @ A1 @ A22 ) @ ( plus_plus_mat_a @ B1 @ B22 ) @ ( plus_plus_mat_a @ C1 @ C22 ) @ ( plus_plus_mat_a @ D1 @ D22 ) ) ) ) ) ) ) ) ) ) ) ).
% add_four_block_mat
thf(fact_751_add__four__block__mat,axiom,
! [A1: mat_complex,Nr1: nat,Nc1: nat,B1: mat_complex,Nc2: nat,C1: mat_complex,Nr2: nat,D1: mat_complex,A22: mat_complex,B22: mat_complex,C22: mat_complex,D22: mat_complex] :
( ( member_mat_complex @ A1 @ ( carrier_mat_complex @ Nr1 @ Nc1 ) )
=> ( ( member_mat_complex @ B1 @ ( carrier_mat_complex @ Nr1 @ Nc2 ) )
=> ( ( member_mat_complex @ C1 @ ( carrier_mat_complex @ Nr2 @ Nc1 ) )
=> ( ( member_mat_complex @ D1 @ ( carrier_mat_complex @ Nr2 @ Nc2 ) )
=> ( ( member_mat_complex @ A22 @ ( carrier_mat_complex @ Nr1 @ Nc1 ) )
=> ( ( member_mat_complex @ B22 @ ( carrier_mat_complex @ Nr1 @ Nc2 ) )
=> ( ( member_mat_complex @ C22 @ ( carrier_mat_complex @ Nr2 @ Nc1 ) )
=> ( ( member_mat_complex @ D22 @ ( carrier_mat_complex @ Nr2 @ Nc2 ) )
=> ( ( plus_p8323303612493835998omplex @ ( four_b559179830521662709omplex @ A1 @ B1 @ C1 @ D1 ) @ ( four_b559179830521662709omplex @ A22 @ B22 @ C22 @ D22 ) )
= ( four_b559179830521662709omplex @ ( plus_p8323303612493835998omplex @ A1 @ A22 ) @ ( plus_p8323303612493835998omplex @ B1 @ B22 ) @ ( plus_p8323303612493835998omplex @ C1 @ C22 ) @ ( plus_p8323303612493835998omplex @ D1 @ D22 ) ) ) ) ) ) ) ) ) ) ) ).
% add_four_block_mat
thf(fact_752_mat__assoc__test_I7_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ ( plus_p8323303612493835998omplex @ A @ B ) @ ( plus_p8323303612493835998omplex @ B @ C ) )
= ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ A @ B ) @ ( times_8009071140041733218omplex @ B @ B ) ) @ ( times_8009071140041733218omplex @ A @ C ) ) @ ( times_8009071140041733218omplex @ B @ C ) ) ) ) ) ) ) ).
% mat_assoc_test(7)
thf(fact_753_hermitian__add,axiom,
! [A: mat_a,N: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_hermitian_a @ A )
=> ( ( complex_hermitian_a @ B )
=> ( complex_hermitian_a @ ( plus_plus_mat_a @ A @ B ) ) ) ) ) ) ).
% hermitian_add
thf(fact_754_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_755_minus__add__minus__mat,axiom,
! [U3: mat_a,Nr: nat,Nc: nat,V3: mat_a,W: mat_a] :
( ( member_mat_a @ U3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ V3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ W @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( minus_minus_mat_a @ U3 @ ( plus_plus_mat_a @ V3 @ W ) )
= ( minus_minus_mat_a @ ( minus_minus_mat_a @ U3 @ V3 ) @ W ) ) ) ) ) ).
% minus_add_minus_mat
thf(fact_756_minus__add__minus__mat,axiom,
! [U3: mat_complex,Nr: nat,Nc: nat,V3: mat_complex,W: mat_complex] :
( ( member_mat_complex @ U3 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ V3 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ W @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( minus_2412168080157227406omplex @ U3 @ ( plus_p8323303612493835998omplex @ V3 @ W ) )
= ( minus_2412168080157227406omplex @ ( minus_2412168080157227406omplex @ U3 @ V3 ) @ W ) ) ) ) ) ).
% minus_add_minus_mat
thf(fact_757_mat__minus__minus,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a,C: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ M ) )
=> ( ( member_mat_a @ C @ ( carrier_mat_a @ N @ M ) )
=> ( ( minus_minus_mat_a @ A @ ( minus_minus_mat_a @ B @ C ) )
= ( plus_plus_mat_a @ ( minus_minus_mat_a @ A @ B ) @ C ) ) ) ) ) ).
% mat_minus_minus
thf(fact_758_mat__minus__minus,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,C: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ M ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ M ) )
=> ( ( minus_2412168080157227406omplex @ A @ ( minus_2412168080157227406omplex @ B @ C ) )
= ( plus_p8323303612493835998omplex @ ( minus_2412168080157227406omplex @ A @ B ) @ C ) ) ) ) ) ).
% mat_minus_minus
thf(fact_759_uminus__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 ) )
=> ( ( uminus_uminus_mat_a @ ( plus_plus_mat_a @ A @ B ) )
= ( plus_plus_mat_a @ ( uminus_uminus_mat_a @ B ) @ ( uminus_uminus_mat_a @ A ) ) ) ) ) ).
% uminus_add_mat
thf(fact_760_uminus__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 ) )
=> ( ( uminus467866341702955550omplex @ ( plus_p8323303612493835998omplex @ A @ B ) )
= ( plus_p8323303612493835998omplex @ ( uminus467866341702955550omplex @ B ) @ ( uminus467866341702955550omplex @ A ) ) ) ) ) ).
% uminus_add_mat
thf(fact_761_mat__assoc__test_I6_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( minus_2412168080157227406omplex @ A @ ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ B @ C ) @ D ) )
= ( minus_2412168080157227406omplex @ ( minus_2412168080157227406omplex @ ( minus_2412168080157227406omplex @ A @ B ) @ C ) @ D ) ) ) ) ) ) ).
% mat_assoc_test(6)
thf(fact_762_mat__assoc__test_I5_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( plus_p8323303612493835998omplex @ A @ ( minus_2412168080157227406omplex @ B @ C ) )
= ( minus_2412168080157227406omplex @ ( plus_p8323303612493835998omplex @ A @ B ) @ C ) ) ) ) ) ) ).
% mat_assoc_test(5)
thf(fact_763_not__sum__squares__lt__zero,axiom,
! [X: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_764_sum__squares__gt__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
= ( ( X != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_765_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_766_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_767_arith__special_I11_J,axiom,
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
= zero_zero_complex ) ).
% arith_special(11)
thf(fact_768_arith__special_I11_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= zero_zero_real ) ).
% arith_special(11)
thf(fact_769_arith__special_I10_J,axiom,
( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= zero_zero_complex ) ).
% arith_special(10)
thf(fact_770_arith__special_I10_J,axiom,
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% arith_special(10)
thf(fact_771_less__add__iff2,axiom,
! [A3: complex,E: complex,C2: complex,B3: complex,D2: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ ( times_times_complex @ A3 @ E ) @ C2 ) @ ( plus_plus_complex @ ( times_times_complex @ B3 @ E ) @ D2 ) )
= ( ord_less_complex @ C2 @ ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B3 @ A3 ) @ E ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_772_less__add__iff2,axiom,
! [A3: real,E: real,C2: real,B3: real,D2: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A3 @ E ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E ) @ D2 ) )
= ( ord_less_real @ C2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B3 @ A3 ) @ E ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_773_less__add__iff1,axiom,
! [A3: complex,E: complex,C2: complex,B3: complex,D2: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ ( times_times_complex @ A3 @ E ) @ C2 ) @ ( plus_plus_complex @ ( times_times_complex @ B3 @ E ) @ D2 ) )
= ( ord_less_complex @ ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A3 @ B3 ) @ E ) @ C2 ) @ D2 ) ) ).
% less_add_iff1
thf(fact_774_less__add__iff1,axiom,
! [A3: real,E: real,C2: real,B3: real,D2: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A3 @ E ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E ) @ D2 ) )
= ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A3 @ B3 ) @ E ) @ C2 ) @ D2 ) ) ).
% less_add_iff1
thf(fact_775_square__diff__one__factored,axiom,
! [X: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
= ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).
% square_diff_one_factored
thf(fact_776_square__diff__one__factored,axiom,
! [X: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_777_mult__four__block__mat,axiom,
! [A1: mat_a,Nr1: nat,N1: nat,B1: mat_a,N22: nat,C1: mat_a,Nr2: nat,D1: mat_a,A22: mat_a,Nc1: nat,B22: mat_a,Nc2: nat,C22: mat_a,D22: mat_a] :
( ( member_mat_a @ A1 @ ( carrier_mat_a @ Nr1 @ N1 ) )
=> ( ( member_mat_a @ B1 @ ( carrier_mat_a @ Nr1 @ N22 ) )
=> ( ( member_mat_a @ C1 @ ( carrier_mat_a @ Nr2 @ N1 ) )
=> ( ( member_mat_a @ D1 @ ( carrier_mat_a @ Nr2 @ N22 ) )
=> ( ( member_mat_a @ A22 @ ( carrier_mat_a @ N1 @ Nc1 ) )
=> ( ( member_mat_a @ B22 @ ( carrier_mat_a @ N1 @ Nc2 ) )
=> ( ( member_mat_a @ C22 @ ( carrier_mat_a @ N22 @ Nc1 ) )
=> ( ( member_mat_a @ D22 @ ( carrier_mat_a @ N22 @ Nc2 ) )
=> ( ( times_times_mat_a @ ( four_block_mat_a @ A1 @ B1 @ C1 @ D1 ) @ ( four_block_mat_a @ A22 @ B22 @ C22 @ D22 ) )
= ( four_block_mat_a @ ( plus_plus_mat_a @ ( times_times_mat_a @ A1 @ A22 ) @ ( times_times_mat_a @ B1 @ C22 ) ) @ ( plus_plus_mat_a @ ( times_times_mat_a @ A1 @ B22 ) @ ( times_times_mat_a @ B1 @ D22 ) ) @ ( plus_plus_mat_a @ ( times_times_mat_a @ C1 @ A22 ) @ ( times_times_mat_a @ D1 @ C22 ) ) @ ( plus_plus_mat_a @ ( times_times_mat_a @ C1 @ B22 ) @ ( times_times_mat_a @ D1 @ D22 ) ) ) ) ) ) ) ) ) ) ) ) ).
% mult_four_block_mat
thf(fact_778_mult__four__block__mat,axiom,
! [A1: mat_complex,Nr1: nat,N1: nat,B1: mat_complex,N22: nat,C1: mat_complex,Nr2: nat,D1: mat_complex,A22: mat_complex,Nc1: nat,B22: mat_complex,Nc2: nat,C22: mat_complex,D22: mat_complex] :
( ( member_mat_complex @ A1 @ ( carrier_mat_complex @ Nr1 @ N1 ) )
=> ( ( member_mat_complex @ B1 @ ( carrier_mat_complex @ Nr1 @ N22 ) )
=> ( ( member_mat_complex @ C1 @ ( carrier_mat_complex @ Nr2 @ N1 ) )
=> ( ( member_mat_complex @ D1 @ ( carrier_mat_complex @ Nr2 @ N22 ) )
=> ( ( member_mat_complex @ A22 @ ( carrier_mat_complex @ N1 @ Nc1 ) )
=> ( ( member_mat_complex @ B22 @ ( carrier_mat_complex @ N1 @ Nc2 ) )
=> ( ( member_mat_complex @ C22 @ ( carrier_mat_complex @ N22 @ Nc1 ) )
=> ( ( member_mat_complex @ D22 @ ( carrier_mat_complex @ N22 @ Nc2 ) )
=> ( ( times_8009071140041733218omplex @ ( four_b559179830521662709omplex @ A1 @ B1 @ C1 @ D1 ) @ ( four_b559179830521662709omplex @ A22 @ B22 @ C22 @ D22 ) )
= ( four_b559179830521662709omplex @ ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ A1 @ A22 ) @ ( times_8009071140041733218omplex @ B1 @ C22 ) ) @ ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ A1 @ B22 ) @ ( times_8009071140041733218omplex @ B1 @ D22 ) ) @ ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ C1 @ A22 ) @ ( times_8009071140041733218omplex @ D1 @ C22 ) ) @ ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ C1 @ B22 ) @ ( times_8009071140041733218omplex @ D1 @ D22 ) ) ) ) ) ) ) ) ) ) ) ) ).
% mult_four_block_mat
thf(fact_779_uminus__l__inv__mat,axiom,
! [A: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ ( uminus_uminus_mat_a @ A ) @ A )
= ( zero_mat_a @ Nr @ Nc ) ) ) ).
% uminus_l_inv_mat
thf(fact_780_uminus__l__inv__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( plus_p8323303612493835998omplex @ ( uminus467866341702955550omplex @ A ) @ A )
= ( zero_mat_complex @ Nr @ Nc ) ) ) ).
% uminus_l_inv_mat
thf(fact_781_uminus__add__minus__mat,axiom,
! [L: mat_a,Nr: nat,Nc: nat,R2: mat_a] :
( ( member_mat_a @ L @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ R2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( uminus_uminus_mat_a @ ( plus_plus_mat_a @ L @ R2 ) )
= ( minus_minus_mat_a @ ( uminus_uminus_mat_a @ L ) @ R2 ) ) ) ) ).
% uminus_add_minus_mat
thf(fact_782_uminus__add__minus__mat,axiom,
! [L: mat_complex,Nr: nat,Nc: nat,R2: mat_complex] :
( ( member_mat_complex @ L @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ R2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( uminus467866341702955550omplex @ ( plus_p8323303612493835998omplex @ L @ R2 ) )
= ( minus_2412168080157227406omplex @ ( uminus467866341702955550omplex @ L ) @ R2 ) ) ) ) ).
