TPTP Problem File: SLH0773^1.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_01081_041896__19300904_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1463 ( 518 unt; 242 typ; 0 def)
% Number of atoms : 3504 (1580 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 14318 ( 236 ~; 89 |; 194 &;12238 @)
% ( 0 <=>;1561 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 7 avg)
% Number of types : 23 ( 22 usr)
% Number of type conns : 558 ( 558 >; 0 *; 0 +; 0 <<)
% Number of symbols : 223 ( 220 usr; 15 con; 0-6 aty)
% Number of variables : 3561 ( 67 ^;3461 !; 33 ?;3561 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:35:40.924
%------------------------------------------------------------------------------
% Could-be-implicit typings (22)
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J_J_J,type,
set_ma3896169189792133775omplex: $tType ).
thf(ty_n_t__Matrix__Omat_It__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J_J,type,
mat_Fo5321781242956565423omplex: $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__Set__Oset_It__Matrix__Omat_It__Real__Oreal_J_J,type,
set_mat_real: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Nat__Onat_J_J,type,
set_mat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Int__Oint_J_J,type,
set_mat_int: $tType ).
thf(ty_n_t__Matrix__Omat_It__Complex__Ocomplex_J,type,
mat_complex: $tType ).
thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
list_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Matrix__Omat_It__Real__Oreal_J,type,
mat_real: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__Matrix__Omat_It__Nat__Onat_J,type,
mat_nat: $tType ).
thf(ty_n_t__Matrix__Omat_It__Int__Oint_J,type,
mat_int: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $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_t__Int__Oint,type,
int: $tType ).
% Explicit typings (220)
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
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_Oeigenvalue_001t__Complex__Ocomplex,type,
char_e7032225803028799586omplex: mat_complex > complex > $o ).
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__multcol_001t__Complex__Ocomplex,type,
column4410001698458707789omplex: nat > complex > mat_complex > mat_complex ).
thf(sy_c_Column__Operations_Omat__swapcols_001t__Complex__Ocomplex,type,
column4357519492343924999omplex: nat > nat > mat_complex > mat_complex ).
thf(sy_c_Complex_Ocomplex_OIm,type,
im: complex > real ).
thf(sy_c_Complex_Ocomplex_ORe,type,
re: complex > real ).
thf(sy_c_Complex_Ocsqrt,type,
csqrt: complex > complex ).
thf(sy_c_Complex_Oimaginary__unit,type,
imaginary_unit: complex ).
thf(sy_c_Complex__Matrix_Odensity__operator,type,
comple5220265106149225959erator: mat_complex > $o ).
thf(sy_c_Complex__Matrix_Ohermitian_001t__Complex__Ocomplex,type,
comple8306762464034002205omplex: mat_complex > $o ).
thf(sy_c_Complex__Matrix_Olowner__le,type,
complex_lowner_le: mat_complex > mat_complex > $o ).
thf(sy_c_Complex__Matrix_Opartial__density__operator,type,
comple1169154605998056944erator: mat_complex > $o ).
thf(sy_c_Complex__Matrix_Opositive,type,
complex_positive: mat_complex > $o ).
thf(sy_c_Complex__Matrix_Otrace_001t__Complex__Ocomplex,type,
comple3184165445352484367omplex: mat_complex > complex ).
thf(sy_c_Complex__Matrix_Otrace_001t__Int__Oint,type,
complex_trace_int: mat_int > int ).
thf(sy_c_Complex__Matrix_Otrace_001t__Real__Oreal,type,
complex_trace_real: mat_real > real ).
thf(sy_c_Complex__Matrix_Ounitary_001t__Complex__Ocomplex,type,
comple6660659447773130958omplex: mat_complex > $o ).
thf(sy_c_Conjugate_Oconjugate__class_Oconjugate_001t__Complex__Ocomplex,type,
conjug1878831970375765195omplex: complex > complex ).
thf(sy_c_Determinant_Oadj__mat_001t__Complex__Ocomplex,type,
adj_mat_complex: mat_complex > mat_complex ).
thf(sy_c_Determinant_Oadj__mat_001t__Int__Oint,type,
adj_mat_int: mat_int > mat_int ).
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__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
det_Fo6681143053013413569omplex: mat_Fo5321781242956565423omplex > formal670952693614245302omplex ).
thf(sy_c_Determinant_Odet_001t__Int__Oint,type,
det_int: mat_int > int ).
thf(sy_c_Determinant_Odet_001t__Real__Oreal,type,
det_real: mat_real > real ).
thf(sy_c_Determinant_Omat__delete_001t__Complex__Ocomplex,type,
mat_delete_complex: mat_complex > nat > nat > mat_complex ).
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__Int__Oint,type,
permut3692553072317293667rt_int: int > nat > ( int > nat ) > int > 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_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
semiri5044797733671781792omplex: nat > complex ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
semiri1406184849735516958ct_int: nat > int ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
semiri1408675320244567234ct_nat: nat > nat ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
semiri2265585572941072030t_real: nat > real ).
thf(sy_c_Formal__Power__Series_Ofps__const_001t__Complex__Ocomplex,type,
formal7822294191640021514omplex: complex > formal670952693614245302omplex ).
thf(sy_c_Formal__Power__Series_Ofps__cos_001t__Complex__Ocomplex,type,
formal7592904152049726030omplex: complex > formal670952693614245302omplex ).
thf(sy_c_Formal__Power__Series_Ofps__exp_001t__Complex__Ocomplex,type,
formal5488582694110793604omplex: complex > formal670952693614245302omplex ).
thf(sy_c_Formal__Power__Series_Ofps__sin_001t__Complex__Ocomplex,type,
formal6336729282288812767omplex: complex > formal670952693614245302omplex ).
thf(sy_c_Formal__Power__Series_Ofps__tan_001t__Complex__Ocomplex,type,
formal6482914284900457064omplex: complex > formal670952693614245302omplex ).
thf(sy_c_Gates_Omat__incr,type,
mat_incr: nat > mat_complex ).
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__Int__Oint,type,
gauss_6494379909522362210at_int: nat > int > nat > nat > mat_int ).
thf(sy_c_Gauss__Jordan__Elimination_Oaddrow__mat_001t__Nat__Onat,type,
gauss_6496870380031412486at_nat: nat > nat > nat > nat > mat_nat ).
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_Oeliminate__entries__gen_001t__Complex__Ocomplex,type,
gauss_2785350030914899391omplex: ( complex > complex > complex ) > ( complex > complex > complex ) > ( nat > complex ) > mat_complex > nat > nat > mat_complex ).
thf(sy_c_Gauss__Jordan__Elimination_Oeliminate__entries__gen_001t__Int__Oint,type,
gauss_76339361826605373en_int: ( int > int > int ) > ( int > int > int ) > ( nat > int ) > mat_int > nat > nat > mat_int ).
thf(sy_c_Gauss__Jordan__Elimination_Oeliminate__entries__gen_001t__Real__Oreal,type,
gauss_9059886599186181437n_real: ( real > real > real ) > ( real > real > real ) > ( nat > real ) > mat_real > nat > nat > mat_real ).
thf(sy_c_Gauss__Jordan__Elimination_Ogauss__jordan__single_001t__Complex__Ocomplex,type,
gauss_4244865067341541924omplex: mat_complex > mat_complex ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__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__Int__Oint,type,
gauss_8882552878057600758en_int: ( int > int > int ) > ( int > int > int ) > int > nat > nat > mat_int > mat_int ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__addrow__gen_001t__Nat__Onat,type,
gauss_8885043348566651034en_nat: ( nat > nat > nat ) > ( nat > nat > nat ) > nat > nat > nat > mat_nat > mat_nat ).
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__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__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
gauss_4832413493682057675omplex: ( formal670952693614245302omplex > formal670952693614245302omplex > formal670952693614245302omplex ) > nat > formal670952693614245302omplex > mat_Fo5321781242956565423omplex > mat_Fo5321781242956565423omplex ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__multrow__gen_001t__Int__Oint,type,
gauss_2407205949817067457en_int: ( int > int > int ) > nat > int > mat_int > mat_int ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__multrow__gen_001t__Nat__Onat,type,
gauss_2409696420326117733en_nat: ( nat > nat > nat ) > nat > nat > mat_nat > mat_nat ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__multrow__gen_001t__Real__Oreal,type,
gauss_1037889766561479105n_real: ( real > real > real ) > nat > real > mat_real > mat_real ).
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_Omultrow__mat_001t__Complex__Ocomplex,type,
gauss_6868829418328711927omplex: nat > nat > complex > mat_complex ).
thf(sy_c_Gauss__Jordan__Elimination_Omultrow__mat_001t__Int__Oint,type,
gauss_3192586071676587637at_int: nat > nat > int > mat_int ).
thf(sy_c_Gauss__Jordan__Elimination_Omultrow__mat_001t__Nat__Onat,type,
gauss_3195076542185637913at_nat: nat > nat > nat > mat_nat ).
thf(sy_c_Gauss__Jordan__Elimination_Omultrow__mat_001t__Real__Oreal,type,
gauss_7241202418770761333t_real: nat > nat > real > mat_real ).
thf(sy_c_Gauss__Jordan__Elimination_Opivot__fun_001t__Complex__Ocomplex,type,
gauss_2609248829700396350omplex: mat_complex > ( nat > nat ) > nat > $o ).
thf(sy_c_Gauss__Jordan__Elimination_Opivot__fun_001t__Int__Oint,type,
gauss_8414077049331371708un_int: mat_int > ( nat > nat ) > nat > $o ).
thf(sy_c_Gauss__Jordan__Elimination_Opivot__fun_001t__Nat__Onat,type,
gauss_8416567519840421984un_nat: mat_nat > ( nat > nat ) > nat > $o ).
thf(sy_c_Gauss__Jordan__Elimination_Opivot__fun_001t__Real__Oreal,type,
gauss_5041415250090615612n_real: mat_real > ( nat > nat ) > nat > $o ).
thf(sy_c_Gauss__Jordan__Elimination_Orow__echelon__form_001t__Complex__Ocomplex,type,
gauss_194721375535881179omplex: mat_complex > $o ).
thf(sy_c_Gauss__Jordan__Elimination_Oswaprows__mat_001t__Complex__Ocomplex,type,
gauss_8970452565587180529omplex: nat > nat > nat > mat_complex ).
thf(sy_c_Gauss__Jordan__Elimination_Oswaprows__mat_001t__Int__Oint,type,
gauss_4917416859360123759at_int: nat > nat > nat > mat_int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex,type,
abs_abs_complex: complex > complex ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
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__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
minus_1072911313905636623omplex: formal670952693614245302omplex > formal670952693614245302omplex > formal670952693614245302omplex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
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__Int__Oint_J,type,
minus_minus_mat_int: mat_int > mat_int > mat_int ).
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__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_Oone__class_Oone_001t__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
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_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
plus_plus_complex: complex > complex > complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
plus_p8472957120637115327omplex: formal670952693614245302omplex > formal670952693614245302omplex > formal670952693614245302omplex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
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__Int__Oint_J,type,
plus_plus_mat_int: mat_int > mat_int > mat_int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_It__Nat__Onat_J,type,
plus_plus_mat_nat: mat_nat > mat_nat > mat_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_It__Real__Oreal_J,type,
plus_plus_mat_real: mat_real > mat_real > mat_real ).
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_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__Int__Oint,type,
times_times_int: int > int > int ).
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__Int__Oint_J,type,
times_times_mat_int: mat_int > mat_int > mat_int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_It__Nat__Onat_J,type,
times_times_mat_nat: mat_nat > mat_nat > mat_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_It__Real__Oreal_J,type,
times_times_mat_real: mat_real > mat_real > mat_real ).
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_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
uminus1482373934393186551omplex: complex > complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
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__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
zero_z1877163951443063103omplex: formal670952693614245302omplex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
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__List_Omonoid__mult__class_Oprod__list_001t__Complex__Ocomplex,type,
groups7979759902575632448omplex: list_complex > complex ).
thf(sy_c_If_001t__Complex__Ocomplex,type,
if_complex: $o > complex > complex > complex ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
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_Ocpx__sq__mat,type,
linear7199532782703566157sq_mat: nat > nat > set_mat_complex > $o ).
thf(sy_c_Linear__Algebra__Complements_Oprojector_001t__Complex__Ocomplex,type,
linear5633924348262549461omplex: mat_complex > $o ).
thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
set_complex2: list_complex > set_complex ).
thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
set_real2: list_real > set_real ).
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__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
carrie9079900694887046380omplex: nat > nat > set_ma3896169189792133775omplex ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Int__Oint,type,
carrier_mat_int: nat > nat > set_mat_int ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Nat__Onat,type,
carrier_mat_nat: nat > nat > set_mat_nat ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Real__Oreal,type,
carrier_mat_real: nat > nat > set_mat_real ).
thf(sy_c_Matrix_Odiag__mat_001t__Complex__Ocomplex,type,
diag_mat_complex: mat_complex > list_complex ).
thf(sy_c_Matrix_Odiag__mat_001t__Int__Oint,type,
diag_mat_int: mat_int > list_int ).
thf(sy_c_Matrix_Odiag__mat_001t__Real__Oreal,type,
diag_mat_real: mat_real > list_real ).
thf(sy_c_Matrix_Odiagonal__mat_001t__Complex__Ocomplex,type,
diagonal_mat_complex: mat_complex > $o ).
thf(sy_c_Matrix_Odim__col_001t__Complex__Ocomplex,type,
dim_col_complex: mat_complex > nat ).
thf(sy_c_Matrix_Odim__col_001t__Int__Oint,type,
dim_col_int: mat_int > nat ).
thf(sy_c_Matrix_Odim__col_001t__Nat__Onat,type,
dim_col_nat: mat_nat > nat ).
thf(sy_c_Matrix_Odim__col_001t__Real__Oreal,type,
dim_col_real: mat_real > nat ).
thf(sy_c_Matrix_Odim__row_001t__Complex__Ocomplex,type,
dim_row_complex: mat_complex > nat ).
thf(sy_c_Matrix_Odim__row_001t__Int__Oint,type,
dim_row_int: mat_int > nat ).
thf(sy_c_Matrix_Odim__row_001t__Nat__Onat,type,
dim_row_nat: mat_nat > nat ).
thf(sy_c_Matrix_Odim__row_001t__Real__Oreal,type,
dim_row_real: mat_real > nat ).
thf(sy_c_Matrix_Oinvertible__mat_001t__Complex__Ocomplex,type,
invert2568027935824841882omplex: mat_complex > $o ).
thf(sy_c_Matrix_Oinverts__mat_001t__Complex__Ocomplex,type,
inverts_mat_complex: mat_complex > mat_complex > $o ).
thf(sy_c_Matrix_Oinverts__mat_001t__Int__Oint,type,
inverts_mat_int: mat_int > mat_int > $o ).
thf(sy_c_Matrix_Omk__diagonal_001t__Complex__Ocomplex,type,
mk_diagonal_complex: list_complex > mat_complex ).
thf(sy_c_Matrix_Oone__mat_001t__Complex__Ocomplex,type,
one_mat_complex: nat > mat_complex ).
thf(sy_c_Matrix_Oone__mat_001t__Int__Oint,type,
one_mat_int: nat > mat_int ).
thf(sy_c_Matrix_Oone__mat_001t__Real__Oreal,type,
one_mat_real: nat > mat_real ).
thf(sy_c_Matrix_Osimilar__mat__wit_001t__Complex__Ocomplex,type,
simila5774310414453981135omplex: mat_complex > mat_complex > mat_complex > mat_complex > $o ).
thf(sy_c_Matrix_Osimilar__mat__wit_001t__Int__Oint,type,
similar_mat_wit_int: mat_int > mat_int > mat_int > mat_int > $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__Int__Oint,type,
smult_mat_int: int > mat_int > mat_int ).
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_Osquare__mat_001t__Complex__Ocomplex,type,
square_mat_complex: mat_complex > $o ).
thf(sy_c_Matrix_Osquare__mat_001t__Int__Oint,type,
square_mat_int: mat_int > $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_001t__Int__Oint,type,
upper_triangular_int: mat_int > $o ).
thf(sy_c_Matrix_Oupper__triangular_001t__Real__Oreal,type,
upper_8570057991637822137r_real: mat_real > $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__Int__Oint,type,
zero_mat_int: nat > nat > mat_int ).
thf(sy_c_Matrix_Ozero__mat_001t__Real__Oreal,type,
zero_mat_real: nat > nat > mat_real ).
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__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
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_NthRoot_Osqrt,type,
sqrt: real > 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__Int__Oint,type,
ord_less_int: int > int > $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__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Path__Connected_Olinepath_001t__Complex__Ocomplex,type,
path_l4128132617387358368omplex: complex > complex > real > complex ).
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__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Projective__Measurements_Ocpx__sq__mat_Oeigen__projector,type,
projec1689266477789839993jector: nat > nat > mat_complex > complex > mat_complex ).
thf(sy_c_Projective__Measurements_Odensity__collapse,type,
projec3470689467825365843llapse: mat_complex > mat_complex > mat_complex ).
thf(sy_c_Projective__Measurements_Oeigvals_001t__Complex__Ocomplex,type,
projec6785268565095433026omplex: mat_complex > list_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_Projective__Measurements_Ospectrum_001t__Complex__Ocomplex,type,
projec527831343749723810omplex: mat_complex > set_complex ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
real_V1022390504157884413omplex: complex > real ).
thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
real_V4546457046886955230omplex: real > complex ).
thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
real_V1803761363581548252l_real: real > real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
divide1717551699836669952omplex: complex > complex > complex ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
divide1348722040316500488omplex: formal670952693614245302omplex > formal670952693614245302omplex > formal670952693614245302omplex ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
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_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_Omat__adjoint_001t__Complex__Ocomplex,type,
schur_5982229384592763574omplex: mat_complex > mat_complex ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
collect_mat_complex: ( mat_complex > $o ) > set_mat_complex ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
set_or1266510415728281911st_int: int > int > set_int ).
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_Oreal__diag__decomp_001t__Complex__Ocomplex,type,
spectr5409772854192057952omplex: mat_complex > mat_complex > mat_complex > $o ).
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_Ounitary__diag_001t__Complex__Ocomplex,type,
spectr532731689276696518omplex: mat_complex > mat_complex > mat_complex > $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_Weierstrass__Theorems_OBernstein,type,
weiers7429072931691461095nstein: nat > nat > real > real ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
member_mat_complex: mat_complex > set_mat_complex > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J_J,type,
member4348805710806261976omplex: mat_Fo5321781242956565423omplex > set_ma3896169189792133775omplex > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Int__Oint_J,type,
member_mat_int: mat_int > set_mat_int > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Nat__Onat_J,type,
member_mat_nat: mat_nat > set_mat_nat > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Real__Oreal_J,type,
member_mat_real: mat_real > set_mat_real > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_A,type,
a: mat_complex ).
thf(sy_v_B,type,
b: mat_complex ).
thf(sy_v_U,type,
u: mat_complex ).
% Relevant facts (1209)
thf(fact_0_assms,axiom,
projec5943904436471448624omplex @ a @ b @ u ).
% assms
thf(fact_1_real__diag__decompD_I1_J,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr5409772854192057952omplex @ A @ B @ U )
=> ( spectr532731689276696518omplex @ A @ B @ U ) ) ).
% real_diag_decompD(1)
thf(fact_2_unitary__diag__imp__unitarily__equiv,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr532731689276696518omplex @ A @ B @ U )
=> ( spectr6340060708231679580omplex @ A @ B @ U ) ) ).
% unitary_diag_imp_unitarily_equiv
thf(fact_3_unitary__diagD_I3_J,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr532731689276696518omplex @ A @ B @ U )
=> ( comple6660659447773130958omplex @ U ) ) ).
% unitary_diagD(3)
thf(fact_4_unitary__diagD_I2_J,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr532731689276696518omplex @ A @ B @ U )
=> ( diagonal_mat_complex @ B ) ) ).
% unitary_diagD(2)
thf(fact_5_unitary__diag__carrier_I2_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr532731689276696518omplex @ A @ B @ U )
=> ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% unitary_diag_carrier(2)
thf(fact_6_unitary__diag__carrier_I1_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr532731689276696518omplex @ A @ B @ U )
=> ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% unitary_diag_carrier(1)
thf(fact_7_hermitian__decomp__unitary,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( projec5943904436471448624omplex @ A @ B @ U )
=> ( comple6660659447773130958omplex @ U ) ) ).
% hermitian_decomp_unitary
thf(fact_8_hermitian__decomp__diag__mat,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( projec5943904436471448624omplex @ A @ B @ U )
=> ( diagonal_mat_complex @ B ) ) ).
% hermitian_decomp_diag_mat
thf(fact_9_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_10_hermitian__decomp__eigenvalues,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( projec5943904436471448624omplex @ A @ B @ U )
=> ( ( diag_mat_complex @ B )
= ( projec6785268565095433026omplex @ A ) ) ) ).
% hermitian_decomp_eigenvalues
thf(fact_11_unitary__diagD_I1_J,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr532731689276696518omplex @ A @ B @ U )
=> ( simila5774310414453981135omplex @ A @ B @ U @ ( schur_5982229384592763574omplex @ U ) ) ) ).
% unitary_diagD(1)
thf(fact_12_cpx__sq__mat_Ohermitian__schur__decomp,axiom,
! [DimR: nat,DimC: nat,Fc_mats: set_mat_complex,A: mat_complex] :
( ( linear7199532782703566157sq_mat @ DimR @ DimC @ Fc_mats )
=> ( ( comple8306762464034002205omplex @ A )
=> ( ( member_mat_complex @ A @ Fc_mats )
=> ~ ! [B3: mat_complex,U3: mat_complex] :
~ ( projec5943904436471448624omplex @ A @ B3 @ U3 ) ) ) ) ).
% cpx_sq_mat.hermitian_schur_decomp
thf(fact_13_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_14_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_15_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_16_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_17_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_18_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_19_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_20_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_21_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_22_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_23_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_24_hermitian__def,axiom,
( comple8306762464034002205omplex
= ( ^ [A2: mat_complex] :
( ( schur_5982229384592763574omplex @ A2 )
= A2 ) ) ) ).
% hermitian_def
thf(fact_25_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_26_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_27_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_28_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_29_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_30_unitary__diagI_H,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 ) )
=> ( ( diagonal_mat_complex @ B )
=> ( ( comple6660659447773130958omplex @ U )
=> ( ( A
= ( spectr5699176650994449695omplex @ U @ B ) )
=> ( spectr532731689276696518omplex @ A @ B @ U ) ) ) ) ) ) ).
% unitary_diagI'
thf(fact_31_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_32_conjugate__eq__unitarily__equiv,axiom,
! [A: mat_complex,N: nat,V: mat_complex,B: mat_complex,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ V @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( ( comple6660659447773130958omplex @ V )
=> ( ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ V @ B ) @ ( schur_5982229384592763574omplex @ V ) )
= B )
=> ( spectr6340060708231679580omplex @ A @ B @ ( times_8009071140041733218omplex @ U @ V ) ) ) ) ) ) ) ).
% conjugate_eq_unitarily_equiv
thf(fact_33_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_34_similar__mat__wit__sym,axiom,
! [A: mat_complex,B: mat_complex,P: mat_complex,Q: mat_complex] :
( ( simila5774310414453981135omplex @ A @ B @ P @ Q )
=> ( simila5774310414453981135omplex @ B @ A @ Q @ P ) ) ).
% similar_mat_wit_sym
thf(fact_35_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_36_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_37_mat__conj__def,axiom,
( spectr5699176650994449695omplex
= ( ^ [U2: mat_complex,V2: mat_complex] : ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ U2 @ V2 ) @ ( schur_5982229384592763574omplex @ U2 ) ) ) ) ).
% mat_conj_def
thf(fact_38_mat__conj__adjoint,axiom,
! [U: mat_complex,V: mat_complex] :
( ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U ) @ V )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ U ) @ V ) @ U ) ) ).
% mat_conj_adjoint
thf(fact_39_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_40_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_41_hermitian__square__hermitian,axiom,
! [A: mat_complex] :
( ( comple8306762464034002205omplex @ A )
=> ( comple8306762464034002205omplex @ ( times_8009071140041733218omplex @ A @ A ) ) ) ).
% hermitian_square_hermitian
thf(fact_42_hermitian__square__similar__mat__wit,axiom,
! [A: mat_complex,N: nat,B: mat_complex,U: mat_complex] :
( ( comple8306762464034002205omplex @ A )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr532731689276696518omplex @ A @ B @ U )
=> ( simila5774310414453981135omplex @ ( times_8009071140041733218omplex @ A @ A ) @ ( times_8009071140041733218omplex @ B @ B ) @ U @ ( schur_5982229384592763574omplex @ U ) ) ) ) ) ).
% hermitian_square_similar_mat_wit
thf(fact_43_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_44_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_45_mem__Collect__eq,axiom,
! [A3: complex,P: complex > $o] :
( ( member_complex @ A3 @ ( collect_complex @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: set_mat_complex] :
( ( collect_mat_complex
@ ^ [X: mat_complex] : ( member_mat_complex @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A: set_complex] :
( ( collect_complex
@ ^ [X: complex] : ( member_complex @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_48_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_49_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_50_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_51_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_52_unitary__mult__conjugate,axiom,
! [A: mat_complex,N: nat,V: mat_complex,U: mat_complex,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ V @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ V )
=> ( ( ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ V ) @ A )
= ( spectr5699176650994449695omplex @ U @ B ) )
=> ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ V @ U ) @ B ) @ ( schur_5982229384592763574omplex @ ( times_8009071140041733218omplex @ V @ U ) ) ) ) ) ) ) ) ) ) ).
% unitary_mult_conjugate
thf(fact_53_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_54_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_55_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_56_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_57_unitarily__equiv__square,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 @ ( times_8009071140041733218omplex @ A @ A ) @ ( times_8009071140041733218omplex @ B @ B ) @ U ) ) ) ).
% unitarily_equiv_square
thf(fact_58_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_59_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_60_unitarily__equiv__conjugate,axiom,
! [A: mat_complex,N: nat,V: mat_complex,U: mat_complex,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ V @ ( 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 @ V ) @ A ) @ B @ U )
=> ( ( comple6660659447773130958omplex @ V )
=> ( spectr6340060708231679580omplex @ A @ B @ ( times_8009071140041733218omplex @ V @ U ) ) ) ) ) ) ) ) ).
% unitarily_equiv_conjugate
thf(fact_61_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_62_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_63_Complex__Matrix_Oadjoint__adjoint,axiom,
! [A: mat_complex] :
( ( schur_5982229384592763574omplex @ ( schur_5982229384592763574omplex @ A ) )
= A ) ).
% Complex_Matrix.adjoint_adjoint
thf(fact_64_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_65_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_66_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_67_unitary__mult__square__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 ) )
=> ( ( ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ U ) @ U )
= ( one_mat_complex @ N ) )
=> ( ( times_8009071140041733218omplex @ A @ A )
= ( spectr5699176650994449695omplex @ U @ ( times_8009071140041733218omplex @ B @ B ) ) ) ) ) ) ) ) ).
% unitary_mult_square_eq
thf(fact_68_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_69_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_70_mk__diagonal__diagonal,axiom,
! [As: list_complex] : ( diagonal_mat_complex @ ( mk_diagonal_complex @ As ) ) ).
% mk_diagonal_diagonal
thf(fact_71_cpx__sq__mat_Ocpx__sq__mat__mult,axiom,
! [DimR: nat,DimC: nat,Fc_mats: set_mat_complex,A: mat_complex,B: mat_complex] :
( ( linear7199532782703566157sq_mat @ DimR @ DimC @ Fc_mats )
=> ( ( member_mat_complex @ A @ Fc_mats )
=> ( ( member_mat_complex @ B @ Fc_mats )
=> ( member_mat_complex @ ( times_8009071140041733218omplex @ A @ B ) @ Fc_mats ) ) ) ) ).
% cpx_sq_mat.cpx_sq_mat_mult
thf(fact_72_cpx__sq__mat_Odim__eq,axiom,
! [DimR: nat,DimC: nat,Fc_mats: set_mat_complex] :
( ( linear7199532782703566157sq_mat @ DimR @ DimC @ Fc_mats )
=> ( DimR = DimC ) ) ).
% cpx_sq_mat.dim_eq
thf(fact_73_mat__conj__smult,axiom,
! [A: mat_complex,N: nat,U: mat_complex,B: mat_complex,X2: 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 @ X2 @ A )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ U @ ( smult_mat_complex @ X2 @ B ) ) @ ( schur_5982229384592763574omplex @ U ) ) ) ) ) ) ) ).
% mat_conj_smult
thf(fact_74_cpx__sq__mat_Omax__mix__density__square,axiom,
! [DimR: nat,DimC: nat,Fc_mats: set_mat_complex] :
( ( linear7199532782703566157sq_mat @ DimR @ DimC @ Fc_mats )
=> ( member_mat_complex @ ( projec8360710381328234318ensity @ DimR ) @ Fc_mats ) ) ).
% cpx_sq_mat.max_mix_density_square
thf(fact_75_adjoint__one,axiom,
! [N: nat] :
( ( schur_5982229384592763574omplex @ ( one_mat_complex @ N ) )
= ( one_mat_complex @ N ) ) ).
% adjoint_one
thf(fact_76_mat__assoc__test_I1_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 @ B ) @ ( times_8009071140041733218omplex @ C @ D ) )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B ) @ C ) @ D ) ) ) ) ) ) ).
% mat_assoc_test(1)
thf(fact_77_mat__assoc__test_I2_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 @ ( schur_5982229384592763574omplex @ ( times_8009071140041733218omplex @ A @ ( schur_5982229384592763574omplex @ B ) ) ) @ C )
= ( times_8009071140041733218omplex @ B @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ A ) @ C ) ) ) ) ) ) ) ).
% mat_assoc_test(2)
thf(fact_78_mat__assoc__test_I3_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 @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ ( one_mat_complex @ N ) ) @ ( one_mat_complex @ N ) ) @ B ) @ ( one_mat_complex @ N ) )
= ( times_8009071140041733218omplex @ A @ B ) ) ) ) ) ) ).
% mat_assoc_test(3)
thf(fact_79_upper__triangular__one,axiom,
! [N: nat] : ( upper_4850907204721561915omplex @ ( one_mat_complex @ N ) ) ).
% upper_triangular_one
thf(fact_80_upper__triangular__one,axiom,
! [N: nat] : ( upper_triangular_int @ ( one_mat_int @ N ) ) ).
