TPTP Problem File: SLH0229^1.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_01923_072834__19330950_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1258 ( 477 unt; 145 typ; 0 def)
% Number of atoms : 3345 (1158 equ; 0 cnn)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 11832 ( 246 ~; 38 |; 121 &;9685 @)
% ( 0 <=>;1742 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 15 ( 14 usr)
% Number of type conns : 632 ( 632 >; 0 *; 0 +; 0 <<)
% Number of symbols : 134 ( 131 usr; 20 con; 0-5 aty)
% Number of variables : 3341 ( 77 ^;3221 !; 43 ?;3341 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:36:16.050
%------------------------------------------------------------------------------
% Could-be-implicit typings (14)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
set_set_mat_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
set_mat_complex: $tType ).
thf(ty_n_t__Matrix__Omat_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
mat_set_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
set_set_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Nat__Onat_J_J,type,
set_mat_nat: $tType ).
thf(ty_n_t__Matrix__Omat_It__Set__Oset_It__Nat__Onat_J_J,type,
mat_set_nat: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
vec_complex: $tType ).
thf(ty_n_t__Matrix__Omat_It__Complex__Ocomplex_J,type,
mat_complex: $tType ).
thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
list_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Matrix__Omat_It__Nat__Onat_J,type,
mat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (131)
thf(sy_c_Complex__Matrix_Ohermitian_001t__Complex__Ocomplex,type,
comple8306762464034002205omplex: mat_complex > $o ).
thf(sy_c_Complex__Matrix_Oouter__prod_001t__Complex__Ocomplex,type,
comple3482886691669134300omplex: vec_complex > vec_complex > mat_complex ).
thf(sy_c_Complex__Matrix_Opositive,type,
complex_positive: mat_complex > $o ).
thf(sy_c_Complex__Matrix_Ounitary_001t__Complex__Ocomplex,type,
comple6660659447773130958omplex: mat_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
finite3207457112153483333omplex: set_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
finite7047982916621727056omplex: set_mat_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Matrix__Omat_It__Nat__Onat_J,type,
finite9067146157826639218at_nat: set_mat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Complex__Ocomplex_J,type,
finite6551019134538273531omplex: set_set_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
finite1349200545324696496omplex: set_set_mat_complex > $o ).
thf(sy_c_Group__On__With_Oab__group__add__on__with_001t__Complex__Ocomplex,type,
group_3842241455253984271omplex: set_complex > ( complex > complex > complex ) > complex > ( complex > complex > complex ) > ( complex > complex ) > $o ).
thf(sy_c_Group__On__With_Oab__group__add__on__with_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
group_7083379013581629702omplex: set_mat_complex > ( mat_complex > mat_complex > mat_complex ) > mat_complex > ( mat_complex > mat_complex > mat_complex ) > ( mat_complex > mat_complex ) > $o ).
thf(sy_c_Group__On__With_Oab__semigroup__add__on__with_001t__Complex__Ocomplex,type,
group_4393537770075021309omplex: set_complex > ( complex > complex > complex ) > $o ).
thf(sy_c_Group__On__With_Oab__semigroup__add__on__with_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
group_8296285978151397272omplex: set_mat_complex > ( mat_complex > mat_complex > mat_complex ) > $o ).
thf(sy_c_Group__On__With_Ocomm__monoid__add__on__with_001t__Complex__Ocomplex,type,
group_2796108508354279923omplex: set_complex > ( complex > complex > complex ) > complex > $o ).
thf(sy_c_Group__On__With_Ocomm__monoid__add__on__with_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
group_5394922976599784994omplex: set_mat_complex > ( mat_complex > mat_complex > mat_complex ) > mat_complex > $o ).
thf(sy_c_Group__On__With_Osemigroup__add__on__with_001t__Complex__Ocomplex,type,
group_6437017389065004156omplex: set_complex > ( complex > complex > complex ) > $o ).
thf(sy_c_Group__On__With_Osemigroup__add__on__with_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
group_6724803037680873049omplex: set_mat_complex > ( mat_complex > mat_complex > mat_complex ) > $o ).
thf(sy_c_Group__On__With_Osum__with_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
group_4775205164212326935omplex: ( complex > complex > complex ) > complex > ( complex > complex ) > set_complex > complex ).
thf(sy_c_Group__On__With_Osum__with_001t__Complex__Ocomplex_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
group_6233491804913373438omplex: ( complex > complex > complex ) > complex > ( mat_complex > complex ) > set_mat_complex > complex ).
thf(sy_c_Group__On__With_Osum__with_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Complex__Ocomplex,type,
group_4310869077125956188omplex: ( mat_complex > mat_complex > mat_complex ) > mat_complex > ( complex > mat_complex ) > set_complex > mat_complex ).
thf(sy_c_Group__On__With_Osum__with_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
group_1588376139278055545omplex: ( mat_complex > mat_complex > mat_complex ) > mat_complex > ( mat_complex > mat_complex ) > set_mat_complex > mat_complex ).
thf(sy_c_Group__On__With_Osum__with_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Nat__Onat,type,
group_3997526426246263166ex_nat: ( mat_complex > mat_complex > mat_complex ) > mat_complex > ( nat > mat_complex ) > set_nat > mat_complex ).
thf(sy_c_Group__On__With_Osum__with_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
group_4914184569911763673omplex: ( mat_complex > mat_complex > mat_complex ) > mat_complex > ( set_mat_complex > mat_complex ) > set_set_mat_complex > mat_complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
minus_minus_complex: complex > complex > complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
minus_2412168080157227406omplex: mat_complex > mat_complex > mat_complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Omat_It__Nat__Onat_J,type,
minus_minus_mat_nat: mat_nat > mat_nat > mat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
minus_811609699411566653omplex: set_complex > set_complex > set_complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
minus_8760755521168068590omplex: set_mat_complex > set_mat_complex > set_mat_complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
plus_plus_complex: complex > complex > complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
plus_p8323303612493835998omplex: mat_complex > mat_complex > mat_complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_It__Nat__Onat_J,type,
plus_plus_mat_nat: mat_nat > mat_nat > mat_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
plus_p3079357308422357842omplex: vec_complex > vec_complex > vec_complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
plus_p7052360327008956141omplex: set_complex > set_complex > set_complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
plus_p4229080058245121342omplex: set_mat_complex > set_mat_complex > set_mat_complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
times_times_complex: complex > complex > complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
times_8009071140041733218omplex: mat_complex > mat_complex > mat_complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_It__Nat__Onat_J,type,
times_times_mat_nat: mat_nat > mat_nat > mat_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Complex__Ocomplex_J,type,
times_6048082448287401577omplex: set_complex > set_complex > set_complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
times_6731331324747250370omplex: set_mat_complex > set_mat_complex > set_mat_complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Nat__Onat_J,type,
times_times_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
uminus1482373934393186551omplex: complex > complex ).
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__Set__Oset_It__Complex__Ocomplex_J,type,
uminus8566677241136511917omplex: set_complex > set_complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
uminus5815530220087396478omplex: set_mat_complex > set_mat_complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Complex__Ocomplex_J,type,
zero_z6614145512433583213omplex: set_complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Nat__Onat_J,type,
zero_zero_set_nat: set_nat ).
thf(sy_c_If_001t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
if_vec_complex: $o > vec_complex > vec_complex > vec_complex ).
thf(sy_c_Linear__Algebra__Complements_Ocpx__sq__mat_Ozero__col_001t__Complex__Ocomplex,type,
linear2567731547100889994omplex: nat > mat_complex > nat > vec_complex ).
thf(sy_c_Linear__Algebra__Complements_Ofixed__carrier__mat_001t__Complex__Ocomplex,type,
linear8738132868031958293omplex: set_mat_complex > nat > nat > $o ).
thf(sy_c_Linear__Algebra__Complements_Ofixed__carrier__mat_Osum__mat_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
linear8664352376190006057omplex: nat > nat > ( complex > mat_complex ) > set_complex > mat_complex ).
thf(sy_c_Linear__Algebra__Complements_Ofixed__carrier__mat_Osum__mat_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Complex__Ocomplex,type,
linear1795808462385993418omplex: nat > nat > ( mat_complex > mat_complex ) > set_mat_complex > mat_complex ).