% uminus_add_minus_mat
thf(fact_783_minus__add__uminus__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 ) )
=> ( ( minus_minus_mat_a @ A @ B )
= ( plus_plus_mat_a @ A @ ( uminus_uminus_mat_a @ B ) ) ) ) ) ).
% minus_add_uminus_mat
thf(fact_784_minus__add__uminus__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 ) )
=> ( ( minus_2412168080157227406omplex @ A @ B )
= ( plus_p8323303612493835998omplex @ A @ ( uminus467866341702955550omplex @ B ) ) ) ) ) ).
% minus_add_uminus_mat
thf(fact_785_add__uminus__minus__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 @ ( uminus_uminus_mat_a @ B ) )
= ( minus_minus_mat_a @ A @ B ) ) ) ) ).
% add_uminus_minus_mat
thf(fact_786_add__uminus__minus__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 @ ( uminus467866341702955550omplex @ B ) )
= ( minus_2412168080157227406omplex @ A @ B ) ) ) ) ).
% add_uminus_minus_mat
thf(fact_787_mat__assoc__test_I4_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( plus_p8323303612493835998omplex @ ( minus_2412168080157227406omplex @ A @ B ) @ ( minus_2412168080157227406omplex @ B @ C ) )
= ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ ( uminus467866341702955550omplex @ B ) ) @ B ) @ ( uminus467866341702955550omplex @ C ) ) ) ) ) ) ) ).
% mat_assoc_test(4)
thf(fact_788_add__scale__eq__noteq,axiom,
! [R2: nat,A3: nat,B3: nat,C2: nat,D2: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A3 = B3 )
& ( C2 != D2 ) )
=> ( ( plus_plus_nat @ A3 @ ( times_times_nat @ R2 @ C2 ) )
!= ( plus_plus_nat @ B3 @ ( times_times_nat @ R2 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_789_add__scale__eq__noteq,axiom,
! [R2: real,A3: real,B3: real,C2: real,D2: real] :
( ( R2 != zero_zero_real )
=> ( ( ( A3 = B3 )
& ( C2 != D2 ) )
=> ( ( plus_plus_real @ A3 @ ( times_times_real @ R2 @ C2 ) )
!= ( plus_plus_real @ B3 @ ( times_times_real @ R2 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_790_add__scale__eq__noteq,axiom,
! [R2: complex,A3: complex,B3: complex,C2: complex,D2: complex] :
( ( R2 != zero_zero_complex )
=> ( ( ( A3 = B3 )
& ( C2 != D2 ) )
=> ( ( plus_plus_complex @ A3 @ ( times_times_complex @ R2 @ C2 ) )
!= ( plus_plus_complex @ B3 @ ( times_times_complex @ R2 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_791_addrow__mat,axiom,
! [A: mat_a,N: nat,Nc: nat,L: nat,A3: a,K: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( ord_less_nat @ L @ N )
=> ( ( gauss_3441994962245461172_gen_a @ plus_plus_a @ times_times_a @ A3 @ K @ L @ A )
= ( times_times_mat_a @ ( gauss_8159914756388622152_mat_a @ N @ A3 @ K @ L ) @ A ) ) ) ) ).
% addrow_mat
thf(fact_792_addrow__mat,axiom,
! [A: mat_real,N: nat,Nc: nat,L: nat,A3: real,K: nat] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ Nc ) )
=> ( ( ord_less_nat @ L @ N )
=> ( ( gauss_4246877906280926838n_real @ plus_plus_real @ times_times_real @ A3 @ K @ L @ A )
= ( times_times_mat_real @ ( gauss_2378325378421436642t_real @ N @ A3 @ K @ L ) @ A ) ) ) ) ).
% addrow_mat
thf(fact_793_addrow__mat,axiom,
! [A: mat_complex,N: nat,Nc: nat,L: nat,A3: complex,K: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( ord_less_nat @ L @ N )
=> ( ( gauss_5252963565656066424omplex @ plus_plus_complex @ times_times_complex @ A3 @ K @ L @ A )
= ( times_8009071140041733218omplex @ ( gauss_947198734564870628omplex @ N @ A3 @ K @ L ) @ A ) ) ) ) ).
% addrow_mat
thf(fact_794_det__addrow,axiom,
! [L: nat,N: nat,K: nat,A: mat_a,A3: a] :
( ( ord_less_nat @ L @ N )
=> ( ( K != L )
=> ( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( det_a @ ( gauss_3441994962245461172_gen_a @ plus_plus_a @ times_times_a @ A3 @ K @ L @ A ) )
= ( det_a @ A ) ) ) ) ) ).
% det_addrow
thf(fact_795_det__addrow,axiom,
! [L: nat,N: nat,K: nat,A: mat_complex,A3: complex] :
( ( ord_less_nat @ L @ N )
=> ( ( K != L )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( det_complex @ ( gauss_5252963565656066424omplex @ plus_plus_complex @ times_times_complex @ A3 @ K @ L @ A ) )
= ( det_complex @ A ) ) ) ) ) ).
% det_addrow
thf(fact_796_det__addrow,axiom,
! [L: nat,N: nat,K: nat,A: mat_real,A3: real] :
( ( ord_less_nat @ L @ N )
=> ( ( K != L )
=> ( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( det_real @ ( gauss_4246877906280926838n_real @ plus_plus_real @ times_times_real @ A3 @ K @ L @ A ) )
= ( det_real @ A ) ) ) ) ) ).
% det_addrow
thf(fact_797_crossproduct__noteq,axiom,
! [A3: complex,B3: complex,C2: complex,D2: complex] :
( ( ( A3 != B3 )
& ( C2 != D2 ) )
= ( ( plus_plus_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B3 @ D2 ) )
!= ( plus_plus_complex @ ( times_times_complex @ A3 @ D2 ) @ ( times_times_complex @ B3 @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_798_crossproduct__noteq,axiom,
! [A3: real,B3: real,C2: real,D2: real] :
( ( ( A3 != B3 )
& ( C2 != D2 ) )
= ( ( plus_plus_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ D2 ) )
!= ( plus_plus_real @ ( times_times_real @ A3 @ D2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_799_addrow__carrier,axiom,
! [Ad: a > a > a,Mul: a > a > a,A3: a,K: nat,L: nat,A: mat_a,N: nat,Nc: nat] :
( ( member_mat_a @ ( gauss_3441994962245461172_gen_a @ Ad @ Mul @ A3 @ K @ L @ A ) @ ( carrier_mat_a @ N @ Nc ) )
= ( member_mat_a @ A @ ( carrier_mat_a @ N @ Nc ) ) ) ).
% addrow_carrier
thf(fact_800_addrow__carrier,axiom,
! [Ad: complex > complex > complex,Mul: complex > complex > complex,A3: complex,K: nat,L: nat,A: mat_complex,N: nat,Nc: nat] :
( ( member_mat_complex @ ( gauss_5252963565656066424omplex @ Ad @ Mul @ A3 @ K @ L @ A ) @ ( carrier_mat_complex @ N @ Nc ) )
= ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ Nc ) ) ) ).
% addrow_carrier
thf(fact_801_index__mat__addrow_I4_J,axiom,
! [Ad: complex > complex > complex,Mul: complex > complex > complex,A3: complex,K: nat,L: nat,A: mat_complex] :
( ( dim_row_complex @ ( gauss_5252963565656066424omplex @ Ad @ Mul @ A3 @ K @ L @ A ) )
= ( dim_row_complex @ A ) ) ).
% index_mat_addrow(4)
thf(fact_802_index__mat__addrow_I5_J,axiom,
! [Ad: complex > complex > complex,Mul: complex > complex > complex,A3: complex,K: nat,L: nat,A: mat_complex] :
( ( dim_col_complex @ ( gauss_5252963565656066424omplex @ Ad @ Mul @ A3 @ K @ L @ A ) )
= ( dim_col_complex @ A ) ) ).
% index_mat_addrow(5)
thf(fact_803_four__block__carrier__mat,axiom,
! [A: mat_a,Nr1: nat,Nc1: nat,D: mat_a,Nr2: nat,Nc2: nat,B: mat_a,C: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ D @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( member_mat_a @ ( four_block_mat_a @ A @ B @ C @ D ) @ ( carrier_mat_a @ ( plus_plus_nat @ Nr1 @ Nr2 ) @ ( plus_plus_nat @ Nc1 @ Nc2 ) ) ) ) ) ).
% four_block_carrier_mat
thf(fact_804_four__block__carrier__mat,axiom,
! [A: mat_complex,Nr1: nat,Nc1: nat,D: mat_complex,Nr2: nat,Nc2: nat,B: mat_complex,C: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr1 @ Nc1 ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ Nr2 @ Nc2 ) )
=> ( member_mat_complex @ ( four_b559179830521662709omplex @ A @ B @ C @ D ) @ ( carrier_mat_complex @ ( plus_plus_nat @ Nr1 @ Nr2 ) @ ( plus_plus_nat @ Nc1 @ Nc2 ) ) ) ) ) ).
% four_block_carrier_mat
thf(fact_805_index__mat__four__block_I2_J,axiom,
! [A: mat_complex,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( dim_row_complex @ ( four_b559179830521662709omplex @ A @ B @ C @ D ) )
= ( plus_plus_nat @ ( dim_row_complex @ A ) @ ( dim_row_complex @ D ) ) ) ).
% index_mat_four_block(2)
thf(fact_806_index__mat__four__block_I3_J,axiom,
! [A: mat_complex,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( dim_col_complex @ ( four_b559179830521662709omplex @ A @ B @ C @ D ) )
= ( plus_plus_nat @ ( dim_col_complex @ A ) @ ( dim_col_complex @ D ) ) ) ).
% index_mat_four_block(3)
thf(fact_807_four__block__zero__mat,axiom,
! [Nr1: nat,Nc1: nat,Nc2: nat,Nr2: nat] :
( ( four_b559179830521662709omplex @ ( zero_mat_complex @ Nr1 @ Nc1 ) @ ( zero_mat_complex @ Nr1 @ Nc2 ) @ ( zero_mat_complex @ Nr2 @ Nc1 ) @ ( zero_mat_complex @ Nr2 @ Nc2 ) )
= ( zero_mat_complex @ ( plus_plus_nat @ Nr1 @ Nr2 ) @ ( plus_plus_nat @ Nc1 @ Nc2 ) ) ) ).
% four_block_zero_mat
thf(fact_808_carrier__append__rows,axiom,
! [A: mat_a,Nr1: nat,Nc: nat,B: mat_a,Nr2: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr1 @ Nc ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ Nr2 @ Nc ) )
=> ( member_mat_a @ ( append_rows_a @ A @ B ) @ ( carrier_mat_a @ ( plus_plus_nat @ Nr1 @ Nr2 ) @ Nc ) ) ) ) ).
% carrier_append_rows
thf(fact_809_carrier__append__rows,axiom,
! [A: mat_complex,Nr1: nat,Nc: nat,B: mat_complex,Nr2: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr1 @ Nc ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ Nr2 @ Nc ) )
=> ( member_mat_complex @ ( append_rows_complex @ A @ B ) @ ( carrier_mat_complex @ ( plus_plus_nat @ Nr1 @ Nr2 ) @ Nc ) ) ) ) ).
% carrier_append_rows
thf(fact_810_four__block__one__mat,axiom,
! [N1: nat,N22: nat] :
( ( four_b559179830521662709omplex @ ( one_mat_complex @ N1 ) @ ( zero_mat_complex @ N1 @ N22 ) @ ( zero_mat_complex @ N22 @ N1 ) @ ( one_mat_complex @ N22 ) )
= ( one_mat_complex @ ( plus_plus_nat @ N1 @ N22 ) ) ) ).
% four_block_one_mat
thf(fact_811_mat__assoc__test_I12_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple3184165445352484367omplex @ ( plus_p8323303612493835998omplex @ A @ ( times_8009071140041733218omplex @ B @ C ) ) )
= ( plus_plus_complex @ ( comple3184165445352484367omplex @ A ) @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ C @ B ) ) ) ) ) ) ) ) ).
% mat_assoc_test(12)
thf(fact_812_crossproduct__eq,axiom,
! [W: complex,Y: complex,X: complex,Z: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ W @ Y ) @ ( times_times_complex @ X @ Z ) )
= ( plus_plus_complex @ ( times_times_complex @ W @ Z ) @ ( times_times_complex @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_813_crossproduct__eq,axiom,
! [W: real,Y: real,X: real,Z: real] :
( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X @ Z ) )
= ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_814_cofactor__def,axiom,
( cofactor_complex
= ( ^ [A2: mat_complex,I2: nat,J: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ I2 @ J ) ) @ ( det_complex @ ( mat_delete_complex @ A2 @ I2 @ J ) ) ) ) ) ).
% cofactor_def
thf(fact_815_cofactor__def,axiom,
( cofactor_real
= ( ^ [A2: mat_real,I2: nat,J: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ I2 @ J ) ) @ ( det_real @ ( mat_delete_real @ A2 @ I2 @ J ) ) ) ) ) ).
% cofactor_def
thf(fact_816_add__col__sub__row__def,axiom,
( column6029646570091773654omplex
= ( ^ [A5: complex,K4: nat,L2: nat,A2: mat_complex] : ( gauss_5252963565656066424omplex @ plus_plus_complex @ times_times_complex @ ( uminus1482373934393186551omplex @ A5 ) @ K4 @ L2 @ ( column896436094548437152omplex @ A5 @ L2 @ K4 @ A2 ) ) ) ) ).
% add_col_sub_row_def
thf(fact_817_add__col__sub__row__def,axiom,
( column3494657893274022100w_real
= ( ^ [A5: real,K4: nat,L2: nat,A2: mat_real] : ( gauss_4246877906280926838n_real @ plus_plus_real @ times_times_real @ ( uminus_uminus_real @ A5 ) @ K4 @ L2 @ ( column5677306341442300318l_real @ A5 @ L2 @ K4 @ A2 ) ) ) ) ).