% upper_triangular_one
thf(fact_81_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_82_smult__smult__times,axiom,
! [A3: real,K: real,A: mat_real] :
( ( smult_mat_real @ A3 @ ( smult_mat_real @ K @ A ) )
= ( smult_mat_real @ ( times_times_real @ A3 @ K ) @ A ) ) ).
% smult_smult_times
thf(fact_83_smult__smult__times,axiom,
! [A3: nat,K: nat,A: mat_nat] :
( ( smult_mat_nat @ A3 @ ( smult_mat_nat @ K @ A ) )
= ( smult_mat_nat @ ( times_times_nat @ A3 @ K ) @ A ) ) ).
% smult_smult_times
thf(fact_84_smult__smult__times,axiom,
! [A3: int,K: int,A: mat_int] :
( ( smult_mat_int @ A3 @ ( smult_mat_int @ K @ A ) )
= ( smult_mat_int @ ( times_times_int @ A3 @ K ) @ A ) ) ).
% smult_smult_times
thf(fact_85_cpx__sq__mat_Ocpx__sq__mat__smult,axiom,
! [DimR: nat,DimC: nat,Fc_mats: set_mat_complex,A: mat_complex,X2: complex] :
( ( linear7199532782703566157sq_mat @ DimR @ DimC @ Fc_mats )
=> ( ( member_mat_complex @ A @ Fc_mats )
=> ( member_mat_complex @ ( smult_mat_complex @ X2 @ A ) @ Fc_mats ) ) ) ).
% cpx_sq_mat.cpx_sq_mat_smult
thf(fact_86_cpx__sq__mat_Oone__mem,axiom,
! [DimR: nat,DimC: nat,Fc_mats: set_mat_complex] :
( ( linear7199532782703566157sq_mat @ DimR @ DimC @ Fc_mats )
=> ( member_mat_complex @ ( one_mat_complex @ DimR ) @ Fc_mats ) ) ).
% cpx_sq_mat.one_mem
thf(fact_87_max__mix__density__carrier,axiom,
! [N: nat] : ( member_mat_complex @ ( projec8360710381328234318ensity @ N ) @ ( carrier_mat_complex @ N @ N ) ) ).
% max_mix_density_carrier
thf(fact_88_one__carrier__mat,axiom,
! [N: nat] : ( member_mat_int @ ( one_mat_int @ N ) @ ( carrier_mat_int @ N @ N ) ) ).
% one_carrier_mat
thf(fact_89_one__carrier__mat,axiom,
! [N: nat] : ( member_mat_complex @ ( one_mat_complex @ N ) @ ( carrier_mat_complex @ N @ N ) ) ).
% one_carrier_mat
thf(fact_90_hermitian__one,axiom,
! [N: nat] : ( comple8306762464034002205omplex @ ( one_mat_complex @ N ) ) ).
% hermitian_one
thf(fact_91_unitary__one,axiom,
! [N: nat] : ( comple6660659447773130958omplex @ ( one_mat_complex @ N ) ) ).
% unitary_one
thf(fact_92_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_93_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_94_diagonal__mat__smult,axiom,
! [A: mat_complex,X2: complex] :
( ( diagonal_mat_complex @ A )
=> ( diagonal_mat_complex @ ( smult_mat_complex @ X2 @ A ) ) ) ).
% diagonal_mat_smult
thf(fact_95_left__mult__one__mat,axiom,
! [A: mat_int,Nr: nat,Nc: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( times_times_mat_int @ ( one_mat_int @ Nr ) @ A )
= A ) ) ).
% left_mult_one_mat
thf(fact_96_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_97_right__mult__one__mat,axiom,
! [A: mat_int,Nr: nat,Nc: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( times_times_mat_int @ A @ ( one_mat_int @ Nc ) )
= A ) ) ).
% right_mult_one_mat
thf(fact_98_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_99_similar__mat__wit__refl,axiom,
! [A: mat_int,N: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( similar_mat_wit_int @ A @ A @ ( one_mat_int @ N ) @ ( one_mat_int @ N ) ) ) ).
% similar_mat_wit_refl
thf(fact_100_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_101_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_102_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_103_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_104_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_105_unitarily__equiv__smult,axiom,
! [A: mat_complex,N: nat,B: mat_complex,U: mat_complex,X2: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( spectr6340060708231679580omplex @ ( smult_mat_complex @ X2 @ A ) @ ( smult_mat_complex @ X2 @ B ) @ U ) ) ) ).
% unitarily_equiv_smult
thf(fact_106_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_107_similar__mat__witI,axiom,
! [P: mat_int,Q: mat_int,N: nat,A: mat_int,B: mat_int] :
( ( ( times_times_mat_int @ P @ Q )
= ( one_mat_int @ N ) )
=> ( ( ( times_times_mat_int @ Q @ P )
= ( one_mat_int @ N ) )
=> ( ( A
= ( times_times_mat_int @ ( times_times_mat_int @ P @ B ) @ Q ) )
=> ( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( ( member_mat_int @ B @ ( carrier_mat_int @ N @ N ) )
=> ( ( member_mat_int @ P @ ( carrier_mat_int @ N @ N ) )
=> ( ( member_mat_int @ Q @ ( carrier_mat_int @ N @ N ) )
=> ( similar_mat_wit_int @ A @ B @ P @ Q ) ) ) ) ) ) ) ) ).
% similar_mat_witI
thf(fact_108_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_109_similar__mat__witD2_I1_J,axiom,
! [A: mat_int,N: nat,M: nat,B: mat_int,P: mat_int,Q: mat_int] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ M ) )
=> ( ( similar_mat_wit_int @ A @ B @ P @ Q )
=> ( ( times_times_mat_int @ P @ Q )
= ( one_mat_int @ N ) ) ) ) ).
% similar_mat_witD2(1)
thf(fact_110_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_111_similar__mat__witD2_I2_J,axiom,
! [A: mat_int,N: nat,M: nat,B: mat_int,P: mat_int,Q: mat_int] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ M ) )
=> ( ( similar_mat_wit_int @ A @ B @ P @ Q )
=> ( ( times_times_mat_int @ Q @ P )
= ( one_mat_int @ N ) ) ) ) ).
% similar_mat_witD2(2)
thf(fact_112_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_113_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_114_projector__def,axiom,
( linear5633924348262549461omplex
= ( ^ [M2: mat_complex] :
( ( comple8306762464034002205omplex @ M2 )
& ( ( times_8009071140041733218omplex @ M2 @ M2 )
= M2 ) ) ) ) ).
% projector_def
thf(fact_115_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_116_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_117_trace__smult,axiom,
! [A: mat_real,N: nat,C2: real] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( complex_trace_real @ ( smult_mat_real @ C2 @ A ) )
= ( times_times_real @ C2 @ ( complex_trace_real @ A ) ) ) ) ).
% trace_smult
thf(fact_118_trace__smult,axiom,
! [A: mat_int,N: nat,C2: int] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( ( complex_trace_int @ ( smult_mat_int @ C2 @ A ) )
= ( times_times_int @ C2 @ ( complex_trace_int @ A ) ) ) ) ).
% trace_smult
thf(fact_119_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_120_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_121_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_122_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_123_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_124_positive__iff__decomp,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_positive @ A )
= ( ? [X: mat_complex] :
( ( member_mat_complex @ X @ ( carrier_mat_complex @ N @ N ) )
& ( ( times_8009071140041733218omplex @ X @ ( schur_5982229384592763574omplex @ X ) )
= A ) ) ) ) ) ).
% positive_iff_decomp
thf(fact_125_positive__one,axiom,
! [N: nat] : ( complex_positive @ ( one_mat_complex @ N ) ) ).
% positive_one
thf(fact_126_projector__positive,axiom,
! [M3: mat_complex] :
( ( linear5633924348262549461omplex @ M3 )
=> ( complex_positive @ M3 ) ) ).
% projector_positive
thf(fact_127_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_128_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_129_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_130_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_131_index__one__mat_I2_J,axiom,
! [N: nat] :
( ( dim_row_complex @ ( one_mat_complex @ N ) )
= N ) ).
% index_one_mat(2)
thf(fact_132_index__one__mat_I2_J,axiom,
! [N: nat] :
( ( dim_row_int @ ( one_mat_int @ N ) )
= N ) ).
% index_one_mat(2)
thf(fact_133_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_134_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_135_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_136_smult__smult__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,K: complex,L: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( smult_mat_complex @ K @ ( smult_mat_complex @ L @ A ) )
= ( smult_mat_complex @ ( times_times_complex @ K @ L ) @ A ) ) ) ).
% smult_smult_mat
thf(fact_137_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_138_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_139_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_140_positive__is__hermitian,axiom,
! [A: mat_complex] :
( ( complex_positive @ A )
=> ( comple8306762464034002205omplex @ A ) ) ).
% positive_is_hermitian
thf(fact_141_left__mult__one__mat_H,axiom,
! [A: mat_int,N: nat] :
( ( ( dim_row_int @ A )
= N )
=> ( ( times_times_mat_int @ ( one_mat_int @ N ) @ A )
= A ) ) ).
% left_mult_one_mat'
thf(fact_142_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_143_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_144_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_145_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_146_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_147_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_148_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_149_mat__assoc__test_I11_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 @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B ) @ C ) @ D ) )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ C @ D ) @ A ) @ B ) ) ) ) ) ) ) ).
% mat_assoc_test(11)
thf(fact_150_mat__assoc__test_I10_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 @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B ) @ C ) )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ B @ C ) @ A ) ) ) ) ) ) ) ).
% mat_assoc_test(10)
thf(fact_151_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_152_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_153_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_154_trace__comm,axiom,
! [A: mat_int,N: nat,B: mat_int] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( ( member_mat_int @ B @ ( carrier_mat_int @ N @ N ) )
=> ( ( complex_trace_int @ ( times_times_mat_int @ A @ B ) )
= ( complex_trace_int @ ( times_times_mat_int @ B @ A ) ) ) ) ) ).
% trace_comm
thf(fact_155_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_156_similar__mat__witD_I1_J,axiom,
! [N: nat,A: mat_int,B: mat_int,P: mat_int,Q: mat_int] :
( ( N
= ( dim_row_int @ A ) )
=> ( ( similar_mat_wit_int @ A @ B @ P @ Q )
=> ( ( times_times_mat_int @ P @ Q )
= ( one_mat_int @ N ) ) ) ) ).
% similar_mat_witD(1)
thf(fact_157_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_158_similar__mat__witD_I2_J,axiom,
! [N: nat,A: mat_int,B: mat_int,P: mat_int,Q: mat_int] :
( ( N
= ( dim_row_int @ A ) )
=> ( ( similar_mat_wit_int @ A @ B @ P @ Q )
=> ( ( times_times_mat_int @ Q @ P )
= ( one_mat_int @ N ) ) ) ) ).
% similar_mat_witD(2)
thf(fact_159_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_160_projector__square__eq,axiom,
! [M3: mat_complex] :
( ( linear5633924348262549461omplex @ M3 )
=> ( ( times_8009071140041733218omplex @ M3 @ M3 )
= M3 ) ) ).
% projector_square_eq
thf(fact_161_inverts__mat__def,axiom,
( inverts_mat_int
= ( ^ [A2: mat_int,B2: mat_int] :
( ( times_times_mat_int @ A2 @ B2 )
= ( one_mat_int @ ( dim_row_int @ A2 ) ) ) ) ) ).
% inverts_mat_def
thf(fact_162_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_163_unitary__operator__keep__trace,axiom,
! [U: mat_complex,N: nat,A: mat_complex] :
( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ U )
=> ( ( comple3184165445352484367omplex @ A )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ U ) @ A ) @ U ) ) ) ) ) ) ).
% unitary_operator_keep_trace
thf(fact_164_projector__hermitian,axiom,
! [M3: mat_complex] :
( ( linear5633924348262549461omplex @ M3 )
=> ( comple8306762464034002205omplex @ M3 ) ) ).
% projector_hermitian
thf(fact_165_positive__if__decomp,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ? [M4: mat_complex] :
( ( times_8009071140041733218omplex @ M4 @ ( schur_5982229384592763574omplex @ M4 ) )
= A )
=> ( complex_positive @ A ) ) ) ).
% positive_if_decomp
thf(fact_166_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_167_mat__incr__mult__adjoint__mat__incr,axiom,
! [N: nat] :
( ( times_8009071140041733218omplex @ ( mat_incr @ N ) @ ( schur_5982229384592763574omplex @ ( mat_incr @ N ) ) )
= ( one_mat_complex @ N ) ) ).
% mat_incr_mult_adjoint_mat_incr
thf(fact_168_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_169_hermitian__smult,axiom,
! [A: mat_complex,N: nat,A3: real] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple8306762464034002205omplex @ A )
=> ( comple8306762464034002205omplex @ ( smult_mat_complex @ ( real_V4546457046886955230omplex @ A3 ) @ A ) ) ) ) ).
% hermitian_smult
thf(fact_170_corthogonal__inv__result,axiom,
! [A: mat_complex] :
( ( schur_549222400177443379omplex @ A )
=> ( inverts_mat_complex @ ( schur_4574106303853392228omplex @ A ) @ A ) ) ).
% corthogonal_inv_result
thf(fact_171_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_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_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_181_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_182_zero__adjoint,axiom,
! [N: nat,M: nat] :
( ( schur_5982229384592763574omplex @ ( zero_mat_complex @ N @ M ) )
= ( zero_mat_complex @ M @ N ) ) ).
% zero_adjoint
thf(fact_183_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_184_zero__hermitian,axiom,
! [N: nat] : ( comple8306762464034002205omplex @ ( zero_mat_complex @ N @ N ) ) ).
% zero_hermitian
thf(fact_185_index__one__mat_I3_J,axiom,
! [N: nat] :
( ( dim_col_complex @ ( one_mat_complex @ N ) )
= N ) ).
% index_one_mat(3)
thf(fact_186_index__one__mat_I3_J,axiom,
! [N: nat] :
( ( dim_col_int @ ( one_mat_int @ N ) )
= N ) ).
% index_one_mat(3)
thf(fact_187_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_188_mat__incr__dim,axiom,
! [N: nat] : ( member_mat_complex @ ( mat_incr @ N ) @ ( carrier_mat_complex @ N @ N ) ) ).
% mat_incr_dim
thf(fact_189_Complex__Matrix_Opositive__zero,axiom,
! [N: nat] : ( complex_positive @ ( zero_mat_complex @ N @ N ) ) ).
% Complex_Matrix.positive_zero
thf(fact_190_upper__triangular__zero,axiom,
! [N: nat] : ( upper_4850907204721561915omplex @ ( zero_mat_complex @ N @ N ) ) ).
% upper_triangular_zero
thf(fact_191_positive__dim__eq,axiom,
! [A: mat_complex] :
( ( complex_positive @ A )
=> ( ( dim_row_complex @ A )
= ( dim_col_complex @ A ) ) ) ).
% positive_dim_eq
thf(fact_192_zero__projector,axiom,
! [N: nat] : ( linear5633924348262549461omplex @ ( zero_mat_complex @ N @ N ) ) ).
% zero_projector
thf(fact_193_unitary__mat__incr,axiom,
! [N: nat] : ( comple6660659447773130958omplex @ ( mat_incr @ N ) ) ).
% unitary_mat_incr
thf(fact_194_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_195_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_196_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_197_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_198_right__mult__one__mat_H,axiom,
! [A: mat_int,N: nat] :
( ( ( dim_col_int @ A )
= N )
=> ( ( times_times_mat_int @ A @ ( one_mat_int @ N ) )
= A ) ) ).
% right_mult_one_mat'
thf(fact_199_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_200_adjoint__dim__col,axiom,
! [A: mat_complex] :
( ( dim_col_complex @ ( schur_5982229384592763574omplex @ A ) )
= ( dim_row_complex @ A ) ) ).
% adjoint_dim_col
thf(fact_201_adjoint__dim__row,axiom,
! [A: mat_complex] :
( ( dim_row_complex @ ( schur_5982229384592763574omplex @ A ) )
= ( dim_col_complex @ A ) ) ).
% adjoint_dim_row
thf(fact_202_diag__mat__diagonal__eq,axiom,
! [A: mat_complex,B: mat_complex] :
( ( ( diag_mat_complex @ A )
= ( diag_mat_complex @ B ) )
=> ( ( diagonal_mat_complex @ A )
=> ( ( diagonal_mat_complex @ B )
=> ( ( ( dim_col_complex @ A )
= ( dim_col_complex @ B ) )
=> ( A = B ) ) ) ) ) ).
% diag_mat_diagonal_eq
thf(fact_203_trace__proj__pos__real,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 ) )
=> ( ( real_V4546457046886955230omplex @ ( re @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R @ P ) ) ) )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R @ P ) ) ) ) ) ) ) ).
% trace_proj_pos_real
thf(fact_204_cpx__sq__mat_Otrace__hermitian__pos__real,axiom,
! [DimR: nat,DimC: nat,Fc_mats: set_mat_complex,A: mat_complex,R: mat_complex] :
( ( linear7199532782703566157sq_mat @ DimR @ DimC @ Fc_mats )
=> ( ( comple8306762464034002205omplex @ A )
=> ( ( complex_positive @ R )
=> ( ( member_mat_complex @ A @ Fc_mats )
=> ( ( member_mat_complex @ R @ Fc_mats )
=> ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R @ A ) )
= ( real_V4546457046886955230omplex @ ( re @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R @ A ) ) ) ) ) ) ) ) ) ) ).
% cpx_sq_mat.trace_hermitian_pos_real
thf(fact_205_of__real__hom_Ohom__mult,axiom,
! [X2: real,Y: real] :
( ( real_V1803761363581548252l_real @ ( times_times_real @ X2 @ Y ) )
= ( times_times_real @ ( real_V1803761363581548252l_real @ X2 ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% of_real_hom.hom_mult
thf(fact_206_of__real__hom_Ohom__mult,axiom,
! [X2: real,Y: real] :
( ( real_V4546457046886955230omplex @ ( times_times_real @ X2 @ Y ) )
= ( times_times_complex @ ( real_V4546457046886955230omplex @ X2 ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).
% of_real_hom.hom_mult
thf(fact_207_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_208_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_209_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 )
= ( ! [X: mat_complex] :
( ( member_mat_complex @ X @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple1169154605998056944erator @ X )
=> ( ord_less_eq_complex @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ A @ X ) ) @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ B @ X ) ) ) ) ) ) ) ) ) ).
% lowner_le_trace
thf(fact_210_adj__mat_I2_J,axiom,
! [A: mat_int,N: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( ( times_times_mat_int @ A @ ( adj_mat_int @ A ) )
= ( smult_mat_int @ ( det_int @ A ) @ ( one_mat_int @ N ) ) ) ) ).
% adj_mat(2)
thf(fact_211_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_212_adj__mat_I3_J,axiom,
! [A: mat_int,N: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( ( times_times_mat_int @ ( adj_mat_int @ A ) @ A )
= ( smult_mat_int @ ( det_int @ A ) @ ( one_mat_int @ N ) ) ) ) ).
% adj_mat(3)
thf(fact_213_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_214_det__mult,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 ) )
=> ( ( det_real @ ( times_times_mat_real @ A @ B ) )
= ( times_times_real @ ( det_real @ A ) @ ( det_real @ B ) ) ) ) ) ).
% det_mult
thf(fact_215_det__mult,axiom,
! [A: mat_int,N: nat,B: mat_int] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( ( member_mat_int @ B @ ( carrier_mat_int @ N @ N ) )
=> ( ( det_int @ ( times_times_mat_int @ A @ B ) )
= ( times_times_int @ ( det_int @ A ) @ ( det_int @ B ) ) ) ) ) ).
% det_mult
thf(fact_216_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_217_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_218_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_219_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_220_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_221_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_222_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_223_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_224_det__upper__triangular,axiom,
! [A: mat_complex,N: nat] :
( ( upper_4850907204721561915omplex @ A )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( det_complex @ A )
= ( groups7979759902575632448omplex @ ( diag_mat_complex @ A ) ) ) ) ) ).
% det_upper_triangular
thf(fact_225_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_226_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_227_trace__minus__linear,axiom,
! [A: mat_int,N: nat,B: mat_int] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( ( member_mat_int @ B @ ( carrier_mat_int @ N @ N ) )
=> ( ( complex_trace_int @ ( minus_minus_mat_int @ A @ B ) )
= ( minus_minus_int @ ( complex_trace_int @ A ) @ ( complex_trace_int @ B ) ) ) ) ) ).
% trace_minus_linear
thf(fact_228_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_229_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_230_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_231_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_232_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_233_mult__hom_Ohom__zero,axiom,
! [C2: complex] :
( ( times_times_complex @ C2 @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_hom.hom_zero
thf(fact_234_mult__hom_Ohom__zero,axiom,
! [C2: real] :
( ( times_times_real @ C2 @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_235_mult__hom_Ohom__zero,axiom,
! [C2: nat] :
( ( times_times_nat @ C2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_236_mult__hom_Ohom__zero,axiom,
! [C2: int] :
( ( times_times_int @ C2 @ zero_zero_int )
= zero_zero_int ) ).
% mult_hom.hom_zero
thf(fact_237_Linear__Algebra__Complements_Osmult__one,axiom,
! [A: mat_complex] :
( ( smult_mat_complex @ one_one_complex @ A )
= A ) ).
% Linear_Algebra_Complements.smult_one
thf(fact_238_Linear__Algebra__Complements_Osmult__one,axiom,
! [A: mat_real] :
( ( smult_mat_real @ one_one_real @ A )
= A ) ).
% Linear_Algebra_Complements.smult_one
thf(fact_239_Linear__Algebra__Complements_Osmult__one,axiom,
! [A: mat_nat] :
( ( smult_mat_nat @ one_one_nat @ A )
= A ) ).
% Linear_Algebra_Complements.smult_one
thf(fact_240_Linear__Algebra__Complements_Osmult__one,axiom,
! [A: mat_int] :
( ( smult_mat_int @ one_one_int @ A )
= A ) ).
% Linear_Algebra_Complements.smult_one
thf(fact_241_of__real__hom_Ohom__mult__eq__zero,axiom,
! [X2: real,Y: real] :
( ( ( times_times_real @ X2 @ Y )
= zero_zero_real )
=> ( ( times_times_real @ ( real_V1803761363581548252l_real @ X2 ) @ ( real_V1803761363581548252l_real @ Y ) )
= zero_zero_real ) ) ).
% of_real_hom.hom_mult_eq_zero
thf(fact_242_of__real__hom_Ohom__mult__eq__zero,axiom,
! [X2: real,Y: real] :
( ( ( times_times_real @ X2 @ Y )
= zero_zero_real )
=> ( ( times_times_complex @ ( real_V4546457046886955230omplex @ X2 ) @ ( real_V4546457046886955230omplex @ Y ) )
= zero_zero_complex ) ) ).
% of_real_hom.hom_mult_eq_zero
thf(fact_243_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_244_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_245_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_246_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_247_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_248_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_249_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_250_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_251_det__one,axiom,
! [N: nat] :
( ( det_complex @ ( one_mat_complex @ N ) )
= one_one_complex ) ).
% det_one
thf(fact_252_det__one,axiom,
! [N: nat] :
( ( det_real @ ( one_mat_real @ N ) )
= one_one_real ) ).
% det_one
thf(fact_253_det__one,axiom,
! [N: nat] :
( ( det_int @ ( one_mat_int @ N ) )
= one_one_int ) ).
% det_one
thf(fact_254_trace__zero,axiom,
! [N: nat] :
( ( comple3184165445352484367omplex @ ( zero_mat_complex @ N @ N ) )
= zero_zero_complex ) ).
% trace_zero
thf(fact_255_trace__zero,axiom,
! [N: nat] :
( ( complex_trace_real @ ( zero_mat_real @ N @ N ) )
= zero_zero_real ) ).
% trace_zero
thf(fact_256_trace__zero,axiom,
! [N: nat] :
( ( complex_trace_int @ ( zero_mat_int @ N @ N ) )
= zero_zero_int ) ).
% trace_zero
thf(fact_257_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_258_density__operator__def,axiom,
( comple5220265106149225959erator
= ( ^ [A2: mat_complex] :
( ( complex_positive @ A2 )
& ( ( comple3184165445352484367omplex @ A2 )
= one_one_complex ) ) ) ) ).
% density_operator_def
thf(fact_259_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_260_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_261_smult__zero,axiom,
! [A: mat_int] :
( ( smult_mat_int @ zero_zero_int @ A )
= ( zero_mat_int @ ( dim_row_int @ A ) @ ( dim_col_int @ A ) ) ) ).
% smult_zero
thf(fact_262_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_263_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_264_mult__left__le__one__le,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X2 ) @ X2 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_265_mult__left__le__one__le,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X2 ) @ X2 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_266_mult__right__le__one__le,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X2 @ Y ) @ X2 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_267_mult__right__le__one__le,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X2 @ Y ) @ X2 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_268_mult__le__one,axiom,
! [A3: real,B4: real] :
( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B4 )
=> ( ( ord_less_eq_real @ B4 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B4 ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_269_mult__le__one,axiom,
! [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B4 ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_270_mult__le__one,axiom,
! [A3: int,B4: int] :
( ( ord_less_eq_int @ A3 @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ B4 ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_271_mult__left__le,axiom,
! [C2: real,A3: real] :
( ( ord_less_eq_real @ C2 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ A3 ) ) ) ).
% mult_left_le
thf(fact_272_mult__left__le,axiom,
! [C2: nat,A3: nat] :
( ( ord_less_eq_nat @ C2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C2 ) @ A3 ) ) ) ).
% mult_left_le
thf(fact_273_mult__left__le,axiom,
! [C2: int,A3: int] :
( ( ord_less_eq_int @ C2 @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C2 ) @ A3 ) ) ) ).
% mult_left_le
thf(fact_274_mult__eq__1,axiom,
! [A3: complex,B4: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ A3 @ one_one_complex )
=> ( ( ord_less_eq_complex @ B4 @ one_one_complex )
=> ( ( ( times_times_complex @ A3 @ B4 )
= one_one_complex )
= ( ( A3 = one_one_complex )
& ( B4 = one_one_complex ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_275_mult__eq__1,axiom,
! [A3: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ( ord_less_eq_real @ B4 @ one_one_real )
=> ( ( ( times_times_real @ A3 @ B4 )
= one_one_real )
= ( ( A3 = one_one_real )
& ( B4 = one_one_real ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_276_mult__eq__1,axiom,
! [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ( ord_less_eq_nat @ B4 @ one_one_nat )
=> ( ( ( times_times_nat @ A3 @ B4 )
= one_one_nat )
= ( ( A3 = one_one_nat )
& ( B4 = one_one_nat ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_277_mult__eq__1,axiom,
! [A3: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ A3 @ one_one_int )
=> ( ( ord_less_eq_int @ B4 @ one_one_int )
=> ( ( ( times_times_int @ A3 @ B4 )
= one_one_int )
= ( ( A3 = one_one_int )
& ( B4 = one_one_int ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_278_poly__cancel__eq__conv,axiom,
! [X2: complex,A3: complex,Y: complex,B4: complex] :
( ( X2 = zero_zero_complex )
=> ( ( A3 != zero_zero_complex )
=> ( ( Y = zero_zero_complex )
= ( ( minus_minus_complex @ ( times_times_complex @ A3 @ Y ) @ ( times_times_complex @ B4 @ X2 ) )
= zero_zero_complex ) ) ) ) ).
% poly_cancel_eq_conv
thf(fact_279_poly__cancel__eq__conv,axiom,
! [X2: real,A3: real,Y: real,B4: real] :
( ( X2 = zero_zero_real )
=> ( ( A3 != zero_zero_real )
=> ( ( Y = zero_zero_real )
= ( ( minus_minus_real @ ( times_times_real @ A3 @ Y ) @ ( times_times_real @ B4 @ X2 ) )
= zero_zero_real ) ) ) ) ).
% poly_cancel_eq_conv
thf(fact_280_positive__scale,axiom,
! [A: mat_complex,N: nat,C2: real] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_positive @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( complex_positive @ ( smult_mat_complex @ ( real_V4546457046886955230omplex @ C2 ) @ A ) ) ) ) ) ).
% positive_scale
thf(fact_281_lowner__le__smult,axiom,
! [C2: real,A: mat_complex,B: mat_complex,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ 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 @ ( real_V4546457046886955230omplex @ C2 ) @ A ) @ ( smult_mat_complex @ ( real_V4546457046886955230omplex @ C2 ) @ B ) ) ) ) ) ) ).
% lowner_le_smult
thf(fact_282_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_283_class__cring_Ofactors__equal,axiom,
! [A3: complex,B4: complex,C2: complex,D2: complex] :
( ( A3 = B4 )
=> ( ( C2 = D2 )
=> ( ( times_times_complex @ A3 @ C2 )
= ( times_times_complex @ B4 @ D2 ) ) ) ) ).
% class_cring.factors_equal
thf(fact_284_class__cring_Ofactors__equal,axiom,
! [A3: real,B4: real,C2: real,D2: real] :
( ( A3 = B4 )
=> ( ( C2 = D2 )
=> ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B4 @ D2 ) ) ) ) ).
% class_cring.factors_equal
thf(fact_285_class__cring_Ofactors__equal,axiom,
! [A3: int,B4: int,C2: int,D2: int] :
( ( A3 = B4 )
=> ( ( C2 = D2 )
=> ( ( times_times_int @ A3 @ C2 )
= ( times_times_int @ B4 @ D2 ) ) ) ) ).
% class_cring.factors_equal
thf(fact_286_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_287_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_288_det__dim__zero,axiom,
! [A: mat_int] :
( ( member_mat_int @ A @ ( carrier_mat_int @ zero_zero_nat @ zero_zero_nat ) )
=> ( ( det_int @ A )
= one_one_int ) ) ).