thf(sy_c_Linear__Algebra__Complements_Ofixed__carrier__mat_Osum__mat_001t__Matrix__Omat_It__Nat__Onat_J_001t__Complex__Ocomplex,type,
linear4360147844572819692omplex: nat > nat > ( mat_nat > mat_complex ) > set_mat_nat > mat_complex ).
thf(sy_c_Linear__Algebra__Complements_Ofixed__carrier__mat_Osum__mat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
linear8108877306658443851omplex: nat > nat > ( nat > mat_complex ) > set_nat > mat_complex ).
thf(sy_c_Linear__Algebra__Complements_Ofixed__carrier__mat_Osum__mat_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Complex__Ocomplex,type,
linear7804121962625877471omplex: nat > nat > ( set_complex > mat_complex ) > set_set_complex > mat_complex ).
thf(sy_c_Linear__Algebra__Complements_Ofixed__carrier__mat_Osum__mat_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J_001t__Complex__Ocomplex,type,
linear5733585793752561962omplex: nat > nat > ( set_mat_complex > mat_complex ) > set_set_mat_complex > mat_complex ).
thf(sy_c_Linear__Algebra__Complements_Orank__1__proj_001t__Complex__Ocomplex,type,
linear1949544614684794075omplex: vec_complex > mat_complex ).
thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
nth_complex: list_complex > nat > complex ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Complex__Ocomplex,type,
carrier_mat_complex: nat > nat > set_mat_complex ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Nat__Onat,type,
carrier_mat_nat: nat > nat > set_mat_nat ).
thf(sy_c_Matrix_Ocol_001t__Complex__Ocomplex,type,
col_complex: mat_complex > nat > vec_complex ).
thf(sy_c_Matrix_Odiag__mat_001t__Complex__Ocomplex,type,
diag_mat_complex: mat_complex > list_complex ).
thf(sy_c_Matrix_Osmult__mat_001t__Complex__Ocomplex,type,
smult_mat_complex: complex > mat_complex > mat_complex ).
thf(sy_c_Matrix_Osmult__mat_001t__Nat__Onat,type,
smult_mat_nat: nat > mat_nat > mat_nat ).
thf(sy_c_Matrix_Osmult__mat_001t__Set__Oset_It__Complex__Ocomplex_J,type,
smult_4557042052056852367omplex: set_complex > mat_set_complex > mat_set_complex ).
thf(sy_c_Matrix_Osmult__mat_001t__Set__Oset_It__Nat__Onat_J,type,
smult_mat_set_nat: set_nat > mat_set_nat > mat_set_nat ).
thf(sy_c_Matrix_Osquare__mat_001t__Complex__Ocomplex,type,
square_mat_complex: mat_complex > $o ).
thf(sy_c_Matrix_Ozero__mat_001t__Complex__Ocomplex,type,
zero_mat_complex: nat > nat > mat_complex ).
thf(sy_c_Matrix_Ozero__mat_001t__Nat__Onat,type,
zero_mat_nat: nat > nat > mat_nat ).
thf(sy_c_Matrix_Ozero__vec_001t__Complex__Ocomplex,type,
zero_vec_complex: nat > vec_complex ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Complex__Ocomplex_M_Eo_J,type,
bot_bot_complex_o: complex > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Matrix__Omat_It__Complex__Ocomplex_J_M_Eo_J,type,
bot_bo2514468519737825834plex_o: mat_complex > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
bot_bot_set_complex: set_complex ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
bot_bo7165004461764951667omplex: set_mat_complex ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Complex__Ocomplex,type,
ord_less_complex: complex > complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
ord_less_mat_complex: mat_complex > mat_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_less_set_complex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
ord_le5598786136212072115omplex: set_mat_complex > set_mat_complex > $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__Matrix__Omat_It__Complex__Ocomplex_J,type,
ord_le1403324449407493959omplex: mat_complex > mat_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_le211207098394363844omplex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
ord_le3632134057777142183omplex: set_mat_complex > set_mat_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Projective__Measurements_Ocpx__sq__mat_Oeigen__projector,type,
projec1689266477789839993jector: nat > nat > mat_complex > complex > mat_complex ).
thf(sy_c_Projective__Measurements_Ohermitian__decomp_001t__Complex__Ocomplex,type,
projec5943904436471448624omplex: mat_complex > mat_complex > mat_complex > $o ).
thf(sy_c_Projective__Measurements_Omax__mix__density,type,
projec8360710381328234318ensity: nat > mat_complex ).
thf(sy_c_Projective__Measurements_Ospectrum_001t__Complex__Ocomplex,type,
projec527831343749723810omplex: mat_complex > set_complex ).
thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
real_V2521375963428798218omplex: set_complex ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
collect_mat_complex: ( mat_complex > $o ) > set_mat_complex ).
thf(sy_c_Set_OCollect_001t__Matrix__Omat_It__Nat__Onat_J,type,
collect_mat_nat: ( mat_nat > $o ) > set_mat_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
collect_set_complex: ( set_complex > $o ) > set_set_complex ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
collec7787716095925712882omplex: ( set_mat_complex > $o ) > set_set_mat_complex ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
image_1468599708987790691omplex: ( complex > complex ) > set_complex > set_complex ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
image_6107471045988054706omplex: ( complex > mat_complex ) > set_complex > set_mat_complex ).
thf(sy_c_Set_Oimage_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Complex__Ocomplex,type,
image_4184848318200637456omplex: ( mat_complex > complex ) > set_mat_complex > set_complex ).
thf(sy_c_Set_Oimage_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
image_23760814813800901omplex: ( mat_complex > mat_complex ) > set_mat_complex > set_mat_complex ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Set__Oset_It__Complex__Ocomplex_J,type,
image_7998606247489673935omplex: ( set_complex > set_complex ) > set_set_complex > set_set_complex ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
image_7857974177869717957omplex: ( set_mat_complex > set_mat_complex ) > set_set_mat_complex > set_set_mat_complex ).
thf(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
insert_complex: complex > set_complex > set_complex ).
thf(sy_c_Set_Oinsert_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
insert_mat_complex: mat_complex > set_mat_complex > set_mat_complex ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
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_Ounitary__diag_001t__Complex__Ocomplex,type,
spectr532731689276696518omplex: mat_complex > mat_complex > mat_complex > $o ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
member_mat_complex: mat_complex > set_mat_complex > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Nat__Onat_J,type,
member_mat_nat: mat_nat > set_mat_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Complex__Ocomplex_J,type,
member_set_complex: set_complex > set_set_complex > $o ).
thf(sy_c_member_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
member3612512168372279472omplex: set_mat_complex > set_set_mat_complex > $o ).
thf(sy_v_A,type,
a: mat_complex ).
thf(sy_v_B,type,
b: mat_complex ).
thf(sy_v_OP____,type,
v_OP____: mat_complex ).
thf(sy_v_U,type,
u: mat_complex ).
thf(sy_v_fc____,type,
fc: set_mat_complex ).
thf(sy_v_i____,type,
i: nat ).
thf(sy_v_j,type,
j: nat ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1109)
thf(fact_0__092_060open_062i_A_092_060noteq_062_Aj_092_060close_062,axiom,
i != j ).
% \<open>i \<noteq> j\<close>
thf(fact_1_dim__eq,axiom,
n = n ).
% dim_eq
thf(fact_2_fc__def,axiom,
( fc
= ( carrier_mat_complex @ n @ n ) ) ).
% fc_def
thf(fact_3_assms_I4_J,axiom,
spectr532731689276696518omplex @ a @ b @ u ).
% assms(4)
thf(fact_4__092_060open_062i_A_060_An_092_060close_062,axiom,
ord_less_nat @ i @ n ).
% \<open>i < n\<close>
thf(fact_5_max__mix__density__square,axiom,
member_mat_complex @ ( projec8360710381328234318ensity @ n ) @ fc ).
% max_mix_density_square
thf(fact_6_square__mats,axiom,
! [A: mat_complex] :
( ( member_mat_complex @ A @ fc )
=> ( square_mat_complex @ A ) ) ).