% add_col_sub_row_def
thf(fact_818_add__col__sub__row__carrier_I3_J,axiom,
! [A: mat_a,N: nat,A3: a,K: nat,L: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( member_mat_a @ ( column3081110322506813142_row_a @ A3 @ K @ L @ A ) @ ( carrier_mat_a @ N @ N ) ) ) ).
% add_col_sub_row_carrier(3)
thf(fact_819_add__col__sub__row__carrier_I3_J,axiom,
! [A: mat_complex,N: nat,A3: complex,K: nat,L: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( member_mat_complex @ ( column6029646570091773654omplex @ A3 @ K @ L @ A ) @ ( carrier_mat_complex @ N @ N ) ) ) ).
% add_col_sub_row_carrier(3)
thf(fact_820_add__col__sub__row__carrier_I1_J,axiom,
! [A3: complex,K: nat,L: nat,A: mat_complex] :
( ( dim_row_complex @ ( column6029646570091773654omplex @ A3 @ K @ L @ A ) )
= ( dim_row_complex @ A ) ) ).
% add_col_sub_row_carrier(1)
thf(fact_821_char__matrix__def,axiom,
( char_c872259621517735348omplex
= ( ^ [A2: mat_complex,E2: complex] : ( plus_p8323303612493835998omplex @ A2 @ ( smult_mat_complex @ ( uminus1482373934393186551omplex @ E2 ) @ ( one_mat_complex @ ( dim_row_complex @ A2 ) ) ) ) ) ) ).
% char_matrix_def
thf(fact_822_char__matrix__def,axiom,
( char_c4597223634827269298x_real
= ( ^ [A2: mat_real,E2: real] : ( plus_plus_mat_real @ A2 @ ( smult_mat_real @ ( uminus_uminus_real @ E2 ) @ ( one_mat_real @ ( dim_row_real @ A2 ) ) ) ) ) ) ).
% char_matrix_def
thf(fact_823_sum__mat__add,axiom,
! [A: mat_a,Nr: nat,Nc: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( comm_m5291664705200495434_mat_a @ ( plus_plus_mat_a @ A @ B ) )
= ( plus_plus_a @ ( comm_m5291664705200495434_mat_a @ A ) @ ( comm_m5291664705200495434_mat_a @ B ) ) ) ) ) ).
% sum_mat_add
thf(fact_824_sum__mat__add,axiom,
! [A: mat_real,Nr: nat,Nc: nat,B: mat_real] :
( ( member_mat_real @ A @ ( carrier_mat_real @ Nr @ Nc ) )
=> ( ( member_mat_real @ B @ ( carrier_mat_real @ Nr @ Nc ) )
=> ( ( comm_m8678487124704766304t_real @ ( plus_plus_mat_real @ A @ B ) )
= ( plus_plus_real @ ( comm_m8678487124704766304t_real @ A ) @ ( comm_m8678487124704766304t_real @ B ) ) ) ) ) ).
% sum_mat_add
thf(fact_825_sum__mat__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 ) )
=> ( ( comm_m3586542329073673570omplex @ ( plus_p8323303612493835998omplex @ A @ B ) )
= ( plus_plus_complex @ ( comm_m3586542329073673570omplex @ A ) @ ( comm_m3586542329073673570omplex @ B ) ) ) ) ) ).
% sum_mat_add
thf(fact_826_swaprows__mat,axiom,
! [A: mat_a,N: nat,Nc: nat,K: nat,L: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( ord_less_nat @ K @ N )
=> ( ( ord_less_nat @ L @ N )
=> ( ( gauss_2482569599970757219rows_a @ K @ L @ A )
= ( times_times_mat_a @ ( gauss_110929411057020027_mat_a @ N @ K @ L ) @ A ) ) ) ) ) ).
% swaprows_mat
thf(fact_827_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_828_char__matrix__closed,axiom,
! [A: mat_a,N: nat,E: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( member_mat_a @ ( char_char_matrix_a @ A @ E ) @ ( carrier_mat_a @ N @ N ) ) ) ).
% char_matrix_closed
thf(fact_829_char__matrix__closed,axiom,
! [A: mat_complex,N: nat,E: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( member_mat_complex @ ( char_c872259621517735348omplex @ A @ E ) @ ( carrier_mat_complex @ N @ N ) ) ) ).
% char_matrix_closed
thf(fact_830_index__mat__swaprows_I3_J,axiom,
! [K: nat,L: nat,A: mat_complex] :
( ( dim_col_complex @ ( gauss_1020679828357514249omplex @ K @ L @ A ) )
= ( dim_col_complex @ A ) ) ).
% index_mat_swaprows(3)
thf(fact_831_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_832_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_833_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_834_sum__mat__0,axiom,
! [Nr: nat,Nc: nat] :
( ( comm_m4056229327131402372at_nat @ ( zero_mat_nat @ Nr @ Nc ) )
= zero_zero_nat ) ).
% sum_mat_0
thf(fact_835_sum__mat__0,axiom,
! [Nr: nat,Nc: nat] :
( ( comm_m8678487124704766304t_real @ ( zero_mat_real @ Nr @ Nc ) )
= zero_zero_real ) ).
% sum_mat_0
thf(fact_836_sum__mat__0,axiom,
! [Nr: nat,Nc: nat] :
( ( comm_m3586542329073673570omplex @ ( zero_mat_complex @ Nr @ Nc ) )
= zero_zero_complex ) ).
% sum_mat_0
thf(fact_837_det__swaprows,axiom,
! [K: nat,N: nat,L: nat,A: mat_a] :
( ( ord_less_nat @ K @ N )
=> ( ( ord_less_nat @ L @ N )
=> ( ( K != L )
=> ( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( det_a @ ( gauss_2482569599970757219rows_a @ K @ L @ A ) )
= ( uminus_uminus_a @ ( det_a @ A ) ) ) ) ) ) ) ).
% det_swaprows
thf(fact_838_det__swaprows,axiom,
! [K: nat,N: nat,L: nat,A: mat_complex] :
( ( ord_less_nat @ K @ N )
=> ( ( ord_less_nat @ L @ N )
=> ( ( K != L )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( det_complex @ ( gauss_1020679828357514249omplex @ K @ L @ A ) )
= ( uminus1482373934393186551omplex @ ( det_complex @ A ) ) ) ) ) ) ) ).
% det_swaprows
thf(fact_839_det__swaprows,axiom,
! [K: nat,N: nat,L: nat,A: mat_real] :
( ( ord_less_nat @ K @ N )
=> ( ( ord_less_nat @ L @ N )
=> ( ( K != L )
=> ( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( det_real @ ( gauss_821192380332421767s_real @ K @ L @ A ) )
= ( uminus_uminus_real @ ( det_real @ A ) ) ) ) ) ) ) ).
% det_swaprows
thf(fact_840_eigenvalue__det,axiom,
! [A: mat_a,N: nat,E: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( char_eigenvalue_a @ A @ E )
= ( ( det_a @ ( char_char_matrix_a @ A @ E ) )
= zero_zero_a ) ) ) ).
% eigenvalue_det
thf(fact_841_eigenvalue__det,axiom,
! [A: mat_real,N: nat,E: real] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( char_eigenvalue_real @ A @ E )
= ( ( det_real @ ( char_c4597223634827269298x_real @ A @ E ) )
= zero_zero_real ) ) ) ).
% eigenvalue_det
thf(fact_842_eigenvalue__det,axiom,
! [A: mat_complex,N: nat,E: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( char_e7032225803028799586omplex @ A @ E )
= ( ( det_complex @ ( char_c872259621517735348omplex @ A @ E ) )
= zero_zero_complex ) ) ) ).
% eigenvalue_det
thf(fact_843_pochhammer__rec__if,axiom,
( comm_s4663373288045622133er_nat
= ( ^ [A5: nat,N3: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ A5 @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A5 @ one_one_nat ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% pochhammer_rec_if
thf(fact_844_pochhammer__rec__if,axiom,
( comm_s2602460028002588243omplex
= ( ^ [A5: complex,N3: nat] : ( if_complex @ ( N3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ A5 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A5 @ one_one_complex ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% pochhammer_rec_if
thf(fact_845_pochhammer__rec__if,axiom,
( comm_s7457072308508201937r_real
= ( ^ [A5: real,N3: nat] : ( if_real @ ( N3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ A5 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A5 @ one_one_real ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% pochhammer_rec_if
thf(fact_846_eigenvalue__imp__nonzero__dim,axiom,
! [A: mat_a,N: nat,Ev: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( char_eigenvalue_a @ A @ Ev )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% eigenvalue_imp_nonzero_dim
thf(fact_847_eigenvalue__imp__nonzero__dim,axiom,
! [A: mat_complex,N: nat,Ev: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( char_e7032225803028799586omplex @ A @ Ev )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% eigenvalue_imp_nonzero_dim
thf(fact_848_pochhammer__rec,axiom,
! [A3: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A3 @ ( suc @ N ) )
= ( times_times_nat @ A3 @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_849_pochhammer__rec,axiom,
! [A3: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ A3 @ ( suc @ N ) )
= ( times_times_complex @ A3 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A3 @ one_one_complex ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_850_pochhammer__rec,axiom,
! [A3: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A3 @ ( suc @ N ) )
= ( times_times_real @ A3 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A3 @ one_one_real ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_851_pochhammer__minus_H,axiom,
! [B3: real,K: nat] :
( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B3 @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B3 ) @ K ) ) ) ).
% pochhammer_minus'
thf(fact_852_pochhammer__minus_H,axiom,
! [B3: complex,K: nat] :
( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B3 @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B3 ) @ K ) ) ) ).
% pochhammer_minus'
thf(fact_853_pochhammer__minus,axiom,
! [B3: real,K: nat] :
( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B3 ) @ K )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B3 @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% pochhammer_minus
thf(fact_854_pochhammer__minus,axiom,
! [B3: complex,K: nat] :
( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B3 ) @ K )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B3 @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% pochhammer_minus
thf(fact_855_pochhammer__absorb__comp,axiom,
! [R2: real,K: nat] :
( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K ) )
= ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_856_pochhammer__absorb__comp,axiom,
! [R2: complex,K: nat] :
( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K ) )
= ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_857_mult__of__nat__commute,axiom,
! [X: nat,Y: complex] :
( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y )
= ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_858_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% of_nat_mult
thf(fact_859_trace__1,axiom,
! [N: nat] :
( ( comple3184165445352484367omplex @ ( one_mat_complex @ N ) )
= ( semiri8010041392384452111omplex @ N ) ) ).
% trace_1
thf(fact_860_Num_Oof__nat__simps_I1_J,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% Num.of_nat_simps(1)
thf(fact_861_Num_Oof__nat__simps_I1_J,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% Num.of_nat_simps(1)
thf(fact_862_Num_Oof__nat__simps_I1_J,axiom,
( ( semiri8010041392384452111omplex @ zero_zero_nat )
= zero_zero_complex ) ).
% Num.of_nat_simps(1)
thf(fact_863_Num_Oof__nat__simps_I4_J,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% Num.of_nat_simps(4)
thf(fact_864_Num_Oof__nat__simps_I4_J,axiom,
! [M: nat,N: nat] :
( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% Num.of_nat_simps(4)
thf(fact_865_Num_Oof__nat__simps_I2_J,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% Num.of_nat_simps(2)
thf(fact_866_Num_Oof__nat__simps_I2_J,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% Num.of_nat_simps(2)
thf(fact_867_Num_Oof__nat__simps_I2_J,axiom,
( ( semiri8010041392384452111omplex @ one_one_nat )
= one_one_complex ) ).
% Num.of_nat_simps(2)
thf(fact_868_Num_Oof__nat__simps_I3_J,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% Num.of_nat_simps(3)
thf(fact_869_Num_Oof__nat__simps_I3_J,axiom,
! [M: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ M ) )
= ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% Num.of_nat_simps(3)
thf(fact_870_Num_Oof__nat__simps_I3_J,axiom,
! [M: nat] :
( ( semiri8010041392384452111omplex @ ( suc @ M ) )
= ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% Num.of_nat_simps(3)
thf(fact_871_pochhammer__Suc,axiom,
! [A3: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A3 @ ( suc @ N ) )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ A3 @ N ) @ ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_872_pochhammer__Suc,axiom,
! [A3: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ A3 @ ( suc @ N ) )
= ( times_times_complex @ ( comm_s2602460028002588243omplex @ A3 @ N ) @ ( plus_plus_complex @ A3 @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_873_pochhammer__rec_H,axiom,
! [Z: real,N: nat] :
( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) )
= ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) ) ).
% pochhammer_rec'
thf(fact_874_pochhammer__rec_H,axiom,
! [Z: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) )
= ( times_times_complex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) ) ).