% det_dim_zero
thf(fact_289_mult__not__zero,axiom,
! [A3: complex,B4: complex] :
( ( ( times_times_complex @ A3 @ B4 )
!= zero_zero_complex )
=> ( ( A3 != zero_zero_complex )
& ( B4 != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_290_mult__not__zero,axiom,
! [A3: real,B4: real] :
( ( ( times_times_real @ A3 @ B4 )
!= zero_zero_real )
=> ( ( A3 != zero_zero_real )
& ( B4 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_291_mult__not__zero,axiom,
! [A3: nat,B4: nat] :
( ( ( times_times_nat @ A3 @ B4 )
!= zero_zero_nat )
=> ( ( A3 != zero_zero_nat )
& ( B4 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_292_mult__not__zero,axiom,
! [A3: int,B4: int] :
( ( ( times_times_int @ A3 @ B4 )
!= zero_zero_int )
=> ( ( A3 != zero_zero_int )
& ( B4 != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_293_mult__zero__left,axiom,
! [A3: complex] :
( ( times_times_complex @ zero_zero_complex @ A3 )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_294_mult__zero__left,axiom,
! [A3: real] :
( ( times_times_real @ zero_zero_real @ A3 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_295_mult__zero__left,axiom,
! [A3: nat] :
( ( times_times_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_296_mult__zero__left,axiom,
! [A3: int] :
( ( times_times_int @ zero_zero_int @ A3 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_297_mult__zero__right,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_298_mult__zero__right,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_299_mult__zero__right,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_300_mult__zero__right,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_301_divisors__zero,axiom,
! [A3: complex,B4: complex] :
( ( ( times_times_complex @ A3 @ B4 )
= zero_zero_complex )
=> ( ( A3 = zero_zero_complex )
| ( B4 = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_302_divisors__zero,axiom,
! [A3: real,B4: real] :
( ( ( times_times_real @ A3 @ B4 )
= zero_zero_real )
=> ( ( A3 = zero_zero_real )
| ( B4 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_303_divisors__zero,axiom,
! [A3: nat,B4: nat] :
( ( ( times_times_nat @ A3 @ B4 )
= zero_zero_nat )
=> ( ( A3 = zero_zero_nat )
| ( B4 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_304_divisors__zero,axiom,
! [A3: int,B4: int] :
( ( ( times_times_int @ A3 @ B4 )
= zero_zero_int )
=> ( ( A3 = zero_zero_int )
| ( B4 = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_305_mult__eq__0__iff,axiom,
! [A3: complex,B4: complex] :
( ( ( times_times_complex @ A3 @ B4 )
= zero_zero_complex )
= ( ( A3 = zero_zero_complex )
| ( B4 = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_306_mult__eq__0__iff,axiom,
! [A3: real,B4: real] :
( ( ( times_times_real @ A3 @ B4 )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( B4 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_307_mult__eq__0__iff,axiom,
! [A3: nat,B4: nat] :
( ( ( times_times_nat @ A3 @ B4 )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
| ( B4 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_308_mult__eq__0__iff,axiom,
! [A3: int,B4: int] :
( ( ( times_times_int @ A3 @ B4 )
= zero_zero_int )
= ( ( A3 = zero_zero_int )
| ( B4 = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_309_no__zero__divisors,axiom,
! [A3: complex,B4: complex] :
( ( A3 != zero_zero_complex )
=> ( ( B4 != zero_zero_complex )
=> ( ( times_times_complex @ A3 @ B4 )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_310_no__zero__divisors,axiom,
! [A3: real,B4: real] :
( ( A3 != zero_zero_real )
=> ( ( B4 != zero_zero_real )
=> ( ( times_times_real @ A3 @ B4 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_311_no__zero__divisors,axiom,
! [A3: nat,B4: nat] :
( ( A3 != zero_zero_nat )
=> ( ( B4 != zero_zero_nat )
=> ( ( times_times_nat @ A3 @ B4 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_312_no__zero__divisors,axiom,
! [A3: int,B4: int] :
( ( A3 != zero_zero_int )
=> ( ( B4 != zero_zero_int )
=> ( ( times_times_int @ A3 @ B4 )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_313_mult__cancel__left,axiom,
! [C2: complex,A3: complex,B4: complex] :
( ( ( times_times_complex @ C2 @ A3 )
= ( times_times_complex @ C2 @ B4 ) )
= ( ( C2 = zero_zero_complex )
| ( A3 = B4 ) ) ) ).
% mult_cancel_left
thf(fact_314_mult__cancel__left,axiom,
! [C2: real,A3: real,B4: real] :
( ( ( times_times_real @ C2 @ A3 )
= ( times_times_real @ C2 @ B4 ) )
= ( ( C2 = zero_zero_real )
| ( A3 = B4 ) ) ) ).
% mult_cancel_left
thf(fact_315_mult__cancel__left,axiom,
! [C2: nat,A3: nat,B4: nat] :
( ( ( times_times_nat @ C2 @ A3 )
= ( times_times_nat @ C2 @ B4 ) )
= ( ( C2 = zero_zero_nat )
| ( A3 = B4 ) ) ) ).
% mult_cancel_left
thf(fact_316_mult__cancel__left,axiom,
! [C2: int,A3: int,B4: int] :
( ( ( times_times_int @ C2 @ A3 )
= ( times_times_int @ C2 @ B4 ) )
= ( ( C2 = zero_zero_int )
| ( A3 = B4 ) ) ) ).
% mult_cancel_left
thf(fact_317_mult__left__cancel,axiom,
! [C2: complex,A3: complex,B4: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ C2 @ A3 )
= ( times_times_complex @ C2 @ B4 ) )
= ( A3 = B4 ) ) ) ).
% mult_left_cancel
thf(fact_318_mult__left__cancel,axiom,
! [C2: real,A3: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A3 )
= ( times_times_real @ C2 @ B4 ) )
= ( A3 = B4 ) ) ) ).
% mult_left_cancel
thf(fact_319_mult__left__cancel,axiom,
! [C2: nat,A3: nat,B4: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A3 )
= ( times_times_nat @ C2 @ B4 ) )
= ( A3 = B4 ) ) ) ).
% mult_left_cancel
thf(fact_320_mult__left__cancel,axiom,
! [C2: int,A3: int,B4: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ C2 @ A3 )
= ( times_times_int @ C2 @ B4 ) )
= ( A3 = B4 ) ) ) ).
% mult_left_cancel
thf(fact_321_mult__cancel__right,axiom,
! [A3: complex,C2: complex,B4: complex] :
( ( ( times_times_complex @ A3 @ C2 )
= ( times_times_complex @ B4 @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( A3 = B4 ) ) ) ).
% mult_cancel_right
thf(fact_322_mult__cancel__right,axiom,
! [A3: real,C2: real,B4: real] :
( ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B4 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A3 = B4 ) ) ) ).
% mult_cancel_right
thf(fact_323_mult__cancel__right,axiom,
! [A3: nat,C2: nat,B4: nat] :
( ( ( times_times_nat @ A3 @ C2 )
= ( times_times_nat @ B4 @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A3 = B4 ) ) ) ).
% mult_cancel_right
thf(fact_324_mult__cancel__right,axiom,
! [A3: int,C2: int,B4: int] :
( ( ( times_times_int @ A3 @ C2 )
= ( times_times_int @ B4 @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( A3 = B4 ) ) ) ).
% mult_cancel_right
thf(fact_325_mult__right__cancel,axiom,
! [C2: complex,A3: complex,B4: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ C2 )
= ( times_times_complex @ B4 @ C2 ) )
= ( A3 = B4 ) ) ) ).
% mult_right_cancel
thf(fact_326_mult__right__cancel,axiom,
! [C2: real,A3: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B4 @ C2 ) )
= ( A3 = B4 ) ) ) ).
% mult_right_cancel
thf(fact_327_mult__right__cancel,axiom,
! [C2: nat,A3: nat,B4: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A3 @ C2 )
= ( times_times_nat @ B4 @ C2 ) )
= ( A3 = B4 ) ) ) ).
% mult_right_cancel
thf(fact_328_mult__right__cancel,axiom,
! [C2: int,A3: int,B4: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ A3 @ C2 )
= ( times_times_int @ B4 @ C2 ) )
= ( A3 = B4 ) ) ) ).
% mult_right_cancel
thf(fact_329_right__diff__distrib_H,axiom,
! [A3: complex,B4: complex,C2: complex] :
( ( times_times_complex @ A3 @ ( minus_minus_complex @ B4 @ C2 ) )
= ( minus_minus_complex @ ( times_times_complex @ A3 @ B4 ) @ ( times_times_complex @ A3 @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_330_right__diff__distrib_H,axiom,
! [A3: real,B4: real,C2: real] :
( ( times_times_real @ A3 @ ( minus_minus_real @ B4 @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A3 @ B4 ) @ ( times_times_real @ A3 @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_331_right__diff__distrib_H,axiom,
! [A3: nat,B4: nat,C2: nat] :
( ( times_times_nat @ A3 @ ( minus_minus_nat @ B4 @ C2 ) )
= ( minus_minus_nat @ ( times_times_nat @ A3 @ B4 ) @ ( times_times_nat @ A3 @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_332_right__diff__distrib_H,axiom,
! [A3: int,B4: int,C2: int] :
( ( times_times_int @ A3 @ ( minus_minus_int @ B4 @ C2 ) )
= ( minus_minus_int @ ( times_times_int @ A3 @ B4 ) @ ( times_times_int @ A3 @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_333_left__diff__distrib_H,axiom,
! [B4: complex,C2: complex,A3: complex] :
( ( times_times_complex @ ( minus_minus_complex @ B4 @ C2 ) @ A3 )
= ( minus_minus_complex @ ( times_times_complex @ B4 @ A3 ) @ ( times_times_complex @ C2 @ A3 ) ) ) ).
% left_diff_distrib'
thf(fact_334_left__diff__distrib_H,axiom,
! [B4: real,C2: real,A3: real] :
( ( times_times_real @ ( minus_minus_real @ B4 @ C2 ) @ A3 )
= ( minus_minus_real @ ( times_times_real @ B4 @ A3 ) @ ( times_times_real @ C2 @ A3 ) ) ) ).
% left_diff_distrib'
thf(fact_335_left__diff__distrib_H,axiom,
! [B4: nat,C2: nat,A3: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B4 @ C2 ) @ A3 )
= ( minus_minus_nat @ ( times_times_nat @ B4 @ A3 ) @ ( times_times_nat @ C2 @ A3 ) ) ) ).
% left_diff_distrib'
thf(fact_336_left__diff__distrib_H,axiom,
! [B4: int,C2: int,A3: int] :
( ( times_times_int @ ( minus_minus_int @ B4 @ C2 ) @ A3 )
= ( minus_minus_int @ ( times_times_int @ B4 @ A3 ) @ ( times_times_int @ C2 @ A3 ) ) ) ).
% left_diff_distrib'
thf(fact_337_right__diff__distrib,axiom,
! [A3: complex,B4: complex,C2: complex] :
( ( times_times_complex @ A3 @ ( minus_minus_complex @ B4 @ C2 ) )
= ( minus_minus_complex @ ( times_times_complex @ A3 @ B4 ) @ ( times_times_complex @ A3 @ C2 ) ) ) ).
% right_diff_distrib
thf(fact_338_right__diff__distrib,axiom,
! [A3: real,B4: real,C2: real] :
( ( times_times_real @ A3 @ ( minus_minus_real @ B4 @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A3 @ B4 ) @ ( times_times_real @ A3 @ C2 ) ) ) ).
% right_diff_distrib
thf(fact_339_right__diff__distrib,axiom,
! [A3: int,B4: int,C2: int] :
( ( times_times_int @ A3 @ ( minus_minus_int @ B4 @ C2 ) )
= ( minus_minus_int @ ( times_times_int @ A3 @ B4 ) @ ( times_times_int @ A3 @ C2 ) ) ) ).
% right_diff_distrib
thf(fact_340_left__diff__distrib,axiom,
! [A3: complex,B4: complex,C2: complex] :
( ( times_times_complex @ ( minus_minus_complex @ A3 @ B4 ) @ C2 )
= ( minus_minus_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B4 @ C2 ) ) ) ).
% left_diff_distrib
thf(fact_341_left__diff__distrib,axiom,
! [A3: real,B4: real,C2: real] :
( ( times_times_real @ ( minus_minus_real @ A3 @ B4 ) @ C2 )
= ( minus_minus_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ C2 ) ) ) ).
% left_diff_distrib
thf(fact_342_left__diff__distrib,axiom,
! [A3: int,B4: int,C2: int] :
( ( times_times_int @ ( minus_minus_int @ A3 @ B4 ) @ C2 )
= ( minus_minus_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ C2 ) ) ) ).
% left_diff_distrib
thf(fact_343_mult__sign__intros_I4_J,axiom,
! [A3: complex,B4: complex] :
( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ B4 @ zero_zero_complex )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A3 @ B4 ) ) ) ) ).
% mult_sign_intros(4)
thf(fact_344_mult__sign__intros_I4_J,axiom,
! [A3: real,B4: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B4 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B4 ) ) ) ) ).
% mult_sign_intros(4)
thf(fact_345_mult__sign__intros_I4_J,axiom,
! [A3: int,B4: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B4 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B4 ) ) ) ) ).
% mult_sign_intros(4)
thf(fact_346_mult__sign__intros_I3_J,axiom,
! [A3: complex,B4: complex] :
( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B4 )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ B4 ) @ zero_zero_complex ) ) ) ).
% mult_sign_intros(3)
thf(fact_347_mult__sign__intros_I3_J,axiom,
! [A3: real,B4: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B4 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B4 ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(3)
thf(fact_348_mult__sign__intros_I3_J,axiom,
! [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B4 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(3)
thf(fact_349_mult__sign__intros_I3_J,axiom,
! [A3: int,B4: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B4 )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ B4 ) @ zero_zero_int ) ) ) ).
% mult_sign_intros(3)
thf(fact_350_mult__sign__intros_I2_J,axiom,
! [A3: complex,B4: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ B4 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ B4 ) @ zero_zero_complex ) ) ) ).
% mult_sign_intros(2)
thf(fact_351_mult__sign__intros_I2_J,axiom,
! [A3: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B4 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B4 ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(2)
thf(fact_352_mult__sign__intros_I2_J,axiom,
! [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B4 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B4 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(2)
thf(fact_353_mult__sign__intros_I2_J,axiom,
! [A3: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ B4 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ B4 ) @ zero_zero_int ) ) ) ).
% mult_sign_intros(2)
thf(fact_354_mult__sign__intros_I1_J,axiom,
! [A3: complex,B4: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B4 )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A3 @ B4 ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_355_mult__sign__intros_I1_J,axiom,
! [A3: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B4 ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_356_mult__sign__intros_I1_J,axiom,
! [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B4 ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_357_mult__sign__intros_I1_J,axiom,
! [A3: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B4 ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_358_mult__mono,axiom,
! [A3: complex,B4: complex,C2: complex,D2: complex] :
( ( ord_less_eq_complex @ A3 @ B4 )
=> ( ( ord_less_eq_complex @ C2 @ D2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B4 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B4 @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_359_mult__mono,axiom,
! [A3: real,B4: real,C2: real,D2: real] :
( ( ord_less_eq_real @ A3 @ B4 )
=> ( ( ord_less_eq_real @ C2 @ D2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_360_mult__mono,axiom,
! [A3: nat,B4: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B4 @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_361_mult__mono,axiom,
! [A3: int,B4: int,C2: int,D2: int] :
( ( ord_less_eq_int @ A3 @ B4 )
=> ( ( ord_less_eq_int @ C2 @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_362_mult__mono_H,axiom,
! [A3: complex,B4: complex,C2: complex,D2: complex] :
( ( ord_less_eq_complex @ A3 @ B4 )
=> ( ( ord_less_eq_complex @ C2 @ D2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B4 @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_363_mult__mono_H,axiom,
! [A3: real,B4: real,C2: real,D2: real] :
( ( ord_less_eq_real @ A3 @ B4 )
=> ( ( ord_less_eq_real @ C2 @ D2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_364_mult__mono_H,axiom,
! [A3: nat,B4: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B4 @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_365_mult__mono_H,axiom,
! [A3: int,B4: int,C2: int,D2: int] :
( ( ord_less_eq_int @ A3 @ B4 )
=> ( ( ord_less_eq_int @ C2 @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_366_zero__le__square,axiom,
! [A3: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ A3 ) ) ).
% zero_le_square
thf(fact_367_zero__le__square,axiom,
! [A3: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ A3 ) ) ).
% zero_le_square
thf(fact_368_split__mult__pos__le,axiom,
! [A3: complex,B4: complex] :
( ( ( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
& ( ord_less_eq_complex @ zero_zero_complex @ B4 ) )
| ( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
& ( ord_less_eq_complex @ B4 @ zero_zero_complex ) ) )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A3 @ B4 ) ) ) ).
% split_mult_pos_le
thf(fact_369_split__mult__pos__le,axiom,
! [A3: real,B4: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B4 ) ) ) ).
% split_mult_pos_le
thf(fact_370_split__mult__pos__le,axiom,
! [A3: int,B4: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ zero_zero_int @ B4 ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ B4 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B4 ) ) ) ).
% split_mult_pos_le
thf(fact_371_mult__left__mono__neg,axiom,
! [B4: complex,A3: complex,C2: complex] :
( ( ord_less_eq_complex @ B4 @ A3 )
=> ( ( ord_less_eq_complex @ C2 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ C2 @ B4 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_372_mult__left__mono__neg,axiom,
! [B4: real,A3: real,C2: real] :
( ( ord_less_eq_real @ B4 @ A3 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B4 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_373_mult__left__mono__neg,axiom,
! [B4: int,A3: int,C2: int] :
( ( ord_less_eq_int @ B4 @ A3 )
=> ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B4 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_374_mult__left__mono,axiom,
! [A3: complex,B4: complex,C2: complex] :
( ( ord_less_eq_complex @ A3 @ B4 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ord_less_eq_complex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ C2 @ B4 ) ) ) ) ).
% mult_left_mono
thf(fact_375_mult__left__mono,axiom,
! [A3: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B4 ) ) ) ) ).
% mult_left_mono
thf(fact_376_mult__left__mono,axiom,
! [A3: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B4 ) ) ) ) ).
% mult_left_mono
thf(fact_377_mult__left__mono,axiom,
! [A3: int,B4: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B4 ) ) ) ) ).
% mult_left_mono
thf(fact_378_mult__right__mono__neg,axiom,
! [B4: complex,A3: complex,C2: complex] :
( ( ord_less_eq_complex @ B4 @ A3 )
=> ( ( ord_less_eq_complex @ C2 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B4 @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_379_mult__right__mono__neg,axiom,
! [B4: real,A3: real,C2: real] :
( ( ord_less_eq_real @ B4 @ A3 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_380_mult__right__mono__neg,axiom,
! [B4: int,A3: int,C2: int] :
( ( ord_less_eq_int @ B4 @ A3 )
=> ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_381_mult__right__mono,axiom,
! [A3: complex,B4: complex,C2: complex] :
( ( ord_less_eq_complex @ A3 @ B4 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B4 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_382_mult__right__mono,axiom,
! [A3: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_383_mult__right__mono,axiom,
! [A3: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B4 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_384_mult__right__mono,axiom,
! [A3: int,B4: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_385_mult__le__0__iff,axiom,
! [A3: real,B4: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ B4 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) ) ) ) ).
% mult_le_0_iff
thf(fact_386_mult__le__0__iff,axiom,
! [A3: int,B4: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ B4 ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ B4 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B4 ) ) ) ) ).
% mult_le_0_iff
thf(fact_387_split__mult__neg__le,axiom,
! [A3: complex,B4: complex] :
( ( ( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
& ( ord_less_eq_complex @ B4 @ zero_zero_complex ) )
| ( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
& ( ord_less_eq_complex @ zero_zero_complex @ B4 ) ) )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ B4 ) @ zero_zero_complex ) ) ).
% split_mult_neg_le
thf(fact_388_split__mult__neg__le,axiom,
! [A3: real,B4: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B4 ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_389_split__mult__neg__le,axiom,
! [A3: nat,B4: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
& ( ord_less_eq_nat @ B4 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B4 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B4 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_390_split__mult__neg__le,axiom,
! [A3: int,B4: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ B4 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B4 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ B4 ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_391_mult__nonneg__nonpos2,axiom,
! [A3: complex,B4: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ B4 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ B4 @ A3 ) @ zero_zero_complex ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_392_mult__nonneg__nonpos2,axiom,
! [A3: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B4 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B4 @ A3 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_393_mult__nonneg__nonpos2,axiom,
! [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B4 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B4 @ A3 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_394_mult__nonneg__nonpos2,axiom,
! [A3: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ B4 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B4 @ A3 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_395_zero__le__mult__iff,axiom,
! [A3: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B4 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_396_zero__le__mult__iff,axiom,
! [A3: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B4 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ zero_zero_int @ B4 ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ B4 @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_397_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: complex,B4: complex,C2: complex] :
( ( ord_less_eq_complex @ A3 @ B4 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ord_less_eq_complex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ C2 @ B4 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_398_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B4 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_399_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B4 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_400_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: int,B4: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B4 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_401_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_402_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_403_mult__cancel__right2,axiom,
! [A3: int,C2: int] :
( ( ( times_times_int @ A3 @ C2 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A3 = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_404_mult__cancel__right1,axiom,
! [C2: complex,B4: complex] :
( ( C2
= ( times_times_complex @ B4 @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( B4 = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_405_mult__cancel__right1,axiom,
! [C2: real,B4: real] :
( ( C2
= ( times_times_real @ B4 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B4 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_406_mult__cancel__right1,axiom,
! [C2: int,B4: int] :
( ( C2
= ( times_times_int @ B4 @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( B4 = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_407_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_408_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_409_mult__cancel__left2,axiom,
! [C2: int,A3: int] :
( ( ( times_times_int @ C2 @ A3 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A3 = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_410_mult__cancel__left1,axiom,
! [C2: complex,B4: complex] :
( ( C2
= ( times_times_complex @ C2 @ B4 ) )
= ( ( C2 = zero_zero_complex )
| ( B4 = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_411_mult__cancel__left1,axiom,
! [C2: real,B4: real] :
( ( C2
= ( times_times_real @ C2 @ B4 ) )
= ( ( C2 = zero_zero_real )
| ( B4 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_412_mult__cancel__left1,axiom,
! [C2: int,B4: int] :
( ( C2
= ( times_times_int @ C2 @ B4 ) )
= ( ( C2 = zero_zero_int )
| ( B4 = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_413_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_414_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_415_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_416_mult__if__delta,axiom,
! [P: $o,Q4: int] :
( ( P
=> ( ( times_times_int @ ( if_int @ P @ one_one_int @ zero_zero_int ) @ Q4 )
= Q4 ) )
& ( ~ P
=> ( ( times_times_int @ ( if_int @ P @ one_one_int @ zero_zero_int ) @ Q4 )
= zero_zero_int ) ) ) ).
% mult_if_delta
thf(fact_417_less__eq__fract__respect,axiom,
! [B4: real,B5: real,D2: real,D3: real,A3: real,A4: real,C2: real,C3: real] :
( ( B4 != zero_zero_real )
=> ( ( B5 != zero_zero_real )
=> ( ( D2 != zero_zero_real )
=> ( ( D3 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ B5 )
= ( times_times_real @ A4 @ B4 ) )
=> ( ( ( times_times_real @ C2 @ D3 )
= ( times_times_real @ C3 @ D2 ) )
=> ( ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A3 @ D2 ) @ ( times_times_real @ B4 @ D2 ) ) @ ( times_times_real @ ( times_times_real @ C2 @ B4 ) @ ( times_times_real @ B4 @ D2 ) ) )
= ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A4 @ D3 ) @ ( times_times_real @ B5 @ D3 ) ) @ ( times_times_real @ ( times_times_real @ C3 @ B5 ) @ ( times_times_real @ B5 @ D3 ) ) ) ) ) ) ) ) ) ) ).
% less_eq_fract_respect
thf(fact_418_less__eq__fract__respect,axiom,
! [B4: int,B5: int,D2: int,D3: int,A3: int,A4: int,C2: int,C3: int] :
( ( B4 != zero_zero_int )
=> ( ( B5 != zero_zero_int )
=> ( ( D2 != zero_zero_int )
=> ( ( D3 != zero_zero_int )
=> ( ( ( times_times_int @ A3 @ B5 )
= ( times_times_int @ A4 @ B4 ) )
=> ( ( ( times_times_int @ C2 @ D3 )
= ( times_times_int @ C3 @ D2 ) )
=> ( ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ B4 @ D2 ) ) @ ( times_times_int @ ( times_times_int @ C2 @ B4 ) @ ( times_times_int @ B4 @ D2 ) ) )
= ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A4 @ D3 ) @ ( times_times_int @ B5 @ D3 ) ) @ ( times_times_int @ ( times_times_int @ C3 @ B5 ) @ ( times_times_int @ B5 @ D3 ) ) ) ) ) ) ) ) ) ) ).
% less_eq_fract_respect
thf(fact_419_vector__space__over__itself_Oscale__right__diff__distrib,axiom,
! [A3: complex,X2: complex,Y: complex] :
( ( times_times_complex @ A3 @ ( minus_minus_complex @ X2 @ Y ) )
= ( minus_minus_complex @ ( times_times_complex @ A3 @ X2 ) @ ( times_times_complex @ A3 @ Y ) ) ) ).
% vector_space_over_itself.scale_right_diff_distrib
thf(fact_420_vector__space__over__itself_Oscale__right__diff__distrib,axiom,
! [A3: real,X2: real,Y: real] :
( ( times_times_real @ A3 @ ( minus_minus_real @ X2 @ Y ) )
= ( minus_minus_real @ ( times_times_real @ A3 @ X2 ) @ ( times_times_real @ A3 @ Y ) ) ) ).
% vector_space_over_itself.scale_right_diff_distrib
thf(fact_421_vector__space__over__itself_Oscale__left__commute,axiom,
! [A3: complex,B4: complex,X2: complex] :
( ( times_times_complex @ A3 @ ( times_times_complex @ B4 @ X2 ) )
= ( times_times_complex @ B4 @ ( times_times_complex @ A3 @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_422_vector__space__over__itself_Oscale__left__commute,axiom,
! [A3: real,B4: real,X2: real] :
( ( times_times_real @ A3 @ ( times_times_real @ B4 @ X2 ) )
= ( times_times_real @ B4 @ ( times_times_real @ A3 @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_423_vector__space__over__itself_Oscale__scale,axiom,
! [A3: complex,B4: complex,X2: complex] :
( ( times_times_complex @ A3 @ ( times_times_complex @ B4 @ X2 ) )
= ( times_times_complex @ ( times_times_complex @ A3 @ B4 ) @ X2 ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_424_vector__space__over__itself_Oscale__scale,axiom,
! [A3: real,B4: real,X2: real] :
( ( times_times_real @ A3 @ ( times_times_real @ B4 @ X2 ) )
= ( times_times_real @ ( times_times_real @ A3 @ B4 ) @ X2 ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_425_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X2: complex,A3: complex,B4: complex] :
( ( X2 != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ X2 )
= ( times_times_complex @ B4 @ X2 ) )
=> ( A3 = B4 ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_426_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X2: real,A3: real,B4: real] :
( ( X2 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ X2 )
= ( times_times_real @ B4 @ X2 ) )
=> ( A3 = B4 ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_427_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A3: complex,X2: complex,B4: complex] :
( ( ( times_times_complex @ A3 @ X2 )
= ( times_times_complex @ B4 @ X2 ) )
= ( ( A3 = B4 )
| ( X2 = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_428_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A3: real,X2: real,B4: real] :
( ( ( times_times_real @ A3 @ X2 )
= ( times_times_real @ B4 @ X2 ) )
= ( ( A3 = B4 )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_429_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A3: complex,X2: complex,Y: complex] :
( ( A3 != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ X2 )
= ( times_times_complex @ A3 @ Y ) )
=> ( X2 = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_430_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A3: real,X2: real,Y: real] :
( ( A3 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ X2 )
= ( times_times_real @ A3 @ Y ) )
=> ( X2 = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_431_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A3: complex,X2: complex,Y: complex] :
( ( ( times_times_complex @ A3 @ X2 )
= ( times_times_complex @ A3 @ Y ) )
= ( ( X2 = Y )
| ( A3 = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_432_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A3: real,X2: real,Y: real] :
( ( ( times_times_real @ A3 @ X2 )
= ( times_times_real @ A3 @ Y ) )
= ( ( X2 = Y )
| ( A3 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_433_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_434_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_435_vector__space__over__itself_Oscale__zero__left,axiom,
! [X2: complex] :
( ( times_times_complex @ zero_zero_complex @ X2 )
= zero_zero_complex ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_436_vector__space__over__itself_Oscale__zero__left,axiom,
! [X2: real] :
( ( times_times_real @ zero_zero_real @ X2 )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_437_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A3: complex,X2: complex] :
( ( ( times_times_complex @ A3 @ X2 )
= zero_zero_complex )
= ( ( A3 = zero_zero_complex )
| ( X2 = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_438_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A3: real,X2: real] :
( ( ( times_times_real @ A3 @ X2 )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_439_vector__space__over__itself_Oscale__one,axiom,
! [X2: complex] :
( ( times_times_complex @ one_one_complex @ X2 )
= X2 ) ).
% vector_space_over_itself.scale_one
thf(fact_440_vector__space__over__itself_Oscale__one,axiom,
! [X2: real] :
( ( times_times_real @ one_one_real @ X2 )
= X2 ) ).
% vector_space_over_itself.scale_one
thf(fact_441_vector__space__over__itself_Oscale__left__diff__distrib,axiom,
! [A3: complex,B4: complex,X2: complex] :
( ( times_times_complex @ ( minus_minus_complex @ A3 @ B4 ) @ X2 )
= ( minus_minus_complex @ ( times_times_complex @ A3 @ X2 ) @ ( times_times_complex @ B4 @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_diff_distrib
thf(fact_442_vector__space__over__itself_Oscale__left__diff__distrib,axiom,
! [A3: real,B4: real,X2: real] :
( ( times_times_real @ ( minus_minus_real @ A3 @ B4 ) @ X2 )
= ( minus_minus_real @ ( times_times_real @ A3 @ X2 ) @ ( times_times_real @ B4 @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_diff_distrib
thf(fact_443_cross3__simps_I12_J,axiom,
! [B4: complex,A3: complex,C2: complex] :
( ( times_times_complex @ B4 @ ( times_times_complex @ A3 @ C2 ) )
= ( times_times_complex @ A3 @ ( times_times_complex @ B4 @ C2 ) ) ) ).
% cross3_simps(12)
thf(fact_444_cross3__simps_I12_J,axiom,
! [B4: real,A3: real,C2: real] :
( ( times_times_real @ B4 @ ( times_times_real @ A3 @ C2 ) )
= ( times_times_real @ A3 @ ( times_times_real @ B4 @ C2 ) ) ) ).
% cross3_simps(12)
thf(fact_445_cross3__simps_I12_J,axiom,
! [B4: nat,A3: nat,C2: nat] :
( ( times_times_nat @ B4 @ ( times_times_nat @ A3 @ C2 ) )
= ( times_times_nat @ A3 @ ( times_times_nat @ B4 @ C2 ) ) ) ).