% square_mats
thf(fact_7_hermitian__schur__decomp,axiom,
! [A: mat_complex] :
( ( comple8306762464034002205omplex @ A )
=> ( ( member_mat_complex @ A @ fc )
=> ~ ! [B: mat_complex,U: mat_complex] :
~ ( projec5943904436471448624omplex @ A @ B @ U ) ) ) ).
% hermitian_schur_decomp
thf(fact_8_carrier__ne,axiom,
fc != bot_bo7165004461764951667omplex ).
% carrier_ne
thf(fact_9_cpx__sq__mat__mult,axiom,
! [A: mat_complex,B2: mat_complex] :
( ( member_mat_complex @ A @ fc )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( member_mat_complex @ ( times_8009071140041733218omplex @ A @ B2 ) @ fc ) ) ) ).
% cpx_sq_mat_mult
thf(fact_10_add__assoc,axiom,
! [A2: mat_complex,B3: mat_complex,C: mat_complex] :
( ( member_mat_complex @ A2 @ fc )
=> ( ( member_mat_complex @ B3 @ fc )
=> ( ( member_mat_complex @ C @ fc )
=> ( ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A2 @ B3 ) @ C )
= ( plus_p8323303612493835998omplex @ A2 @ ( plus_p8323303612493835998omplex @ B3 @ C ) ) ) ) ) ) ).
% add_assoc
thf(fact_11_add__commute,axiom,
! [A2: mat_complex,B3: mat_complex] :
( ( member_mat_complex @ A2 @ fc )
=> ( ( member_mat_complex @ B3 @ fc )
=> ( ( plus_p8323303612493835998omplex @ A2 @ B3 )
= ( plus_p8323303612493835998omplex @ B3 @ A2 ) ) ) ) ).
% add_commute
thf(fact_12_add__mem,axiom,
! [A2: mat_complex,B3: mat_complex] :
( ( member_mat_complex @ A2 @ fc )
=> ( ( member_mat_complex @ B3 @ fc )
=> ( member_mat_complex @ ( plus_p8323303612493835998omplex @ A2 @ B3 ) @ fc ) ) ) ).
% add_mem
thf(fact_13_uminus__mem,axiom,
! [A2: mat_complex] :
( ( member_mat_complex @ A2 @ fc )
=> ( member_mat_complex @ ( uminus467866341702955550omplex @ A2 ) @ fc ) ) ).
% uminus_mem
thf(fact_14_fixed__carrier__mat__axioms,axiom,
linear8738132868031958293omplex @ fc @ n @ n ).
% fixed_carrier_mat_axioms
thf(fact_15_cpx__sq__mat__smult,axiom,
! [A: mat_complex,X: complex] :
( ( member_mat_complex @ A @ fc )
=> ( member_mat_complex @ ( smult_mat_complex @ X @ A ) @ fc ) ) ).
% cpx_sq_mat_smult
thf(fact_16_smult__mem,axiom,
! [A: mat_complex,A2: complex] :
( ( member_mat_complex @ A @ fc )
=> ( member_mat_complex @ ( smult_mat_complex @ A2 @ A ) @ fc ) ) ).
% smult_mem
thf(fact_17_unitary__diag__carrier_I2_J,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,U2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr532731689276696518omplex @ A @ B2 @ U2 )
=> ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% unitary_diag_carrier(2)
thf(fact_18_assms_I2_J,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% assms(2)
thf(fact_19_assms_I3_J,axiom,
comple8306762464034002205omplex @ a ).
% assms(3)
thf(fact_20_assms_I5_J,axiom,
ord_less_nat @ j @ n ).
% assms(5)
thf(fact_21_assms_I1_J,axiom,
member_mat_complex @ a @ ( carrier_mat_complex @ n @ n ) ).
% assms(1)
thf(fact_22_mult__add__distrib__right,axiom,
! [A: mat_complex,B2: mat_complex,C2: mat_complex] :
( ( member_mat_complex @ A @ fc )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( ( member_mat_complex @ C2 @ fc )
=> ( ( times_8009071140041733218omplex @ A @ ( plus_p8323303612493835998omplex @ B2 @ C2 ) )
= ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ A @ B2 ) @ ( times_8009071140041733218omplex @ A @ C2 ) ) ) ) ) ) ).
% mult_add_distrib_right
thf(fact_23_mult__add__distrib__left,axiom,
! [A: mat_complex,B2: mat_complex,C2: mat_complex] :
( ( member_mat_complex @ A @ fc )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( ( member_mat_complex @ C2 @ fc )
=> ( ( times_8009071140041733218omplex @ ( plus_p8323303612493835998omplex @ B2 @ C2 ) @ A )
= ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ B2 @ A ) @ ( times_8009071140041733218omplex @ C2 @ A ) ) ) ) ) ) ).
% mult_add_distrib_left
thf(fact_24_cm,axiom,
member_mat_complex @ v_OP____ @ ( carrier_mat_complex @ n @ n ) ).
% cm
thf(fact_25_hermitian__square__hermitian,axiom,
! [A: mat_complex] :
( ( comple8306762464034002205omplex @ A )
=> ( comple8306762464034002205omplex @ ( times_8009071140041733218omplex @ A @ A ) ) ) ).
% hermitian_square_hermitian
thf(fact_26_unitary__diag__carrier_I1_J,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,U2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr532731689276696518omplex @ A @ B2 @ U2 )
=> ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% unitary_diag_carrier(1)
thf(fact_27_semigroup__add__on__with__axioms,axiom,
group_6724803037680873049omplex @ fc @ plus_p8323303612493835998omplex ).
% semigroup_add_on_with_axioms
thf(fact_28_ab__semigroup__add__on__with__axioms,axiom,
group_8296285978151397272omplex @ fc @ plus_p8323303612493835998omplex ).
% ab_semigroup_add_on_with_axioms
thf(fact_29_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_30_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_31_local_Oab__left__minus,axiom,
! [A2: mat_complex] :
( ( member_mat_complex @ A2 @ fc )
=> ( ( plus_p8323303612493835998omplex @ ( uminus467866341702955550omplex @ A2 ) @ A2 )
= ( zero_mat_complex @ n @ n ) ) ) ).
% local.ab_left_minus
thf(fact_32_ab__diff__conv__add__uminus,axiom,
! [A2: mat_complex,B3: mat_complex] :
( ( member_mat_complex @ A2 @ fc )
=> ( ( member_mat_complex @ B3 @ fc )
=> ( ( minus_2412168080157227406omplex @ A2 @ B3 )
= ( plus_p8323303612493835998omplex @ A2 @ ( uminus467866341702955550omplex @ B3 ) ) ) ) ) ).
% ab_diff_conv_add_uminus
thf(fact_33_uminus__add__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( uminus467866341702955550omplex @ ( plus_p8323303612493835998omplex @ A @ B2 ) )
= ( plus_p8323303612493835998omplex @ ( uminus467866341702955550omplex @ B2 ) @ ( uminus467866341702955550omplex @ A ) ) ) ) ) ).
% uminus_add_mat
thf(fact_34_mat__assoc__test_I7_J,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,C2: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ ( plus_p8323303612493835998omplex @ B2 @ C2 ) )
= ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ A @ B2 ) @ ( times_8009071140041733218omplex @ B2 @ B2 ) ) @ ( times_8009071140041733218omplex @ A @ C2 ) ) @ ( times_8009071140041733218omplex @ B2 @ C2 ) ) ) ) ) ) ) ).
% mat_assoc_test(7)
thf(fact_35_hermitian__add,axiom,
! [A: mat_complex,N: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple8306762464034002205omplex @ A )
=> ( ( comple8306762464034002205omplex @ B2 )
=> ( comple8306762464034002205omplex @ ( plus_p8323303612493835998omplex @ A @ B2 ) ) ) ) ) ) ).
% hermitian_add
thf(fact_36_add__smult__distrib__left__mat,axiom,
! [A: mat_nat,Nr: nat,Nc: nat,B2: mat_nat,K: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( smult_mat_nat @ K @ ( plus_plus_mat_nat @ A @ B2 ) )
= ( plus_plus_mat_nat @ ( smult_mat_nat @ K @ A ) @ ( smult_mat_nat @ K @ B2 ) ) ) ) ) ).