% pochhammer_rec'
thf(fact_875_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
= zero_zero_real ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_876_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
= zero_zero_complex ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_877_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
= zero_zero_real )
= ( ord_less_nat @ N @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_878_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
= zero_zero_complex )
= ( ord_less_nat @ N @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_879_pochhammer__eq__0__iff,axiom,
! [A3: real,N: nat] :
( ( ( comm_s7457072308508201937r_real @ A3 @ N )
= zero_zero_real )
= ( ? [K4: nat] :
( ( ord_less_nat @ K4 @ N )
& ( A3
= ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K4 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_880_pochhammer__eq__0__iff,axiom,
! [A3: complex,N: nat] :
( ( ( comm_s2602460028002588243omplex @ A3 @ N )
= zero_zero_complex )
= ( ? [K4: nat] :
( ( ord_less_nat @ K4 @ N )
& ( A3
= ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K4 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_881_pochhammer__product_H,axiom,
! [Z: real,N: nat,M: nat] :
( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N @ M ) )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_882_pochhammer__product_H,axiom,
! [Z: complex,N: nat,M: nat] :
( ( comm_s2602460028002588243omplex @ Z @ ( plus_plus_nat @ N @ M ) )
= ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ N ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_883_ex__less__of__nat__mult,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N2: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_884_fps__power__first__eq_H,axiom,
! [A3: formal_Power_fps_nat,N: nat] :
( ( ( formal3720337525774269570th_nat @ A3 @ one_one_nat )
= one_one_nat )
=> ( ( formal3720337525774269570th_nat @ ( power_568658719666546786ps_nat @ A3 @ N ) @ one_one_nat )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( power_power_nat @ ( formal3720337525774269570th_nat @ A3 @ zero_zero_nat ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fps_power_first_eq'
thf(fact_885_fps__power__first__eq_H,axiom,
! [A3: formal3361831859752904756s_real,N: nat] :
( ( ( formal2580924720334399070h_real @ A3 @ one_one_nat )
= one_one_real )
=> ( ( formal2580924720334399070h_real @ ( power_1846127563762588094s_real @ A3 @ N ) @ one_one_nat )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( formal2580924720334399070h_real @ A3 @ zero_zero_nat ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fps_power_first_eq'
thf(fact_886_fps__power__first__eq_H,axiom,
! [A3: formal670952693614245302omplex,N: nat] :
( ( ( formal3666518339620930912omplex @ A3 @ one_one_nat )
= one_one_complex )
=> ( ( formal3666518339620930912omplex @ ( power_8487976900264310848omplex @ A3 @ N ) @ one_one_nat )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ ( formal3666518339620930912omplex @ A3 @ zero_zero_nat ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fps_power_first_eq'
thf(fact_887_fps__power__first,axiom,
! [A3: formal670952693614245302omplex,N: nat] :
( ( formal3666518339620930912omplex @ ( power_8487976900264310848omplex @ A3 @ N ) @ one_one_nat )
= ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ ( formal3666518339620930912omplex @ A3 @ zero_zero_nat ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ ( formal3666518339620930912omplex @ A3 @ one_one_nat ) ) ) ).
% fps_power_first
thf(fact_888_fps__neg__nth,axiom,
! [F: formal3495187508964346165omplex,N: nat] :
( ( formal6566246685975971957omplex @ ( uminus8384360544929349932omplex @ F ) @ N )
= ( uminus467866341702955550omplex @ ( formal6566246685975971957omplex @ F @ N ) ) ) ).
% fps_neg_nth
thf(fact_889_fps__neg__nth,axiom,
! [F: formal670952693614245302omplex,N: nat] :
( ( formal3666518339620930912omplex @ ( uminus9178514011183859839omplex @ F ) @ N )
= ( uminus1482373934393186551omplex @ ( formal3666518339620930912omplex @ F @ N ) ) ) ).
% fps_neg_nth
thf(fact_890_fps__neg__nth,axiom,
! [F: formal3361831859752904756s_real,N: nat] :
( ( formal2580924720334399070h_real @ ( uminus8389970968385878141s_real @ F ) @ N )
= ( uminus_uminus_real @ ( formal2580924720334399070h_real @ F @ N ) ) ) ).
% fps_neg_nth
thf(fact_891_fps__mult__of__nat__nth_I1_J,axiom,
! [K: nat,F: formal670952693614245302omplex,N: nat] :
( ( formal3666518339620930912omplex @ ( times_1444617028055533883omplex @ ( semiri8948773824294531479omplex @ K ) @ F ) @ N )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( formal3666518339620930912omplex @ F @ N ) ) ) ).
% fps_mult_of_nat_nth(1)
thf(fact_892_fps__mult__of__nat__nth_I2_J,axiom,
! [F: formal670952693614245302omplex,K: nat,N: nat] :
( ( formal3666518339620930912omplex @ ( times_1444617028055533883omplex @ F @ ( semiri8948773824294531479omplex @ K ) ) @ N )
= ( times_times_complex @ ( formal3666518339620930912omplex @ F @ N ) @ ( semiri8010041392384452111omplex @ K ) ) ) ).
% fps_mult_of_nat_nth(2)
thf(fact_893_fps__is__right__unit__iff__zeroth__is__right__unit,axiom,
! [F: formal670952693614245302omplex] :
( ( ? [G: formal670952693614245302omplex] :
( one_on1590755018477040891omplex
= ( times_1444617028055533883omplex @ G @ F ) ) )
= ( ? [K4: complex] :
( one_one_complex
= ( times_times_complex @ K4 @ ( formal3666518339620930912omplex @ F @ zero_zero_nat ) ) ) ) ) ).
% fps_is_right_unit_iff_zeroth_is_right_unit
thf(fact_894_fps__is__right__unit__iff__zeroth__is__right__unit,axiom,
! [F: formal3361831859752904756s_real] :
( ( ? [G: formal3361831859752904756s_real] :
( one_on8598947968683843321s_real
= ( times_7561426564079326009s_real @ G @ F ) ) )
= ( ? [K4: real] :
( one_one_real
= ( times_times_real @ K4 @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) ) ) ) ) ).
% fps_is_right_unit_iff_zeroth_is_right_unit
thf(fact_895_fps__is__left__unit__iff__zeroth__is__left__unit,axiom,
! [F: formal670952693614245302omplex] :
( ( ? [G: formal670952693614245302omplex] :
( one_on1590755018477040891omplex
= ( times_1444617028055533883omplex @ F @ G ) ) )
= ( ? [K4: complex] :
( one_one_complex
= ( times_times_complex @ ( formal3666518339620930912omplex @ F @ zero_zero_nat ) @ K4 ) ) ) ) ).
% fps_is_left_unit_iff_zeroth_is_left_unit
thf(fact_896_fps__is__left__unit__iff__zeroth__is__left__unit,axiom,
! [F: formal3361831859752904756s_real] :
( ( ? [G: formal3361831859752904756s_real] :
( one_on8598947968683843321s_real
= ( times_7561426564079326009s_real @ F @ G ) ) )
= ( ? [K4: real] :
( one_one_real
= ( times_times_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) @ K4 ) ) ) ) ).
% fps_is_left_unit_iff_zeroth_is_left_unit
thf(fact_897_fps__mult__nth__1_H,axiom,
! [F: formal670952693614245302omplex,G2: formal670952693614245302omplex] :
( ( formal3666518339620930912omplex @ ( times_1444617028055533883omplex @ F @ G2 ) @ ( suc @ zero_zero_nat ) )
= ( plus_plus_complex @ ( times_times_complex @ ( formal3666518339620930912omplex @ F @ zero_zero_nat ) @ ( formal3666518339620930912omplex @ G2 @ ( suc @ zero_zero_nat ) ) ) @ ( times_times_complex @ ( formal3666518339620930912omplex @ F @ ( suc @ zero_zero_nat ) ) @ ( formal3666518339620930912omplex @ G2 @ zero_zero_nat ) ) ) ) ).
% fps_mult_nth_1'
thf(fact_898_fps__mult__nth__1_H,axiom,
! [F: formal3361831859752904756s_real,G2: formal3361831859752904756s_real] :
( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ F @ G2 ) @ ( suc @ zero_zero_nat ) )
= ( plus_plus_real @ ( times_times_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) @ ( formal2580924720334399070h_real @ G2 @ ( suc @ zero_zero_nat ) ) ) @ ( times_times_real @ ( formal2580924720334399070h_real @ F @ ( suc @ zero_zero_nat ) ) @ ( formal2580924720334399070h_real @ G2 @ zero_zero_nat ) ) ) ) ).
% fps_mult_nth_1'
thf(fact_899_fps__mult__nth__1,axiom,
! [F: formal670952693614245302omplex,G2: formal670952693614245302omplex] :
( ( formal3666518339620930912omplex @ ( times_1444617028055533883omplex @ F @ G2 ) @ one_one_nat )
= ( plus_plus_complex @ ( times_times_complex @ ( formal3666518339620930912omplex @ F @ zero_zero_nat ) @ ( formal3666518339620930912omplex @ G2 @ one_one_nat ) ) @ ( times_times_complex @ ( formal3666518339620930912omplex @ F @ one_one_nat ) @ ( formal3666518339620930912omplex @ G2 @ zero_zero_nat ) ) ) ) ).
% fps_mult_nth_1
thf(fact_900_fps__mult__nth__1,axiom,
! [F: formal3361831859752904756s_real,G2: formal3361831859752904756s_real] :
( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ F @ G2 ) @ one_one_nat )
= ( plus_plus_real @ ( times_times_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) @ ( formal2580924720334399070h_real @ G2 @ one_one_nat ) ) @ ( times_times_real @ ( formal2580924720334399070h_real @ F @ one_one_nat ) @ ( formal2580924720334399070h_real @ G2 @ zero_zero_nat ) ) ) ) ).
% fps_mult_nth_1
thf(fact_901_fps__power__first__eq,axiom,
! [A3: formal_Power_fps_nat,N: nat] :
( ( ( formal3720337525774269570th_nat @ A3 @ zero_zero_nat )
= one_one_nat )
=> ( ( formal3720337525774269570th_nat @ ( power_568658719666546786ps_nat @ A3 @ N ) @ one_one_nat )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( formal3720337525774269570th_nat @ A3 @ one_one_nat ) ) ) ) ).
% fps_power_first_eq
thf(fact_902_fps__power__first__eq,axiom,
! [A3: formal3361831859752904756s_real,N: nat] :
( ( ( formal2580924720334399070h_real @ A3 @ zero_zero_nat )
= one_one_real )
=> ( ( formal2580924720334399070h_real @ ( power_1846127563762588094s_real @ A3 @ N ) @ one_one_nat )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( formal2580924720334399070h_real @ A3 @ one_one_nat ) ) ) ) ).
% fps_power_first_eq
thf(fact_903_fps__power__first__eq,axiom,
! [A3: formal670952693614245302omplex,N: nat] :
( ( ( formal3666518339620930912omplex @ A3 @ zero_zero_nat )
= one_one_complex )
=> ( ( formal3666518339620930912omplex @ ( power_8487976900264310848omplex @ A3 @ N ) @ one_one_nat )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( formal3666518339620930912omplex @ A3 @ one_one_nat ) ) ) ) ).
% fps_power_first_eq
thf(fact_904_radical__mult__distrib,axiom,
! [K: nat,R2: nat > real > real,A3: formal3361831859752904756s_real,B3: formal3361831859752904756s_real] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( power_power_real @ ( R2 @ K @ ( formal2580924720334399070h_real @ A3 @ zero_zero_nat ) ) @ K )
= ( formal2580924720334399070h_real @ A3 @ zero_zero_nat ) )
=> ( ( ( power_power_real @ ( R2 @ K @ ( formal2580924720334399070h_real @ B3 @ zero_zero_nat ) ) @ K )
= ( formal2580924720334399070h_real @ B3 @ zero_zero_nat ) )
=> ( ( ( formal2580924720334399070h_real @ A3 @ zero_zero_nat )
!= zero_zero_real )
=> ( ( ( formal2580924720334399070h_real @ B3 @ zero_zero_nat )
!= zero_zero_real )
=> ( ( ( R2 @ K @ ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ A3 @ B3 ) @ zero_zero_nat ) )
= ( times_times_real @ ( R2 @ K @ ( formal2580924720334399070h_real @ A3 @ zero_zero_nat ) ) @ ( R2 @ K @ ( formal2580924720334399070h_real @ B3 @ zero_zero_nat ) ) ) )
= ( ( formal8604817403481219167l_real @ R2 @ K @ ( times_7561426564079326009s_real @ A3 @ B3 ) )
= ( times_7561426564079326009s_real @ ( formal8604817403481219167l_real @ R2 @ K @ A3 ) @ ( formal8604817403481219167l_real @ R2 @ K @ B3 ) ) ) ) ) ) ) ) ) ).
% radical_mult_distrib
thf(fact_905_radical__mult__distrib,axiom,
! [K: nat,R2: nat > complex > complex,A3: formal670952693614245302omplex,B3: formal670952693614245302omplex] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( power_power_complex @ ( R2 @ K @ ( formal3666518339620930912omplex @ A3 @ zero_zero_nat ) ) @ K )
= ( formal3666518339620930912omplex @ A3 @ zero_zero_nat ) )
=> ( ( ( power_power_complex @ ( R2 @ K @ ( formal3666518339620930912omplex @ B3 @ zero_zero_nat ) ) @ K )
= ( formal3666518339620930912omplex @ B3 @ zero_zero_nat ) )
=> ( ( ( formal3666518339620930912omplex @ A3 @ zero_zero_nat )
!= zero_zero_complex )
=> ( ( ( formal3666518339620930912omplex @ B3 @ zero_zero_nat )
!= zero_zero_complex )
=> ( ( ( R2 @ K @ ( formal3666518339620930912omplex @ ( times_1444617028055533883omplex @ A3 @ B3 ) @ zero_zero_nat ) )
= ( times_times_complex @ ( R2 @ K @ ( formal3666518339620930912omplex @ A3 @ zero_zero_nat ) ) @ ( R2 @ K @ ( formal3666518339620930912omplex @ B3 @ zero_zero_nat ) ) ) )
= ( ( formal878984433073253857omplex @ R2 @ K @ ( times_1444617028055533883omplex @ A3 @ B3 ) )
= ( times_1444617028055533883omplex @ ( formal878984433073253857omplex @ R2 @ K @ A3 ) @ ( formal878984433073253857omplex @ R2 @ K @ B3 ) ) ) ) ) ) ) ) ) ).
% radical_mult_distrib
thf(fact_906_fps__XDp__nth,axiom,
! [C2: real,A3: formal3361831859752904756s_real,N: nat] :
( ( formal2580924720334399070h_real @ ( formal2839450981996073129p_real @ C2 @ A3 ) @ N )
= ( times_times_real @ ( plus_plus_real @ C2 @ ( semiri5074537144036343181t_real @ N ) ) @ ( formal2580924720334399070h_real @ A3 @ N ) ) ) ).