% cross3_simps(12)
thf(fact_446_cross3__simps_I12_J,axiom,
! [B4: int,A3: int,C2: int] :
( ( times_times_int @ B4 @ ( times_times_int @ A3 @ C2 ) )
= ( times_times_int @ A3 @ ( times_times_int @ B4 @ C2 ) ) ) ).
% cross3_simps(12)
thf(fact_447_cross3__simps_I11_J,axiom,
( times_times_complex
= ( ^ [A5: complex,B6: complex] : ( times_times_complex @ B6 @ A5 ) ) ) ).
% cross3_simps(11)
thf(fact_448_cross3__simps_I11_J,axiom,
( times_times_real
= ( ^ [A5: real,B6: real] : ( times_times_real @ B6 @ A5 ) ) ) ).
% cross3_simps(11)
thf(fact_449_cross3__simps_I11_J,axiom,
( times_times_nat
= ( ^ [A5: nat,B6: nat] : ( times_times_nat @ B6 @ A5 ) ) ) ).
% cross3_simps(11)
thf(fact_450_cross3__simps_I11_J,axiom,
( times_times_int
= ( ^ [A5: int,B6: int] : ( times_times_int @ B6 @ A5 ) ) ) ).
% cross3_simps(11)
thf(fact_451_cross3__simps_I10_J,axiom,
! [A3: complex,B4: complex,C2: complex] :
( ( times_times_complex @ ( times_times_complex @ A3 @ B4 ) @ C2 )
= ( times_times_complex @ A3 @ ( times_times_complex @ B4 @ C2 ) ) ) ).
% cross3_simps(10)
thf(fact_452_cross3__simps_I10_J,axiom,
! [A3: real,B4: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A3 @ B4 ) @ C2 )
= ( times_times_real @ A3 @ ( times_times_real @ B4 @ C2 ) ) ) ).
% cross3_simps(10)
thf(fact_453_cross3__simps_I10_J,axiom,
! [A3: nat,B4: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B4 ) @ C2 )
= ( times_times_nat @ A3 @ ( times_times_nat @ B4 @ C2 ) ) ) ).
% cross3_simps(10)
thf(fact_454_cross3__simps_I10_J,axiom,
! [A3: int,B4: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A3 @ B4 ) @ C2 )
= ( times_times_int @ A3 @ ( times_times_int @ B4 @ C2 ) ) ) ).
% cross3_simps(10)
thf(fact_455_inf__period_I1_J,axiom,
! [P: complex > $o,D: complex,Q: complex > $o] :
( ! [X3: complex,K2: complex] :
( ( P @ X3 )
= ( P @ ( minus_minus_complex @ X3 @ ( times_times_complex @ K2 @ D ) ) ) )
=> ( ! [X3: complex,K2: complex] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_complex @ X3 @ ( times_times_complex @ K2 @ D ) ) ) )
=> ! [X4: complex,K3: complex] :
( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P @ ( minus_minus_complex @ X4 @ ( times_times_complex @ K3 @ D ) ) )
& ( Q @ ( minus_minus_complex @ X4 @ ( times_times_complex @ K3 @ D ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_456_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_457_inf__period_I1_J,axiom,
! [P: int > $o,D: int,Q: int > $o] :
( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ! [X4: int,K3: int] :
( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) )
& ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_458_inf__period_I2_J,axiom,
! [P: complex > $o,D: complex,Q: complex > $o] :
( ! [X3: complex,K2: complex] :
( ( P @ X3 )
= ( P @ ( minus_minus_complex @ X3 @ ( times_times_complex @ K2 @ D ) ) ) )
=> ( ! [X3: complex,K2: complex] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_complex @ X3 @ ( times_times_complex @ K2 @ D ) ) ) )
=> ! [X4: complex,K3: complex] :
( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P @ ( minus_minus_complex @ X4 @ ( times_times_complex @ K3 @ D ) ) )
| ( Q @ ( minus_minus_complex @ X4 @ ( times_times_complex @ K3 @ D ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_459_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_460_inf__period_I2_J,axiom,
! [P: int > $o,D: int,Q: int > $o] :
( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ! [X4: int,K3: int] :
( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) )
| ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_461_more__arith__simps_I6_J,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ one_one_complex )
= A3 ) ).
% more_arith_simps(6)
thf(fact_462_more__arith__simps_I6_J,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% more_arith_simps(6)
thf(fact_463_more__arith__simps_I6_J,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% more_arith_simps(6)
thf(fact_464_more__arith__simps_I6_J,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ one_one_int )
= A3 ) ).
% more_arith_simps(6)
thf(fact_465_more__arith__simps_I5_J,axiom,
! [A3: complex] :
( ( times_times_complex @ one_one_complex @ A3 )
= A3 ) ).
% more_arith_simps(5)
thf(fact_466_more__arith__simps_I5_J,axiom,
! [A3: real] :
( ( times_times_real @ one_one_real @ A3 )
= A3 ) ).
% more_arith_simps(5)
thf(fact_467_more__arith__simps_I5_J,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% more_arith_simps(5)
thf(fact_468_more__arith__simps_I5_J,axiom,
! [A3: int] :
( ( times_times_int @ one_one_int @ A3 )
= A3 ) ).
% more_arith_simps(5)
thf(fact_469_mult_Ocomm__neutral,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ one_one_complex )
= A3 ) ).
% mult.comm_neutral
thf(fact_470_mult_Ocomm__neutral,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% mult.comm_neutral
thf(fact_471_mult_Ocomm__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.comm_neutral
thf(fact_472_mult_Ocomm__neutral,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ one_one_int )
= A3 ) ).
% mult.comm_neutral
thf(fact_473_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: complex,B4: complex,C2: complex] :
( ( times_times_complex @ ( times_times_complex @ A3 @ B4 ) @ C2 )
= ( times_times_complex @ A3 @ ( times_times_complex @ B4 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_474_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: real,B4: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A3 @ B4 ) @ C2 )
= ( times_times_real @ A3 @ ( times_times_real @ B4 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_475_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: nat,B4: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B4 ) @ C2 )
= ( times_times_nat @ A3 @ ( times_times_nat @ B4 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_476_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: int,B4: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A3 @ B4 ) @ C2 )
= ( times_times_int @ A3 @ ( times_times_int @ B4 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_477_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_478_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_479_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_480_comm__monoid__mult__class_Omult__1,axiom,
! [A3: int] :
( ( times_times_int @ one_one_int @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_481_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_482_mult__delta__right,axiom,
! [B4: $o,X2: complex,Y: complex] :
( ( B4
=> ( ( times_times_complex @ X2 @ ( if_complex @ B4 @ Y @ zero_zero_complex ) )
= ( times_times_complex @ X2 @ Y ) ) )
& ( ~ B4
=> ( ( times_times_complex @ X2 @ ( if_complex @ B4 @ Y @ zero_zero_complex ) )
= zero_zero_complex ) ) ) ).
% mult_delta_right
thf(fact_483_mult__delta__right,axiom,
! [B4: $o,X2: real,Y: real] :
( ( B4
=> ( ( times_times_real @ X2 @ ( if_real @ B4 @ Y @ zero_zero_real ) )
= ( times_times_real @ X2 @ Y ) ) )
& ( ~ B4
=> ( ( times_times_real @ X2 @ ( if_real @ B4 @ Y @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_484_mult__delta__right,axiom,
! [B4: $o,X2: nat,Y: nat] :
( ( B4
=> ( ( times_times_nat @ X2 @ ( if_nat @ B4 @ Y @ zero_zero_nat ) )
= ( times_times_nat @ X2 @ Y ) ) )
& ( ~ B4
=> ( ( times_times_nat @ X2 @ ( if_nat @ B4 @ Y @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_485_mult__delta__right,axiom,
! [B4: $o,X2: int,Y: int] :
( ( B4
=> ( ( times_times_int @ X2 @ ( if_int @ B4 @ Y @ zero_zero_int ) )
= ( times_times_int @ X2 @ Y ) ) )
& ( ~ B4
=> ( ( times_times_int @ X2 @ ( if_int @ B4 @ Y @ zero_zero_int ) )
= zero_zero_int ) ) ) ).
% mult_delta_right
thf(fact_486_mult__delta__left,axiom,
! [B4: $o,X2: complex,Y: complex] :
( ( B4
=> ( ( times_times_complex @ ( if_complex @ B4 @ X2 @ zero_zero_complex ) @ Y )
= ( times_times_complex @ X2 @ Y ) ) )
& ( ~ B4
=> ( ( times_times_complex @ ( if_complex @ B4 @ X2 @ zero_zero_complex ) @ Y )
= zero_zero_complex ) ) ) ).
% mult_delta_left
thf(fact_487_mult__delta__left,axiom,
! [B4: $o,X2: real,Y: real] :
( ( B4
=> ( ( times_times_real @ ( if_real @ B4 @ X2 @ zero_zero_real ) @ Y )
= ( times_times_real @ X2 @ Y ) ) )
& ( ~ B4
=> ( ( times_times_real @ ( if_real @ B4 @ X2 @ zero_zero_real ) @ Y )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_488_mult__delta__left,axiom,
! [B4: $o,X2: nat,Y: nat] :
( ( B4
=> ( ( times_times_nat @ ( if_nat @ B4 @ X2 @ zero_zero_nat ) @ Y )
= ( times_times_nat @ X2 @ Y ) ) )
& ( ~ B4
=> ( ( times_times_nat @ ( if_nat @ B4 @ X2 @ zero_zero_nat ) @ Y )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_489_mult__delta__left,axiom,
! [B4: $o,X2: int,Y: int] :
( ( B4
=> ( ( times_times_int @ ( if_int @ B4 @ X2 @ zero_zero_int ) @ Y )
= ( times_times_int @ X2 @ Y ) ) )
& ( ~ B4
=> ( ( times_times_int @ ( if_int @ B4 @ X2 @ zero_zero_int ) @ Y )
= zero_zero_int ) ) ) ).
% mult_delta_left
thf(fact_490_density__collapse__def,axiom,
( projec3470689467825365843llapse
= ( ^ [R2: mat_complex,P4: mat_complex] :
( if_mat_complex
@ ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R2 @ P4 ) )
= zero_zero_complex )
@ ( projec8360710381328234318ensity @ ( dim_row_complex @ R2 ) )
@ ( smult_mat_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ one_one_real ) @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R2 @ P4 ) ) ) @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P4 @ R2 ) @ P4 ) ) ) ) ) ).
% density_collapse_def
thf(fact_491_upper__triangular__imp__det__eq__0__iff,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( upper_4850907204721561915omplex @ A )
=> ( ( ( det_complex @ A )
= zero_zero_complex )
= ( member_complex @ zero_zero_complex @ ( set_complex2 @ ( diag_mat_complex @ A ) ) ) ) ) ) ).
% upper_triangular_imp_det_eq_0_iff
thf(fact_492_upper__triangular__imp__det__eq__0__iff,axiom,
! [A: mat_real,N: nat] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( upper_8570057991637822137r_real @ A )
=> ( ( ( det_real @ A )
= zero_zero_real )
= ( member_real @ zero_zero_real @ ( set_real2 @ ( diag_mat_real @ A ) ) ) ) ) ) ).
% upper_triangular_imp_det_eq_0_iff
thf(fact_493_upper__triangular__imp__det__eq__0__iff,axiom,
! [A: mat_int,N: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( ( upper_triangular_int @ A )
=> ( ( ( det_int @ A )
= zero_zero_int )
= ( member_int @ zero_zero_int @ ( set_int2 @ ( diag_mat_int @ A ) ) ) ) ) ) ).
% upper_triangular_imp_det_eq_0_iff
thf(fact_494_nonzero__mult__div__cancel__right,axiom,
! [B4: complex,A3: complex] :
( ( B4 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ B4 ) @ B4 )
= A3 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_495_nonzero__mult__div__cancel__right,axiom,
! [B4: nat,A3: nat] :
( ( B4 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A3 @ B4 ) @ B4 )
= A3 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_496_nonzero__mult__div__cancel__right,axiom,
! [B4: real,A3: real] :
( ( B4 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A3 @ B4 ) @ B4 )
= A3 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_497_nonzero__mult__div__cancel__right,axiom,
! [B4: int,A3: int] :
( ( B4 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A3 @ B4 ) @ B4 )
= A3 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_498_nonzero__mult__div__cancel__right,axiom,
! [B4: formal670952693614245302omplex,A3: formal670952693614245302omplex] :
( ( B4 != zero_z1877163951443063103omplex )
=> ( ( divide1348722040316500488omplex @ ( times_1444617028055533883omplex @ A3 @ B4 ) @ B4 )
= A3 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_499_nonzero__mult__div__cancel__left,axiom,
! [A3: complex,B4: complex] :
( ( A3 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ B4 ) @ A3 )
= B4 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_500_nonzero__mult__div__cancel__left,axiom,
! [A3: nat,B4: nat] :
( ( A3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A3 @ B4 ) @ A3 )
= B4 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_501_nonzero__mult__div__cancel__left,axiom,
! [A3: real,B4: real] :
( ( A3 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A3 @ B4 ) @ A3 )
= B4 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_502_nonzero__mult__div__cancel__left,axiom,
! [A3: int,B4: int] :
( ( A3 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A3 @ B4 ) @ A3 )
= B4 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_503_nonzero__mult__div__cancel__left,axiom,
! [A3: formal670952693614245302omplex,B4: formal670952693614245302omplex] :
( ( A3 != zero_z1877163951443063103omplex )
=> ( ( divide1348722040316500488omplex @ ( times_1444617028055533883omplex @ A3 @ B4 ) @ A3 )
= B4 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_504_frac__le__eq,axiom,
! [Y: real,Z: real,X2: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% frac_le_eq
thf(fact_505_divide__diff__eq__iff,axiom,
! [Z: complex,X2: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X2 @ Z ) @ Y )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ X2 @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_506_divide__diff__eq__iff,axiom,
! [Z: real,X2: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Z ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ X2 @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_507_diff__divide__eq__iff,axiom,
! [Z: complex,X2: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ X2 @ ( divide1717551699836669952omplex @ Y @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X2 @ Z ) @ Y ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_508_diff__divide__eq__iff,axiom,
! [Z: real,X2: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ X2 @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ Y ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_509_diff__frac__eq,axiom,
! [Y: complex,Z: complex,X2: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X2 @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X2 @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_510_diff__frac__eq,axiom,
! [Y: real,Z: real,X2: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_511_add__divide__eq__if__simps_I4_J,axiom,
! [Z: complex,A3: complex,B4: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( minus_minus_complex @ A3 @ ( divide1717551699836669952omplex @ B4 @ Z ) )
= A3 ) )
& ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ A3 @ ( divide1717551699836669952omplex @ B4 @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A3 @ Z ) @ B4 ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_512_add__divide__eq__if__simps_I4_J,axiom,
! [Z: real,A3: real,B4: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ A3 @ ( divide_divide_real @ B4 @ Z ) )
= A3 ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ A3 @ ( divide_divide_real @ B4 @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A3 @ Z ) @ B4 ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_513_nonzero__divide__mult__cancel__right,axiom,
! [B4: complex,A3: complex] :
( ( B4 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ B4 @ ( times_times_complex @ A3 @ B4 ) )
= ( divide1717551699836669952omplex @ one_one_complex @ A3 ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_514_nonzero__divide__mult__cancel__right,axiom,
! [B4: real,A3: real] :
( ( B4 != zero_zero_real )
=> ( ( divide_divide_real @ B4 @ ( times_times_real @ A3 @ B4 ) )
= ( divide_divide_real @ one_one_real @ A3 ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_515_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_516_times__divide__eq__left,axiom,
! [B4: complex,C2: complex,A3: complex] :
( ( times_times_complex @ ( divide1717551699836669952omplex @ B4 @ C2 ) @ A3 )
= ( divide1717551699836669952omplex @ ( times_times_complex @ B4 @ A3 ) @ C2 ) ) ).
% times_divide_eq_left
thf(fact_517_times__divide__eq__left,axiom,
! [B4: real,C2: real,A3: real] :
( ( times_times_real @ ( divide_divide_real @ B4 @ C2 ) @ A3 )
= ( divide_divide_real @ ( times_times_real @ B4 @ A3 ) @ C2 ) ) ).
% times_divide_eq_left
thf(fact_518_divide__divide__eq__left,axiom,
! [A3: complex,B4: complex,C2: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A3 @ B4 ) @ C2 )
= ( divide1717551699836669952omplex @ A3 @ ( times_times_complex @ B4 @ C2 ) ) ) ).
% divide_divide_eq_left
thf(fact_519_divide__divide__eq__left,axiom,
! [A3: real,B4: real,C2: real] :
( ( divide_divide_real @ ( divide_divide_real @ A3 @ B4 ) @ C2 )
= ( divide_divide_real @ A3 @ ( times_times_real @ B4 @ C2 ) ) ) ).
% divide_divide_eq_left
thf(fact_520_times__divide__times__eq,axiom,
! [X2: complex,Y: complex,Z: complex,W: complex] :
( ( times_times_complex @ ( divide1717551699836669952omplex @ X2 @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ X2 @ Z ) @ ( times_times_complex @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_521_times__divide__times__eq,axiom,
! [X2: real,Y: real,Z: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X2 @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_522_divide__divide__eq__right,axiom,
! [A3: complex,B4: complex,C2: complex] :
( ( divide1717551699836669952omplex @ A3 @ ( divide1717551699836669952omplex @ B4 @ C2 ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ C2 ) @ B4 ) ) ).
% divide_divide_eq_right
thf(fact_523_divide__divide__eq__right,axiom,
! [A3: real,B4: real,C2: real] :
( ( divide_divide_real @ A3 @ ( divide_divide_real @ B4 @ C2 ) )
= ( divide_divide_real @ ( times_times_real @ A3 @ C2 ) @ B4 ) ) ).
% divide_divide_eq_right
thf(fact_524_divide__divide__times__eq,axiom,
! [X2: complex,Y: complex,Z: complex,W: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X2 @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ X2 @ W ) @ ( times_times_complex @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_525_divide__divide__times__eq,axiom,
! [X2: real,Y: real,Z: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X2 @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X2 @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_526_times__divide__eq__right,axiom,
! [A3: complex,B4: complex,C2: complex] :
( ( times_times_complex @ A3 @ ( divide1717551699836669952omplex @ B4 @ C2 ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ B4 ) @ C2 ) ) ).
% times_divide_eq_right
thf(fact_527_times__divide__eq__right,axiom,
! [A3: real,B4: real,C2: real] :
( ( times_times_real @ A3 @ ( divide_divide_real @ B4 @ C2 ) )
= ( divide_divide_real @ ( times_times_real @ A3 @ B4 ) @ C2 ) ) ).
% times_divide_eq_right
thf(fact_528_divide__divide__eq__left_H,axiom,
! [A3: complex,B4: complex,C2: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A3 @ B4 ) @ C2 )
= ( divide1717551699836669952omplex @ A3 @ ( times_times_complex @ C2 @ B4 ) ) ) ).
% divide_divide_eq_left'
thf(fact_529_divide__divide__eq__left_H,axiom,
! [A3: real,B4: real,C2: real] :
( ( divide_divide_real @ ( divide_divide_real @ A3 @ B4 ) @ C2 )
= ( divide_divide_real @ A3 @ ( times_times_real @ C2 @ B4 ) ) ) ).
% divide_divide_eq_left'
thf(fact_530_frac__eq__eq,axiom,
! [Y: complex,Z: complex,X2: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ X2 @ Y )
= ( divide1717551699836669952omplex @ W @ Z ) )
= ( ( times_times_complex @ X2 @ Z )
= ( times_times_complex @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_531_frac__eq__eq,axiom,
! [Y: real,Z: real,X2: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ( divide_divide_real @ X2 @ Y )
= ( divide_divide_real @ W @ Z ) )
= ( ( times_times_real @ X2 @ Z )
= ( times_times_real @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_532_divide__eq__eq,axiom,
! [B4: complex,C2: complex,A3: complex] :
( ( ( divide1717551699836669952omplex @ B4 @ C2 )
= A3 )
= ( ( ( C2 != zero_zero_complex )
=> ( B4
= ( times_times_complex @ A3 @ C2 ) ) )
& ( ( C2 = zero_zero_complex )
=> ( A3 = zero_zero_complex ) ) ) ) ).
% divide_eq_eq
thf(fact_533_divide__eq__eq,axiom,
! [B4: real,C2: real,A3: real] :
( ( ( divide_divide_real @ B4 @ C2 )
= A3 )
= ( ( ( C2 != zero_zero_real )
=> ( B4
= ( times_times_real @ A3 @ C2 ) ) )
& ( ( C2 = zero_zero_real )
=> ( A3 = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_534_eq__divide__eq,axiom,
! [A3: complex,B4: complex,C2: complex] :
( ( A3
= ( divide1717551699836669952omplex @ B4 @ C2 ) )
= ( ( ( C2 != zero_zero_complex )
=> ( ( times_times_complex @ A3 @ C2 )
= B4 ) )
& ( ( C2 = zero_zero_complex )
=> ( A3 = zero_zero_complex ) ) ) ) ).
% eq_divide_eq
thf(fact_535_eq__divide__eq,axiom,
! [A3: real,B4: real,C2: real] :
( ( A3
= ( divide_divide_real @ B4 @ C2 ) )
= ( ( ( C2 != zero_zero_real )
=> ( ( times_times_real @ A3 @ C2 )
= B4 ) )
& ( ( C2 = zero_zero_real )
=> ( A3 = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_536_divide__eq__imp,axiom,
! [C2: complex,B4: complex,A3: complex] :
( ( C2 != zero_zero_complex )
=> ( ( B4
= ( times_times_complex @ A3 @ C2 ) )
=> ( ( divide1717551699836669952omplex @ B4 @ C2 )
= A3 ) ) ) ).
% divide_eq_imp
thf(fact_537_divide__eq__imp,axiom,
! [C2: real,B4: real,A3: real] :
( ( C2 != zero_zero_real )
=> ( ( B4
= ( times_times_real @ A3 @ C2 ) )
=> ( ( divide_divide_real @ B4 @ C2 )
= A3 ) ) ) ).
% divide_eq_imp
thf(fact_538_eq__divide__imp,axiom,
! [C2: complex,A3: complex,B4: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ C2 )
= B4 )
=> ( A3
= ( divide1717551699836669952omplex @ B4 @ C2 ) ) ) ) ).
% eq_divide_imp
thf(fact_539_eq__divide__imp,axiom,
! [C2: real,A3: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ C2 )
= B4 )
=> ( A3
= ( divide_divide_real @ B4 @ C2 ) ) ) ) ).
% eq_divide_imp
thf(fact_540_nonzero__divide__eq__eq,axiom,
! [C2: complex,B4: complex,A3: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ B4 @ C2 )
= A3 )
= ( B4
= ( times_times_complex @ A3 @ C2 ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_541_nonzero__divide__eq__eq,axiom,
! [C2: real,B4: real,A3: real] :
( ( C2 != zero_zero_real )
=> ( ( ( divide_divide_real @ B4 @ C2 )
= A3 )
= ( B4
= ( times_times_real @ A3 @ C2 ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_542_nonzero__eq__divide__eq,axiom,
! [C2: complex,A3: complex,B4: complex] :
( ( C2 != zero_zero_complex )
=> ( ( A3
= ( divide1717551699836669952omplex @ B4 @ C2 ) )
= ( ( times_times_complex @ A3 @ C2 )
= B4 ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_543_nonzero__eq__divide__eq,axiom,
! [C2: real,A3: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( A3
= ( divide_divide_real @ B4 @ C2 ) )
= ( ( times_times_real @ A3 @ C2 )
= B4 ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_544_mult__divide__mult__cancel__left,axiom,
! [C2: complex,A3: complex,B4: complex] :
( ( C2 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ C2 @ B4 ) )
= ( divide1717551699836669952omplex @ A3 @ B4 ) ) ) ).
% mult_divide_mult_cancel_left
thf(fact_545_mult__divide__mult__cancel__left,axiom,
! [C2: real,A3: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B4 ) )
= ( divide_divide_real @ A3 @ B4 ) ) ) ).
% mult_divide_mult_cancel_left
thf(fact_546_mult__divide__mult__cancel__right,axiom,
! [C2: complex,A3: complex,B4: complex] :
( ( C2 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B4 @ C2 ) )
= ( divide1717551699836669952omplex @ A3 @ B4 ) ) ) ).
% mult_divide_mult_cancel_right
thf(fact_547_mult__divide__mult__cancel__right,axiom,
! [C2: real,A3: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ C2 ) )
= ( divide_divide_real @ A3 @ B4 ) ) ) ).
% mult_divide_mult_cancel_right
thf(fact_548_mult__divide__mult__cancel__left__if,axiom,
! [C2: complex,A3: complex,B4: complex] :
( ( ( C2 = zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ C2 @ B4 ) )
= zero_zero_complex ) )
& ( ( C2 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ C2 @ B4 ) )
= ( divide1717551699836669952omplex @ A3 @ B4 ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_549_mult__divide__mult__cancel__left__if,axiom,
! [C2: real,A3: real,B4: real] :
( ( ( C2 = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B4 ) )
= zero_zero_real ) )
& ( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B4 ) )
= ( divide_divide_real @ A3 @ B4 ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_550_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C2: complex,A3: complex,B4: complex] :
( ( C2 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ B4 @ C2 ) )
= ( divide1717551699836669952omplex @ A3 @ B4 ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_551_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C2: real,A3: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ B4 @ C2 ) )
= ( divide_divide_real @ A3 @ B4 ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_552_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C2: complex,A3: complex,B4: complex] :
( ( C2 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ C2 @ B4 ) )
= ( divide1717551699836669952omplex @ A3 @ B4 ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_553_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C2: real,A3: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ C2 @ B4 ) )
= ( divide_divide_real @ A3 @ B4 ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_554_nonzero__divide__mult__cancel__left,axiom,
! [A3: complex,B4: complex] :
( ( A3 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A3 @ ( times_times_complex @ A3 @ B4 ) )
= ( divide1717551699836669952omplex @ one_one_complex @ B4 ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_555_nonzero__divide__mult__cancel__left,axiom,
! [A3: real,B4: real] :
( ( A3 != zero_zero_real )
=> ( ( divide_divide_real @ A3 @ ( times_times_real @ A3 @ B4 ) )
= ( divide_divide_real @ one_one_real @ B4 ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_556_spectrum__def,axiom,
( projec527831343749723810omplex
= ( ^ [M2: mat_complex] : ( set_complex2 @ ( projec6785268565095433026omplex @ M2 ) ) ) ) ).
% spectrum_def
thf(fact_557_div__mult__mult1,axiom,
! [C2: nat,A3: nat,B4: nat] :
( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B4 ) )
= ( divide_divide_nat @ A3 @ B4 ) ) ) ).
% div_mult_mult1
thf(fact_558_div__mult__mult1,axiom,
! [C2: int,A3: int,B4: int] :
( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B4 ) )
= ( divide_divide_int @ A3 @ B4 ) ) ) ).
% div_mult_mult1
thf(fact_559_div__mult__mult1,axiom,
! [C2: formal670952693614245302omplex,A3: formal670952693614245302omplex,B4: formal670952693614245302omplex] :
( ( C2 != zero_z1877163951443063103omplex )
=> ( ( divide1348722040316500488omplex @ ( times_1444617028055533883omplex @ C2 @ A3 ) @ ( times_1444617028055533883omplex @ C2 @ B4 ) )
= ( divide1348722040316500488omplex @ A3 @ B4 ) ) ) ).
% div_mult_mult1
thf(fact_560_div__mult__mult2,axiom,
! [C2: nat,A3: nat,B4: nat] :
( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B4 @ C2 ) )
= ( divide_divide_nat @ A3 @ B4 ) ) ) ).
% div_mult_mult2
thf(fact_561_div__mult__mult2,axiom,
! [C2: int,A3: int,B4: int] :
( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ C2 ) )
= ( divide_divide_int @ A3 @ B4 ) ) ) ).
% div_mult_mult2
thf(fact_562_div__mult__mult2,axiom,
! [C2: formal670952693614245302omplex,A3: formal670952693614245302omplex,B4: formal670952693614245302omplex] :
( ( C2 != zero_z1877163951443063103omplex )
=> ( ( divide1348722040316500488omplex @ ( times_1444617028055533883omplex @ A3 @ C2 ) @ ( times_1444617028055533883omplex @ B4 @ C2 ) )
= ( divide1348722040316500488omplex @ A3 @ B4 ) ) ) ).