% add_smult_distrib_left_mat
thf(fact_37_add__smult__distrib__left__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,B2: mat_complex,K: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( smult_mat_complex @ K @ ( plus_p8323303612493835998omplex @ A @ B2 ) )
= ( plus_p8323303612493835998omplex @ ( smult_mat_complex @ K @ A ) @ ( smult_mat_complex @ K @ B2 ) ) ) ) ) ).
% add_smult_distrib_left_mat
thf(fact_38_mult__smult__distrib,axiom,
! [A: mat_nat,Nr: nat,N: nat,B2: mat_nat,Nc: nat,K: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ N ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ N @ Nc ) )
=> ( ( times_times_mat_nat @ A @ ( smult_mat_nat @ K @ B2 ) )
= ( smult_mat_nat @ K @ ( times_times_mat_nat @ A @ B2 ) ) ) ) ) ).
% mult_smult_distrib
thf(fact_39_mult__smult__distrib,axiom,
! [A: mat_complex,Nr: nat,N: nat,B2: mat_complex,Nc: nat,K: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ A @ ( smult_mat_complex @ K @ B2 ) )
= ( smult_mat_complex @ K @ ( times_8009071140041733218omplex @ A @ B2 ) ) ) ) ) ).
% mult_smult_distrib
thf(fact_40_mult__smult__assoc__mat,axiom,
! [A: mat_nat,Nr: nat,N: nat,B2: mat_nat,Nc: nat,K: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ N ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ N @ Nc ) )
=> ( ( times_times_mat_nat @ ( smult_mat_nat @ K @ A ) @ B2 )
= ( smult_mat_nat @ K @ ( times_times_mat_nat @ A @ B2 ) ) ) ) ) ).
% mult_smult_assoc_mat
thf(fact_41_mult__smult__assoc__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,B2: mat_complex,Nc: nat,K: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( smult_mat_complex @ K @ A ) @ B2 )
= ( smult_mat_complex @ K @ ( times_8009071140041733218omplex @ A @ B2 ) ) ) ) ) ).
% mult_smult_assoc_mat
thf(fact_42_add__mult__distrib__mat,axiom,
! [A: mat_nat,Nr: nat,N: nat,B2: mat_nat,C2: mat_nat,Nc: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ N ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ Nr @ N ) )
=> ( ( member_mat_nat @ C2 @ ( carrier_mat_nat @ N @ Nc ) )
=> ( ( times_times_mat_nat @ ( plus_plus_mat_nat @ A @ B2 ) @ C2 )
= ( plus_plus_mat_nat @ ( times_times_mat_nat @ A @ C2 ) @ ( times_times_mat_nat @ B2 @ C2 ) ) ) ) ) ) ).
% add_mult_distrib_mat
thf(fact_43_add__mult__distrib__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,B2: mat_complex,C2: mat_complex,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ C2 )
= ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ A @ C2 ) @ ( times_8009071140041733218omplex @ B2 @ C2 ) ) ) ) ) ) ).
% add_mult_distrib_mat
thf(fact_44_zero__mem,axiom,
member_mat_complex @ ( zero_mat_complex @ n @ n ) @ fc ).
% zero_mem
thf(fact_45_add__zero,axiom,
! [A2: mat_complex] :
( ( member_mat_complex @ A2 @ fc )
=> ( ( plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ A2 )
= A2 ) ) ).
% add_zero
thf(fact_46_sum__mat__empty,axiom,
! [A: nat > mat_complex] :
( ( linear8108877306658443851omplex @ n @ n @ A @ bot_bot_set_nat )
= ( zero_mat_complex @ n @ n ) ) ).
% sum_mat_empty
thf(fact_47_sum__mat__empty,axiom,
! [A: mat_complex > mat_complex] :
( ( linear1795808462385993418omplex @ n @ n @ A @ bot_bo7165004461764951667omplex )
= ( zero_mat_complex @ n @ n ) ) ).
% sum_mat_empty
thf(fact_48_sum__mat__empty,axiom,
! [A: complex > mat_complex] :
( ( linear8664352376190006057omplex @ n @ n @ A @ bot_bot_set_complex )
= ( zero_mat_complex @ n @ n ) ) ).
% sum_mat_empty
thf(fact_49_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_50_mem__Collect__eq,axiom,
! [A2: mat_nat,P: mat_nat > $o] :
( ( member_mat_nat @ A2 @ ( collect_mat_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_51_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_52_mem__Collect__eq,axiom,
! [A2: set_complex,P: set_complex > $o] :
( ( member_set_complex @ A2 @ ( collect_set_complex @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
! [A2: set_mat_complex,P: set_mat_complex > $o] :
( ( member3612512168372279472omplex @ A2 @ ( collec7787716095925712882omplex @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
! [A2: mat_complex,P: mat_complex > $o] :
( ( member_mat_complex @ A2 @ ( collect_mat_complex @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_55_mem__Collect__eq,axiom,
! [A2: complex,P: complex > $o] :
( ( member_complex @ A2 @ ( collect_complex @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A: set_mat_nat] :
( ( collect_mat_nat
@ ^ [X2: mat_nat] : ( member_mat_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A: set_set_complex] :
( ( collect_set_complex
@ ^ [X2: set_complex] : ( member_set_complex @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A: set_set_mat_complex] :
( ( collec7787716095925712882omplex
@ ^ [X2: set_mat_complex] : ( member3612512168372279472omplex @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A: set_mat_complex] :
( ( collect_mat_complex
@ ^ [X2: mat_complex] : ( member_mat_complex @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
! [A: set_complex] :
( ( collect_complex
@ ^ [X2: complex] : ( member_complex @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_62_Collect__cong,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X3: complex] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_complex @ P )
= ( collect_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_63_Collect__cong,axiom,
! [P: mat_complex > $o,Q: mat_complex > $o] :
( ! [X3: mat_complex] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_mat_complex @ P )
= ( collect_mat_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_64_zero__carrier__mat,axiom,
! [Nr: nat,Nc: nat] : ( member_mat_nat @ ( zero_mat_nat @ Nr @ Nc ) @ ( carrier_mat_nat @ Nr @ Nc ) ) ).
% zero_carrier_mat
thf(fact_65_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_66_minus__carrier__mat_H,axiom,
! [A: mat_nat,Nr: nat,Nc: nat,B2: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( member_mat_nat @ ( minus_minus_mat_nat @ A @ B2 ) @ ( carrier_mat_nat @ Nr @ Nc ) ) ) ) ).
% minus_carrier_mat'
thf(fact_67_minus__carrier__mat_H,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( minus_2412168080157227406omplex @ A @ B2 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% minus_carrier_mat'
thf(fact_68_minus__carrier__mat,axiom,
! [B2: mat_nat,Nr: nat,Nc: nat,A: mat_nat] :
( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( member_mat_nat @ ( minus_minus_mat_nat @ A @ B2 ) @ ( carrier_mat_nat @ Nr @ Nc ) ) ) ).
% minus_carrier_mat
thf(fact_69_minus__carrier__mat,axiom,
! [B2: mat_complex,Nr: nat,Nc: nat,A: mat_complex] :
( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( minus_2412168080157227406omplex @ A @ B2 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% minus_carrier_mat
thf(fact_70_smult__zero__mat,axiom,
! [K: nat,Nr: nat,Nc: nat] :
( ( smult_mat_nat @ K @ ( zero_mat_nat @ Nr @ Nc ) )
= ( zero_mat_nat @ Nr @ Nc ) ) ).
% smult_zero_mat
thf(fact_71_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_72_zero__hermitian,axiom,
! [N: nat] : ( comple8306762464034002205omplex @ ( zero_mat_complex @ N @ N ) ) ).
% zero_hermitian
thf(fact_73_right__mult__zero__mat,axiom,
! [A: mat_nat,Nr: nat,N: nat,Nc: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ N ) )
=> ( ( times_times_mat_nat @ A @ ( zero_mat_nat @ N @ Nc ) )
= ( zero_mat_nat @ Nr @ Nc ) ) ) ).