% fps_XDp_nth
thf(fact_907_fps__XDp__nth,axiom,
! [C2: complex,A3: formal670952693614245302omplex,N: nat] :
( ( formal3666518339620930912omplex @ ( formal5989188765539143467omplex @ C2 @ A3 ) @ N )
= ( times_times_complex @ ( plus_plus_complex @ C2 @ ( semiri8010041392384452111omplex @ N ) ) @ ( formal3666518339620930912omplex @ A3 @ N ) ) ) ).
% fps_XDp_nth
thf(fact_908_fps__ln__nth,axiom,
! [N: nat,C2: real] :
( ( ( N = zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( formal8688746759596762231n_real @ C2 ) @ N )
= zero_zero_real ) )
& ( ( N != zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( formal8688746759596762231n_real @ C2 ) @ N )
= ( times_times_real @ ( divide_divide_real @ one_one_real @ C2 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% fps_ln_nth
thf(fact_909_fps__ln__nth,axiom,
! [N: nat,C2: complex] :
( ( ( N = zero_zero_nat )
=> ( ( formal3666518339620930912omplex @ ( formal6928690614366948857omplex @ C2 ) @ N )
= zero_zero_complex ) )
& ( ( N != zero_zero_nat )
=> ( ( formal3666518339620930912omplex @ ( formal6928690614366948857omplex @ C2 ) @ N )
= ( times_times_complex @ ( divide1717551699836669952omplex @ one_one_complex @ C2 ) @ ( divide1717551699836669952omplex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ) ) ) ).
% fps_ln_nth
thf(fact_910_div__0,axiom,
! [A3: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% div_0
thf(fact_911_div__0,axiom,
! [A3: real] :
( ( divide_divide_real @ zero_zero_real @ A3 )
= zero_zero_real ) ).
% div_0
thf(fact_912_div__0,axiom,
! [A3: complex] :
( ( divide1717551699836669952omplex @ zero_zero_complex @ A3 )
= zero_zero_complex ) ).
% div_0
thf(fact_913_div__by__0,axiom,
! [A3: nat] :
( ( divide_divide_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_914_div__by__0,axiom,
! [A3: real] :
( ( divide_divide_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_915_div__by__0,axiom,
! [A3: complex] :
( ( divide1717551699836669952omplex @ A3 @ zero_zero_complex )
= zero_zero_complex ) ).
% div_by_0
thf(fact_916_div__by__1,axiom,
! [A3: nat] :
( ( divide_divide_nat @ A3 @ one_one_nat )
= A3 ) ).
% div_by_1
thf(fact_917_div__by__1,axiom,
! [A3: real] :
( ( divide_divide_real @ A3 @ one_one_real )
= A3 ) ).
% div_by_1
thf(fact_918_div__by__1,axiom,
! [A3: complex] :
( ( divide1717551699836669952omplex @ A3 @ one_one_complex )
= A3 ) ).
% div_by_1
thf(fact_919_nonzero__mult__div__cancel__left,axiom,
! [A3: nat,B3: nat] :
( ( A3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A3 @ B3 ) @ A3 )
= B3 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_920_nonzero__mult__div__cancel__left,axiom,
! [A3: real,B3: real] :
( ( A3 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A3 @ B3 ) @ A3 )
= B3 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_921_nonzero__mult__div__cancel__left,axiom,
! [A3: complex,B3: complex] :
( ( A3 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ B3 ) @ A3 )
= B3 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_922_nonzero__mult__div__cancel__right,axiom,
! [B3: nat,A3: nat] :
( ( B3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_923_nonzero__mult__div__cancel__right,axiom,
! [B3: real,A3: real] :
( ( B3 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_924_nonzero__mult__div__cancel__right,axiom,
! [B3: complex,A3: complex] :
( ( B3 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_925_div__self,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
=> ( ( divide_divide_nat @ A3 @ A3 )
= one_one_nat ) ) ).
% div_self
thf(fact_926_div__self,axiom,
! [A3: real] :
( ( A3 != zero_zero_real )
=> ( ( divide_divide_real @ A3 @ A3 )
= one_one_real ) ) ).
% div_self
thf(fact_927_div__self,axiom,
! [A3: complex] :
( ( A3 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A3 @ A3 )
= one_one_complex ) ) ).
% div_self
thf(fact_928_distrib__left__NO__MATCH,axiom,
! [X: complex,Y: complex,A3: complex,B3: complex,C2: complex] :
( ( nO_MAT8947977539597988553omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ A3 )
=> ( ( times_times_complex @ A3 @ ( plus_plus_complex @ B3 @ C2 ) )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ B3 ) @ ( times_times_complex @ A3 @ C2 ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_929_distrib__left__NO__MATCH,axiom,
! [X: complex,Y: complex,A3: real,B3: real,C2: real] :
( ( nO_MAT9165723751696580935x_real @ ( divide1717551699836669952omplex @ X @ Y ) @ A3 )
=> ( ( times_times_real @ A3 @ ( plus_plus_real @ B3 @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A3 @ B3 ) @ ( times_times_real @ A3 @ C2 ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_930_distrib__right__NO__MATCH,axiom,
! [X: complex,Y: complex,C2: complex,A3: complex,B3: complex] :
( ( nO_MAT8947977539597988553omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ C2 )
=> ( ( times_times_complex @ ( plus_plus_complex @ A3 @ B3 ) @ C2 )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B3 @ C2 ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_931_distrib__right__NO__MATCH,axiom,
! [X: complex,Y: complex,C2: real,A3: real,B3: real] :
( ( nO_MAT9165723751696580935x_real @ ( divide1717551699836669952omplex @ X @ Y ) @ C2 )
=> ( ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_932_vector__space__over__itself_Oscale__right__distrib__NO__MATCH,axiom,
! [X: real,Y: real,A3: real] :
( ( nO_MATCH_real_real @ ( divide_divide_real @ X @ Y ) @ A3 )
=> ( ( times_times_real @ A3 @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ A3 @ X ) @ ( times_times_real @ A3 @ Y ) ) ) ) ).
% vector_space_over_itself.scale_right_distrib_NO_MATCH
thf(fact_933_vector__space__over__itself_Oscale__right__distrib__NO__MATCH,axiom,
! [X: complex,Y: complex,A3: complex] :
( ( nO_MAT8947977539597988553omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ A3 )
=> ( ( times_times_complex @ A3 @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ X ) @ ( times_times_complex @ A3 @ Y ) ) ) ) ).
% vector_space_over_itself.scale_right_distrib_NO_MATCH
thf(fact_934_left__diff__distrib__NO__MATCH,axiom,
! [X: complex,Y: complex,C2: real,A3: real,B3: real] :
( ( nO_MAT9165723751696580935x_real @ ( divide1717551699836669952omplex @ X @ Y ) @ C2 )
=> ( ( times_times_real @ ( minus_minus_real @ A3 @ B3 ) @ C2 )
= ( minus_minus_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_935_right__diff__distrib__NO__MATCH,axiom,
! [X: complex,Y: complex,A3: real,B3: real,C2: real] :
( ( nO_MAT9165723751696580935x_real @ ( divide1717551699836669952omplex @ X @ Y ) @ A3 )
=> ( ( times_times_real @ A3 @ ( minus_minus_real @ B3 @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A3 @ B3 ) @ ( times_times_real @ A3 @ C2 ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_936_vector__space__over__itself_Oscale__right__diff__distrib__NO__MATCH,axiom,
! [X: real,Y: real,A3: real] :
( ( nO_MATCH_real_real @ ( divide_divide_real @ X @ Y ) @ A3 )
=> ( ( times_times_real @ A3 @ ( minus_minus_real @ X @ Y ) )
= ( minus_minus_real @ ( times_times_real @ A3 @ X ) @ ( times_times_real @ A3 @ Y ) ) ) ) ).
% vector_space_over_itself.scale_right_diff_distrib_NO_MATCH
thf(fact_937_vector__space__over__itself_Oscale__right__diff__distrib__NO__MATCH,axiom,
! [X: complex,Y: complex,A3: complex] :
( ( nO_MAT8947977539597988553omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ A3 )
=> ( ( times_times_complex @ A3 @ ( minus_minus_complex @ X @ Y ) )
= ( minus_minus_complex @ ( times_times_complex @ A3 @ X ) @ ( times_times_complex @ A3 @ Y ) ) ) ) ).
% vector_space_over_itself.scale_right_diff_distrib_NO_MATCH
thf(fact_938_minus__divide__diff__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_939_minus__divide__diff__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_940_add__divide__eq__if__simps_I5_J,axiom,
! [Z: complex,A3: complex,B3: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A3 @ Z ) @ B3 )
= ( uminus1482373934393186551omplex @ B3 ) ) )
& ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A3 @ Z ) @ B3 )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ A3 @ ( times_times_complex @ B3 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_941_add__divide__eq__if__simps_I5_J,axiom,
! [Z: real,A3: real,B3: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A3 @ Z ) @ B3 )
= ( uminus_uminus_real @ B3 ) ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A3 @ Z ) @ B3 )
= ( divide_divide_real @ ( minus_minus_real @ A3 @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_942_add__divide__eq__if__simps_I6_J,axiom,
! [Z: complex,A3: complex,B3: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ Z ) ) @ B3 )
= ( uminus1482373934393186551omplex @ B3 ) ) )
& ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ Z ) ) @ B3 )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A3 ) @ ( times_times_complex @ B3 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_943_add__divide__eq__if__simps_I6_J,axiom,
! [Z: real,A3: real,B3: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A3 @ Z ) ) @ B3 )
= ( uminus_uminus_real @ B3 ) ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A3 @ Z ) ) @ B3 )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A3 ) @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_944_times__divide__eq__left,axiom,
! [B3: complex,C2: complex,A3: complex] :
( ( times_times_complex @ ( divide1717551699836669952omplex @ B3 @ C2 ) @ A3 )
= ( divide1717551699836669952omplex @ ( times_times_complex @ B3 @ A3 ) @ C2 ) ) ).
% times_divide_eq_left
thf(fact_945_divide__divide__eq__left,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A3 @ B3 ) @ C2 )
= ( divide1717551699836669952omplex @ A3 @ ( times_times_complex @ B3 @ C2 ) ) ) ).
% divide_divide_eq_left
thf(fact_946_times__divide__times__eq,axiom,
! [X: complex,Y: complex,Z: complex,W: complex] :
( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_947_divide__divide__eq__right,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( divide1717551699836669952omplex @ A3 @ ( divide1717551699836669952omplex @ B3 @ C2 ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ C2 ) @ B3 ) ) ).
% divide_divide_eq_right
thf(fact_948_divide__divide__times__eq,axiom,
! [X: complex,Y: complex,Z: complex,W: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_949_times__divide__eq__right,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( times_times_complex @ A3 @ ( divide1717551699836669952omplex @ B3 @ C2 ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ B3 ) @ C2 ) ) ).
% times_divide_eq_right
thf(fact_950_divide__divide__eq__left_H,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A3 @ B3 ) @ C2 )
= ( divide1717551699836669952omplex @ A3 @ ( times_times_complex @ C2 @ B3 ) ) ) ).
% divide_divide_eq_left'
thf(fact_951_divide__minus__right,axiom,
! [A3: complex,B3: complex] :
( ( divide1717551699836669952omplex @ A3 @ ( uminus1482373934393186551omplex @ B3 ) )
= ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ B3 ) ) ) ).
% divide_minus_right
thf(fact_952_divide__minus__right,axiom,
! [A3: real,B3: real] :
( ( divide_divide_real @ A3 @ ( uminus_uminus_real @ B3 ) )
= ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) ) ) ).
% divide_minus_right
thf(fact_953_minus__divide__divide,axiom,
! [A3: complex,B3: complex] :
( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A3 ) @ ( uminus1482373934393186551omplex @ B3 ) )
= ( divide1717551699836669952omplex @ A3 @ B3 ) ) ).
% minus_divide_divide
thf(fact_954_minus__divide__divide,axiom,
! [A3: real,B3: real] :
( ( divide_divide_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) )
= ( divide_divide_real @ A3 @ B3 ) ) ).
% minus_divide_divide
thf(fact_955_divide__minus__left,axiom,
! [A3: complex,B3: complex] :
( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A3 ) @ B3 )
= ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ B3 ) ) ) ).
% divide_minus_left
thf(fact_956_divide__minus__left,axiom,
! [A3: real,B3: real] :
( ( divide_divide_real @ ( uminus_uminus_real @ A3 ) @ B3 )
= ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) ) ) ).
% divide_minus_left
thf(fact_957_frac__eq__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y )
= ( divide_divide_real @ W @ Z ) )
= ( ( times_times_real @ X @ Z )
= ( times_times_real @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_958_frac__eq__eq,axiom,
! [Y: complex,Z: complex,X: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ X @ Y )
= ( divide1717551699836669952omplex @ W @ Z ) )
= ( ( times_times_complex @ X @ Z )
= ( times_times_complex @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_959_divide__eq__eq,axiom,
! [B3: real,C2: real,A3: real] :
( ( ( divide_divide_real @ B3 @ C2 )
= A3 )
= ( ( ( C2 != zero_zero_real )
=> ( B3
= ( times_times_real @ A3 @ C2 ) ) )
& ( ( C2 = zero_zero_real )
=> ( A3 = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_960_divide__eq__eq,axiom,
! [B3: complex,C2: complex,A3: complex] :
( ( ( divide1717551699836669952omplex @ B3 @ C2 )
= A3 )
= ( ( ( C2 != zero_zero_complex )
=> ( B3
= ( times_times_complex @ A3 @ C2 ) ) )
& ( ( C2 = zero_zero_complex )
=> ( A3 = zero_zero_complex ) ) ) ) ).