% div_mult_mult2
thf(fact_563_div__mult__mult1__if,axiom,
! [C2: nat,A3: nat,B4: nat] :
( ( ( C2 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B4 ) )
= zero_zero_nat ) )
& ( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B4 ) )
= ( divide_divide_nat @ A3 @ B4 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_564_div__mult__mult1__if,axiom,
! [C2: int,A3: int,B4: int] :
( ( ( C2 = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B4 ) )
= zero_zero_int ) )
& ( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B4 ) )
= ( divide_divide_int @ A3 @ B4 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_565_div__mult__mult1__if,axiom,
! [C2: formal670952693614245302omplex,A3: formal670952693614245302omplex,B4: formal670952693614245302omplex] :
( ( ( C2 = zero_z1877163951443063103omplex )
=> ( ( divide1348722040316500488omplex @ ( times_1444617028055533883omplex @ C2 @ A3 ) @ ( times_1444617028055533883omplex @ C2 @ B4 ) )
= zero_z1877163951443063103omplex ) )
& ( ( C2 != zero_z1877163951443063103omplex )
=> ( ( divide1348722040316500488omplex @ ( times_1444617028055533883omplex @ C2 @ A3 ) @ ( times_1444617028055533883omplex @ C2 @ B4 ) )
= ( divide1348722040316500488omplex @ A3 @ B4 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_566_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_567_times__nat_Osimps_I1_J,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% times_nat.simps(1)
thf(fact_568_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_569_mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel1
thf(fact_570_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_571_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_572_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_573_mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel1
thf(fact_574_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_575_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_576_less__eq__div__iff__mult__less__eq,axiom,
! [Q4: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q4 )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q4 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q4 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_577_div__less__iff__less__mult,axiom,
! [Q4: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q4 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q4 ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q4 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_578_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_579_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_580_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B4 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_581_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: real,B4: real,C2: real] :
( ( ord_less_real @ A3 @ B4 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B4 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_582_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: int,B4: int,C2: int] :
( ( ord_less_int @ A3 @ B4 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B4 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_583_mult__less__cancel__right__disj,axiom,
! [A3: real,C2: real,B4: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A3 @ B4 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B4 @ A3 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_584_mult__less__cancel__right__disj,axiom,
! [A3: int,C2: int,B4: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A3 @ B4 ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B4 @ A3 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_585_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B4 @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_586_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: real,B4: real,C2: real] :
( ( ord_less_real @ A3 @ B4 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_587_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: int,B4: int,C2: int] :
( ( ord_less_int @ A3 @ B4 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_588_mult__strict__right__mono__neg,axiom,
! [B4: real,A3: real,C2: real] :
( ( ord_less_real @ B4 @ A3 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_589_mult__strict__right__mono__neg,axiom,
! [B4: int,A3: int,C2: int] :
( ( ord_less_int @ B4 @ A3 )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_590_mult__less__cancel__left__disj,axiom,
! [C2: real,A3: real,B4: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B4 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A3 @ B4 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B4 @ A3 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_591_mult__less__cancel__left__disj,axiom,
! [C2: int,A3: int,B4: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B4 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A3 @ B4 ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B4 @ A3 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_592_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B4 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_593_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: real,B4: real,C2: real] :
( ( ord_less_real @ A3 @ B4 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B4 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_594_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: int,B4: int,C2: int] :
( ( ord_less_int @ A3 @ B4 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B4 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_595_mult__strict__left__mono__neg,axiom,
! [B4: real,A3: real,C2: real] :
( ( ord_less_real @ B4 @ A3 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B4 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_596_mult__strict__left__mono__neg,axiom,
! [B4: int,A3: int,C2: int] :
( ( ord_less_int @ B4 @ A3 )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B4 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_597_mult__less__cancel__left__pos,axiom,
! [C2: real,A3: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B4 ) )
= ( ord_less_real @ A3 @ B4 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_598_mult__less__cancel__left__pos,axiom,
! [C2: int,A3: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B4 ) )
= ( ord_less_int @ A3 @ B4 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_599_mult__less__cancel__left__neg,axiom,
! [C2: real,A3: real,B4: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B4 ) )
= ( ord_less_real @ B4 @ A3 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_600_mult__less__cancel__left__neg,axiom,
! [C2: int,A3: int,B4: int] :
( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B4 ) )
= ( ord_less_int @ B4 @ A3 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_601_zero__less__mult__pos2,axiom,
! [B4: nat,A3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B4 @ A3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B4 ) ) ) ).
% zero_less_mult_pos2
thf(fact_602_zero__less__mult__pos2,axiom,
! [B4: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B4 @ A3 ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ B4 ) ) ) ).
% zero_less_mult_pos2
thf(fact_603_zero__less__mult__pos2,axiom,
! [B4: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B4 @ A3 ) )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ord_less_int @ zero_zero_int @ B4 ) ) ) ).
% zero_less_mult_pos2
thf(fact_604_zero__less__mult__pos,axiom,
! [A3: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B4 ) ) ) ).
% zero_less_mult_pos
thf(fact_605_zero__less__mult__pos,axiom,
! [A3: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B4 ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ B4 ) ) ) ).
% zero_less_mult_pos
thf(fact_606_zero__less__mult__pos,axiom,
! [A3: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B4 ) )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ord_less_int @ zero_zero_int @ B4 ) ) ) ).
% zero_less_mult_pos
thf(fact_607_zero__less__mult__iff,axiom,
! [A3: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B4 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ zero_zero_real @ B4 ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ B4 @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_608_zero__less__mult__iff,axiom,
! [A3: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B4 ) )
= ( ( ( ord_less_int @ zero_zero_int @ A3 )
& ( ord_less_int @ zero_zero_int @ B4 ) )
| ( ( ord_less_int @ A3 @ zero_zero_int )
& ( ord_less_int @ B4 @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_609_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B4 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B4 @ A3 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_610_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B4 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B4 @ A3 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_611_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B4 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B4 @ A3 ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_612_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A3: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B4 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B4 ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_613_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A3: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ B4 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B4 ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_614_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A3: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B4 ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_615_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A3: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B4 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B4 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_616_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A3: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B4 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ B4 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_617_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A3: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B4 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A3 @ B4 ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_618_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A3: nat,B4: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B4 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B4 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_619_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A3: real,B4: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B4 )
=> ( ord_less_real @ ( times_times_real @ A3 @ B4 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_620_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A3: int,B4: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ord_less_int @ ( times_times_int @ A3 @ B4 ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_621_mult__less__0__iff,axiom,
! [A3: real,B4: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ B4 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ B4 @ zero_zero_real ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B4 ) ) ) ) ).
% mult_less_0_iff
thf(fact_622_mult__less__0__iff,axiom,
! [A3: int,B4: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ B4 ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A3 )
& ( ord_less_int @ B4 @ zero_zero_int ) )
| ( ( ord_less_int @ A3 @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B4 ) ) ) ) ).
% mult_less_0_iff
thf(fact_623_not__square__less__zero,axiom,
! [A3: real] :
~ ( ord_less_real @ ( times_times_real @ A3 @ A3 ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_624_not__square__less__zero,axiom,
! [A3: int] :
~ ( ord_less_int @ ( times_times_int @ A3 @ A3 ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_625_mult__neg__neg,axiom,
! [A3: real,B4: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ B4 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B4 ) ) ) ) ).
% mult_neg_neg
thf(fact_626_mult__neg__neg,axiom,
! [A3: int,B4: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ B4 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B4 ) ) ) ) ).
% mult_neg_neg
thf(fact_627_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_628_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_629_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_630_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_631_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_632_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_633_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_634_cpx__sq__mat_Onpos,axiom,
! [DimR: nat,DimC: nat,Fc_mats: set_mat_complex] :
( ( linear7199532782703566157sq_mat @ DimR @ DimC @ Fc_mats )
=> ( ord_less_nat @ zero_zero_nat @ DimR ) ) ).
% cpx_sq_mat.npos
thf(fact_635_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A3: real,B4: real,C2: real,D2: real] :
( ( ord_less_real @ A3 @ B4 )
=> ( ( ord_less_eq_real @ C2 @ D2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_636_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A3: nat,B4: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B4 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_637_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A3: int,B4: int,C2: int,D2: int] :
( ( ord_less_int @ A3 @ B4 )
=> ( ( ord_less_eq_int @ C2 @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_638_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A3: real,B4: real,C2: real,D2: real] :
( ( ord_less_eq_real @ A3 @ B4 )
=> ( ( ord_less_real @ C2 @ D2 )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_639_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A3: nat,B4: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B4 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_640_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A3: int,B4: int,C2: int,D2: int] :
( ( ord_less_eq_int @ A3 @ B4 )
=> ( ( ord_less_int @ C2 @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_641_mult__right__le__imp__le,axiom,
! [A3: real,C2: real,B4: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ C2 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A3 @ B4 ) ) ) ).
% mult_right_le_imp_le
thf(fact_642_mult__right__le__imp__le,axiom,
! [A3: nat,C2: nat,B4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B4 @ C2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A3 @ B4 ) ) ) ).
% mult_right_le_imp_le
thf(fact_643_mult__right__le__imp__le,axiom,
! [A3: int,C2: int,B4: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ C2 ) )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A3 @ B4 ) ) ) ).
% mult_right_le_imp_le
thf(fact_644_mult__left__le__imp__le,axiom,
! [C2: real,A3: real,B4: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B4 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A3 @ B4 ) ) ) ).
% mult_left_le_imp_le
thf(fact_645_mult__left__le__imp__le,axiom,
! [C2: nat,A3: nat,B4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A3 @ B4 ) ) ) ).
% mult_left_le_imp_le
thf(fact_646_mult__left__le__imp__le,axiom,
! [C2: int,A3: int,B4: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B4 ) )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A3 @ B4 ) ) ) ).
% mult_left_le_imp_le
thf(fact_647_mult__le__cancel__left__pos,axiom,
! [C2: real,A3: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B4 ) )
= ( ord_less_eq_real @ A3 @ B4 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_648_mult__le__cancel__left__pos,axiom,
! [C2: int,A3: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B4 ) )
= ( ord_less_eq_int @ A3 @ B4 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_649_mult__le__cancel__left__neg,axiom,
! [C2: real,A3: real,B4: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B4 ) )
= ( ord_less_eq_real @ B4 @ A3 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_650_mult__le__cancel__left__neg,axiom,
! [C2: int,A3: int,B4: int] :
( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B4 ) )
= ( ord_less_eq_int @ B4 @ A3 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_651_mult__less__cancel__right,axiom,
! [A3: real,C2: real,B4: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A3 @ B4 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B4 @ A3 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_652_mult__less__cancel__right,axiom,
! [A3: int,C2: int,B4: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ C2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A3 @ B4 ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B4 @ A3 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_653_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A3: real,B4: real,C2: real,D2: real] :
( ( ord_less_real @ A3 @ B4 )
=> ( ( ord_less_real @ C2 @ D2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_654_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A3: nat,B4: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B4 )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B4 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_655_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A3: int,B4: int,C2: int,D2: int] :
( ( ord_less_int @ A3 @ B4 )
=> ( ( ord_less_int @ C2 @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_656_mult__right__less__imp__less,axiom,
! [A3: real,C2: real,B4: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ C2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A3 @ B4 ) ) ) ).
% mult_right_less_imp_less
thf(fact_657_mult__right__less__imp__less,axiom,
! [A3: nat,C2: nat,B4: nat] :
( ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B4 @ C2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A3 @ B4 ) ) ) ).
% mult_right_less_imp_less
thf(fact_658_mult__right__less__imp__less,axiom,
! [A3: int,C2: int,B4: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ C2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A3 @ B4 ) ) ) ).
% mult_right_less_imp_less
thf(fact_659_mult__less__cancel__left,axiom,
! [C2: real,A3: real,B4: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B4 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A3 @ B4 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B4 @ A3 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_660_mult__less__cancel__left,axiom,
! [C2: int,A3: int,B4: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B4 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A3 @ B4 ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B4 @ A3 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_661_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A3: real,B4: real,C2: real,D2: real] :
( ( ord_less_real @ A3 @ B4 )
=> ( ( ord_less_real @ C2 @ D2 )
=> ( ( ord_less_real @ zero_zero_real @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_662_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A3: nat,B4: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B4 )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B4 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_663_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A3: int,B4: int,C2: int,D2: int] :
( ( ord_less_int @ A3 @ B4 )
=> ( ( ord_less_int @ C2 @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_664_mult__left__less__imp__less,axiom,
! [C2: real,A3: real,B4: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B4 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A3 @ B4 ) ) ) ).
% mult_left_less_imp_less
thf(fact_665_mult__left__less__imp__less,axiom,
! [C2: nat,A3: nat,B4: nat] :
( ( ord_less_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B4 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A3 @ B4 ) ) ) ).
% mult_left_less_imp_less
thf(fact_666_mult__left__less__imp__less,axiom,
! [C2: int,A3: int,B4: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B4 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A3 @ B4 ) ) ) ).
% mult_left_less_imp_less
thf(fact_667_mult__le__cancel__right,axiom,
! [A3: real,C2: real,B4: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A3 @ B4 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B4 @ A3 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_668_mult__le__cancel__right,axiom,
! [A3: int,C2: int,B4: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A3 @ B4 ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B4 @ A3 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_669_mult__le__cancel__left,axiom,
! [C2: real,A3: real,B4: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B4 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A3 @ B4 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B4 @ A3 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_670_mult__le__cancel__left,axiom,
! [C2: int,A3: int,B4: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B4 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A3 @ B4 ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B4 @ A3 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_671_divide__less__eq,axiom,
! [B4: real,C2: real,A3: real] :
( ( ord_less_real @ ( divide_divide_real @ B4 @ C2 ) @ A3 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ B4 @ ( 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 ) @ B4 ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A3 ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_672_less__divide__eq,axiom,
! [A3: real,B4: real,C2: real] :
( ( ord_less_real @ A3 @ ( divide_divide_real @ B4 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ B4 ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B4 @ ( times_times_real @ A3 @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ A3 @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_673_neg__divide__less__eq,axiom,
! [C2: real,B4: real,A3: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B4 @ C2 ) @ A3 )
= ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ B4 ) ) ) ).
% neg_divide_less_eq
thf(fact_674_neg__less__divide__eq,axiom,
! [C2: real,A3: real,B4: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ A3 @ ( divide_divide_real @ B4 @ C2 ) )
= ( ord_less_real @ B4 @ ( times_times_real @ A3 @ C2 ) ) ) ) ).
% neg_less_divide_eq
thf(fact_675_pos__divide__less__eq,axiom,
! [C2: real,B4: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( divide_divide_real @ B4 @ C2 ) @ A3 )
= ( ord_less_real @ B4 @ ( times_times_real @ A3 @ C2 ) ) ) ) ).
% pos_divide_less_eq
thf(fact_676_pos__less__divide__eq,axiom,
! [C2: real,A3: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ A3 @ ( divide_divide_real @ B4 @ C2 ) )
= ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ B4 ) ) ) ).
% pos_less_divide_eq
thf(fact_677_mult__imp__div__pos__less,axiom,
! [Y: real,X2: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X2 @ ( times_times_real @ Z @ Y ) )
=> ( ord_less_real @ ( divide_divide_real @ X2 @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_less
thf(fact_678_mult__imp__less__div__pos,axiom,
! [Y: real,Z: real,X2: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( times_times_real @ Z @ Y ) @ X2 )
=> ( ord_less_real @ Z @ ( divide_divide_real @ X2 @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_679_divide__strict__left__mono,axiom,
! [B4: real,A3: real,C2: real] :
( ( ord_less_real @ B4 @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B4 ) )
=> ( ord_less_real @ ( divide_divide_real @ C2 @ A3 ) @ ( divide_divide_real @ C2 @ B4 ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_680_divide__strict__left__mono__neg,axiom,
! [A3: real,B4: real,C2: real] :
( ( ord_less_real @ A3 @ B4 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B4 ) )
=> ( ord_less_real @ ( divide_divide_real @ C2 @ A3 ) @ ( divide_divide_real @ C2 @ B4 ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_681_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_682_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_683_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_684_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_685_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_686_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_687_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_688_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_689_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_690_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_691_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_692_div__mult2__eq,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q4 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q4 ) ) ).
% div_mult2_eq
thf(fact_693_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_694_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_695_mult__less__cancel__right2,axiom,
! [A3: real,C2: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ C2 )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A3 @ one_one_real ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A3 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_696_mult__less__cancel__right2,axiom,
! [A3: int,C2: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ C2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A3 @ one_one_int ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A3 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_697_mult__less__cancel__right1,axiom,
! [C2: real,B4: real] :
( ( ord_less_real @ C2 @ ( times_times_real @ B4 @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ one_one_real @ B4 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B4 @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_698_mult__less__cancel__right1,axiom,
! [C2: int,B4: int] :
( ( ord_less_int @ C2 @ ( times_times_int @ B4 @ C2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ one_one_int @ B4 ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B4 @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_699_mult__less__cancel__left2,axiom,
! [C2: real,A3: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ C2 )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A3 @ one_one_real ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A3 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_700_mult__less__cancel__left2,axiom,
! [C2: int,A3: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ C2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A3 @ one_one_int ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A3 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_701_mult__less__cancel__left1,axiom,
! [C2: real,B4: real] :
( ( ord_less_real @ C2 @ ( times_times_real @ C2 @ B4 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ one_one_real @ B4 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B4 @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_702_mult__less__cancel__left1,axiom,
! [C2: int,B4: int] :
( ( ord_less_int @ C2 @ ( times_times_int @ C2 @ B4 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ one_one_int @ B4 ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B4 @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_703_mult__le__cancel__right2,axiom,
! [A3: real,C2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ C2 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A3 @ one_one_real ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A3 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_704_mult__le__cancel__right2,axiom,
! [A3: int,C2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ C2 ) @ C2 )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A3 @ one_one_int ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A3 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_705_mult__le__cancel__right1,axiom,
! [C2: real,B4: real] :
( ( ord_less_eq_real @ C2 @ ( times_times_real @ B4 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ one_one_real @ B4 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B4 @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_706_mult__le__cancel__right1,axiom,
! [C2: int,B4: int] :
( ( ord_less_eq_int @ C2 @ ( times_times_int @ B4 @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ one_one_int @ B4 ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B4 @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_707_mult__le__cancel__left2,axiom,
! [C2: real,A3: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ C2 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A3 @ one_one_real ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A3 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_708_mult__le__cancel__left2,axiom,
! [C2: int,A3: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A3 ) @ C2 )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A3 @ one_one_int ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A3 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_709_mult__le__cancel__left1,axiom,
! [C2: real,B4: real] :
( ( ord_less_eq_real @ C2 @ ( times_times_real @ C2 @ B4 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ one_one_real @ B4 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B4 @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_710_mult__le__cancel__left1,axiom,
! [C2: int,B4: int] :
( ( ord_less_eq_int @ C2 @ ( times_times_int @ C2 @ B4 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ one_one_int @ B4 ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B4 @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_711_field__le__mult__one__interval,axiom,
! [X2: real,Y: real] :
( ! [Z2: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ Z2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z2 @ X2 ) @ Y ) ) )
=> ( ord_less_eq_real @ X2 @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_712_divide__left__mono__neg,axiom,
! [A3: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B4 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B4 ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C2 @ A3 ) @ ( divide_divide_real @ C2 @ B4 ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_713_mult__imp__le__div__pos,axiom,
! [Y: real,Z: real,X2: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y ) @ X2 )
=> ( ord_less_eq_real @ Z @ ( divide_divide_real @ X2 @ Y ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_714_mult__imp__div__pos__le,axiom,
! [Y: real,X2: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X2 @ ( times_times_real @ Z @ Y ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_le
thf(fact_715_pos__le__divide__eq,axiom,
! [C2: real,A3: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ A3 @ ( divide_divide_real @ B4 @ C2 ) )
= ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ B4 ) ) ) ).
% pos_le_divide_eq
thf(fact_716_pos__divide__le__eq,axiom,
! [C2: real,B4: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B4 @ C2 ) @ A3 )
= ( ord_less_eq_real @ B4 @ ( times_times_real @ A3 @ C2 ) ) ) ) ).
% pos_divide_le_eq
thf(fact_717_neg__le__divide__eq,axiom,
! [C2: real,A3: real,B4: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ A3 @ ( divide_divide_real @ B4 @ C2 ) )
= ( ord_less_eq_real @ B4 @ ( times_times_real @ A3 @ C2 ) ) ) ) ).
% neg_le_divide_eq
thf(fact_718_neg__divide__le__eq,axiom,
! [C2: real,B4: real,A3: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B4 @ C2 ) @ A3 )
= ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ B4 ) ) ) ).
% neg_divide_le_eq
thf(fact_719_divide__left__mono,axiom,
! [B4: real,A3: real,C2: real] :
( ( ord_less_eq_real @ B4 @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B4 ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C2 @ A3 ) @ ( divide_divide_real @ C2 @ B4 ) ) ) ) ) ).
% divide_left_mono
thf(fact_720_le__divide__eq,axiom,
! [A3: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A3 @ ( divide_divide_real @ B4 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ B4 ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B4 @ ( times_times_real @ A3 @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_721_divide__le__eq,axiom,
! [B4: real,C2: real,A3: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B4 @ C2 ) @ A3 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ B4 @ ( times_times_real @ A3 @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ B4 ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_722_frac__less__eq,axiom,
! [Y: real,Z: real,X2: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ X2 @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% frac_less_eq
thf(fact_723_max__mix__is__density,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( comple5220265106149225959erator @ ( projec8360710381328234318ensity @ N ) ) ) ).
% max_mix_is_density
thf(fact_724_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_725_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_726_det__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( det_int @ ( zero_mat_int @ N @ N ) )
= zero_zero_int ) ) ).
% det_zero
thf(fact_727_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_728_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_729_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_730_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_731_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_732_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_733_mult__le__cancel__iff2,axiom,
! [Z: real,X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ X2 ) @ ( times_times_real @ Z @ Y ) )
= ( ord_less_eq_real @ X2 @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_734_mult__le__cancel__iff2,axiom,
! [Z: int,X2: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z @ X2 ) @ ( times_times_int @ Z @ Y ) )
= ( ord_less_eq_int @ X2 @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_735_mult__le__cancel__iff1,axiom,
! [Z: real,X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y @ Z ) )
= ( ord_less_eq_real @ X2 @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_736_mult__le__cancel__iff1,axiom,
! [Z: int,X2: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ X2 @ Z ) @ ( times_times_int @ Y @ Z ) )
= ( ord_less_eq_int @ X2 @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_737_mult__less__iff1,axiom,
! [Z: real,X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y @ Z ) )
= ( ord_less_real @ X2 @ Y ) ) ) ).
% mult_less_iff1
thf(fact_738_mult__less__iff1,axiom,
! [Z: int,X2: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_int @ ( times_times_int @ X2 @ Z ) @ ( times_times_int @ Y @ Z ) )
= ( ord_less_int @ X2 @ Y ) ) ) ).
% mult_less_iff1
thf(fact_739_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_740_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_741_det__multrow__div,axiom,
! [K: nat,N: nat,A: mat_int,A3: int] :
( ( ord_less_nat @ K @ N )
=> ( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( ( A3 != zero_zero_int )
=> ( ( divide_divide_int @ ( det_int @ ( gauss_2407205949817067457en_int @ times_times_int @ K @ A3 @ A ) ) @ A3 )
= ( det_int @ A ) ) ) ) ) ).
% det_multrow_div
thf(fact_742_det__multrow__div,axiom,
! [K: nat,N: nat,A: mat_Fo5321781242956565423omplex,A3: formal670952693614245302omplex] :
( ( ord_less_nat @ K @ N )
=> ( ( member4348805710806261976omplex @ A @ ( carrie9079900694887046380omplex @ N @ N ) )
=> ( ( A3 != zero_z1877163951443063103omplex )
=> ( ( divide1348722040316500488omplex @ ( det_Fo6681143053013413569omplex @ ( gauss_4832413493682057675omplex @ times_1444617028055533883omplex @ K @ A3 @ A ) ) @ A3 )
= ( det_Fo6681143053013413569omplex @ A ) ) ) ) ) ).
% det_multrow_div
thf(fact_743_not__real__square__gt__zero,axiom,
! [X2: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X2 @ X2 ) ) )
= ( X2 = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_744_mat__delete__dim_I2_J,axiom,
! [A: mat_complex,I: nat,J: nat] :
( ( dim_col_complex @ ( mat_delete_complex @ A @ I @ J ) )
= ( minus_minus_nat @ ( dim_col_complex @ A ) @ one_one_nat ) ) ).
% mat_delete_dim(2)
thf(fact_745_mat__delete__dim_I1_J,axiom,
! [A: mat_complex,I: nat,J: nat] :
( ( dim_row_complex @ ( mat_delete_complex @ A @ I @ J ) )
= ( minus_minus_nat @ ( dim_row_complex @ A ) @ one_one_nat ) ) ).
% mat_delete_dim(1)
thf(fact_746_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_747_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_748_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_749_real__divide__square__eq,axiom,
! [R3: real,A3: real] :
( ( divide_divide_real @ ( times_times_real @ R3 @ A3 ) @ ( times_times_real @ R3 @ R3 ) )
= ( divide_divide_real @ A3 @ R3 ) ) ).
% real_divide_square_eq
thf(fact_750_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_751_det__multrow,axiom,
! [K: nat,N: nat,A: mat_real,A3: real] :
( ( ord_less_nat @ K @ N )
=> ( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( det_real @ ( gauss_1037889766561479105n_real @ times_times_real @ K @ A3 @ A ) )
= ( times_times_real @ A3 @ ( det_real @ A ) ) ) ) ) ).
% det_multrow
thf(fact_752_det__multrow,axiom,
! [K: nat,N: nat,A: mat_int,A3: int] :
( ( ord_less_nat @ K @ N )
=> ( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( ( det_int @ ( gauss_2407205949817067457en_int @ times_times_int @ K @ A3 @ A ) )
= ( times_times_int @ A3 @ ( det_int @ A ) ) ) ) ) ).
% det_multrow
thf(fact_753_mat__delete__carrier,axiom,
! [A: mat_complex,M: nat,N: nat,I: nat,J: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ M @ N ) )
=> ( member_mat_complex @ ( mat_delete_complex @ A @ I @ J ) @ ( carrier_mat_complex @ ( minus_minus_nat @ M @ one_one_nat ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% mat_delete_carrier
thf(fact_754_pivot__fun__multrow,axiom,
! [A: mat_complex,F: nat > nat,Jj: nat,Nr: nat,Nc: nat,I0: nat,A3: complex] :
( ( gauss_2609248829700396350omplex @ A @ F @ Jj )
=> ( ( ( dim_row_complex @ A )
= Nr )
=> ( ( ( dim_col_complex @ A )
= Nc )
=> ( ( ( F @ I0 )
= Jj )
=> ( ( ord_less_eq_nat @ Jj @ Nc )
=> ( gauss_2609248829700396350omplex @ ( gauss_2324787009747932227omplex @ times_times_complex @ I0 @ A3 @ A ) @ F @ Jj ) ) ) ) ) ) ).
% pivot_fun_multrow
thf(fact_755_pivot__fun__multrow,axiom,
! [A: mat_real,F: nat > nat,Jj: nat,Nr: nat,Nc: nat,I0: nat,A3: real] :
( ( gauss_5041415250090615612n_real @ A @ F @ Jj )
=> ( ( ( dim_row_real @ A )
= Nr )
=> ( ( ( dim_col_real @ A )
= Nc )
=> ( ( ( F @ I0 )
= Jj )
=> ( ( ord_less_eq_nat @ Jj @ Nc )
=> ( gauss_5041415250090615612n_real @ ( gauss_1037889766561479105n_real @ times_times_real @ I0 @ A3 @ A ) @ F @ Jj ) ) ) ) ) ) ).
% pivot_fun_multrow
thf(fact_756_pivot__fun__multrow,axiom,
! [A: mat_nat,F: nat > nat,Jj: nat,Nr: nat,Nc: nat,I0: nat,A3: nat] :
( ( gauss_8416567519840421984un_nat @ A @ F @ Jj )
=> ( ( ( dim_row_nat @ A )
= Nr )
=> ( ( ( dim_col_nat @ A )
= Nc )
=> ( ( ( F @ I0 )
= Jj )
=> ( ( ord_less_eq_nat @ Jj @ Nc )
=> ( gauss_8416567519840421984un_nat @ ( gauss_2409696420326117733en_nat @ times_times_nat @ I0 @ A3 @ A ) @ F @ Jj ) ) ) ) ) ) ).
% pivot_fun_multrow
thf(fact_757_pivot__fun__multrow,axiom,
! [A: mat_int,F: nat > nat,Jj: nat,Nr: nat,Nc: nat,I0: nat,A3: int] :
( ( gauss_8414077049331371708un_int @ A @ F @ Jj )
=> ( ( ( dim_row_int @ A )
= Nr )
=> ( ( ( dim_col_int @ A )
= Nc )
=> ( ( ( F @ I0 )
= Jj )
=> ( ( ord_less_eq_nat @ Jj @ Nc )
=> ( gauss_8414077049331371708un_int @ ( gauss_2407205949817067457en_int @ times_times_int @ I0 @ A3 @ A ) @ F @ Jj ) ) ) ) ) ) ).
% pivot_fun_multrow
thf(fact_758_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_759_multrow__mat,axiom,
! [A: mat_real,N: nat,Nc: nat,K: nat,A3: real] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ Nc ) )
=> ( ( gauss_1037889766561479105n_real @ times_times_real @ K @ A3 @ A )
= ( times_times_mat_real @ ( gauss_7241202418770761333t_real @ N @ K @ A3 ) @ A ) ) ) ).
% multrow_mat
thf(fact_760_multrow__mat,axiom,
! [A: mat_nat,N: nat,Nc: nat,K: nat,A3: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ Nc ) )
=> ( ( gauss_2409696420326117733en_nat @ times_times_nat @ K @ A3 @ A )
= ( times_times_mat_nat @ ( gauss_3195076542185637913at_nat @ N @ K @ A3 ) @ A ) ) ) ).
% multrow_mat
thf(fact_761_multrow__mat,axiom,
! [A: mat_int,N: nat,Nc: nat,K: nat,A3: int] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( gauss_2407205949817067457en_int @ times_times_int @ K @ A3 @ A )
= ( times_times_mat_int @ ( gauss_3192586071676587637at_int @ N @ K @ A3 ) @ A ) ) ) ).
% multrow_mat
thf(fact_762_max__mix__density__def,axiom,
( projec8360710381328234318ensity
= ( ^ [N2: nat] : ( smult_mat_complex @ ( real_V4546457046886955230omplex @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ ( one_mat_complex @ N2 ) ) ) ) ).
% max_mix_density_def
thf(fact_763_cpx__sq__mat_Oeigen__projector__carrier,axiom,
! [DimR: nat,DimC: nat,Fc_mats: set_mat_complex,A: mat_complex,A3: complex] :
( ( linear7199532782703566157sq_mat @ DimR @ DimC @ Fc_mats )
=> ( ( member_mat_complex @ A @ Fc_mats )
=> ( ( member_complex @ A3 @ ( projec527831343749723810omplex @ A ) )
=> ( ( comple8306762464034002205omplex @ A )
=> ( member_mat_complex @ ( projec1689266477789839993jector @ DimR @ DimC @ A @ A3 ) @ Fc_mats ) ) ) ) ) ).
% cpx_sq_mat.eigen_projector_carrier
thf(fact_764_mult__of__nat__commute,axiom,
! [X2: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_765_mult__of__nat__commute,axiom,
! [X2: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_766_mult__of__nat__commute,axiom,
! [X2: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X2 ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_767_mult__of__nat__commute,axiom,
! [X2: nat,Y: complex] :
( ( times_times_complex @ ( semiri8010041392384452111omplex @ X2 ) @ Y )
= ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_768_cpx__sq__mat_Oeigen__projector_Ocong,axiom,
projec1689266477789839993jector = projec1689266477789839993jector ).
% cpx_sq_mat.eigen_projector.cong
thf(fact_769_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_770_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_771_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_772_Num_Oof__nat__simps_I5_J,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% Num.of_nat_simps(5)
thf(fact_773_Num_Oof__nat__simps_I5_J,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% Num.of_nat_simps(5)
thf(fact_774_Num_Oof__nat__simps_I5_J,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% Num.of_nat_simps(5)
thf(fact_775_Num_Oof__nat__simps_I5_J,axiom,
! [M: nat,N: nat] :
( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% Num.of_nat_simps(5)
thf(fact_776_row__echelon__form__def,axiom,
( gauss_194721375535881179omplex
= ( ^ [A2: mat_complex] :
? [F2: nat > nat] : ( gauss_2609248829700396350omplex @ A2 @ F2 @ ( dim_col_complex @ A2 ) ) ) ) ).