% right_mult_zero_mat
thf(fact_74_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_75_left__mult__zero__mat,axiom,
! [A: mat_nat,N: nat,Nc: nat,Nr: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ Nc ) )
=> ( ( times_times_mat_nat @ ( zero_mat_nat @ Nr @ N ) @ A )
= ( zero_mat_nat @ Nr @ Nc ) ) ) ).
% left_mult_zero_mat
thf(fact_76_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_77_Complex__Matrix_Oright__add__zero__mat,axiom,
! [A: mat_nat,Nr: nat,Nc: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( plus_plus_mat_nat @ A @ ( zero_mat_nat @ Nr @ Nc ) )
= A ) ) ).
% Complex_Matrix.right_add_zero_mat
thf(fact_78_Complex__Matrix_Oright__add__zero__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( plus_p8323303612493835998omplex @ A @ ( zero_mat_complex @ Nr @ Nc ) )
= A ) ) ).
% Complex_Matrix.right_add_zero_mat
thf(fact_79_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_80_left__add__zero__mat,axiom,
! [A: mat_nat,Nr: nat,Nc: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( plus_plus_mat_nat @ ( zero_mat_nat @ Nr @ Nc ) @ A )
= A ) ) ).
% left_add_zero_mat
thf(fact_81_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_82_mult__minus__distrib__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,B2: mat_complex,Nc: nat,C2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ A @ ( minus_2412168080157227406omplex @ B2 @ C2 ) )
= ( minus_2412168080157227406omplex @ ( times_8009071140041733218omplex @ A @ B2 ) @ ( times_8009071140041733218omplex @ A @ C2 ) ) ) ) ) ) ).
% mult_minus_distrib_mat
thf(fact_83_minus__mult__distrib__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,B2: mat_complex,C2: mat_complex,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( minus_2412168080157227406omplex @ A @ B2 ) @ C2 )
= ( minus_2412168080157227406omplex @ ( times_8009071140041733218omplex @ A @ C2 ) @ ( times_8009071140041733218omplex @ B2 @ C2 ) ) ) ) ) ) ).
% minus_mult_distrib_mat
thf(fact_84_mat__minus__minus,axiom,
! [A: mat_complex,N: nat,M: nat,B2: mat_complex,C2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ M ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ M ) )
=> ( ( minus_2412168080157227406omplex @ A @ ( minus_2412168080157227406omplex @ B2 @ C2 ) )
= ( plus_p8323303612493835998omplex @ ( minus_2412168080157227406omplex @ A @ B2 ) @ C2 ) ) ) ) ) ).
% mat_minus_minus
thf(fact_85_minus__add__minus__mat,axiom,
! [U3: mat_complex,Nr: nat,Nc: nat,V: mat_complex,W: mat_complex] :
( ( member_mat_complex @ U3 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ V @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ W @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( minus_2412168080157227406omplex @ U3 @ ( plus_p8323303612493835998omplex @ V @ W ) )
= ( minus_2412168080157227406omplex @ ( minus_2412168080157227406omplex @ U3 @ V ) @ W ) ) ) ) ) ).
% minus_add_minus_mat
thf(fact_86_smult__distrib__left__minus__mat,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,C: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( smult_mat_complex @ C @ ( minus_2412168080157227406omplex @ B2 @ A ) )
= ( minus_2412168080157227406omplex @ ( smult_mat_complex @ C @ B2 ) @ ( smult_mat_complex @ C @ A ) ) ) ) ) ).
% smult_distrib_left_minus_mat
thf(fact_87_hermitian__minus,axiom,
! [A: mat_complex,N: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple8306762464034002205omplex @ A )
=> ( ( comple8306762464034002205omplex @ B2 )
=> ( comple8306762464034002205omplex @ ( minus_2412168080157227406omplex @ A @ B2 ) ) ) ) ) ) ).
% hermitian_minus
thf(fact_88_mat__assoc__test_I9_J,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,C2: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ ( minus_2412168080157227406omplex @ B2 @ C2 ) ) @ D )
= ( minus_2412168080157227406omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B2 ) @ D ) @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ C2 ) @ D ) ) ) ) ) ) ) ).
% mat_assoc_test(9)
thf(fact_89_mat__assoc__test_I5_J,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,C2: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( plus_p8323303612493835998omplex @ A @ ( minus_2412168080157227406omplex @ B2 @ C2 ) )
= ( minus_2412168080157227406omplex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ C2 ) ) ) ) ) ) ).
% mat_assoc_test(5)
thf(fact_90_mat__assoc__test_I6_J,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,C2: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( minus_2412168080157227406omplex @ A @ ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ B2 @ C2 ) @ D ) )
= ( minus_2412168080157227406omplex @ ( minus_2412168080157227406omplex @ ( minus_2412168080157227406omplex @ A @ B2 ) @ C2 ) @ D ) ) ) ) ) ) ).
% mat_assoc_test(6)
thf(fact_91_uminus__l__inv__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( plus_p8323303612493835998omplex @ ( uminus467866341702955550omplex @ A ) @ A )
= ( zero_mat_complex @ Nr @ Nc ) ) ) ).
% uminus_l_inv_mat
thf(fact_92_uminus__add__minus__mat,axiom,
! [L: mat_complex,Nr: nat,Nc: nat,R: mat_complex] :
( ( member_mat_complex @ L @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ R @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( uminus467866341702955550omplex @ ( plus_p8323303612493835998omplex @ L @ R ) )
= ( minus_2412168080157227406omplex @ ( uminus467866341702955550omplex @ L ) @ R ) ) ) ) ).
% uminus_add_minus_mat
thf(fact_93_minus__add__uminus__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( minus_2412168080157227406omplex @ A @ B2 )
= ( plus_p8323303612493835998omplex @ A @ ( uminus467866341702955550omplex @ B2 ) ) ) ) ) ).
% minus_add_uminus_mat
thf(fact_94_add__uminus__minus__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( plus_p8323303612493835998omplex @ A @ ( uminus467866341702955550omplex @ B2 ) )
= ( minus_2412168080157227406omplex @ A @ B2 ) ) ) ) ).
% add_uminus_minus_mat
thf(fact_95_mat__assoc__test_I4_J,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,C2: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( plus_p8323303612493835998omplex @ ( minus_2412168080157227406omplex @ A @ B2 ) @ ( minus_2412168080157227406omplex @ B2 @ C2 ) )
= ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ ( uminus467866341702955550omplex @ B2 ) ) @ B2 ) @ ( uminus467866341702955550omplex @ C2 ) ) ) ) ) ) ) ).
% mat_assoc_test(4)
thf(fact_96_uminus__uminus__mat,axiom,
! [A: mat_complex] :
( ( uminus467866341702955550omplex @ ( uminus467866341702955550omplex @ A ) )
= A ) ).
% uminus_uminus_mat
thf(fact_97_uminus__eq__mat,axiom,
! [A: mat_complex,B2: mat_complex] :
( ( ( uminus467866341702955550omplex @ A )
= ( uminus467866341702955550omplex @ B2 ) )
= ( A = B2 ) ) ).
% uminus_eq_mat
thf(fact_98_mult__carrier__mat,axiom,
! [A: mat_nat,Nr: nat,N: nat,B2: mat_nat,Nc: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ N ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ N @ Nc ) )
=> ( member_mat_nat @ ( times_times_mat_nat @ A @ B2 ) @ ( carrier_mat_nat @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_99_mult__carrier__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,B2: mat_complex,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( member_mat_complex @ ( times_8009071140041733218omplex @ A @ B2 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_100_assoc__mult__mat,axiom,
! [A: mat_nat,N_1: nat,N_2: nat,B2: mat_nat,N_3: nat,C2: mat_nat,N_4: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N_1 @ N_2 ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ N_2 @ N_3 ) )
=> ( ( member_mat_nat @ C2 @ ( carrier_mat_nat @ N_3 @ N_4 ) )
=> ( ( times_times_mat_nat @ ( times_times_mat_nat @ A @ B2 ) @ C2 )
= ( times_times_mat_nat @ A @ ( times_times_mat_nat @ B2 @ C2 ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_101_assoc__mult__mat,axiom,
! [A: mat_complex,N_1: nat,N_2: nat,B2: mat_complex,N_3: nat,C2: mat_complex,N_4: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N_1 @ N_2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N_2 @ N_3 ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N_3 @ N_4 ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B2 ) @ C2 )
= ( times_8009071140041733218omplex @ A @ ( times_8009071140041733218omplex @ B2 @ C2 ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_102_smult__smult__times,axiom,
! [A2: set_nat,K: set_nat,A: mat_set_nat] :
( ( smult_mat_set_nat @ A2 @ ( smult_mat_set_nat @ K @ A ) )
= ( smult_mat_set_nat @ ( times_times_set_nat @ A2 @ K ) @ A ) ) ).