% divide_eq_eq
thf(fact_961_eq__divide__eq,axiom,
! [A3: real,B3: real,C2: real] :
( ( A3
= ( divide_divide_real @ B3 @ C2 ) )
= ( ( ( C2 != zero_zero_real )
=> ( ( times_times_real @ A3 @ C2 )
= B3 ) )
& ( ( C2 = zero_zero_real )
=> ( A3 = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_962_eq__divide__eq,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( A3
= ( divide1717551699836669952omplex @ B3 @ C2 ) )
= ( ( ( C2 != zero_zero_complex )
=> ( ( times_times_complex @ A3 @ C2 )
= B3 ) )
& ( ( C2 = zero_zero_complex )
=> ( A3 = zero_zero_complex ) ) ) ) ).
% eq_divide_eq
thf(fact_963_divide__eq__imp,axiom,
! [C2: real,B3: real,A3: real] :
( ( C2 != zero_zero_real )
=> ( ( B3
= ( times_times_real @ A3 @ C2 ) )
=> ( ( divide_divide_real @ B3 @ C2 )
= A3 ) ) ) ).
% divide_eq_imp
thf(fact_964_divide__eq__imp,axiom,
! [C2: complex,B3: complex,A3: complex] :
( ( C2 != zero_zero_complex )
=> ( ( B3
= ( times_times_complex @ A3 @ C2 ) )
=> ( ( divide1717551699836669952omplex @ B3 @ C2 )
= A3 ) ) ) ).
% divide_eq_imp
thf(fact_965_eq__divide__imp,axiom,
! [C2: real,A3: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ C2 )
= B3 )
=> ( A3
= ( divide_divide_real @ B3 @ C2 ) ) ) ) ).
% eq_divide_imp
thf(fact_966_eq__divide__imp,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ C2 )
= B3 )
=> ( A3
= ( divide1717551699836669952omplex @ B3 @ C2 ) ) ) ) ).
% eq_divide_imp
thf(fact_967_nonzero__divide__eq__eq,axiom,
! [C2: real,B3: real,A3: real] :
( ( C2 != zero_zero_real )
=> ( ( ( divide_divide_real @ B3 @ C2 )
= A3 )
= ( B3
= ( times_times_real @ A3 @ C2 ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_968_nonzero__divide__eq__eq,axiom,
! [C2: complex,B3: complex,A3: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ B3 @ C2 )
= A3 )
= ( B3
= ( times_times_complex @ A3 @ C2 ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_969_nonzero__eq__divide__eq,axiom,
! [C2: real,A3: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( A3
= ( divide_divide_real @ B3 @ C2 ) )
= ( ( times_times_real @ A3 @ C2 )
= B3 ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_970_nonzero__eq__divide__eq,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( C2 != zero_zero_complex )
=> ( ( A3
= ( divide1717551699836669952omplex @ B3 @ C2 ) )
= ( ( times_times_complex @ A3 @ C2 )
= B3 ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_971_mult__divide__mult__cancel__left,axiom,
! [C2: real,A3: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) )
= ( divide_divide_real @ A3 @ B3 ) ) ) ).
% mult_divide_mult_cancel_left
thf(fact_972_mult__divide__mult__cancel__left,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( C2 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ C2 @ B3 ) )
= ( divide1717551699836669952omplex @ A3 @ B3 ) ) ) ).
% mult_divide_mult_cancel_left
thf(fact_973_mult__divide__mult__cancel__right,axiom,
! [C2: real,A3: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) )
= ( divide_divide_real @ A3 @ B3 ) ) ) ).
% mult_divide_mult_cancel_right
thf(fact_974_mult__divide__mult__cancel__right,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( C2 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B3 @ C2 ) )
= ( divide1717551699836669952omplex @ A3 @ B3 ) ) ) ).
% mult_divide_mult_cancel_right
thf(fact_975_mult__divide__mult__cancel__left__if,axiom,
! [C2: real,A3: real,B3: real] :
( ( ( C2 = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) )
= zero_zero_real ) )
& ( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) )
= ( divide_divide_real @ A3 @ B3 ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_976_mult__divide__mult__cancel__left__if,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( ( C2 = zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ C2 @ B3 ) )
= zero_zero_complex ) )
& ( ( C2 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ C2 @ B3 ) )
= ( divide1717551699836669952omplex @ A3 @ B3 ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_977_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C2: real,A3: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ B3 @ C2 ) )
= ( divide_divide_real @ A3 @ B3 ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_978_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( C2 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ B3 @ C2 ) )
= ( divide1717551699836669952omplex @ A3 @ B3 ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_979_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C2: real,A3: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ C2 @ B3 ) )
= ( divide_divide_real @ A3 @ B3 ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_980_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( C2 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ C2 @ B3 ) )
= ( divide1717551699836669952omplex @ A3 @ B3 ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_981_nonzero__minus__divide__right,axiom,
! [B3: complex,A3: complex] :
( ( B3 != zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ B3 ) )
= ( divide1717551699836669952omplex @ A3 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_982_nonzero__minus__divide__right,axiom,
! [B3: real,A3: real] :
( ( B3 != zero_zero_real )
=> ( ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) )
= ( divide_divide_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_983_nonzero__minus__divide__divide,axiom,
! [B3: complex,A3: complex] :
( ( B3 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A3 ) @ ( uminus1482373934393186551omplex @ B3 ) )
= ( divide1717551699836669952omplex @ A3 @ B3 ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_984_nonzero__minus__divide__divide,axiom,
! [B3: real,A3: real] :
( ( B3 != zero_zero_real )
=> ( ( divide_divide_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) )
= ( divide_divide_real @ A3 @ B3 ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_985_divide__minus1,axiom,
! [X: complex] :
( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ X ) ) ).
% divide_minus1
thf(fact_986_divide__minus1,axiom,
! [X: real] :
( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ X ) ) ).
% divide_minus1
thf(fact_987_divide__strict__left__mono__neg,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
=> ( ord_less_real @ ( divide_divide_real @ C2 @ A3 ) @ ( divide_divide_real @ C2 @ B3 ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_988_divide__strict__left__mono,axiom,
! [B3: real,A3: real,C2: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
=> ( ord_less_real @ ( divide_divide_real @ C2 @ A3 ) @ ( divide_divide_real @ C2 @ B3 ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_989_mult__imp__less__div__pos,axiom,
! [Y: real,Z: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( times_times_real @ Z @ Y ) @ X )
=> ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_990_mult__imp__div__pos__less,axiom,
! [Y: real,X: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y ) )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_less
thf(fact_991_pos__less__divide__eq,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ A3 @ ( divide_divide_real @ B3 @ C2 ) )
= ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ B3 ) ) ) ).
% pos_less_divide_eq
thf(fact_992_pos__divide__less__eq,axiom,
! [C2: real,B3: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( divide_divide_real @ B3 @ C2 ) @ A3 )
= ( ord_less_real @ B3 @ ( times_times_real @ A3 @ C2 ) ) ) ) ).
% pos_divide_less_eq
thf(fact_993_neg__less__divide__eq,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ A3 @ ( divide_divide_real @ B3 @ C2 ) )
= ( ord_less_real @ B3 @ ( times_times_real @ A3 @ C2 ) ) ) ) ).
% neg_less_divide_eq
thf(fact_994_neg__divide__less__eq,axiom,
! [C2: real,B3: real,A3: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B3 @ C2 ) @ A3 )
= ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ B3 ) ) ) ).
% neg_divide_less_eq
thf(fact_995_less__divide__eq,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ ( divide_divide_real @ B3 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ B3 ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B3 @ ( times_times_real @ A3 @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ A3 @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_996_divide__less__eq,axiom,
! [B3: real,C2: real,A3: real] :
( ( ord_less_real @ ( divide_divide_real @ B3 @ C2 ) @ A3 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ B3 @ ( times_times_real @ A3 @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ B3 ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A3 ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_997_add__divide__eq__if__simps_I2_J,axiom,
! [Z: real,A3: real,B3: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A3 @ Z ) @ B3 )
= B3 ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A3 @ Z ) @ B3 )
= ( divide_divide_real @ ( plus_plus_real @ A3 @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_998_add__divide__eq__if__simps_I2_J,axiom,
! [Z: complex,A3: complex,B3: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A3 @ Z ) @ B3 )
= B3 ) )
& ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A3 @ Z ) @ B3 )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ A3 @ ( times_times_complex @ B3 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_999_add__divide__eq__if__simps_I1_J,axiom,
! [Z: real,A3: real,B3: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ A3 @ ( divide_divide_real @ B3 @ Z ) )
= A3 ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ A3 @ ( divide_divide_real @ B3 @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A3 @ Z ) @ B3 ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1000_add__divide__eq__if__simps_I1_J,axiom,
! [Z: complex,A3: complex,B3: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( plus_plus_complex @ A3 @ ( divide1717551699836669952omplex @ B3 @ Z ) )
= A3 ) )
& ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ A3 @ ( divide1717551699836669952omplex @ B3 @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A3 @ Z ) @ B3 ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1001_add__frac__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_1002_add__frac__eq,axiom,
! [Y: complex,Z: complex,X: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_1003_add__frac__num,axiom,
! [Y: real,X: real,Z: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_1004_add__frac__num,axiom,
! [Y: complex,X: complex,Z: complex] :
( ( Y != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_1005_add__num__frac,axiom,
! [Y: real,Z: real,X: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y ) )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_1006_add__num__frac,axiom,
! [Y: complex,Z: complex,X: complex] :
( ( Y != zero_zero_complex )
=> ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X @ Y ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_1007_add__divide__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_1008_add__divide__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_1009_divide__add__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_1010_divide__add__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_1011_nonzero__divide__mult__cancel__right,axiom,
! [B3: real,A3: real] :
( ( B3 != zero_zero_real )
=> ( ( divide_divide_real @ B3 @ ( times_times_real @ A3 @ B3 ) )
= ( divide_divide_real @ one_one_real @ A3 ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1012_nonzero__divide__mult__cancel__right,axiom,
! [B3: complex,A3: complex] :
( ( B3 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ B3 @ ( times_times_complex @ A3 @ B3 ) )
= ( divide1717551699836669952omplex @ one_one_complex @ A3 ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1013_nonzero__divide__mult__cancel__left,axiom,
! [A3: real,B3: real] :
( ( A3 != zero_zero_real )
=> ( ( divide_divide_real @ A3 @ ( times_times_real @ A3 @ B3 ) )
= ( divide_divide_real @ one_one_real @ B3 ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1014_nonzero__divide__mult__cancel__left,axiom,
! [A3: complex,B3: complex] :
( ( A3 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A3 @ ( times_times_complex @ A3 @ B3 ) )
= ( divide1717551699836669952omplex @ one_one_complex @ B3 ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1015_divide__diff__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_1016_divide__diff__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_1017_diff__divide__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_1018_diff__divide__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_1019_diff__frac__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1020_diff__frac__eq,axiom,
! [Y: complex,Z: complex,X: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1021_add__divide__eq__if__simps_I4_J,axiom,
! [Z: real,A3: real,B3: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ A3 @ ( divide_divide_real @ B3 @ Z ) )
= A3 ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ A3 @ ( divide_divide_real @ B3 @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A3 @ Z ) @ B3 ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1022_add__divide__eq__if__simps_I4_J,axiom,
! [Z: complex,A3: complex,B3: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( minus_minus_complex @ A3 @ ( divide1717551699836669952omplex @ B3 @ Z ) )
= A3 ) )
& ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ A3 @ ( divide1717551699836669952omplex @ B3 @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A3 @ Z ) @ B3 ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1023_nonzero__neg__divide__eq__eq2,axiom,
! [B3: complex,C2: complex,A3: complex] :
( ( B3 != zero_zero_complex )
=> ( ( C2
= ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ B3 ) ) )
= ( ( times_times_complex @ C2 @ B3 )
= ( uminus1482373934393186551omplex @ A3 ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_1024_nonzero__neg__divide__eq__eq2,axiom,
! [B3: real,C2: real,A3: real] :
( ( B3 != zero_zero_real )
=> ( ( C2
= ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) ) )
= ( ( times_times_real @ C2 @ B3 )
= ( uminus_uminus_real @ A3 ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_1025_nonzero__neg__divide__eq__eq,axiom,
! [B3: complex,A3: complex,C2: complex] :
( ( B3 != zero_zero_complex )
=> ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ B3 ) )
= C2 )
= ( ( uminus1482373934393186551omplex @ A3 )
= ( times_times_complex @ C2 @ B3 ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_1026_nonzero__neg__divide__eq__eq,axiom,
! [B3: real,A3: real,C2: real] :
( ( B3 != zero_zero_real )
=> ( ( ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) )
= C2 )
= ( ( uminus_uminus_real @ A3 )
= ( times_times_real @ C2 @ B3 ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_1027_minus__divide__eq__eq,axiom,
! [B3: complex,C2: complex,A3: complex] :
( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B3 @ C2 ) )
= A3 )
= ( ( ( C2 != zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ B3 )
= ( times_times_complex @ A3 @ C2 ) ) )
& ( ( C2 = zero_zero_complex )
=> ( A3 = zero_zero_complex ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_1028_minus__divide__eq__eq,axiom,
! [B3: real,C2: real,A3: real] :
( ( ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C2 ) )
= A3 )
= ( ( ( C2 != zero_zero_real )
=> ( ( uminus_uminus_real @ B3 )
= ( times_times_real @ A3 @ C2 ) ) )
& ( ( C2 = zero_zero_real )
=> ( A3 = zero_zero_real ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_1029_eq__minus__divide__eq,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( A3
= ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B3 @ C2 ) ) )
= ( ( ( C2 != zero_zero_complex )
=> ( ( times_times_complex @ A3 @ C2 )
= ( uminus1482373934393186551omplex @ B3 ) ) )
& ( ( C2 = zero_zero_complex )
=> ( A3 = zero_zero_complex ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_1030_eq__minus__divide__eq,axiom,
! [A3: real,B3: real,C2: real] :
( ( A3
= ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C2 ) ) )
= ( ( ( C2 != zero_zero_real )
=> ( ( times_times_real @ A3 @ C2 )
= ( uminus_uminus_real @ B3 ) ) )
& ( ( C2 = zero_zero_real )
=> ( A3 = zero_zero_real ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_1031_divide__eq__minus__1__iff,axiom,
! [A3: complex,B3: complex] :
( ( ( divide1717551699836669952omplex @ A3 @ B3 )
= ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( ( B3 != zero_zero_complex )
& ( A3
= ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_1032_divide__eq__minus__1__iff,axiom,
! [A3: real,B3: real] :
( ( ( divide_divide_real @ A3 @ B3 )
= ( uminus_uminus_real @ one_one_real ) )
= ( ( B3 != zero_zero_real )
& ( A3
= ( uminus_uminus_real @ B3 ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_1033_frac__less__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% frac_less_eq
thf(fact_1034_less__minus__divide__eq,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C2 ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( uminus_uminus_real @ B3 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A3 @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ A3 @ zero_zero_real ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_1035_minus__divide__less__eq,axiom,
! [B3: real,C2: real,A3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C2 ) ) @ A3 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A3 @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( uminus_uminus_real @ B3 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A3 ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_1036_neg__less__minus__divide__eq,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ A3 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C2 ) ) )
= ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A3 @ C2 ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_1037_neg__minus__divide__less__eq,axiom,
! [C2: real,B3: real,A3: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C2 ) ) @ A3 )
= ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_1038_pos__less__minus__divide__eq,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ A3 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C2 ) ) )
= ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_1039_pos__minus__divide__less__eq,axiom,
! [C2: real,B3: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C2 ) ) @ A3 )
= ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A3 @ C2 ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_1040_add__divide__eq__if__simps_I3_J,axiom,
! [Z: complex,A3: complex,B3: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ Z ) ) @ B3 )
= B3 ) )
& ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ Z ) ) @ B3 )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A3 ) @ ( times_times_complex @ B3 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_1041_add__divide__eq__if__simps_I3_J,axiom,
! [Z: real,A3: real,B3: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A3 @ Z ) ) @ B3 )
= B3 ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A3 @ Z ) ) @ B3 )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_1042_minus__divide__add__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_1043_minus__divide__add__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_1044_div__mult__self1,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( B3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ ( times_times_nat @ C2 @ B3 ) ) @ B3 )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).