% row_echelon_form_def
thf(fact_777_pivot__funD_I1_J,axiom,
! [A: mat_complex,Nr: nat,F: nat > nat,Nc: nat,I: nat] :
( ( ( dim_row_complex @ A )
= Nr )
=> ( ( gauss_2609248829700396350omplex @ A @ F @ Nc )
=> ( ( ord_less_nat @ I @ Nr )
=> ( ord_less_eq_nat @ ( F @ I ) @ Nc ) ) ) ) ).
% pivot_funD(1)
thf(fact_778_real__archimedian__rdiv__eq__0,axiom,
! [X2: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ! [M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ X2 ) @ C2 ) )
=> ( X2 = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_779_reals__Archimedean3,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ! [Y2: real] :
? [N3: nat] : ( ord_less_real @ Y2 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X2 ) ) ) ).
% reals_Archimedean3
thf(fact_780_ex__less__of__nat__mult,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ? [N3: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X2 ) ) ) ).
% ex_less_of_nat_mult
thf(fact_781_div__mult2__eq_H,axiom,
! [A3: nat,M: nat,N: nat] :
( ( divide_divide_nat @ A3 @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A3 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% div_mult2_eq'
thf(fact_782_div__mult2__eq_H,axiom,
! [A3: int,M: nat,N: nat] :
( ( divide_divide_int @ A3 @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( divide_divide_int @ ( divide_divide_int @ A3 @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% div_mult2_eq'
thf(fact_783_int__ops_I7_J,axiom,
! [A3: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A3 @ B4 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ).
% int_ops(7)
thf(fact_784_trace__1,axiom,
! [N: nat] :
( ( comple3184165445352484367omplex @ ( one_mat_complex @ N ) )
= ( semiri8010041392384452111omplex @ N ) ) ).
% trace_1
thf(fact_785_pivot__fun__eliminate__entries,axiom,
! [A: mat_complex,F: nat > nat,Jj: nat,Nr: nat,Nc: nat,L: nat,Vs: nat > complex,J: nat] :
( ( gauss_2609248829700396350omplex @ A @ F @ Jj )
=> ( ( ( dim_row_complex @ A )
= Nr )
=> ( ( ( dim_col_complex @ A )
= Nc )
=> ( ( ( F @ L )
= Jj )
=> ( ( ord_less_nat @ L @ Nr )
=> ( ( ord_less_eq_nat @ Jj @ Nc )
=> ( gauss_2609248829700396350omplex @ ( gauss_2785350030914899391omplex @ minus_minus_complex @ times_times_complex @ Vs @ A @ L @ J ) @ F @ Jj ) ) ) ) ) ) ) ).
% pivot_fun_eliminate_entries
thf(fact_786_pivot__fun__eliminate__entries,axiom,
! [A: mat_real,F: nat > nat,Jj: nat,Nr: nat,Nc: nat,L: nat,Vs: nat > real,J: nat] :
( ( gauss_5041415250090615612n_real @ A @ F @ Jj )
=> ( ( ( dim_row_real @ A )
= Nr )
=> ( ( ( dim_col_real @ A )
= Nc )
=> ( ( ( F @ L )
= Jj )
=> ( ( ord_less_nat @ L @ Nr )
=> ( ( ord_less_eq_nat @ Jj @ Nc )
=> ( gauss_5041415250090615612n_real @ ( gauss_9059886599186181437n_real @ minus_minus_real @ times_times_real @ Vs @ A @ L @ J ) @ F @ Jj ) ) ) ) ) ) ) ).
% pivot_fun_eliminate_entries
thf(fact_787_pivot__fun__eliminate__entries,axiom,
! [A: mat_int,F: nat > nat,Jj: nat,Nr: nat,Nc: nat,L: nat,Vs: nat > int,J: nat] :
( ( gauss_8414077049331371708un_int @ A @ F @ Jj )
=> ( ( ( dim_row_int @ A )
= Nr )
=> ( ( ( dim_col_int @ A )
= Nc )
=> ( ( ( F @ L )
= Jj )
=> ( ( ord_less_nat @ L @ Nr )
=> ( ( ord_less_eq_nat @ Jj @ Nc )
=> ( gauss_8414077049331371708un_int @ ( gauss_76339361826605373en_int @ minus_minus_int @ times_times_int @ Vs @ A @ L @ J ) @ F @ Jj ) ) ) ) ) ) ) ).
% pivot_fun_eliminate_entries
thf(fact_788_pivot__fun__swaprows,axiom,
! [A: mat_complex,F: nat > nat,Jj: nat,Nr: nat,Nc: nat,L: nat,K: nat] :
( ( gauss_2609248829700396350omplex @ A @ F @ Jj )
=> ( ( ( dim_row_complex @ A )
= Nr )
=> ( ( ( dim_col_complex @ A )
= Nc )
=> ( ( ( F @ L )
= Jj )
=> ( ( ( F @ K )
= Jj )
=> ( ( ord_less_nat @ L @ Nr )
=> ( ( ord_less_nat @ K @ Nr )
=> ( ( ord_less_eq_nat @ Jj @ Nc )
=> ( gauss_2609248829700396350omplex @ ( gauss_1020679828357514249omplex @ L @ K @ A ) @ F @ Jj ) ) ) ) ) ) ) ) ) ).
% pivot_fun_swaprows
thf(fact_789_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_790_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_791_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_792_trace__id,axiom,
! [N: nat] :
( ( complex_trace_int @ ( one_mat_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% trace_id
thf(fact_793_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_794_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_795_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_796_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_797_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_798_plusinfinity,axiom,
! [D2: int,P2: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D2 ) ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [X_1: int] : ( P2 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_799_minusinfinity,axiom,
! [D2: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D2 ) ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_800_decr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_801_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_802_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_803_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_804_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_805_zdiv__zmult2__eq,axiom,
! [C2: int,A3: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( divide_divide_int @ A3 @ ( times_times_int @ B4 @ C2 ) )
= ( divide_divide_int @ ( divide_divide_int @ A3 @ B4 ) @ C2 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_806_carrier__eliminate__entries_I1_J,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,Minus: complex > complex > complex,Times: complex > complex > complex,V3: nat > complex,I: nat,Bs: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( gauss_2785350030914899391omplex @ Minus @ Times @ V3 @ A @ I @ Bs ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% carrier_eliminate_entries(1)
thf(fact_807_dim__eliminate__entries__gen_I1_J,axiom,
! [Minus: complex > complex > complex,Times: complex > complex > complex,V3: nat > complex,B: mat_complex,I: nat,As: nat] :
( ( dim_row_complex @ ( gauss_2785350030914899391omplex @ Minus @ Times @ V3 @ B @ I @ As ) )
= ( dim_row_complex @ B ) ) ).
% dim_eliminate_entries_gen(1)
thf(fact_808_dim__eliminate__entries__gen_I2_J,axiom,
! [Minus: complex > complex > complex,Times: complex > complex > complex,V3: nat > complex,B: mat_complex,I: nat,As: nat] :
( ( dim_col_complex @ ( gauss_2785350030914899391omplex @ Minus @ Times @ V3 @ B @ I @ As ) )
= ( dim_col_complex @ B ) ) ).
% dim_eliminate_entries_gen(2)
thf(fact_809_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_810_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_811_gbinomial__trinomial__revision,axiom,
! [K: nat,M: nat,A3: real] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( times_times_real @ ( gbinomial_real @ A3 @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
= ( times_times_real @ ( gbinomial_real @ A3 @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_812_gbinomial__trinomial__revision,axiom,
! [K: nat,M: nat,A3: complex] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( times_times_complex @ ( gbinomial_complex @ A3 @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
= ( times_times_complex @ ( gbinomial_complex @ A3 @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A3 @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_813_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_814_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_815_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri2265585572941072030t_real @ N )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_816_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri1406184849735516958ct_int @ N )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_817_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri5044797733671781792omplex @ N )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_818_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri1408675320244567234ct_nat @ N )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_819_fact__num__eq__if,axiom,
( semiri2265585572941072030t_real
= ( ^ [M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M6 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_820_fact__num__eq__if,axiom,
( semiri1406184849735516958ct_int
= ( ^ [M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_821_fact__num__eq__if,axiom,
( semiri5044797733671781792omplex
= ( ^ [M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M6 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_822_fact__num__eq__if,axiom,
( semiri1408675320244567234ct_nat
= ( ^ [M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M6 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_823_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_824_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_825_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_826_swaprows__mat__inv,axiom,
! [K: nat,N: nat,L: nat] :
( ( ord_less_nat @ K @ N )
=> ( ( ord_less_nat @ L @ N )
=> ( ( times_times_mat_int @ ( gauss_4917416859360123759at_int @ N @ K @ L ) @ ( gauss_4917416859360123759at_int @ N @ K @ L ) )
= ( one_mat_int @ N ) ) ) ) ).
% swaprows_mat_inv
thf(fact_827_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_828_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_829_binomial__fact,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
= ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% binomial_fact
thf(fact_830_binomial__fact,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
= ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% binomial_fact
thf(fact_831_fact__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) )
= ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% fact_binomial
thf(fact_832_fact__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) ) )
= ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% fact_binomial
thf(fact_833_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I2: int,J2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
& ( ord_less_int @ J2 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J2 ) ) )
=> ( P @ I2 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I2: int,J2: int] :
( ( ( ord_less_int @ K @ J2 )
& ( ord_less_eq_int @ J2 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J2 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_zdiv
thf(fact_834_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_835_trace__add__linear,axiom,
! [A: mat_int,N: nat,B: mat_int] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( ( member_mat_int @ B @ ( carrier_mat_int @ N @ N ) )
=> ( ( complex_trace_int @ ( plus_plus_mat_int @ A @ B ) )
= ( plus_plus_int @ ( complex_trace_int @ A ) @ ( complex_trace_int @ B ) ) ) ) ) ).
% trace_add_linear
thf(fact_836_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_837_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_838_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_839_add__smult__distrib__right__mat,axiom,
! [A: mat_nat,Nr: nat,Nc: nat,K: nat,L: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( smult_mat_nat @ ( plus_plus_nat @ K @ L ) @ A )
= ( plus_plus_mat_nat @ ( smult_mat_nat @ K @ A ) @ ( smult_mat_nat @ L @ A ) ) ) ) ).
% add_smult_distrib_right_mat
thf(fact_840_add__smult__distrib__right__mat,axiom,
! [A: mat_int,Nr: nat,Nc: nat,K: int,L: int] :
( ( member_mat_int @ A @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( smult_mat_int @ ( plus_plus_int @ K @ L ) @ A )
= ( plus_plus_mat_int @ ( smult_mat_int @ K @ A ) @ ( smult_mat_int @ L @ A ) ) ) ) ).
% add_smult_distrib_right_mat
thf(fact_841_vector__space__over__itself_Oscale__right__distrib,axiom,
! [A3: complex,X2: complex,Y: complex] :
( ( times_times_complex @ A3 @ ( plus_plus_complex @ X2 @ Y ) )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ X2 ) @ ( times_times_complex @ A3 @ Y ) ) ) ).
% vector_space_over_itself.scale_right_distrib
thf(fact_842_vector__space__over__itself_Oscale__right__distrib,axiom,
! [A3: real,X2: real,Y: real] :
( ( times_times_real @ A3 @ ( plus_plus_real @ X2 @ Y ) )
= ( plus_plus_real @ ( times_times_real @ A3 @ X2 ) @ ( times_times_real @ A3 @ Y ) ) ) ).
% vector_space_over_itself.scale_right_distrib
thf(fact_843_vector__space__over__itself_Oscale__left__distrib,axiom,
! [A3: complex,B4: complex,X2: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A3 @ B4 ) @ X2 )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ X2 ) @ ( times_times_complex @ B4 @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_distrib
thf(fact_844_vector__space__over__itself_Oscale__left__distrib,axiom,
! [A3: real,B4: real,X2: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B4 ) @ X2 )
= ( plus_plus_real @ ( times_times_real @ A3 @ X2 ) @ ( times_times_real @ B4 @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_distrib
thf(fact_845_mult__hom_Ohom__add,axiom,
! [C2: complex,X2: complex,Y: complex] :
( ( times_times_complex @ C2 @ ( plus_plus_complex @ X2 @ Y ) )
= ( plus_plus_complex @ ( times_times_complex @ C2 @ X2 ) @ ( times_times_complex @ C2 @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_846_mult__hom_Ohom__add,axiom,
! [C2: real,X2: real,Y: real] :
( ( times_times_real @ C2 @ ( plus_plus_real @ X2 @ Y ) )
= ( plus_plus_real @ ( times_times_real @ C2 @ X2 ) @ ( times_times_real @ C2 @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_847_mult__hom_Ohom__add,axiom,
! [C2: nat,X2: nat,Y: nat] :
( ( times_times_nat @ C2 @ ( plus_plus_nat @ X2 @ Y ) )
= ( plus_plus_nat @ ( times_times_nat @ C2 @ X2 ) @ ( times_times_nat @ C2 @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_848_mult__hom_Ohom__add,axiom,
! [C2: int,X2: int,Y: int] :
( ( times_times_int @ C2 @ ( plus_plus_int @ X2 @ Y ) )
= ( plus_plus_int @ ( times_times_int @ C2 @ X2 ) @ ( times_times_int @ C2 @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_849_combine__common__factor,axiom,
! [A3: complex,E: complex,B4: complex,C2: complex] :
( ( plus_plus_complex @ ( times_times_complex @ A3 @ E ) @ ( plus_plus_complex @ ( times_times_complex @ B4 @ E ) @ C2 ) )
= ( plus_plus_complex @ ( times_times_complex @ ( plus_plus_complex @ A3 @ B4 ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_850_combine__common__factor,axiom,
! [A3: real,E: real,B4: real,C2: real] :
( ( plus_plus_real @ ( times_times_real @ A3 @ E ) @ ( plus_plus_real @ ( times_times_real @ B4 @ E ) @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A3 @ B4 ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_851_combine__common__factor,axiom,
! [A3: nat,E: nat,B4: nat,C2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A3 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B4 @ E ) @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A3 @ B4 ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_852_combine__common__factor,axiom,
! [A3: int,E: int,B4: int,C2: int] :
( ( plus_plus_int @ ( times_times_int @ A3 @ E ) @ ( plus_plus_int @ ( times_times_int @ B4 @ E ) @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A3 @ B4 ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_853_distrib__right,axiom,
! [A3: complex,B4: complex,C2: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A3 @ B4 ) @ C2 )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B4 @ C2 ) ) ) ).
% distrib_right
thf(fact_854_distrib__right,axiom,
! [A3: real,B4: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B4 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ C2 ) ) ) ).
% distrib_right
thf(fact_855_distrib__right,axiom,
! [A3: nat,B4: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A3 @ B4 ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B4 @ C2 ) ) ) ).
% distrib_right
thf(fact_856_distrib__right,axiom,
! [A3: int,B4: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A3 @ B4 ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ C2 ) ) ) ).
% distrib_right
thf(fact_857_distrib__left,axiom,
! [A3: complex,B4: complex,C2: complex] :
( ( times_times_complex @ A3 @ ( plus_plus_complex @ B4 @ C2 ) )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ B4 ) @ ( times_times_complex @ A3 @ C2 ) ) ) ).
% distrib_left
thf(fact_858_distrib__left,axiom,
! [A3: real,B4: real,C2: real] :
( ( times_times_real @ A3 @ ( plus_plus_real @ B4 @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A3 @ B4 ) @ ( times_times_real @ A3 @ C2 ) ) ) ).
% distrib_left
thf(fact_859_distrib__left,axiom,
! [A3: nat,B4: nat,C2: nat] :
( ( times_times_nat @ A3 @ ( plus_plus_nat @ B4 @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ B4 ) @ ( times_times_nat @ A3 @ C2 ) ) ) ).
% distrib_left
thf(fact_860_distrib__left,axiom,
! [A3: int,B4: int,C2: int] :
( ( times_times_int @ A3 @ ( plus_plus_int @ B4 @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ A3 @ B4 ) @ ( times_times_int @ A3 @ C2 ) ) ) ).
% distrib_left
thf(fact_861_comm__semiring__class_Odistrib,axiom,
! [A3: complex,B4: complex,C2: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A3 @ B4 ) @ C2 )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B4 @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_862_comm__semiring__class_Odistrib,axiom,
! [A3: real,B4: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B4 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_863_comm__semiring__class_Odistrib,axiom,
! [A3: nat,B4: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A3 @ B4 ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B4 @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_864_comm__semiring__class_Odistrib,axiom,
! [A3: int,B4: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A3 @ B4 ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_865_ring__class_Oring__distribs_I1_J,axiom,
! [A3: complex,B4: complex,C2: complex] :
( ( times_times_complex @ A3 @ ( plus_plus_complex @ B4 @ C2 ) )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ B4 ) @ ( times_times_complex @ A3 @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_866_ring__class_Oring__distribs_I1_J,axiom,
! [A3: real,B4: real,C2: real] :
( ( times_times_real @ A3 @ ( plus_plus_real @ B4 @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A3 @ B4 ) @ ( times_times_real @ A3 @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_867_ring__class_Oring__distribs_I1_J,axiom,
! [A3: int,B4: int,C2: int] :
( ( times_times_int @ A3 @ ( plus_plus_int @ B4 @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ A3 @ B4 ) @ ( times_times_int @ A3 @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_868_ring__class_Oring__distribs_I2_J,axiom,
! [A3: complex,B4: complex,C2: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A3 @ B4 ) @ C2 )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B4 @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_869_ring__class_Oring__distribs_I2_J,axiom,
! [A3: real,B4: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B4 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_870_ring__class_Oring__distribs_I2_J,axiom,
! [A3: int,B4: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A3 @ B4 ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_871_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_872_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_873_sum__squares__eq__zero__iff,axiom,
! [X2: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_874_sum__squares__eq__zero__iff,axiom,
! [X2: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_875_mult__hom_Ohom__add__eq__zero,axiom,
! [X2: complex,Y: complex,C2: complex] :
( ( ( plus_plus_complex @ X2 @ Y )
= zero_zero_complex )
=> ( ( plus_plus_complex @ ( times_times_complex @ C2 @ X2 ) @ ( times_times_complex @ C2 @ Y ) )
= zero_zero_complex ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_876_mult__hom_Ohom__add__eq__zero,axiom,
! [X2: real,Y: real,C2: real] :
( ( ( plus_plus_real @ X2 @ Y )
= zero_zero_real )
=> ( ( plus_plus_real @ ( times_times_real @ C2 @ X2 ) @ ( times_times_real @ C2 @ Y ) )
= zero_zero_real ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_877_mult__hom_Ohom__add__eq__zero,axiom,
! [X2: nat,Y: nat,C2: nat] :
( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
=> ( ( plus_plus_nat @ ( times_times_nat @ C2 @ X2 ) @ ( times_times_nat @ C2 @ Y ) )
= zero_zero_nat ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_878_mult__hom_Ohom__add__eq__zero,axiom,
! [X2: int,Y: int,C2: int] :
( ( ( plus_plus_int @ X2 @ Y )
= zero_zero_int )
=> ( ( plus_plus_int @ ( times_times_int @ C2 @ X2 ) @ ( times_times_int @ C2 @ Y ) )
= zero_zero_int ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_879_eq__add__iff1,axiom,
! [A3: complex,E: complex,C2: complex,B4: complex,D2: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ A3 @ E ) @ C2 )
= ( plus_plus_complex @ ( times_times_complex @ B4 @ E ) @ D2 ) )
= ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A3 @ B4 ) @ E ) @ C2 )
= D2 ) ) ).
% eq_add_iff1
thf(fact_880_eq__add__iff1,axiom,
! [A3: real,E: real,C2: real,B4: real,D2: real] :
( ( ( plus_plus_real @ ( times_times_real @ A3 @ E ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ B4 @ E ) @ D2 ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A3 @ B4 ) @ E ) @ C2 )
= D2 ) ) ).
% eq_add_iff1
thf(fact_881_eq__add__iff1,axiom,
! [A3: int,E: int,C2: int,B4: int,D2: int] :
( ( ( plus_plus_int @ ( times_times_int @ A3 @ E ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ B4 @ E ) @ D2 ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A3 @ B4 ) @ E ) @ C2 )
= D2 ) ) ).
% eq_add_iff1
thf(fact_882_eq__add__iff2,axiom,
! [A3: complex,E: complex,C2: complex,B4: complex,D2: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ A3 @ E ) @ C2 )
= ( plus_plus_complex @ ( times_times_complex @ B4 @ E ) @ D2 ) )
= ( C2
= ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B4 @ A3 ) @ E ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_883_eq__add__iff2,axiom,
! [A3: real,E: real,C2: real,B4: real,D2: real] :
( ( ( plus_plus_real @ ( times_times_real @ A3 @ E ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ B4 @ E ) @ D2 ) )
= ( C2
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B4 @ A3 ) @ E ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_884_eq__add__iff2,axiom,
! [A3: int,E: int,C2: int,B4: int,D2: int] :
( ( ( plus_plus_int @ ( times_times_int @ A3 @ E ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ B4 @ E ) @ D2 ) )
= ( C2
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B4 @ A3 ) @ E ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_885_square__diff__square__factored,axiom,
! [X2: complex,Y: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X2 @ X2 ) @ ( times_times_complex @ Y @ Y ) )
= ( times_times_complex @ ( plus_plus_complex @ X2 @ Y ) @ ( minus_minus_complex @ X2 @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_886_square__diff__square__factored,axiom,
! [X2: real,Y: real] :
( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) )
= ( times_times_real @ ( plus_plus_real @ X2 @ Y ) @ ( minus_minus_real @ X2 @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_887_square__diff__square__factored,axiom,
! [X2: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X2 @ Y ) @ ( minus_minus_int @ X2 @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_888_mult__diff__mult,axiom,
! [X2: complex,Y: complex,A3: complex,B4: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X2 @ Y ) @ ( times_times_complex @ A3 @ B4 ) )
= ( plus_plus_complex @ ( times_times_complex @ X2 @ ( minus_minus_complex @ Y @ B4 ) ) @ ( times_times_complex @ ( minus_minus_complex @ X2 @ A3 ) @ B4 ) ) ) ).
% mult_diff_mult
thf(fact_889_mult__diff__mult,axiom,
! [X2: real,Y: real,A3: real,B4: real] :
( ( minus_minus_real @ ( times_times_real @ X2 @ Y ) @ ( times_times_real @ A3 @ B4 ) )
= ( plus_plus_real @ ( times_times_real @ X2 @ ( minus_minus_real @ Y @ B4 ) ) @ ( times_times_real @ ( minus_minus_real @ X2 @ A3 ) @ B4 ) ) ) ).
% mult_diff_mult
thf(fact_890_mult__diff__mult,axiom,
! [X2: int,Y: int,A3: int,B4: int] :
( ( minus_minus_int @ ( times_times_int @ X2 @ Y ) @ ( times_times_int @ A3 @ B4 ) )
= ( plus_plus_int @ ( times_times_int @ X2 @ ( minus_minus_int @ Y @ B4 ) ) @ ( times_times_int @ ( minus_minus_int @ X2 @ A3 ) @ B4 ) ) ) ).
% mult_diff_mult
thf(fact_891_choose__mult,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
= ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_892_binomial__fact__lemma,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
= ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% binomial_fact_lemma
thf(fact_893_binomial__absorb__comp,axiom,
! [N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% binomial_absorb_comp
thf(fact_894_sum__squares__le__zero__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_895_sum__squares__le__zero__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_896_sum__squares__ge__zero,axiom,
! [X2: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_897_sum__squares__ge__zero,axiom,
! [X2: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_898_sum__squares__gt__zero__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) ) )
= ( ( X2 != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_899_sum__squares__gt__zero__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X2 != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_900_not__sum__squares__lt__zero,axiom,
! [X2: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_901_not__sum__squares__lt__zero,axiom,
! [X2: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_902_ordered__ring__class_Ole__add__iff1,axiom,
! [A3: complex,E: complex,C2: complex,B4: complex,D2: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ ( times_times_complex @ A3 @ E ) @ C2 ) @ ( plus_plus_complex @ ( times_times_complex @ B4 @ E ) @ D2 ) )
= ( ord_less_eq_complex @ ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A3 @ B4 ) @ E ) @ C2 ) @ D2 ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_903_ordered__ring__class_Ole__add__iff1,axiom,
! [A3: real,E: real,C2: real,B4: real,D2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A3 @ E ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B4 @ E ) @ D2 ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A3 @ B4 ) @ E ) @ C2 ) @ D2 ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_904_ordered__ring__class_Ole__add__iff1,axiom,
! [A3: int,E: int,C2: int,B4: int,D2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A3 @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B4 @ E ) @ D2 ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A3 @ B4 ) @ E ) @ C2 ) @ D2 ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_905_ordered__ring__class_Ole__add__iff2,axiom,
! [A3: complex,E: complex,C2: complex,B4: complex,D2: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ ( times_times_complex @ A3 @ E ) @ C2 ) @ ( plus_plus_complex @ ( times_times_complex @ B4 @ E ) @ D2 ) )
= ( ord_less_eq_complex @ C2 @ ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B4 @ A3 ) @ E ) @ D2 ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_906_ordered__ring__class_Ole__add__iff2,axiom,
! [A3: real,E: real,C2: real,B4: real,D2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A3 @ E ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B4 @ E ) @ D2 ) )
= ( ord_less_eq_real @ C2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B4 @ A3 ) @ E ) @ D2 ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_907_ordered__ring__class_Ole__add__iff2,axiom,
! [A3: int,E: int,C2: int,B4: int,D2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A3 @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B4 @ E ) @ D2 ) )
= ( ord_less_eq_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B4 @ A3 ) @ E ) @ D2 ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_908_less__add__iff2,axiom,
! [A3: complex,E: complex,C2: complex,B4: complex,D2: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ ( times_times_complex @ A3 @ E ) @ C2 ) @ ( plus_plus_complex @ ( times_times_complex @ B4 @ E ) @ D2 ) )
= ( ord_less_complex @ C2 @ ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B4 @ A3 ) @ E ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_909_less__add__iff2,axiom,
! [A3: real,E: real,C2: real,B4: real,D2: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A3 @ E ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B4 @ E ) @ D2 ) )
= ( ord_less_real @ C2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B4 @ A3 ) @ E ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_910_less__add__iff2,axiom,
! [A3: int,E: int,C2: int,B4: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A3 @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B4 @ E ) @ D2 ) )
= ( ord_less_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B4 @ A3 ) @ E ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_911_less__add__iff1,axiom,
! [A3: complex,E: complex,C2: complex,B4: complex,D2: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ ( times_times_complex @ A3 @ E ) @ C2 ) @ ( plus_plus_complex @ ( times_times_complex @ B4 @ E ) @ D2 ) )
= ( ord_less_complex @ ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A3 @ B4 ) @ E ) @ C2 ) @ D2 ) ) ).
% less_add_iff1
thf(fact_912_less__add__iff1,axiom,
! [A3: real,E: real,C2: real,B4: real,D2: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A3 @ E ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B4 @ E ) @ D2 ) )
= ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A3 @ B4 ) @ E ) @ C2 ) @ D2 ) ) ).
% less_add_iff1
thf(fact_913_less__add__iff1,axiom,
! [A3: int,E: int,C2: int,B4: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A3 @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B4 @ E ) @ D2 ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A3 @ B4 ) @ E ) @ C2 ) @ D2 ) ) ).
% less_add_iff1
thf(fact_914_divide__add__eq__iff,axiom,
! [Z: complex,X2: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Z ) @ Y )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_915_divide__add__eq__iff,axiom,
! [Z: real,X2: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Z ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_916_add__divide__eq__iff,axiom,
! [Z: complex,X2: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ X2 @ ( divide1717551699836669952omplex @ Y @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X2 @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_917_add__divide__eq__iff,axiom,
! [Z: real,X2: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ X2 @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_918_add__num__frac,axiom,
! [Y: complex,Z: complex,X2: complex] :
( ( Y != zero_zero_complex )
=> ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X2 @ Y ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_919_add__num__frac,axiom,
! [Y: real,Z: real,X2: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ Z @ ( divide_divide_real @ X2 @ Y ) )
= ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_920_add__frac__num,axiom,
! [Y: complex,X2: complex,Z: complex] :
( ( Y != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Y ) @ Z )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_921_add__frac__num,axiom,
! [Y: real,X2: real,Z: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y ) @ Z )
= ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_922_add__frac__eq,axiom,
! [Y: complex,Z: complex,X2: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X2 @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_923_add__frac__eq,axiom,
! [Y: real,Z: real,X2: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_924_add__divide__eq__if__simps_I1_J,axiom,
! [Z: complex,A3: complex,B4: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( plus_plus_complex @ A3 @ ( divide1717551699836669952omplex @ B4 @ Z ) )
= A3 ) )
& ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ A3 @ ( divide1717551699836669952omplex @ B4 @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A3 @ Z ) @ B4 ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_925_add__divide__eq__if__simps_I1_J,axiom,
! [Z: real,A3: real,B4: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ A3 @ ( divide_divide_real @ B4 @ Z ) )
= A3 ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ A3 @ ( divide_divide_real @ B4 @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A3 @ Z ) @ B4 ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_926_add__divide__eq__if__simps_I2_J,axiom,
! [Z: complex,A3: complex,B4: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A3 @ Z ) @ B4 )
= B4 ) )
& ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A3 @ Z ) @ B4 )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ A3 @ ( times_times_complex @ B4 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_927_add__divide__eq__if__simps_I2_J,axiom,
! [Z: real,A3: real,B4: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A3 @ Z ) @ B4 )
= B4 ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A3 @ Z ) @ B4 )
= ( divide_divide_real @ ( plus_plus_real @ A3 @ ( times_times_real @ B4 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_928_div__mult__self4,axiom,
! [B4: nat,C2: nat,A3: nat] :
( ( B4 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B4 @ C2 ) @ A3 ) @ B4 )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A3 @ B4 ) ) ) ) ).
% div_mult_self4
thf(fact_929_div__mult__self4,axiom,
! [B4: int,C2: int,A3: int] :
( ( B4 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B4 @ C2 ) @ A3 ) @ B4 )
= ( plus_plus_int @ C2 @ ( divide_divide_int @ A3 @ B4 ) ) ) ) ).