% smult_smult_times
thf(fact_103_smult__smult__times,axiom,
! [A2: set_complex,K: set_complex,A: mat_set_complex] :
( ( smult_4557042052056852367omplex @ A2 @ ( smult_4557042052056852367omplex @ K @ A ) )
= ( smult_4557042052056852367omplex @ ( times_6048082448287401577omplex @ A2 @ K ) @ A ) ) ).
% smult_smult_times
thf(fact_104_smult__smult__times,axiom,
! [A2: complex,K: complex,A: mat_complex] :
( ( smult_mat_complex @ A2 @ ( smult_mat_complex @ K @ A ) )
= ( smult_mat_complex @ ( times_times_complex @ A2 @ K ) @ A ) ) ).
% smult_smult_times
thf(fact_105_smult__smult__times,axiom,
! [A2: nat,K: nat,A: mat_nat] :
( ( smult_mat_nat @ A2 @ ( smult_mat_nat @ K @ A ) )
= ( smult_mat_nat @ ( times_times_nat @ A2 @ K ) @ A ) ) ).
% smult_smult_times
thf(fact_106_add__carrier__mat_H,axiom,
! [A: mat_nat,Nr: nat,Nc: nat,B2: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( member_mat_nat @ ( plus_plus_mat_nat @ A @ B2 ) @ ( carrier_mat_nat @ Nr @ Nc ) ) ) ) ).
% add_carrier_mat'
thf(fact_107_add__carrier__mat_H,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% add_carrier_mat'
thf(fact_108_swap__plus__mat,axiom,
! [A: mat_nat,N: nat,B2: mat_nat,C2: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ N ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ N @ N ) )
=> ( ( member_mat_nat @ C2 @ ( carrier_mat_nat @ N @ N ) )
=> ( ( plus_plus_mat_nat @ ( plus_plus_mat_nat @ A @ B2 ) @ C2 )
= ( plus_plus_mat_nat @ ( plus_plus_mat_nat @ A @ C2 ) @ B2 ) ) ) ) ) ).
% swap_plus_mat
thf(fact_109_swap__plus__mat,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,C2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ C2 )
= ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ C2 ) @ B2 ) ) ) ) ) ).
% swap_plus_mat
thf(fact_110_add__carrier__mat,axiom,
! [B2: mat_nat,Nr: nat,Nc: nat,A: mat_nat] :
( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( member_mat_nat @ ( plus_plus_mat_nat @ A @ B2 ) @ ( carrier_mat_nat @ Nr @ Nc ) ) ) ).
% add_carrier_mat
thf(fact_111_add__carrier__mat,axiom,
! [B2: mat_complex,Nr: nat,Nc: nat,A: mat_complex] :
( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% add_carrier_mat
thf(fact_112_assoc__add__mat,axiom,
! [A: mat_nat,Nr: nat,Nc: nat,B2: mat_nat,C2: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( member_mat_nat @ C2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( plus_plus_mat_nat @ ( plus_plus_mat_nat @ A @ B2 ) @ C2 )
= ( plus_plus_mat_nat @ A @ ( plus_plus_mat_nat @ B2 @ C2 ) ) ) ) ) ) ).
% assoc_add_mat
thf(fact_113_assoc__add__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,B2: mat_complex,C2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ C2 )
= ( plus_p8323303612493835998omplex @ A @ ( plus_p8323303612493835998omplex @ B2 @ C2 ) ) ) ) ) ) ).
% assoc_add_mat
thf(fact_114_comm__add__mat,axiom,
! [A: mat_nat,Nr: nat,Nc: nat,B2: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( plus_plus_mat_nat @ A @ B2 )
= ( plus_plus_mat_nat @ B2 @ A ) ) ) ) ).
% comm_add_mat
thf(fact_115_comm__add__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( plus_p8323303612493835998omplex @ A @ B2 )
= ( plus_p8323303612493835998omplex @ B2 @ A ) ) ) ) ).
% comm_add_mat
thf(fact_116_smult__carrier__mat,axiom,
! [A: mat_nat,Nr: nat,Nc: nat,K: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( member_mat_nat @ ( smult_mat_nat @ K @ A ) @ ( carrier_mat_nat @ Nr @ Nc ) ) ) ).
% smult_carrier_mat
thf(fact_117_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_118_mat__assoc__test_I1_J,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,C2: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B2 ) @ ( times_8009071140041733218omplex @ C2 @ D ) )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B2 ) @ C2 ) @ D ) ) ) ) ) ) ).
% mat_assoc_test(1)
thf(fact_119_uminus__carrier__iff__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ ( uminus467866341702955550omplex @ A ) @ ( carrier_mat_complex @ Nr @ Nc ) )
= ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% uminus_carrier_iff_mat
thf(fact_120_uminus__carrier__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( uminus467866341702955550omplex @ A ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% uminus_carrier_mat
thf(fact_121_mat__assoc__test_I13_J,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,C2: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( plus_p8323303612493835998omplex @ A @ B2 )
= ( plus_p8323303612493835998omplex @ B2 @ A ) ) ) ) ) ) ).
% mat_assoc_test(13)
thf(fact_122_mat__assoc__test_I14_J,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,C2: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ C2 )
= ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ C2 @ B2 ) @ A ) ) ) ) ) ) ).
% mat_assoc_test(14)
thf(fact_123_mat__assoc__test_I15_J,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,C2: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ ( plus_p8323303612493835998omplex @ C2 @ D ) )
= ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ C2 ) @ ( plus_p8323303612493835998omplex @ B2 @ D ) ) ) ) ) ) ) ).
% mat_assoc_test(15)
thf(fact_124_fixed__carrier__mat__def,axiom,
( linear8738132868031958293omplex
= ( ^ [Fc_mats: set_mat_complex,DimR: nat,DimC: nat] :
( Fc_mats
= ( carrier_mat_complex @ DimR @ DimC ) ) ) ) ).
% fixed_carrier_mat_def
thf(fact_125_fixed__carrier__mat_Ofc__mats__carrier,axiom,
! [Fc_mats2: set_mat_complex,DimR2: nat,DimC2: nat] :
( ( linear8738132868031958293omplex @ Fc_mats2 @ DimR2 @ DimC2 )
=> ( Fc_mats2
= ( carrier_mat_complex @ DimR2 @ DimC2 ) ) ) ).
% fixed_carrier_mat.fc_mats_carrier
thf(fact_126_fixed__carrier__mat_Ointro,axiom,
! [Fc_mats2: set_mat_complex,DimR2: nat,DimC2: nat] :
( ( Fc_mats2
= ( carrier_mat_complex @ DimR2 @ DimC2 ) )
=> ( linear8738132868031958293omplex @ Fc_mats2 @ DimR2 @ DimC2 ) ) ).
% fixed_carrier_mat.intro
thf(fact_127_fixed__carrier__mat_Osmult__mem,axiom,
! [Fc_mats2: set_mat_complex,DimR2: nat,DimC2: nat,A: mat_complex,A2: complex] :
( ( linear8738132868031958293omplex @ Fc_mats2 @ DimR2 @ DimC2 )
=> ( ( member_mat_complex @ A @ Fc_mats2 )
=> ( member_mat_complex @ ( smult_mat_complex @ A2 @ A ) @ Fc_mats2 ) ) ) ).