% div_mult_self1
thf(fact_1045_div__mult__self2,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( B3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ ( times_times_nat @ B3 @ C2 ) ) @ B3 )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).
% div_mult_self2
thf(fact_1046_div__mult__self3,axiom,
! [B3: nat,C2: nat,A3: nat] :
( ( B3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C2 @ B3 ) @ A3 ) @ B3 )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).
% div_mult_self3
thf(fact_1047_div__mult__mult1__if,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ( C2 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B3 ) )
= zero_zero_nat ) )
& ( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B3 ) )
= ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1048_div__mult__mult2,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ C2 ) )
= ( divide_divide_nat @ A3 @ B3 ) ) ) ).
% div_mult_mult2
thf(fact_1049_div__mult__mult1,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B3 ) )
= ( divide_divide_nat @ A3 @ B3 ) ) ) ).
% div_mult_mult1
thf(fact_1050_div__mult__self4,axiom,
! [B3: nat,C2: nat,A3: nat] :
( ( B3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B3 @ C2 ) @ A3 ) @ B3 )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).
% div_mult_self4
thf(fact_1051_det__multrow__div,axiom,
! [K: nat,N: nat,A: mat_a,A3: a] :
( ( ord_less_nat @ K @ N )
=> ( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( A3 != zero_zero_a )
=> ( ( divide_divide_a @ ( det_a @ ( gauss_5154200947219177641_gen_a @ times_times_a @ K @ A3 @ A ) ) @ A3 )
= ( det_a @ A ) ) ) ) ) ).
% det_multrow_div
thf(fact_1052_det__multrow__div,axiom,
! [K: nat,N: nat,A: mat_real,A3: real] :
( ( ord_less_nat @ K @ N )
=> ( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( A3 != zero_zero_real )
=> ( ( divide_divide_real @ ( det_real @ ( gauss_1037889766561479105n_real @ times_times_real @ K @ A3 @ A ) ) @ A3 )
= ( det_real @ A ) ) ) ) ) ).
% det_multrow_div
thf(fact_1053_det__multrow__div,axiom,
! [K: nat,N: nat,A: mat_complex,A3: complex] :
( ( ord_less_nat @ K @ N )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( A3 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( det_complex @ ( gauss_2324787009747932227omplex @ times_times_complex @ K @ A3 @ A ) ) @ A3 )
= ( det_complex @ A ) ) ) ) ) ).
% det_multrow_div
thf(fact_1054_multrow__carrier,axiom,
! [Mul: a > a > a,K: nat,A3: a,A: mat_a,N: nat,Nc: nat] :
( ( member_mat_a @ ( gauss_5154200947219177641_gen_a @ Mul @ K @ A3 @ A ) @ ( carrier_mat_a @ N @ Nc ) )
= ( member_mat_a @ A @ ( carrier_mat_a @ N @ Nc ) ) ) ).
% multrow_carrier
thf(fact_1055_multrow__carrier,axiom,
! [Mul: complex > complex > complex,K: nat,A3: complex,A: mat_complex,N: nat,Nc: nat] :
( ( member_mat_complex @ ( gauss_2324787009747932227omplex @ Mul @ K @ A3 @ A ) @ ( carrier_mat_complex @ N @ Nc ) )
= ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ Nc ) ) ) ).
% multrow_carrier
thf(fact_1056_index__mat__multrow_I4_J,axiom,
! [Mul: complex > complex > complex,K: nat,A3: complex,A: mat_complex] :
( ( dim_row_complex @ ( gauss_2324787009747932227omplex @ Mul @ K @ A3 @ A ) )
= ( dim_row_complex @ A ) ) ).
% index_mat_multrow(4)
thf(fact_1057_index__mat__multrow_I5_J,axiom,
! [Mul: complex > complex > complex,K: nat,A3: complex,A: mat_complex] :
( ( dim_col_complex @ ( gauss_2324787009747932227omplex @ Mul @ K @ A3 @ A ) )
= ( dim_col_complex @ A ) ) ).
% index_mat_multrow(5)
thf(fact_1058_det__multrow,axiom,
! [K: nat,N: nat,A: mat_a,A3: a] :
( ( ord_less_nat @ K @ N )
=> ( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( det_a @ ( gauss_5154200947219177641_gen_a @ times_times_a @ K @ A3 @ A ) )
= ( times_times_a @ A3 @ ( det_a @ A ) ) ) ) ) ).
% det_multrow
thf(fact_1059_det__multrow,axiom,
! [K: nat,N: nat,A: mat_complex,A3: complex] :
( ( ord_less_nat @ K @ N )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( det_complex @ ( gauss_2324787009747932227omplex @ times_times_complex @ K @ A3 @ A ) )
= ( times_times_complex @ A3 @ ( det_complex @ A ) ) ) ) ) ).
% det_multrow
thf(fact_1060_multrow__mat,axiom,
! [A: mat_a,N: nat,Nc: nat,K: nat,A3: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( gauss_5154200947219177641_gen_a @ times_times_a @ K @ A3 @ A )
= ( times_times_mat_a @ ( gauss_5015385051186949877_mat_a @ N @ K @ A3 ) @ A ) ) ) ).
% multrow_mat
thf(fact_1061_multrow__mat,axiom,
! [A: mat_complex,N: nat,Nc: nat,K: nat,A3: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( gauss_2324787009747932227omplex @ times_times_complex @ K @ A3 @ A )
= ( times_8009071140041733218omplex @ ( gauss_6868829418328711927omplex @ N @ K @ A3 ) @ A ) ) ) ).
% multrow_mat
thf(fact_1062_gbinomial__absorption_H,axiom,
! [K: nat,A3: real] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( gbinomial_real @ A3 @ K )
= ( times_times_real @ ( divide_divide_real @ A3 @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A3 @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_1063_gbinomial__absorption_H,axiom,
! [K: nat,A3: complex] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( gbinomial_complex @ A3 @ K )
= ( times_times_complex @ ( divide1717551699836669952omplex @ A3 @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A3 @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_1064_index__mat__multrow__mat_I2_J,axiom,
! [N: nat,K: nat,A3: complex] :
( ( dim_row_complex @ ( gauss_6868829418328711927omplex @ N @ K @ A3 ) )
= N ) ).
% index_mat_multrow_mat(2)
thf(fact_1065_index__mat__multrow__mat_I3_J,axiom,
! [N: nat,K: nat,A3: complex] :
( ( dim_col_complex @ ( gauss_6868829418328711927omplex @ N @ K @ A3 ) )
= N ) ).
% index_mat_multrow_mat(3)
thf(fact_1066_multrow__mat__carrier,axiom,
! [N: nat,K: nat,A3: a] : ( member_mat_a @ ( gauss_5015385051186949877_mat_a @ N @ K @ A3 ) @ ( carrier_mat_a @ N @ N ) ) ).
% multrow_mat_carrier
thf(fact_1067_multrow__mat__carrier,axiom,
! [N: nat,K: nat,A3: complex] : ( member_mat_complex @ ( gauss_6868829418328711927omplex @ N @ K @ A3 ) @ ( carrier_mat_complex @ N @ N ) ) ).
% multrow_mat_carrier
thf(fact_1068_gbinomial__absorb__comp,axiom,
! [A3: real,K: nat] :
( ( times_times_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A3 @ K ) )
= ( times_times_real @ A3 @ ( gbinomial_real @ ( minus_minus_real @ A3 @ one_one_real ) @ K ) ) ) ).
% gbinomial_absorb_comp
thf(fact_1069_gbinomial__absorb__comp,axiom,
! [A3: complex,K: nat] :
( ( times_times_complex @ ( minus_minus_complex @ A3 @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ A3 @ K ) )
= ( times_times_complex @ A3 @ ( gbinomial_complex @ ( minus_minus_complex @ A3 @ one_one_complex ) @ K ) ) ) ).
% gbinomial_absorb_comp
thf(fact_1070_gbinomial__mult__1_H,axiom,
! [A3: real,K: nat] :
( ( times_times_real @ ( gbinomial_real @ A3 @ K ) @ A3 )
= ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A3 @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A3 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_1071_gbinomial__mult__1_H,axiom,
! [A3: complex,K: nat] :
( ( times_times_complex @ ( gbinomial_complex @ A3 @ K ) @ A3 )
= ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A3 @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A3 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_1072_gbinomial__mult__1,axiom,
! [A3: real,K: nat] :
( ( times_times_real @ A3 @ ( gbinomial_real @ A3 @ K ) )
= ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A3 @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A3 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_1073_gbinomial__mult__1,axiom,
! [A3: complex,K: nat] :
( ( times_times_complex @ A3 @ ( gbinomial_complex @ A3 @ K ) )
= ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A3 @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A3 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_1074_Suc__times__gbinomial,axiom,
! [K: nat,A3: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A3 @ one_one_real ) @ ( suc @ K ) ) )
= ( times_times_real @ ( plus_plus_real @ A3 @ one_one_real ) @ ( gbinomial_real @ A3 @ K ) ) ) ).
% Suc_times_gbinomial
thf(fact_1075_Suc__times__gbinomial,axiom,
! [K: nat,A3: complex] :
( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A3 @ one_one_complex ) @ ( suc @ K ) ) )
= ( times_times_complex @ ( plus_plus_complex @ A3 @ one_one_complex ) @ ( gbinomial_complex @ A3 @ K ) ) ) ).
% Suc_times_gbinomial
thf(fact_1076_gbinomial__absorption,axiom,
! [K: nat,A3: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A3 @ ( suc @ K ) ) )
= ( times_times_real @ A3 @ ( gbinomial_real @ ( minus_minus_real @ A3 @ one_one_real ) @ K ) ) ) ).
% gbinomial_absorption
thf(fact_1077_gbinomial__absorption,axiom,
! [K: nat,A3: complex] :
( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A3 @ ( suc @ K ) ) )
= ( times_times_complex @ A3 @ ( gbinomial_complex @ ( minus_minus_complex @ A3 @ one_one_complex ) @ K ) ) ) ).
% gbinomial_absorption
thf(fact_1078_gbinomial__negated__upper,axiom,
( gbinomial_real
= ( ^ [A5: real,K4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K4 ) @ ( gbinomial_real @ ( minus_minus_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ K4 ) @ A5 ) @ one_one_real ) @ K4 ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_1079_gbinomial__negated__upper,axiom,
( gbinomial_complex
= ( ^ [A5: complex,K4: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K4 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K4 ) @ A5 ) @ one_one_complex ) @ K4 ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_1080_gbinomial__index__swap,axiom,
! [K: nat,N: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ K ) )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ N ) ) ) ).
% gbinomial_index_swap
thf(fact_1081_gbinomial__index__swap,axiom,
! [K: nat,N: nat] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ K ) )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N ) ) ) ).
% gbinomial_index_swap
thf(fact_1082_gbinomial__rec,axiom,
! [A3: real,K: nat] :
( ( gbinomial_real @ ( plus_plus_real @ A3 @ one_one_real ) @ ( suc @ K ) )
= ( times_times_real @ ( gbinomial_real @ A3 @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A3 @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).
% gbinomial_rec
thf(fact_1083_gbinomial__rec,axiom,
! [A3: complex,K: nat] :
( ( gbinomial_complex @ ( plus_plus_complex @ A3 @ one_one_complex ) @ ( suc @ K ) )
= ( times_times_complex @ ( gbinomial_complex @ A3 @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A3 @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).
% gbinomial_rec
thf(fact_1084_gbinomial__factors,axiom,
! [A3: real,K: nat] :
( ( gbinomial_real @ ( plus_plus_real @ A3 @ one_one_real ) @ ( suc @ K ) )
= ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A3 @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A3 @ K ) ) ) ).
% gbinomial_factors
thf(fact_1085_gbinomial__factors,axiom,
! [A3: complex,K: nat] :
( ( gbinomial_complex @ ( plus_plus_complex @ A3 @ one_one_complex ) @ ( suc @ K ) )
= ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A3 @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A3 @ K ) ) ) ).