% div_mult_self4
thf(fact_930_div__mult__self4,axiom,
! [B4: formal670952693614245302omplex,C2: formal670952693614245302omplex,A3: formal670952693614245302omplex] :
( ( B4 != zero_z1877163951443063103omplex )
=> ( ( divide1348722040316500488omplex @ ( plus_p8472957120637115327omplex @ ( times_1444617028055533883omplex @ B4 @ C2 ) @ A3 ) @ B4 )
= ( plus_p8472957120637115327omplex @ C2 @ ( divide1348722040316500488omplex @ A3 @ B4 ) ) ) ) ).
% div_mult_self4
thf(fact_931_div__mult__self3,axiom,
! [B4: nat,C2: nat,A3: nat] :
( ( B4 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C2 @ B4 ) @ A3 ) @ B4 )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A3 @ B4 ) ) ) ) ).
% div_mult_self3
thf(fact_932_div__mult__self3,axiom,
! [B4: int,C2: int,A3: int] :
( ( B4 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C2 @ B4 ) @ A3 ) @ B4 )
= ( plus_plus_int @ C2 @ ( divide_divide_int @ A3 @ B4 ) ) ) ) ).
% div_mult_self3
thf(fact_933_div__mult__self3,axiom,
! [B4: formal670952693614245302omplex,C2: formal670952693614245302omplex,A3: formal670952693614245302omplex] :
( ( B4 != zero_z1877163951443063103omplex )
=> ( ( divide1348722040316500488omplex @ ( plus_p8472957120637115327omplex @ ( times_1444617028055533883omplex @ C2 @ B4 ) @ A3 ) @ B4 )
= ( plus_p8472957120637115327omplex @ C2 @ ( divide1348722040316500488omplex @ A3 @ B4 ) ) ) ) ).
% div_mult_self3
thf(fact_934_div__mult__self2,axiom,
! [B4: nat,A3: nat,C2: nat] :
( ( B4 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ ( times_times_nat @ B4 @ C2 ) ) @ B4 )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A3 @ B4 ) ) ) ) ).
% div_mult_self2
thf(fact_935_div__mult__self2,axiom,
! [B4: int,A3: int,C2: int] :
( ( B4 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A3 @ ( times_times_int @ B4 @ C2 ) ) @ B4 )
= ( plus_plus_int @ C2 @ ( divide_divide_int @ A3 @ B4 ) ) ) ) ).
% div_mult_self2
thf(fact_936_div__mult__self2,axiom,
! [B4: formal670952693614245302omplex,A3: formal670952693614245302omplex,C2: formal670952693614245302omplex] :
( ( B4 != zero_z1877163951443063103omplex )
=> ( ( divide1348722040316500488omplex @ ( plus_p8472957120637115327omplex @ A3 @ ( times_1444617028055533883omplex @ B4 @ C2 ) ) @ B4 )
= ( plus_p8472957120637115327omplex @ C2 @ ( divide1348722040316500488omplex @ A3 @ B4 ) ) ) ) ).
% div_mult_self2
thf(fact_937_div__mult__self1,axiom,
! [B4: nat,A3: nat,C2: nat] :
( ( B4 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ ( times_times_nat @ C2 @ B4 ) ) @ B4 )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A3 @ B4 ) ) ) ) ).
% div_mult_self1
thf(fact_938_div__mult__self1,axiom,
! [B4: int,A3: int,C2: int] :
( ( B4 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A3 @ ( times_times_int @ C2 @ B4 ) ) @ B4 )
= ( plus_plus_int @ C2 @ ( divide_divide_int @ A3 @ B4 ) ) ) ) ).
% div_mult_self1
thf(fact_939_div__mult__self1,axiom,
! [B4: formal670952693614245302omplex,A3: formal670952693614245302omplex,C2: formal670952693614245302omplex] :
( ( B4 != zero_z1877163951443063103omplex )
=> ( ( divide1348722040316500488omplex @ ( plus_p8472957120637115327omplex @ A3 @ ( times_1444617028055533883omplex @ C2 @ B4 ) ) @ B4 )
= ( plus_p8472957120637115327omplex @ C2 @ ( divide1348722040316500488omplex @ A3 @ B4 ) ) ) ) ).
% div_mult_self1
thf(fact_940_square__diff__one__factored,axiom,
! [X2: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X2 @ X2 ) @ one_one_complex )
= ( times_times_complex @ ( plus_plus_complex @ X2 @ one_one_complex ) @ ( minus_minus_complex @ X2 @ one_one_complex ) ) ) ).
% square_diff_one_factored
thf(fact_941_square__diff__one__factored,axiom,
! [X2: real] :
( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X2 @ one_one_real ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_942_square__diff__one__factored,axiom,
! [X2: int] :
( ( minus_minus_int @ ( times_times_int @ X2 @ X2 ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X2 @ one_one_int ) @ ( minus_minus_int @ X2 @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_943_binomial__fact_H,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% binomial_fact'
thf(fact_944_convex__bound__le,axiom,
! [X2: real,A3: real,Y: real,U4: real,V3: real] :
( ( ord_less_eq_real @ X2 @ A3 )
=> ( ( ord_less_eq_real @ Y @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ U4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ V3 )
=> ( ( ( plus_plus_real @ U4 @ V3 )
= one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U4 @ X2 ) @ ( times_times_real @ V3 @ Y ) ) @ A3 ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_945_convex__bound__le,axiom,
! [X2: int,A3: int,Y: int,U4: int,V3: int] :
( ( ord_less_eq_int @ X2 @ A3 )
=> ( ( ord_less_eq_int @ Y @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ V3 )
=> ( ( ( plus_plus_int @ U4 @ V3 )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U4 @ X2 ) @ ( times_times_int @ V3 @ Y ) ) @ A3 ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_946_incr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus_int @ X4 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_947_times__binomial__minus1__eq,axiom,
! [K: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_948_convex__bound__lt,axiom,
! [X2: real,A3: real,Y: real,U4: real,V3: real] :
( ( ord_less_real @ X2 @ A3 )
=> ( ( ord_less_real @ Y @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ U4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ V3 )
=> ( ( ( plus_plus_real @ U4 @ V3 )
= one_one_real )
=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U4 @ X2 ) @ ( times_times_real @ V3 @ Y ) ) @ A3 ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_949_convex__bound__lt,axiom,
! [X2: int,A3: int,Y: int,U4: int,V3: int] :
( ( ord_less_int @ X2 @ A3 )
=> ( ( ord_less_int @ Y @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ V3 )
=> ( ( ( plus_plus_int @ U4 @ V3 )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U4 @ X2 ) @ ( times_times_int @ V3 @ Y ) ) @ A3 ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_950_linordered__field__class_Oscaling__mono,axiom,
! [U4: real,V3: real,R3: real,S: real] :
( ( ord_less_eq_real @ U4 @ V3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ R3 )
=> ( ( ord_less_eq_real @ R3 @ S )
=> ( ord_less_eq_real @ ( plus_plus_real @ U4 @ ( divide_divide_real @ ( times_times_real @ R3 @ ( minus_minus_real @ V3 @ U4 ) ) @ S ) ) @ V3 ) ) ) ) ).
% linordered_field_class.scaling_mono
thf(fact_951_int__div__pos__eq,axiom,
! [A3: int,B4: int,Q4: int,R3: int] :
( ( A3
= ( plus_plus_int @ ( times_times_int @ B4 @ Q4 ) @ R3 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R3 )
=> ( ( ord_less_int @ R3 @ B4 )
=> ( ( divide_divide_int @ A3 @ B4 )
= Q4 ) ) ) ) ).
% int_div_pos_eq
thf(fact_952_int__div__neg__eq,axiom,
! [A3: int,B4: int,Q4: int,R3: int] :
( ( A3
= ( plus_plus_int @ ( times_times_int @ B4 @ Q4 ) @ R3 ) )
=> ( ( ord_less_eq_int @ R3 @ zero_zero_int )
=> ( ( ord_less_int @ B4 @ R3 )
=> ( ( divide_divide_int @ A3 @ B4 )
= Q4 ) ) ) ) ).
% int_div_neg_eq
thf(fact_953_linepath__le__1,axiom,
! [A3: real,B4: real,U4: real] :
( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ( ord_less_eq_real @ B4 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ U4 )
=> ( ( ord_less_eq_real @ U4 @ one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ one_one_real @ U4 ) @ A3 ) @ ( times_times_real @ U4 @ B4 ) ) @ one_one_real ) ) ) ) ) ).
% linepath_le_1
thf(fact_954_linepath__le__1,axiom,
! [A3: int,B4: int,U4: int] :
( ( ord_less_eq_int @ A3 @ one_one_int )
=> ( ( ord_less_eq_int @ B4 @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ U4 )
=> ( ( ord_less_eq_int @ U4 @ one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ one_one_int @ U4 ) @ A3 ) @ ( times_times_int @ U4 @ B4 ) ) @ one_one_int ) ) ) ) ) ).
% linepath_le_1
thf(fact_955_affine__ineq,axiom,
! [X2: real,V3: real,U4: real] :
( ( ord_less_eq_real @ X2 @ one_one_real )
=> ( ( ord_less_eq_real @ V3 @ U4 )
=> ( ord_less_eq_real @ ( plus_plus_real @ V3 @ ( times_times_real @ X2 @ U4 ) ) @ ( plus_plus_real @ U4 @ ( times_times_real @ X2 @ V3 ) ) ) ) ) ).
% affine_ineq
thf(fact_956_affine__ineq,axiom,
! [X2: int,V3: int,U4: int] :
( ( ord_less_eq_int @ X2 @ one_one_int )
=> ( ( ord_less_eq_int @ V3 @ U4 )
=> ( ord_less_eq_int @ ( plus_plus_int @ V3 @ ( times_times_int @ X2 @ U4 ) ) @ ( plus_plus_int @ U4 @ ( times_times_int @ X2 @ V3 ) ) ) ) ) ).
% affine_ineq
thf(fact_957_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_958_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_959_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_960_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_961_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_962_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_963_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_964_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_965_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_966_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_967_left__add__mult__distrib,axiom,
! [I: nat,U4: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U4 ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_968_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_969_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_970_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_971_sum__le__prod1,axiom,
! [A3: real,B4: real] :
( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ( ord_less_eq_real @ B4 @ one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ B4 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ A3 @ B4 ) ) ) ) ) ).
% sum_le_prod1
thf(fact_972_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_973_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_974_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_975_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_976_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 ) ) ).
% Matrix.right_add_zero_mat
thf(fact_977_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_978_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_979_choose__mult__lemma,axiom,
! [M: nat,R3: nat,K: nat] :
( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R3 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
= ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R3 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R3 ) @ M ) ) ) ).
% choose_mult_lemma
thf(fact_980_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_981_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_982_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_983_minus__add__minus__mat,axiom,
! [U4: mat_complex,Nr: nat,Nc: nat,V3: mat_complex,W: mat_complex] :
( ( member_mat_complex @ U4 @ ( 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 @ U4 @ ( plus_p8323303612493835998omplex @ V3 @ W ) )
= ( minus_2412168080157227406omplex @ ( minus_2412168080157227406omplex @ U4 @ V3 ) @ W ) ) ) ) ) ).
% minus_add_minus_mat
thf(fact_984_cpx__sq__mat_Omult__add__distrib__right,axiom,
! [DimR: nat,DimC: nat,Fc_mats: set_mat_complex,A: mat_complex,B: mat_complex,C: mat_complex] :
( ( linear7199532782703566157sq_mat @ DimR @ DimC @ Fc_mats )
=> ( ( member_mat_complex @ A @ Fc_mats )
=> ( ( member_mat_complex @ B @ Fc_mats )
=> ( ( member_mat_complex @ C @ Fc_mats )
=> ( ( times_8009071140041733218omplex @ A @ ( plus_p8323303612493835998omplex @ B @ C ) )
= ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ A @ B ) @ ( times_8009071140041733218omplex @ A @ C ) ) ) ) ) ) ) ).
% cpx_sq_mat.mult_add_distrib_right
thf(fact_985_cpx__sq__mat_Omult__add__distrib__left,axiom,
! [DimR: nat,DimC: nat,Fc_mats: set_mat_complex,A: mat_complex,B: mat_complex,C: mat_complex] :
( ( linear7199532782703566157sq_mat @ DimR @ DimC @ Fc_mats )
=> ( ( member_mat_complex @ A @ Fc_mats )
=> ( ( member_mat_complex @ B @ Fc_mats )
=> ( ( member_mat_complex @ C @ Fc_mats )
=> ( ( times_8009071140041733218omplex @ ( plus_p8323303612493835998omplex @ B @ C ) @ A )
= ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ B @ A ) @ ( times_8009071140041733218omplex @ C @ A ) ) ) ) ) ) ) ).
% cpx_sq_mat.mult_add_distrib_left
thf(fact_986_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_987_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_988_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_989_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_990_segment__bound__lemma,axiom,
! [B: real,X2: real,Y: real,U4: real] :
( ( ord_less_eq_real @ B @ X2 )
=> ( ( ord_less_eq_real @ B @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ U4 )
=> ( ( ord_less_eq_real @ U4 @ one_one_real )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ one_one_real @ U4 ) @ X2 ) @ ( times_times_real @ U4 @ Y ) ) ) ) ) ) ) ).
% segment_bound_lemma
thf(fact_991_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U4: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U4 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U4 ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_992_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U4: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U4 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U4 ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_993_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U4: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U4 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U4 ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_994_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U4: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U4 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U4 ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_995_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U4: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U4 ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U4 ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_996_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U4: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U4 ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U4 ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_997_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U4: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U4 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U4 ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_998_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U4: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U4 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U4 ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_999_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M6: nat,N2: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1000_split__div,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I2: nat,J2: nat] :
( ( ( ord_less_nat @ J2 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J2 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_div
thf(fact_1001_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_1002_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1003_pivot__bound,axiom,
! [A: mat_complex,Nr: nat,F: nat > nat,N: nat,I: nat,J: nat] :
( ( ( dim_row_complex @ A )
= Nr )
=> ( ( gauss_2609248829700396350omplex @ A @ F @ N )
=> ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ Nr )
=> ( ( ( F @ ( plus_plus_nat @ I @ J ) )
= N )
| ( ord_less_eq_nat @ ( plus_plus_nat @ J @ ( F @ I ) ) @ ( F @ ( plus_plus_nat @ I @ J ) ) ) ) ) ) ) ).
% pivot_bound
thf(fact_1004_square__bound__lemma,axiom,
! [X2: real] : ( ord_less_real @ X2 @ ( times_times_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( plus_plus_real @ one_one_real @ X2 ) ) ) ).
% square_bound_lemma
thf(fact_1005_add__scale__eq__noteq,axiom,
! [R3: complex,A3: complex,B4: complex,C2: complex,D2: complex] :
( ( R3 != zero_zero_complex )
=> ( ( ( A3 = B4 )
& ( C2 != D2 ) )
=> ( ( plus_plus_complex @ A3 @ ( times_times_complex @ R3 @ C2 ) )
!= ( plus_plus_complex @ B4 @ ( times_times_complex @ R3 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1006_add__scale__eq__noteq,axiom,
! [R3: real,A3: real,B4: real,C2: real,D2: real] :
( ( R3 != zero_zero_real )
=> ( ( ( A3 = B4 )
& ( C2 != D2 ) )
=> ( ( plus_plus_real @ A3 @ ( times_times_real @ R3 @ C2 ) )
!= ( plus_plus_real @ B4 @ ( times_times_real @ R3 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1007_add__scale__eq__noteq,axiom,
! [R3: nat,A3: nat,B4: nat,C2: nat,D2: nat] :
( ( R3 != zero_zero_nat )
=> ( ( ( A3 = B4 )
& ( C2 != D2 ) )
=> ( ( plus_plus_nat @ A3 @ ( times_times_nat @ R3 @ C2 ) )
!= ( plus_plus_nat @ B4 @ ( times_times_nat @ R3 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1008_add__scale__eq__noteq,axiom,
! [R3: int,A3: int,B4: int,C2: int,D2: int] :
( ( R3 != zero_zero_int )
=> ( ( ( A3 = B4 )
& ( C2 != D2 ) )
=> ( ( plus_plus_int @ A3 @ ( times_times_int @ R3 @ C2 ) )
!= ( plus_plus_int @ B4 @ ( times_times_int @ R3 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1009_real__binomial__eq__mult__binomial__Suc,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
= ( times_times_real @ ( divide_divide_real @ ( minus_minus_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ K ) ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( suc @ N ) @ K ) ) ) ) ) ).
% real_binomial_eq_mult_binomial_Suc
thf(fact_1010_of__nat__binomial__eq__mult__binomial__Suc,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
= ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ K ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N @ one_one_nat ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( suc @ N ) @ K ) ) ) ) ) ).
% of_nat_binomial_eq_mult_binomial_Suc
thf(fact_1011_of__nat__binomial__eq__mult__binomial__Suc,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
= ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ K ) ) @ ( semiri8010041392384452111omplex @ ( plus_plus_nat @ N @ one_one_nat ) ) ) @ ( semiri8010041392384452111omplex @ ( binomial @ ( suc @ N ) @ K ) ) ) ) ) ).
% of_nat_binomial_eq_mult_binomial_Suc
thf(fact_1012_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1013_unit__vecs__first_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [N3: nat] : ( P @ N3 @ zero_zero_nat )
=> ( ! [N3: nat,I3: nat] :
( ( P @ N3 @ I3 )
=> ( P @ N3 @ ( suc @ I3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% unit_vecs_first.induct
thf(fact_1014_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1015_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1016_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1017_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1018_times__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% times_nat.simps(2)
thf(fact_1019_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_1020_Suc__times__binomial__eq,axiom,
! [N: nat,K: nat] :
( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
= ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% Suc_times_binomial_eq
thf(fact_1021_Suc__times__binomial,axiom,
! [K: nat,N: nat] :
( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
= ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% Suc_times_binomial
thf(fact_1022_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1023_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1024_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1025_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1026_fact__Suc,axiom,
! [N: nat] :
( ( semiri2265585572941072030t_real @ ( suc @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% fact_Suc
thf(fact_1027_fact__Suc,axiom,
! [N: nat] :
( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% fact_Suc
thf(fact_1028_fact__Suc,axiom,
! [N: nat] :
( ( semiri5044797733671781792omplex @ ( suc @ N ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% fact_Suc
thf(fact_1029_fact__Suc,axiom,
! [N: nat] :
( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% fact_Suc
thf(fact_1030_Suc__times__binomial__add,axiom,
! [A3: nat,B4: nat] :
( ( times_times_nat @ ( suc @ A3 ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A3 @ B4 ) ) @ ( suc @ A3 ) ) )
= ( times_times_nat @ ( suc @ B4 ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A3 @ B4 ) ) @ A3 ) ) ) ).
% Suc_times_binomial_add
thf(fact_1031_binomial__Suc__Suc__eq__times,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% binomial_Suc_Suc_eq_times
thf(fact_1032_pivot__funD_I3_J,axiom,
! [A: mat_complex,Nr: nat,F: nat > nat,Nc: nat,I: nat] :
( ( ( dim_row_complex @ A )
= Nr )
=> ( ( gauss_2609248829700396350omplex @ A @ F @ Nc )
=> ( ( ord_less_nat @ I @ Nr )
=> ( ( ord_less_nat @ ( suc @ I ) @ Nr )
=> ( ( ord_less_nat @ ( F @ I ) @ ( F @ ( suc @ I ) ) )
| ( ( F @ ( suc @ I ) )
= Nc ) ) ) ) ) ) ).
% pivot_funD(3)
thf(fact_1033_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_1034_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_1035_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_1036_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_1037_div__nat__eqI,axiom,
! [N: nat,Q4: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
=> ( ( divide_divide_nat @ M @ N )
= Q4 ) ) ) ).
% div_nat_eqI
thf(fact_1038_binomial__absorption,axiom,
! [K: nat,N: nat] :
( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% binomial_absorption
thf(fact_1039_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_1040_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_1041_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_1042_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_1043_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q5: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q5 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q5 ) ) )
& ( P @ Q5 ) ) ) ) ).
% split_div'
thf(fact_1044_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_1045_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_1046_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_1047_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_1048_crossproduct__eq,axiom,
! [W: complex,Y: complex,X2: complex,Z: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ W @ Y ) @ ( times_times_complex @ X2 @ Z ) )
= ( plus_plus_complex @ ( times_times_complex @ W @ Z ) @ ( times_times_complex @ X2 @ Y ) ) )
= ( ( W = X2 )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_1049_crossproduct__eq,axiom,
! [W: real,Y: real,X2: real,Z: real] :
( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X2 @ Z ) )
= ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X2 @ Y ) ) )
= ( ( W = X2 )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_1050_crossproduct__eq,axiom,
! [W: nat,Y: nat,X2: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X2 @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X2 @ Y ) ) )
= ( ( W = X2 )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_1051_crossproduct__eq,axiom,
! [W: int,Y: int,X2: int,Z: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X2 @ Z ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X2 @ Y ) ) )
= ( ( W = X2 )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_1052_crossproduct__noteq,axiom,
! [A3: complex,B4: complex,C2: complex,D2: complex] :
( ( ( A3 != B4 )
& ( C2 != D2 ) )
= ( ( plus_plus_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B4 @ D2 ) )
!= ( plus_plus_complex @ ( times_times_complex @ A3 @ D2 ) @ ( times_times_complex @ B4 @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_1053_crossproduct__noteq,axiom,
! [A3: real,B4: real,C2: real,D2: real] :
( ( ( A3 != B4 )
& ( C2 != D2 ) )
= ( ( plus_plus_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B4 @ D2 ) )
!= ( plus_plus_real @ ( times_times_real @ A3 @ D2 ) @ ( times_times_real @ B4 @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_1054_crossproduct__noteq,axiom,
! [A3: nat,B4: nat,C2: nat,D2: nat] :
( ( ( A3 != B4 )
& ( C2 != D2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B4 @ D2 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A3 @ D2 ) @ ( times_times_nat @ B4 @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_1055_crossproduct__noteq,axiom,
! [A3: int,B4: int,C2: int,D2: int] :
( ( ( A3 != B4 )
& ( C2 != D2 ) )
= ( ( plus_plus_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B4 @ D2 ) )
!= ( plus_plus_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ B4 @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_1056_permutation__insert__expand,axiom,
( permut138581522262023397omplex
= ( ^ [I2: complex,J2: nat,P5: complex > nat,I4: complex] : ( if_nat @ ( ord_less_complex @ I4 @ I2 ) @ ( if_nat @ ( ord_less_nat @ ( P5 @ I4 ) @ J2 ) @ ( P5 @ I4 ) @ ( suc @ ( P5 @ I4 ) ) ) @ ( if_nat @ ( I4 = I2 ) @ J2 @ ( if_nat @ ( ord_less_nat @ ( P5 @ ( minus_minus_complex @ I4 @ one_one_complex ) ) @ J2 ) @ ( P5 @ ( minus_minus_complex @ I4 @ one_one_complex ) ) @ ( suc @ ( P5 @ ( minus_minus_complex @ I4 @ one_one_complex ) ) ) ) ) ) ) ) ).
% permutation_insert_expand
thf(fact_1057_permutation__insert__expand,axiom,
( permut3695043542826343943rt_nat
= ( ^ [I2: nat,J2: nat,P5: nat > nat,I4: nat] : ( if_nat @ ( ord_less_nat @ I4 @ I2 ) @ ( if_nat @ ( ord_less_nat @ ( P5 @ I4 ) @ J2 ) @ ( P5 @ I4 ) @ ( suc @ ( P5 @ I4 ) ) ) @ ( if_nat @ ( I4 = I2 ) @ J2 @ ( if_nat @ ( ord_less_nat @ ( P5 @ ( minus_minus_nat @ I4 @ one_one_nat ) ) @ J2 ) @ ( P5 @ ( minus_minus_nat @ I4 @ one_one_nat ) ) @ ( suc @ ( P5 @ ( minus_minus_nat @ I4 @ one_one_nat ) ) ) ) ) ) ) ) ).
% permutation_insert_expand
thf(fact_1058_permutation__insert__expand,axiom,
( permut4060954620988167523t_real
= ( ^ [I2: real,J2: nat,P5: real > nat,I4: real] : ( if_nat @ ( ord_less_real @ I4 @ I2 ) @ ( if_nat @ ( ord_less_nat @ ( P5 @ I4 ) @ J2 ) @ ( P5 @ I4 ) @ ( suc @ ( P5 @ I4 ) ) ) @ ( if_nat @ ( I4 = I2 ) @ J2 @ ( if_nat @ ( ord_less_nat @ ( P5 @ ( minus_minus_real @ I4 @ one_one_real ) ) @ J2 ) @ ( P5 @ ( minus_minus_real @ I4 @ one_one_real ) ) @ ( suc @ ( P5 @ ( minus_minus_real @ I4 @ one_one_real ) ) ) ) ) ) ) ) ).
% permutation_insert_expand
thf(fact_1059_permutation__insert__expand,axiom,
( permut3692553072317293667rt_int
= ( ^ [I2: int,J2: nat,P5: int > nat,I4: int] : ( if_nat @ ( ord_less_int @ I4 @ I2 ) @ ( if_nat @ ( ord_less_nat @ ( P5 @ I4 ) @ J2 ) @ ( P5 @ I4 ) @ ( suc @ ( P5 @ I4 ) ) ) @ ( if_nat @ ( I4 = I2 ) @ J2 @ ( if_nat @ ( ord_less_nat @ ( P5 @ ( minus_minus_int @ I4 @ one_one_int ) ) @ J2 ) @ ( P5 @ ( minus_minus_int @ I4 @ one_one_int ) ) @ ( suc @ ( P5 @ ( minus_minus_int @ I4 @ one_one_int ) ) ) ) ) ) ) ) ).
% permutation_insert_expand
thf(fact_1060_fact__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
= ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% fact_diff_Suc
thf(fact_1061_fact__cancel,axiom,
! [N: nat,C2: real] :
( ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ C2 ) @ ( semiri2265585572941072030t_real @ ( suc @ N ) ) )
= ( divide_divide_real @ C2 @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% fact_cancel
thf(fact_1062_fact__cancel,axiom,
! [N: nat,C2: complex] :
( ( divide1717551699836669952omplex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ C2 ) @ ( semiri5044797733671781792omplex @ ( suc @ N ) ) )
= ( divide1717551699836669952omplex @ C2 @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% fact_cancel
thf(fact_1063_delete__index__def,axiom,
( delete_index
= ( ^ [I2: nat,I4: nat] : ( if_nat @ ( ord_less_nat @ I4 @ I2 ) @ I4 @ ( minus_minus_nat @ I4 @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% delete_index_def
thf(fact_1064_permutation__delete__expand,axiom,
( permutation_delete
= ( ^ [P5: nat > nat,I2: nat,J2: nat] : ( if_nat @ ( ord_less_nat @ ( P5 @ ( if_nat @ ( ord_less_nat @ J2 @ I2 ) @ J2 @ ( suc @ J2 ) ) ) @ ( P5 @ I2 ) ) @ ( P5 @ ( if_nat @ ( ord_less_nat @ J2 @ I2 ) @ J2 @ ( suc @ J2 ) ) ) @ ( minus_minus_nat @ ( P5 @ ( if_nat @ ( ord_less_nat @ J2 @ I2 ) @ J2 @ ( suc @ J2 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% permutation_delete_expand
thf(fact_1065_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_1066_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_1067_det__addrow,axiom,
! [L: nat,N: nat,K: nat,A: mat_int,A3: int] :
( ( ord_less_nat @ L @ N )
=> ( ( K != L )
=> ( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( ( det_int @ ( gauss_8882552878057600758en_int @ plus_plus_int @ times_times_int @ A3 @ K @ L @ A ) )
= ( det_int @ A ) ) ) ) ) ).
% det_addrow
thf(fact_1068_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_1069_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_1070_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_1071_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_1072_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_1073_addrow__mat,axiom,
! [A: mat_nat,N: nat,Nc: nat,L: nat,A3: nat,K: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ Nc ) )
=> ( ( ord_less_nat @ L @ N )
=> ( ( gauss_8885043348566651034en_nat @ plus_plus_nat @ times_times_nat @ A3 @ K @ L @ A )
= ( times_times_mat_nat @ ( gauss_6496870380031412486at_nat @ N @ A3 @ K @ L ) @ A ) ) ) ) ).
% addrow_mat
thf(fact_1074_addrow__mat,axiom,
! [A: mat_int,N: nat,Nc: nat,L: nat,A3: int,K: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( ord_less_nat @ L @ N )
=> ( ( gauss_8882552878057600758en_int @ plus_plus_int @ times_times_int @ A3 @ K @ L @ A )
= ( times_times_mat_int @ ( gauss_6494379909522362210at_int @ N @ A3 @ K @ L ) @ A ) ) ) ) ).
% addrow_mat
thf(fact_1075_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_1076_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_1077_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_1078_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_1079_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_1080_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_1081_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_1082_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_1083_det__addrow__mat,axiom,
! [K: nat,L: nat,N: nat,A3: int] :
( ( K != L )
=> ( ( det_int @ ( gauss_6494379909522362210at_int @ N @ A3 @ K @ L ) )
= one_one_int ) ) ).
% det_addrow_mat
thf(fact_1084_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_1085_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_1086_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_1087_Linear__Algebra__Complements_Otrace__add,axiom,
! [A: mat_int,B: mat_int] :
( ( square_mat_int @ A )
=> ( ( square_mat_int @ B )
=> ( ( ( dim_row_int @ A )
= ( dim_row_int @ B ) )
=> ( ( complex_trace_int @ ( plus_plus_mat_int @ A @ B ) )
= ( plus_plus_int @ ( complex_trace_int @ A ) @ ( complex_trace_int @ B ) ) ) ) ) ) ).
% Linear_Algebra_Complements.trace_add
thf(fact_1088_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_1089_Re__linepath,axiom,
! [A3: real,B4: real,X2: real] :
( ( re @ ( path_l4128132617387358368omplex @ ( real_V4546457046886955230omplex @ A3 ) @ ( real_V4546457046886955230omplex @ B4 ) @ X2 ) )
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 ) @ A3 ) @ ( times_times_real @ X2 @ B4 ) ) ) ).
% Re_linepath
thf(fact_1090_cpx__sq__mat_Osquare__mats,axiom,
! [DimR: nat,DimC: nat,Fc_mats: set_mat_complex,A: mat_complex] :
( ( linear7199532782703566157sq_mat @ DimR @ DimC @ Fc_mats )
=> ( ( member_mat_complex @ A @ Fc_mats )
=> ( square_mat_complex @ A ) ) ) ).
% cpx_sq_mat.square_mats
thf(fact_1091_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_1092_square__mat_Osimps,axiom,
( square_mat_complex
= ( ^ [A2: mat_complex] :
( ( dim_col_complex @ A2 )
= ( dim_row_complex @ A2 ) ) ) ) ).