% fixed_carrier_mat.smult_mem
thf(fact_128_mult__add__distrib__mat,axiom,
! [A: mat_nat,Nr: nat,N: nat,B2: mat_nat,Nc: nat,C2: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ N ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ N @ Nc ) )
=> ( ( member_mat_nat @ C2 @ ( carrier_mat_nat @ N @ Nc ) )
=> ( ( times_times_mat_nat @ A @ ( plus_plus_mat_nat @ B2 @ C2 ) )
= ( plus_plus_mat_nat @ ( times_times_mat_nat @ A @ B2 ) @ ( times_times_mat_nat @ A @ C2 ) ) ) ) ) ) ).
% mult_add_distrib_mat
thf(fact_129_mult__add__distrib__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,B2: mat_complex,Nc: nat,C2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ A @ ( plus_p8323303612493835998omplex @ B2 @ C2 ) )
= ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ A @ B2 ) @ ( times_8009071140041733218omplex @ A @ C2 ) ) ) ) ) ) ).
% mult_add_distrib_mat
thf(fact_130_ab__group__add__on__with__axioms,axiom,
group_7083379013581629702omplex @ fc @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ minus_2412168080157227406omplex @ uminus467866341702955550omplex ).
% ab_group_add_on_with_axioms
thf(fact_131_comm__monoid__add__on__with__axioms,axiom,
group_5394922976599784994omplex @ fc @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) ).
% comm_monoid_add_on_with_axioms
thf(fact_132_less__add__iff2,axiom,
! [A2: complex,E: complex,C: complex,B3: complex,D2: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ ( times_times_complex @ A2 @ E ) @ C ) @ ( plus_plus_complex @ ( times_times_complex @ B3 @ E ) @ D2 ) )
= ( ord_less_complex @ C @ ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B3 @ A2 ) @ E ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_133_less__add__iff1,axiom,
! [A2: complex,E: complex,C: complex,B3: complex,D2: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ ( times_times_complex @ A2 @ E ) @ C ) @ ( plus_plus_complex @ ( times_times_complex @ B3 @ E ) @ D2 ) )
= ( ord_less_complex @ ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A2 @ B3 ) @ E ) @ C ) @ D2 ) ) ).
% less_add_iff1
thf(fact_134_pth__2,axiom,
( minus_minus_complex
= ( ^ [X2: complex,Y: complex] : ( plus_plus_complex @ X2 @ ( uminus1482373934393186551omplex @ Y ) ) ) ) ).
% pth_2
thf(fact_135_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_complex
= ( ^ [A3: complex,B4: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_136_uminus__add__conv__diff,axiom,
! [A2: complex,B3: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B3 )
= ( minus_minus_complex @ B3 @ A2 ) ) ).
% uminus_add_conv_diff
thf(fact_137_diff__conv__add__uminus,axiom,
( minus_minus_complex
= ( ^ [A3: complex,B4: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_138_sum__mat__carrier,axiom,
! [I: set_mat_nat,A: mat_nat > mat_complex] :
( ! [I2: mat_nat] :
( ( member_mat_nat @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( member_mat_complex @ ( linear4360147844572819692omplex @ n @ n @ A @ I ) @ ( carrier_mat_complex @ n @ n ) ) ) ).
% sum_mat_carrier
thf(fact_139_sum__mat__carrier,axiom,
! [I: set_nat,A: nat > mat_complex] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( member_mat_complex @ ( linear8108877306658443851omplex @ n @ n @ A @ I ) @ ( carrier_mat_complex @ n @ n ) ) ) ).
% sum_mat_carrier
thf(fact_140_sum__mat__carrier,axiom,
! [I: set_set_complex,A: set_complex > mat_complex] :
( ! [I2: set_complex] :
( ( member_set_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( member_mat_complex @ ( linear7804121962625877471omplex @ n @ n @ A @ I ) @ ( carrier_mat_complex @ n @ n ) ) ) ).
% sum_mat_carrier
thf(fact_141_sum__mat__carrier,axiom,
! [I: set_set_mat_complex,A: set_mat_complex > mat_complex] :
( ! [I2: set_mat_complex] :
( ( member3612512168372279472omplex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( member_mat_complex @ ( linear5733585793752561962omplex @ n @ n @ A @ I ) @ ( carrier_mat_complex @ n @ n ) ) ) ).
% sum_mat_carrier
thf(fact_142_sum__mat__carrier,axiom,
! [I: set_mat_complex,A: mat_complex > mat_complex] :
( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( member_mat_complex @ ( linear1795808462385993418omplex @ n @ n @ A @ I ) @ ( carrier_mat_complex @ n @ n ) ) ) ).
% sum_mat_carrier
thf(fact_143_sum__mat__carrier,axiom,
! [I: set_complex,A: complex > mat_complex] :
( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( member_mat_complex @ ( linear8664352376190006057omplex @ n @ n @ A @ I ) @ ( carrier_mat_complex @ n @ n ) ) ) ).
% sum_mat_carrier
thf(fact_144_fixed__carrier__mat_Osum__mat_Ocong,axiom,
linear8664352376190006057omplex = linear8664352376190006057omplex ).
% fixed_carrier_mat.sum_mat.cong
thf(fact_145_fixed__carrier__mat_Osum__mat_Ocong,axiom,
linear1795808462385993418omplex = linear1795808462385993418omplex ).
% fixed_carrier_mat.sum_mat.cong
thf(fact_146_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_147_zero__reorient,axiom,
! [X: set_complex] :
( ( zero_z6614145512433583213omplex = X )
= ( X = zero_z6614145512433583213omplex ) ) ).
% zero_reorient
thf(fact_148_zero__reorient,axiom,
! [X: set_nat] :
( ( zero_zero_set_nat = X )
= ( X = zero_zero_set_nat ) ) ).
% zero_reorient
thf(fact_149_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_150_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
= ( X = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_151_Groups_Omult__ac_I3_J,axiom,
! [B3: set_nat,A2: set_nat,C: set_nat] :
( ( times_times_set_nat @ B3 @ ( times_times_set_nat @ A2 @ C ) )
= ( times_times_set_nat @ A2 @ ( times_times_set_nat @ B3 @ C ) ) ) ).
% Groups.mult_ac(3)
thf(fact_152_Groups_Omult__ac_I3_J,axiom,
! [B3: set_complex,A2: set_complex,C: set_complex] :
( ( times_6048082448287401577omplex @ B3 @ ( times_6048082448287401577omplex @ A2 @ C ) )
= ( times_6048082448287401577omplex @ A2 @ ( times_6048082448287401577omplex @ B3 @ C ) ) ) ).
% Groups.mult_ac(3)
thf(fact_153_Groups_Omult__ac_I3_J,axiom,
! [B3: complex,A2: complex,C: complex] :
( ( times_times_complex @ B3 @ ( times_times_complex @ A2 @ C ) )
= ( times_times_complex @ A2 @ ( times_times_complex @ B3 @ C ) ) ) ).
% Groups.mult_ac(3)
thf(fact_154_Groups_Omult__ac_I3_J,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( times_times_nat @ B3 @ ( times_times_nat @ A2 @ C ) )
= ( times_times_nat @ A2 @ ( times_times_nat @ B3 @ C ) ) ) ).
% Groups.mult_ac(3)
thf(fact_155_Groups_Omult__ac_I2_J,axiom,
( times_times_set_nat
= ( ^ [A3: set_nat,B4: set_nat] : ( times_times_set_nat @ B4 @ A3 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_156_Groups_Omult__ac_I2_J,axiom,
( times_6048082448287401577omplex
= ( ^ [A3: set_complex,B4: set_complex] : ( times_6048082448287401577omplex @ B4 @ A3 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_157_Groups_Omult__ac_I2_J,axiom,
( times_times_complex
= ( ^ [A3: complex,B4: complex] : ( times_times_complex @ B4 @ A3 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_158_Groups_Omult__ac_I2_J,axiom,
( times_times_nat
= ( ^ [A3: nat,B4: nat] : ( times_times_nat @ B4 @ A3 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_159_Groups_Omult__ac_I1_J,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( times_times_set_nat @ ( times_times_set_nat @ A2 @ B3 ) @ C )
= ( times_times_set_nat @ A2 @ ( times_times_set_nat @ B3 @ C ) ) ) ).