% gbinomial_factors
thf(fact_1086_gbinomial__minus,axiom,
! [A3: real,K: nat] :
( ( gbinomial_real @ ( uminus_uminus_real @ A3 ) @ K )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% gbinomial_minus
thf(fact_1087_gbinomial__minus,axiom,
! [A3: complex,K: nat] :
( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A3 ) @ K )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ A3 @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% gbinomial_minus
thf(fact_1088_gbinomial__minus_H,axiom,
! [A3: real,B3: nat] :
( ( gbinomial_real @ ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ B3 ) ) @ B3 )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ B3 ) @ ( gbinomial_real @ ( uminus_uminus_real @ ( plus_plus_real @ A3 @ one_one_real ) ) @ B3 ) ) ) ).
% gbinomial_minus'
thf(fact_1089_gbinomial__minus_H,axiom,
! [A3: complex,B3: nat] :
( ( gbinomial_complex @ ( plus_plus_complex @ A3 @ ( semiri8010041392384452111omplex @ B3 ) ) @ B3 )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ B3 ) @ ( gbinomial_complex @ ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A3 @ one_one_complex ) ) @ B3 ) ) ) ).
% gbinomial_minus'
thf(fact_1090_multcol__mat,axiom,
! [A: mat_a,Nr: nat,N: nat,K: nat,A3: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( column_mat_multcol_a @ K @ A3 @ A )
= ( times_times_mat_a @ A @ ( gauss_5015385051186949877_mat_a @ N @ K @ A3 ) ) ) ) ).
% multcol_mat
thf(fact_1091_multcol__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,K: nat,A3: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( column4410001698458707789omplex @ K @ A3 @ A )
= ( times_8009071140041733218omplex @ A @ ( gauss_6868829418328711927omplex @ N @ K @ A3 ) ) ) ) ).
% multcol_mat
thf(fact_1092_index__mat__multcol_I4_J,axiom,
! [K: nat,A3: complex,A: mat_complex] :
( ( dim_row_complex @ ( column4410001698458707789omplex @ K @ A3 @ A ) )
= ( dim_row_complex @ A ) ) ).
% index_mat_multcol(4)
thf(fact_1093_gbinomial__pochhammer,axiom,
( gbinomial_complex
= ( ^ [A5: complex,K4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K4 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A5 ) @ K4 ) ) @ ( semiri5044797733671781792omplex @ K4 ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_1094_gbinomial__pochhammer,axiom,
( gbinomial_real
= ( ^ [A5: real,K4: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K4 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ A5 ) @ K4 ) ) @ ( semiri2265585572941072030t_real @ K4 ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_1095_fps__XD__Suc,axiom,
! [A3: formal670952693614245302omplex,N: nat] :
( ( formal3666518339620930912omplex @ ( formal1655152611307539683omplex @ A3 ) @ ( suc @ N ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( formal3666518339620930912omplex @ A3 @ ( suc @ N ) ) ) ) ).
% fps_XD_Suc
thf(fact_1096_fct__bound,axiom,
! [F: complex > real] :
( ( ( plus_plus_real @ ( F @ ( uminus1482373934393186551omplex @ one_one_complex ) ) @ ( F @ one_one_complex ) )
= one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ one_one_complex ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ ( uminus1482373934393186551omplex @ one_one_complex ) ) )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_real @ ( F @ one_one_complex ) @ ( F @ ( uminus1482373934393186551omplex @ one_one_complex ) ) ) )
& ( ord_less_eq_real @ ( minus_minus_real @ ( F @ one_one_complex ) @ ( F @ ( uminus1482373934393186551omplex @ one_one_complex ) ) ) @ one_one_real ) ) ) ) ) ).
% fct_bound
thf(fact_1097_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_1098_positive__proj__trace,axiom,
! [P: mat_complex,R: mat_complex,N: nat] :
( ( linear5633924348262549461omplex @ P )
=> ( ( complex_positive @ R )
=> ( ( member_mat_complex @ P @ ( 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 @ P ) ) ) ) ) ) ) ).
% positive_proj_trace
thf(fact_1099_positive__one,axiom,
! [N: nat] : ( complex_positive @ ( one_mat_complex @ N ) ) ).
% positive_one
thf(fact_1100_Complex__Matrix_Opositive__zero,axiom,
! [N: nat] : ( complex_positive @ ( zero_mat_complex @ N @ N ) ) ).
% Complex_Matrix.positive_zero
thf(fact_1101_projector__positive,axiom,
! [M3: mat_complex] :
( ( linear5633924348262549461omplex @ M3 )
=> ( complex_positive @ M3 ) ) ).
% projector_positive
thf(fact_1102_positive__is__hermitian,axiom,
! [A: mat_complex] :
( ( complex_positive @ A )
=> ( comple8306762464034002205omplex @ A ) ) ).
% positive_is_hermitian
thf(fact_1103_positive__dim__eq,axiom,
! [A: mat_complex] :
( ( complex_positive @ A )
=> ( ( dim_row_complex @ A )
= ( dim_col_complex @ A ) ) ) ).
% positive_dim_eq
thf(fact_1104_positive__antisym,axiom,
! [A: mat_complex] :
( ( complex_positive @ A )
=> ( ( complex_positive @ ( uminus467866341702955550omplex @ A ) )
=> ( A
= ( zero_mat_complex @ ( dim_col_complex @ A ) @ ( dim_col_complex @ A ) ) ) ) ) ).
% positive_antisym
thf(fact_1105_Complex__Matrix_Opositive__add,axiom,
! [A: mat_complex,B: mat_complex,N: nat] :
( ( complex_positive @ A )
=> ( ( complex_positive @ B )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( complex_positive @ ( plus_p8323303612493835998omplex @ A @ B ) ) ) ) ) ) ).
% Complex_Matrix.positive_add
thf(fact_1106_positive__close__under__left__right__mult__adjoint,axiom,
! [M3: mat_complex,N: nat,A: mat_complex] :
( ( member_mat_complex @ M3 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_positive @ A )
=> ( complex_positive @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ M3 @ A ) @ ( schur_5982229384592763574omplex @ M3 ) ) ) ) ) ) ).
% positive_close_under_left_right_mult_adjoint
thf(fact_1107_positive__only__if__decomp,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_positive @ A )
=> ? [X3: mat_complex] :
( ( member_mat_complex @ X3 @ ( carrier_mat_complex @ N @ N ) )
& ( ( times_8009071140041733218omplex @ X3 @ ( schur_5982229384592763574omplex @ X3 ) )
= A ) ) ) ) ).
% positive_only_if_decomp
thf(fact_1108_positive__iff__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_iff_decomp
thf(fact_1109_positive__if__decomp,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ? [M5: mat_complex] :
( ( times_8009071140041733218omplex @ M5 @ ( schur_5982229384592763574omplex @ M5 ) )
= A )
=> ( complex_positive @ A ) ) ) ).
% positive_if_decomp
thf(fact_1110_density__operator__def,axiom,
( comple5220265106149225959erator
= ( ^ [A2: mat_complex] :
( ( complex_positive @ A2 )
& ( ( comple3184165445352484367omplex @ A2 )
= one_one_complex ) ) ) ) ).
% density_operator_def
thf(fact_1111_positive__trace,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_positive @ A )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( comple3184165445352484367omplex @ A ) ) ) ) ).
% positive_trace
thf(fact_1112_positive__smult,axiom,
! [A: mat_complex,N: nat,C2: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_positive @ A )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( complex_positive @ ( smult_mat_complex @ C2 @ A ) ) ) ) ) ).
% positive_smult
thf(fact_1113_partial__density__operator__def,axiom,
( comple1169154605998056944erator
= ( ^ [A2: mat_complex] :
( ( complex_positive @ A2 )
& ( ord_less_eq_complex @ ( comple3184165445352484367omplex @ A2 ) @ one_one_complex ) ) ) ) ).
% partial_density_operator_def
thf(fact_1114_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_1115_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_1116_lowner__le__trans__positiveI,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_positive @ A )
=> ( ( complex_lowner_le @ A @ B )
=> ( complex_positive @ B ) ) ) ) ).
% lowner_le_trans_positiveI
thf(fact_1117_zero__lowner__le__positiveD,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_lowner_le @ ( zero_mat_complex @ N @ N ) @ A )
=> ( complex_positive @ A ) ) ) ).
% zero_lowner_le_positiveD
thf(fact_1118_zero__lowner__le__positiveI,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_positive @ A )
=> ( complex_lowner_le @ ( zero_mat_complex @ N @ N ) @ A ) ) ) ).
% zero_lowner_le_positiveI
thf(fact_1119_lowner__le__add,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_lowner_le @ A @ B )
=> ( ( complex_lowner_le @ C @ D )
=> ( complex_lowner_le @ ( plus_p8323303612493835998omplex @ A @ C ) @ ( plus_p8323303612493835998omplex @ B @ D ) ) ) ) ) ) ) ) ).
% lowner_le_add
thf(fact_1120_lowner__le__minus,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_lowner_le @ A @ B )
=> ( ( complex_lowner_le @ C @ D )
=> ( complex_lowner_le @ ( minus_2412168080157227406omplex @ A @ D ) @ ( minus_2412168080157227406omplex @ B @ C ) ) ) ) ) ) ) ) ).
% lowner_le_minus
thf(fact_1121_lowner__le__refl,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( complex_lowner_le @ A @ A ) ) ).
% lowner_le_refl
thf(fact_1122_lowner__le__trans,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_lowner_le @ A @ B )
=> ( ( complex_lowner_le @ B @ C )
=> ( complex_lowner_le @ A @ C ) ) ) ) ) ) ).
% lowner_le_trans
thf(fact_1123_lowner__le__antisym,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 )
=> ( ( complex_lowner_le @ B @ A )
=> ( A = B ) ) ) ) ) ).
% lowner_le_antisym
thf(fact_1124_lowner__le__swap,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 )
=> ( complex_lowner_le @ ( uminus467866341702955550omplex @ B ) @ ( uminus467866341702955550omplex @ A ) ) ) ) ) ).
% lowner_le_swap
thf(fact_1125_lowner__le__keep__under__measurement,axiom,
! [M3: mat_complex,N: nat,A: mat_complex,B: mat_complex] :
( ( member_mat_complex @ M3 @ ( 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 @ M3 ) @ A ) @ M3 ) @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ M3 ) @ B ) @ M3 ) ) ) ) ) ) ).
% lowner_le_keep_under_measurement
thf(fact_1126_lowner__le__imp__trace__le,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 )
=> ( ord_less_eq_complex @ ( comple3184165445352484367omplex @ A ) @ ( comple3184165445352484367omplex @ B ) ) ) ) ) ).
% lowner_le_imp_trace_le
thf(fact_1127_lowner__le__def,axiom,
( complex_lowner_le
= ( ^ [A2: mat_complex,B2: mat_complex] :
( ( ( dim_row_complex @ A2 )
= ( dim_row_complex @ B2 ) )
& ( ( dim_col_complex @ A2 )
= ( dim_col_complex @ B2 ) )
& ( complex_positive @ ( minus_2412168080157227406omplex @ B2 @ A2 ) ) ) ) ) ).
% lowner_le_def
thf(fact_1128_lowner__le__smultc,axiom,
! [C2: complex,A: mat_complex,B: mat_complex,N: nat] :
( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ( complex_lowner_le @ A @ B )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( complex_lowner_le @ ( smult_mat_complex @ C2 @ A ) @ ( smult_mat_complex @ C2 @ B ) ) ) ) ) ) ).
% lowner_le_smultc
thf(fact_1129_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 )
= ( ! [X2: mat_complex] :
( ( member_mat_complex @ X2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple1169154605998056944erator @ X2 )
=> ( ord_less_eq_complex @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ A @ X2 ) ) @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ B @ X2 ) ) ) ) ) ) ) ) ) ).
% lowner_le_trace
thf(fact_1130_inverts__mat__sym,axiom,
! [A: mat_complex,B: mat_complex] :
( ( inverts_mat_complex @ A @ B )
=> ( ( ( dim_row_complex @ B )
= ( dim_col_complex @ A ) )
=> ( ( square_mat_complex @ B )
=> ( inverts_mat_complex @ B @ A ) ) ) ) ).
% inverts_mat_sym
thf(fact_1131_fct__bound_H,axiom,
! [F: complex > real] :
( ( ( plus_plus_real @ ( F @ ( uminus1482373934393186551omplex @ one_one_complex ) ) @ ( F @ one_one_complex ) )
= one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ one_one_complex ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ ( uminus1482373934393186551omplex @ one_one_complex ) ) )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ one_one_complex ) @ ( F @ ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) @ one_one_real ) ) ) ) ).
% fct_bound'
thf(fact_1132_density__collapse__def,axiom,
( projec3470689467825365843llapse
= ( ^ [R3: mat_complex,P5: mat_complex] :
( if_mat_complex
@ ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R3 @ P5 ) )
= zero_zero_complex )
@ ( projec8360710381328234318ensity @ ( dim_row_complex @ R3 ) )
@ ( smult_mat_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ one_one_real ) @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R3 @ P5 ) ) ) @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P5 @ R3 ) @ P5 ) ) ) ) ) ).
% density_collapse_def
% Helper facts (9)
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_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
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 ) ).
thf(help_If_3_1_If_001t__Matrix__Omat_It__Complex__Ocomplex_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Matrix__Omat_It__Complex__Ocomplex_J_T,axiom,
! [X: mat_complex,Y: mat_complex] :
( ( if_mat_complex @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Matrix__Omat_It__Complex__Ocomplex_J_T,axiom,
! [X: mat_complex,Y: mat_complex] :
( ( if_mat_complex @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
spectr4825054497075562704quiv_a @ a2 @ b @ u ).
%------------------------------------------------------------------------------