% square_mat.simps
thf(fact_1093_square__mat_Oelims_I1_J,axiom,
! [X2: mat_complex,Y: $o] :
( ( ( square_mat_complex @ X2 )
= Y )
=> ( Y
= ( ( dim_col_complex @ X2 )
= ( dim_row_complex @ X2 ) ) ) ) ).
% square_mat.elims(1)
thf(fact_1094_square__mat_Oelims_I2_J,axiom,
! [X2: mat_complex] :
( ( square_mat_complex @ X2 )
=> ( ( dim_col_complex @ X2 )
= ( dim_row_complex @ X2 ) ) ) ).
% square_mat.elims(2)
thf(fact_1095_square__mat_Oelims_I3_J,axiom,
! [X2: mat_complex] :
( ~ ( square_mat_complex @ X2 )
=> ( ( dim_col_complex @ X2 )
!= ( dim_row_complex @ X2 ) ) ) ).
% square_mat.elims(3)
thf(fact_1096_linepath__of__real,axiom,
! [A3: real,B4: real,X2: real] :
( ( path_l4128132617387358368omplex @ ( real_V4546457046886955230omplex @ A3 ) @ ( real_V4546457046886955230omplex @ B4 ) @ X2 )
= ( real_V4546457046886955230omplex @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 ) @ A3 ) @ ( times_times_real @ X2 @ B4 ) ) ) ) ).
% linepath_of_real
thf(fact_1097_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_1098_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_1099_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_1100_incr__lemma,axiom,
! [D2: int,Z: int,X2: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ord_less_int @ Z @ ( plus_plus_int @ X2 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Z ) ) @ one_one_int ) @ D2 ) ) ) ) ).
% incr_lemma
thf(fact_1101_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_1102_abs__mult__less,axiom,
! [A3: real,C2: real,B4: real,D2: real] :
( ( ord_less_real @ ( abs_abs_real @ A3 ) @ C2 )
=> ( ( ord_less_real @ ( abs_abs_real @ B4 ) @ D2 )
=> ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B4 ) ) @ ( times_times_real @ C2 @ D2 ) ) ) ) ).
% abs_mult_less
thf(fact_1103_abs__mult__less,axiom,
! [A3: int,C2: int,B4: int,D2: int] :
( ( ord_less_int @ ( abs_abs_int @ A3 ) @ C2 )
=> ( ( ord_less_int @ ( abs_abs_int @ B4 ) @ D2 )
=> ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B4 ) ) @ ( times_times_int @ C2 @ D2 ) ) ) ) ).
% abs_mult_less
thf(fact_1104_abs__mult__self__eq,axiom,
! [A3: real] :
( ( times_times_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ A3 ) )
= ( times_times_real @ A3 @ A3 ) ) ).
% abs_mult_self_eq
thf(fact_1105_abs__mult__self__eq,axiom,
! [A3: int] :
( ( times_times_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ A3 ) )
= ( times_times_int @ A3 @ A3 ) ) ).
% abs_mult_self_eq
thf(fact_1106_abs__mult,axiom,
! [A3: complex,B4: complex] :
( ( abs_abs_complex @ ( times_times_complex @ A3 @ B4 ) )
= ( times_times_complex @ ( abs_abs_complex @ A3 ) @ ( abs_abs_complex @ B4 ) ) ) ).
% abs_mult
thf(fact_1107_abs__mult,axiom,
! [A3: real,B4: real] :
( ( abs_abs_real @ ( times_times_real @ A3 @ B4 ) )
= ( times_times_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B4 ) ) ) ).
% abs_mult
thf(fact_1108_abs__mult,axiom,
! [A3: int,B4: int] :
( ( abs_abs_int @ ( times_times_int @ A3 @ B4 ) )
= ( times_times_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B4 ) ) ) ).
% abs_mult
thf(fact_1109_abs__mult__pos_H,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( times_times_real @ X2 @ ( abs_abs_real @ Y ) )
= ( abs_abs_real @ ( times_times_real @ X2 @ Y ) ) ) ) ).
% abs_mult_pos'
thf(fact_1110_abs__mult__pos_H,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( times_times_int @ X2 @ ( abs_abs_int @ Y ) )
= ( abs_abs_int @ ( times_times_int @ X2 @ Y ) ) ) ) ).
% abs_mult_pos'
thf(fact_1111_abs__eq__mult,axiom,
! [A3: real,B4: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
| ( ord_less_eq_real @ A3 @ zero_zero_real ) )
& ( ( ord_less_eq_real @ zero_zero_real @ B4 )
| ( ord_less_eq_real @ B4 @ zero_zero_real ) ) )
=> ( ( abs_abs_real @ ( times_times_real @ A3 @ B4 ) )
= ( times_times_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B4 ) ) ) ) ).
% abs_eq_mult
thf(fact_1112_abs__eq__mult,axiom,
! [A3: int,B4: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
| ( ord_less_eq_int @ A3 @ zero_zero_int ) )
& ( ( ord_less_eq_int @ zero_zero_int @ B4 )
| ( ord_less_eq_int @ B4 @ zero_zero_int ) ) )
=> ( ( abs_abs_int @ ( times_times_int @ A3 @ B4 ) )
= ( times_times_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B4 ) ) ) ) ).
% abs_eq_mult
thf(fact_1113_abs__mult__pos,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( times_times_real @ ( abs_abs_real @ Y ) @ X2 )
= ( abs_abs_real @ ( times_times_real @ Y @ X2 ) ) ) ) ).
% abs_mult_pos
thf(fact_1114_abs__mult__pos,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( times_times_int @ ( abs_abs_int @ Y ) @ X2 )
= ( abs_abs_int @ ( times_times_int @ Y @ X2 ) ) ) ) ).
% abs_mult_pos
thf(fact_1115_decr__lemma,axiom,
! [D2: int,X2: int,Z: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ord_less_int @ ( minus_minus_int @ X2 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Z ) ) @ one_one_int ) @ D2 ) ) @ Z ) ) ).
% decr_lemma
thf(fact_1116_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_1117_cpmi,axiom,
! [D: int,P: int > $o,P2: int > $o,B: set_int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B )
=> ( X3
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ( ? [X5: int] : ( P @ X5 ) )
= ( ? [X: int] :
( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
& ( P2 @ X ) )
| ? [X: int] :
( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
& ? [Y3: int] :
( ( member_int @ Y3 @ B )
& ( P @ ( plus_plus_int @ Y3 @ X ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_1118_more__arith__simps_I8_J,axiom,
! [A3: complex,B4: complex] :
( ( times_times_complex @ A3 @ ( uminus1482373934393186551omplex @ B4 ) )
= ( uminus1482373934393186551omplex @ ( times_times_complex @ A3 @ B4 ) ) ) ).
% more_arith_simps(8)
thf(fact_1119_more__arith__simps_I8_J,axiom,
! [A3: real,B4: real] :
( ( times_times_real @ A3 @ ( uminus_uminus_real @ B4 ) )
= ( uminus_uminus_real @ ( times_times_real @ A3 @ B4 ) ) ) ).
% more_arith_simps(8)
thf(fact_1120_more__arith__simps_I8_J,axiom,
! [A3: int,B4: int] :
( ( times_times_int @ A3 @ ( uminus_uminus_int @ B4 ) )
= ( uminus_uminus_int @ ( times_times_int @ A3 @ B4 ) ) ) ).
% more_arith_simps(8)
thf(fact_1121_more__arith__simps_I7_J,axiom,
! [A3: complex,B4: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ A3 ) @ B4 )
= ( uminus1482373934393186551omplex @ ( times_times_complex @ A3 @ B4 ) ) ) ).
% more_arith_simps(7)
thf(fact_1122_more__arith__simps_I7_J,axiom,
! [A3: real,B4: real] :
( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ B4 )
= ( uminus_uminus_real @ ( times_times_real @ A3 @ B4 ) ) ) ).
% more_arith_simps(7)
thf(fact_1123_more__arith__simps_I7_J,axiom,
! [A3: int,B4: int] :
( ( times_times_int @ ( uminus_uminus_int @ A3 ) @ B4 )
= ( uminus_uminus_int @ ( times_times_int @ A3 @ B4 ) ) ) ).
% more_arith_simps(7)
thf(fact_1124_vector__space__over__itself_Oscale__minus__left,axiom,
! [A3: complex,X2: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ A3 ) @ X2 )
= ( uminus1482373934393186551omplex @ ( times_times_complex @ A3 @ X2 ) ) ) ).
% vector_space_over_itself.scale_minus_left
thf(fact_1125_vector__space__over__itself_Oscale__minus__left,axiom,
! [A3: real,X2: real] :
( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ X2 )
= ( uminus_uminus_real @ ( times_times_real @ A3 @ X2 ) ) ) ).
% vector_space_over_itself.scale_minus_left
thf(fact_1126_vector__space__over__itself_Oscale__minus__right,axiom,
! [A3: complex,X2: complex] :
( ( times_times_complex @ A3 @ ( uminus1482373934393186551omplex @ X2 ) )
= ( uminus1482373934393186551omplex @ ( times_times_complex @ A3 @ X2 ) ) ) ).
% vector_space_over_itself.scale_minus_right
thf(fact_1127_vector__space__over__itself_Oscale__minus__right,axiom,
! [A3: real,X2: real] :
( ( times_times_real @ A3 @ ( uminus_uminus_real @ X2 ) )
= ( uminus_uminus_real @ ( times_times_real @ A3 @ X2 ) ) ) ).
% vector_space_over_itself.scale_minus_right
thf(fact_1128_square__eq__iff,axiom,
! [A3: complex,B4: complex] :
( ( ( times_times_complex @ A3 @ A3 )
= ( times_times_complex @ B4 @ B4 ) )
= ( ( A3 = B4 )
| ( A3
= ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% square_eq_iff
thf(fact_1129_square__eq__iff,axiom,
! [A3: real,B4: real] :
( ( ( times_times_real @ A3 @ A3 )
= ( times_times_real @ B4 @ B4 ) )
= ( ( A3 = B4 )
| ( A3
= ( uminus_uminus_real @ B4 ) ) ) ) ).
% square_eq_iff
thf(fact_1130_square__eq__iff,axiom,
! [A3: int,B4: int] :
( ( ( times_times_int @ A3 @ A3 )
= ( times_times_int @ B4 @ B4 ) )
= ( ( A3 = B4 )
| ( A3
= ( uminus_uminus_int @ B4 ) ) ) ) ).
% square_eq_iff
thf(fact_1131_minus__mult__minus,axiom,
! [A3: complex,B4: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ A3 ) @ ( uminus1482373934393186551omplex @ B4 ) )
= ( times_times_complex @ A3 @ B4 ) ) ).
% minus_mult_minus
thf(fact_1132_minus__mult__minus,axiom,
! [A3: real,B4: real] :
( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B4 ) )
= ( times_times_real @ A3 @ B4 ) ) ).
% minus_mult_minus
thf(fact_1133_minus__mult__minus,axiom,
! [A3: int,B4: int] :
( ( times_times_int @ ( uminus_uminus_int @ A3 ) @ ( uminus_uminus_int @ B4 ) )
= ( times_times_int @ A3 @ B4 ) ) ).
% minus_mult_minus
thf(fact_1134_minus__mult__commute,axiom,
! [A3: complex,B4: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ A3 ) @ B4 )
= ( times_times_complex @ A3 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ).
% minus_mult_commute
thf(fact_1135_minus__mult__commute,axiom,
! [A3: real,B4: real] :
( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ B4 )
= ( times_times_real @ A3 @ ( uminus_uminus_real @ B4 ) ) ) ).
% minus_mult_commute
thf(fact_1136_minus__mult__commute,axiom,
! [A3: int,B4: int] :
( ( times_times_int @ ( uminus_uminus_int @ A3 ) @ B4 )
= ( times_times_int @ A3 @ ( uminus_uminus_int @ B4 ) ) ) ).
% minus_mult_commute
thf(fact_1137_square__eq__1__iff,axiom,
! [X2: complex] :
( ( ( times_times_complex @ X2 @ X2 )
= one_one_complex )
= ( ( X2 = one_one_complex )
| ( X2
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% square_eq_1_iff
thf(fact_1138_square__eq__1__iff,axiom,
! [X2: real] :
( ( ( times_times_real @ X2 @ X2 )
= one_one_real )
= ( ( X2 = one_one_real )
| ( X2
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% square_eq_1_iff
thf(fact_1139_square__eq__1__iff,axiom,
! [X2: int] :
( ( ( times_times_int @ X2 @ X2 )
= one_one_int )
= ( ( X2 = one_one_int )
| ( X2
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% square_eq_1_iff
thf(fact_1140_mult__minus1__right,axiom,
! [Z: complex] :
( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ Z ) ) ).
% mult_minus1_right
thf(fact_1141_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_1142_mult__minus1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1_right
thf(fact_1143_mult__minus1,axiom,
! [Z: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
= ( uminus1482373934393186551omplex @ Z ) ) ).
% mult_minus1
thf(fact_1144_mult__minus1,axiom,
! [Z: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1
thf(fact_1145_mult__minus1,axiom,
! [Z: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1
thf(fact_1146_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_1147_nonzero__neg__divide__eq__eq2,axiom,
! [B4: complex,C2: complex,A3: complex] :
( ( B4 != zero_zero_complex )
=> ( ( C2
= ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ B4 ) ) )
= ( ( times_times_complex @ C2 @ B4 )
= ( uminus1482373934393186551omplex @ A3 ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_1148_nonzero__neg__divide__eq__eq2,axiom,
! [B4: real,C2: real,A3: real] :
( ( B4 != zero_zero_real )
=> ( ( C2
= ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B4 ) ) )
= ( ( times_times_real @ C2 @ B4 )
= ( uminus_uminus_real @ A3 ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_1149_nonzero__neg__divide__eq__eq,axiom,
! [B4: complex,A3: complex,C2: complex] :
( ( B4 != zero_zero_complex )
=> ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ B4 ) )
= C2 )
= ( ( uminus1482373934393186551omplex @ A3 )
= ( times_times_complex @ C2 @ B4 ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_1150_nonzero__neg__divide__eq__eq,axiom,
! [B4: real,A3: real,C2: real] :
( ( B4 != zero_zero_real )
=> ( ( ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B4 ) )
= C2 )
= ( ( uminus_uminus_real @ A3 )
= ( times_times_real @ C2 @ B4 ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_1151_minus__divide__eq__eq,axiom,
! [B4: complex,C2: complex,A3: complex] :
( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B4 @ C2 ) )
= A3 )
= ( ( ( C2 != zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ B4 )
= ( times_times_complex @ A3 @ C2 ) ) )
& ( ( C2 = zero_zero_complex )
=> ( A3 = zero_zero_complex ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_1152_minus__divide__eq__eq,axiom,
! [B4: real,C2: real,A3: real] :
( ( ( uminus_uminus_real @ ( divide_divide_real @ B4 @ C2 ) )
= A3 )
= ( ( ( C2 != zero_zero_real )
=> ( ( uminus_uminus_real @ B4 )
= ( times_times_real @ A3 @ C2 ) ) )
& ( ( C2 = zero_zero_real )
=> ( A3 = zero_zero_real ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_1153_eq__minus__divide__eq,axiom,
! [A3: complex,B4: complex,C2: complex] :
( ( A3
= ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B4 @ C2 ) ) )
= ( ( ( C2 != zero_zero_complex )
=> ( ( times_times_complex @ A3 @ C2 )
= ( uminus1482373934393186551omplex @ B4 ) ) )
& ( ( C2 = zero_zero_complex )
=> ( A3 = zero_zero_complex ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_1154_eq__minus__divide__eq,axiom,
! [A3: real,B4: real,C2: real] :
( ( A3
= ( uminus_uminus_real @ ( divide_divide_real @ B4 @ C2 ) ) )
= ( ( ( C2 != zero_zero_real )
=> ( ( times_times_real @ A3 @ C2 )
= ( uminus_uminus_real @ B4 ) ) )
& ( ( C2 = zero_zero_real )
=> ( A3 = zero_zero_real ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_1155_pos__minus__divide__less__eq,axiom,
! [C2: real,B4: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B4 @ C2 ) ) @ A3 )
= ( ord_less_real @ ( uminus_uminus_real @ B4 ) @ ( times_times_real @ A3 @ C2 ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_1156_pos__less__minus__divide__eq,axiom,
! [C2: real,A3: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ A3 @ ( uminus_uminus_real @ ( divide_divide_real @ B4 @ C2 ) ) )
= ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_1157_neg__minus__divide__less__eq,axiom,
! [C2: real,B4: real,A3: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B4 @ C2 ) ) @ A3 )
= ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_1158_neg__less__minus__divide__eq,axiom,
! [C2: real,A3: real,B4: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ A3 @ ( uminus_uminus_real @ ( divide_divide_real @ B4 @ C2 ) ) )
= ( ord_less_real @ ( uminus_uminus_real @ B4 ) @ ( times_times_real @ A3 @ C2 ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_1159_minus__divide__less__eq,axiom,
! [B4: real,C2: real,A3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B4 @ C2 ) ) @ A3 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( uminus_uminus_real @ B4 ) @ ( 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 @ B4 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A3 ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_1160_less__minus__divide__eq,axiom,
! [A3: real,B4: real,C2: real] :
( ( ord_less_real @ A3 @ ( uminus_uminus_real @ ( divide_divide_real @ B4 @ C2 ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( uminus_uminus_real @ B4 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( uminus_uminus_real @ B4 ) @ ( 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_1161_minus__divide__add__eq__iff,axiom,
! [Z: complex,X2: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X2 @ Z ) ) @ Y )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_1162_minus__divide__add__eq__iff,axiom,
! [Z: real,X2: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X2 @ Z ) ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_1163_add__divide__eq__if__simps_I3_J,axiom,
! [Z: complex,A3: complex,B4: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ Z ) ) @ B4 )
= B4 ) )
& ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ Z ) ) @ B4 )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A3 ) @ ( times_times_complex @ B4 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_1164_add__divide__eq__if__simps_I3_J,axiom,
! [Z: real,A3: real,B4: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A3 @ Z ) ) @ B4 )
= B4 ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A3 @ Z ) ) @ B4 )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ ( times_times_real @ B4 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_1165_add__divide__eq__if__simps_I6_J,axiom,
! [Z: complex,A3: complex,B4: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ Z ) ) @ B4 )
= ( uminus1482373934393186551omplex @ B4 ) ) )
& ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ Z ) ) @ B4 )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A3 ) @ ( times_times_complex @ B4 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_1166_add__divide__eq__if__simps_I6_J,axiom,
! [Z: real,A3: real,B4: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A3 @ Z ) ) @ B4 )
= ( uminus_uminus_real @ B4 ) ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A3 @ Z ) ) @ B4 )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A3 ) @ ( times_times_real @ B4 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_1167_add__divide__eq__if__simps_I5_J,axiom,
! [Z: complex,A3: complex,B4: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A3 @ Z ) @ B4 )
= ( uminus1482373934393186551omplex @ B4 ) ) )
& ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A3 @ Z ) @ B4 )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ A3 @ ( times_times_complex @ B4 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_1168_add__divide__eq__if__simps_I5_J,axiom,
! [Z: real,A3: real,B4: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A3 @ Z ) @ B4 )
= ( uminus_uminus_real @ B4 ) ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A3 @ Z ) @ B4 )
= ( divide_divide_real @ ( minus_minus_real @ A3 @ ( times_times_real @ B4 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_1169_square__continuous,axiom,
! [E: real,X2: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [Y2: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ Y2 @ X2 ) ) @ D4 )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( times_times_real @ Y2 @ Y2 ) @ ( times_times_real @ X2 @ X2 ) ) ) @ E ) ) ) ) ).
% square_continuous
thf(fact_1170_periodic__finite__ex,axiom,
! [D2: int,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D2 ) ) ) )
=> ( ( ? [X5: int] : ( P @ X5 ) )
= ( ? [X: int] :
( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
& ( P @ X ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_1171_cppi,axiom,
! [D: int,P: int > $o,P2: int > $o,A: set_int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A )
=> ( X3
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ( ? [X5: int] : ( P @ X5 ) )
= ( ? [X: int] :
( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
& ( P2 @ X ) )
| ? [X: int] :
( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
& ? [Y3: int] :
( ( member_int @ Y3 @ A )
& ( P @ ( minus_minus_int @ Y3 @ X ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_1172_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_1173_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1174_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1175_real__minus__mult__self__le,axiom,
! [U4: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U4 @ U4 ) ) @ ( times_times_real @ X2 @ X2 ) ) ).
% real_minus_mult_self_le
thf(fact_1176_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_1177_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_1178_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_1179_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_1180_Bernoulli__inequality,axiom,
! [X2: real,N: nat] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X2 ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_1181_linear__plus__1__le__power,axiom,
! [X2: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X2 @ one_one_real ) @ N ) ) ) ).
% linear_plus_1_le_power
thf(fact_1182_Bernstein__def,axiom,
( weiers7429072931691461095nstein
= ( ^ [N2: nat,K4: nat,X: real] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K4 ) ) @ ( power_power_real @ X @ K4 ) ) @ ( power_power_real @ ( minus_minus_real @ one_one_real @ X ) @ ( minus_minus_nat @ N2 @ K4 ) ) ) ) ) ).
% Bernstein_def
thf(fact_1183_spectrum__eigenvalues,axiom,
! [A: mat_complex,N: nat,K: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_complex @ K @ ( projec527831343749723810omplex @ A ) )
=> ( char_e7032225803028799586omplex @ A @ K ) ) ) ).
% spectrum_eigenvalues
thf(fact_1184_reduce__poly__simple,axiom,
! [B4: complex,N: nat] :
( ( B4 != zero_zero_complex )
=> ( ( N != zero_zero_nat )
=> ? [Z2: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ B4 @ ( power_power_complex @ Z2 @ N ) ) ) ) @ one_one_real ) ) ) ).
% reduce_poly_simple
thf(fact_1185_unitary__eigenvalues__norm__square,axiom,
! [U: mat_complex,N: nat,K: complex] :
( ( comple6660659447773130958omplex @ U )
=> ( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( char_e7032225803028799586omplex @ U @ K )
=> ( ( times_times_complex @ ( conjug1878831970375765195omplex @ K ) @ K )
= one_one_complex ) ) ) ) ).
% unitary_eigenvalues_norm_square
thf(fact_1186_Re__complex__div__lemma,axiom,
! [Z: complex] :
( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ imaginary_unit @ Z ) ) ) ) )
= ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real ) ) ).
% Re_complex_div_lemma
thf(fact_1187_complex__mat__decomposition__to__hermitian,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ? [B3: mat_complex,C4: mat_complex] :
( ( comple8306762464034002205omplex @ B3 )
& ( comple8306762464034002205omplex @ C4 )
& ( A
= ( plus_p8323303612493835998omplex @ B3 @ ( smult_mat_complex @ imaginary_unit @ C4 ) ) )
& ( member_mat_complex @ B3 @ ( carrier_mat_complex @ N @ N ) )
& ( member_mat_complex @ C4 @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% complex_mat_decomposition_to_hermitian
thf(fact_1188_divide__i,axiom,
! [X2: complex] :
( ( divide1717551699836669952omplex @ X2 @ imaginary_unit )
= ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X2 ) ) ).
% divide_i
thf(fact_1189_i__times__eq__iff,axiom,
! [W: complex,Z: complex] :
( ( ( times_times_complex @ imaginary_unit @ W )
= Z )
= ( W
= ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) ) ) ).
% i_times_eq_iff
thf(fact_1190_complex__i__mult__minus,axiom,
! [X2: complex] :
( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X2 ) )
= ( uminus1482373934393186551omplex @ X2 ) ) ).
% complex_i_mult_minus
thf(fact_1191_i__squared,axiom,
( ( times_times_complex @ imaginary_unit @ imaginary_unit )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% i_squared
thf(fact_1192_Im__complex__div__lemma,axiom,
! [Z: complex] :
( ( ( im @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ imaginary_unit @ Z ) ) ) )
= zero_zero_real )
= ( ( re @ Z )
= zero_zero_real ) ) ).
% Im_complex_div_lemma
thf(fact_1193_Im__i__times,axiom,
! [Z: complex] :
( ( im @ ( times_times_complex @ imaginary_unit @ Z ) )
= ( re @ Z ) ) ).
% Im_i_times
thf(fact_1194_times__complex_Osel_I1_J,axiom,
! [X2: complex,Y: complex] :
( ( re @ ( times_times_complex @ X2 @ Y ) )
= ( minus_minus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y ) ) ) ) ).
% times_complex.sel(1)
thf(fact_1195_times__complex_Osel_I2_J,axiom,
! [X2: complex,Y: complex] :
( ( im @ ( times_times_complex @ X2 @ Y ) )
= ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y ) ) ) ) ).
% times_complex.sel(2)
thf(fact_1196_Re__i__times,axiom,
! [Z: complex] :
( ( re @ ( times_times_complex @ imaginary_unit @ Z ) )
= ( uminus_uminus_real @ ( im @ Z ) ) ) ).
% Re_i_times
thf(fact_1197_complex__axes,axiom,
! [Y: complex,X2: complex,Z: complex] :
( ( ( im @ ( divide1717551699836669952omplex @ Y @ X2 ) )
!= zero_zero_real )
=> ~ ! [A6: real,B7: real] :
( Z
!= ( plus_plus_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ A6 ) @ X2 ) @ ( times_times_complex @ ( real_V4546457046886955230omplex @ B7 ) @ Y ) ) ) ) ).
% complex_axes
thf(fact_1198_complex__eq,axiom,
! [A3: complex] :
( A3
= ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A3 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A3 ) ) ) ) ) ).
% complex_eq
thf(fact_1199_csqrt__minus,axiom,
! [X2: complex] :
( ( ( ord_less_real @ ( im @ X2 ) @ zero_zero_real )
| ( ( ( im @ X2 )
= zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ ( re @ X2 ) ) ) )
=> ( ( csqrt @ ( uminus1482373934393186551omplex @ X2 ) )
= ( times_times_complex @ imaginary_unit @ ( csqrt @ X2 ) ) ) ) ).
% csqrt_minus
thf(fact_1200_csqrt__of__real__nonpos,axiom,
! [X2: complex] :
( ( ( im @ X2 )
= zero_zero_real )
=> ( ( ord_less_eq_real @ ( re @ X2 ) @ zero_zero_real )
=> ( ( csqrt @ X2 )
= ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X2 ) ) ) ) ) ) ) ) ).
% csqrt_of_real_nonpos
thf(fact_1201_le__real__sqrt__sumsq,axiom,
! [X2: real,Y: real] : ( ord_less_eq_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) ) ) ) ).
% le_real_sqrt_sumsq
thf(fact_1202_real__sqrt__abs2,axiom,
! [X2: real] :
( ( sqrt @ ( times_times_real @ X2 @ X2 ) )
= ( abs_abs_real @ X2 ) ) ).
% real_sqrt_abs2
thf(fact_1203_real__sqrt__mult__self,axiom,
! [A3: real] :
( ( times_times_real @ ( sqrt @ A3 ) @ ( sqrt @ A3 ) )
= ( abs_abs_real @ A3 ) ) ).
% real_sqrt_mult_self
thf(fact_1204_real__sqrt__mult,axiom,
! [X2: real,Y: real] :
( ( sqrt @ ( times_times_real @ X2 @ Y ) )
= ( times_times_real @ ( sqrt @ X2 ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_mult
thf(fact_1205_fps__demoivre,axiom,
! [A3: complex,N: nat] :
( ( power_8487976900264310848omplex @ ( plus_p8472957120637115327omplex @ ( formal7592904152049726030omplex @ A3 ) @ ( times_1444617028055533883omplex @ ( formal7822294191640021514omplex @ imaginary_unit ) @ ( formal6336729282288812767omplex @ A3 ) ) ) @ N )
= ( plus_p8472957120637115327omplex @ ( formal7592904152049726030omplex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ A3 ) ) @ ( times_1444617028055533883omplex @ ( formal7822294191640021514omplex @ imaginary_unit ) @ ( formal6336729282288812767omplex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ A3 ) ) ) ) ) ).
% fps_demoivre
thf(fact_1206_fps__exp__ii__sin__cos,axiom,
! [C2: complex] :
( ( formal5488582694110793604omplex @ ( times_times_complex @ imaginary_unit @ C2 ) )
= ( plus_p8472957120637115327omplex @ ( formal7592904152049726030omplex @ C2 ) @ ( times_1444617028055533883omplex @ ( formal7822294191640021514omplex @ imaginary_unit ) @ ( formal6336729282288812767omplex @ C2 ) ) ) ) ).
% fps_exp_ii_sin_cos
thf(fact_1207_fps__exp__minus__ii__sin__cos,axiom,
! [C2: complex] :
( ( formal5488582694110793604omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ C2 ) ) )
= ( minus_1072911313905636623omplex @ ( formal7592904152049726030omplex @ C2 ) @ ( times_1444617028055533883omplex @ ( formal7822294191640021514omplex @ imaginary_unit ) @ ( formal6336729282288812767omplex @ C2 ) ) ) ) ).
% fps_exp_minus_ii_sin_cos
thf(fact_1208_fps__tan__fps__exp__ii,axiom,
( formal6482914284900457064omplex
= ( ^ [C5: complex] : ( divide1348722040316500488omplex @ ( minus_1072911313905636623omplex @ ( formal5488582694110793604omplex @ ( times_times_complex @ imaginary_unit @ C5 ) ) @ ( formal5488582694110793604omplex @ ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ C5 ) ) ) @ ( times_1444617028055533883omplex @ ( formal7822294191640021514omplex @ imaginary_unit ) @ ( plus_p8472957120637115327omplex @ ( formal5488582694110793604omplex @ ( times_times_complex @ imaginary_unit @ C5 ) ) @ ( formal5488582694110793604omplex @ ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ C5 ) ) ) ) ) ) ) ).
% fps_tan_fps_exp_ii
% Helper facts (11)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X2: complex,Y: complex] :
( ( if_complex @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X2: complex,Y: complex] :
( ( if_complex @ $true @ X2 @ Y )
= X2 ) ).
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,
! [X2: mat_complex,Y: mat_complex] :
( ( if_mat_complex @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Matrix__Omat_It__Complex__Ocomplex_J_T,axiom,
! [X2: mat_complex,Y: mat_complex] :
( ( if_mat_complex @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
spectr5409772854192057952omplex @ a @ b @ u ).
%------------------------------------------------------------------------------