% Groups.mult_ac(1)
thf(fact_160_Groups_Omult__ac_I1_J,axiom,
! [A2: set_complex,B3: set_complex,C: set_complex] :
( ( times_6048082448287401577omplex @ ( times_6048082448287401577omplex @ A2 @ B3 ) @ C )
= ( times_6048082448287401577omplex @ A2 @ ( times_6048082448287401577omplex @ B3 @ C ) ) ) ).
% Groups.mult_ac(1)
thf(fact_161_Groups_Omult__ac_I1_J,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( times_times_complex @ ( times_times_complex @ A2 @ B3 ) @ C )
= ( times_times_complex @ A2 @ ( times_times_complex @ B3 @ C ) ) ) ).
% Groups.mult_ac(1)
thf(fact_162_Groups_Omult__ac_I1_J,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B3 ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B3 @ C ) ) ) ).
% Groups.mult_ac(1)
thf(fact_163_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( times_times_set_nat @ ( times_times_set_nat @ A2 @ B3 ) @ C )
= ( times_times_set_nat @ A2 @ ( times_times_set_nat @ B3 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_164_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: set_complex,B3: set_complex,C: set_complex] :
( ( times_6048082448287401577omplex @ ( times_6048082448287401577omplex @ A2 @ B3 ) @ C )
= ( times_6048082448287401577omplex @ A2 @ ( times_6048082448287401577omplex @ B3 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_165_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( times_times_complex @ ( times_times_complex @ A2 @ B3 ) @ C )
= ( times_times_complex @ A2 @ ( times_times_complex @ B3 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_166_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B3 ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B3 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_167_Groups_Oadd__ac_I3_J,axiom,
! [B3: set_complex,A2: set_complex,C: set_complex] :
( ( plus_p7052360327008956141omplex @ B3 @ ( plus_p7052360327008956141omplex @ A2 @ C ) )
= ( plus_p7052360327008956141omplex @ A2 @ ( plus_p7052360327008956141omplex @ B3 @ C ) ) ) ).
% Groups.add_ac(3)
thf(fact_168_Groups_Oadd__ac_I3_J,axiom,
! [B3: set_nat,A2: set_nat,C: set_nat] :
( ( plus_plus_set_nat @ B3 @ ( plus_plus_set_nat @ A2 @ C ) )
= ( plus_plus_set_nat @ A2 @ ( plus_plus_set_nat @ B3 @ C ) ) ) ).
% Groups.add_ac(3)
thf(fact_169_Groups_Oadd__ac_I3_J,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% Groups.add_ac(3)
thf(fact_170_Groups_Oadd__ac_I2_J,axiom,
( plus_p7052360327008956141omplex
= ( ^ [A3: set_complex,B4: set_complex] : ( plus_p7052360327008956141omplex @ B4 @ A3 ) ) ) ).
% Groups.add_ac(2)
thf(fact_171_Groups_Oadd__ac_I2_J,axiom,
( plus_plus_set_nat
= ( ^ [A3: set_nat,B4: set_nat] : ( plus_plus_set_nat @ B4 @ A3 ) ) ) ).
% Groups.add_ac(2)
thf(fact_172_Groups_Oadd__ac_I2_J,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B4: nat] : ( plus_plus_nat @ B4 @ A3 ) ) ) ).
% Groups.add_ac(2)
thf(fact_173_Groups_Oadd__ac_I1_J,axiom,
! [A2: set_complex,B3: set_complex,C: set_complex] :
( ( plus_p7052360327008956141omplex @ ( plus_p7052360327008956141omplex @ A2 @ B3 ) @ C )
= ( plus_p7052360327008956141omplex @ A2 @ ( plus_p7052360327008956141omplex @ B3 @ C ) ) ) ).
% Groups.add_ac(1)
thf(fact_174_Groups_Oadd__ac_I1_J,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( plus_plus_set_nat @ ( plus_plus_set_nat @ A2 @ B3 ) @ C )
= ( plus_plus_set_nat @ A2 @ ( plus_plus_set_nat @ B3 @ C ) ) ) ).
% Groups.add_ac(1)
thf(fact_175_Groups_Oadd__ac_I1_J,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B3 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% Groups.add_ac(1)
thf(fact_176_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: set_complex,B3: set_complex,C: set_complex] :
( ( plus_p7052360327008956141omplex @ ( plus_p7052360327008956141omplex @ A2 @ B3 ) @ C )
= ( plus_p7052360327008956141omplex @ A2 @ ( plus_p7052360327008956141omplex @ B3 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_177_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( plus_plus_set_nat @ ( plus_plus_set_nat @ A2 @ B3 ) @ C )
= ( plus_plus_set_nat @ A2 @ ( plus_plus_set_nat @ B3 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_178_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B3 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_179_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( I3 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I3 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_180_group__cancel_Oadd1,axiom,
! [A: set_complex,K: set_complex,A2: set_complex,B3: set_complex] :
( ( A
= ( plus_p7052360327008956141omplex @ K @ A2 ) )
=> ( ( plus_p7052360327008956141omplex @ A @ B3 )
= ( plus_p7052360327008956141omplex @ K @ ( plus_p7052360327008956141omplex @ A2 @ B3 ) ) ) ) ).
% group_cancel.add1
thf(fact_181_group__cancel_Oadd1,axiom,
! [A: set_nat,K: set_nat,A2: set_nat,B3: set_nat] :
( ( A
= ( plus_plus_set_nat @ K @ A2 ) )
=> ( ( plus_plus_set_nat @ A @ B3 )
= ( plus_plus_set_nat @ K @ ( plus_plus_set_nat @ A2 @ B3 ) ) ) ) ).
% group_cancel.add1
thf(fact_182_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B3: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% group_cancel.add1
thf(fact_183_group__cancel_Oadd2,axiom,
! [B2: set_complex,K: set_complex,B3: set_complex,A2: set_complex] :
( ( B2
= ( plus_p7052360327008956141omplex @ K @ B3 ) )
=> ( ( plus_p7052360327008956141omplex @ A2 @ B2 )
= ( plus_p7052360327008956141omplex @ K @ ( plus_p7052360327008956141omplex @ A2 @ B3 ) ) ) ) ).
% group_cancel.add2
thf(fact_184_group__cancel_Oadd2,axiom,
! [B2: set_nat,K: set_nat,B3: set_nat,A2: set_nat] :
( ( B2
= ( plus_plus_set_nat @ K @ B3 ) )
=> ( ( plus_plus_set_nat @ A2 @ B2 )
= ( plus_plus_set_nat @ K @ ( plus_plus_set_nat @ A2 @ B3 ) ) ) ) ).
% group_cancel.add2
thf(fact_185_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B3: nat,A2: nat] :
( ( B2
= ( plus_plus_nat @ K @ B3 ) )
=> ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% group_cancel.add2
thf(fact_186_add__left__cancel,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B3 )
= ( plus_plus_nat @ A2 @ C ) )
= ( B3 = C ) ) ).
% add_left_cancel
thf(fact_187_add__left__imp__eq,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B3 )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B3 = C ) ) ).
% add_left_imp_eq
thf(fact_188_add__right__cancel,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B3 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B3 = C ) ) ).
% add_right_cancel
thf(fact_189_add__right__imp__eq,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B3 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B3 = C ) ) ).
% add_right_imp_eq
thf(fact_190_diff__eq__diff__eq,axiom,
! [A2: complex,B3: complex,C: complex,D2: complex] :
( ( ( minus_minus_complex @ A2 @ B3 )
= ( minus_minus_complex @ C @ D2 ) )
=> ( ( A2 = B3 )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_191_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: complex,C: complex,B3: complex] :
( ( minus_minus_complex @ ( minus_minus_complex @ A2 @ C ) @ B3 )
= ( minus_minus_complex @ ( minus_minus_complex @ A2 @ B3 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_192_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B3 )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B3 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_193_minus__minus,axiom,
! [A2: complex] :
( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A2 ) )
= A2 ) ).
% minus_minus
thf(fact_194_equation__minus__iff,axiom,
! [A2: complex,B3: complex] :
( ( A2
= ( uminus1482373934393186551omplex @ B3 ) )
= ( B3
= ( uminus1482373934393186551omplex @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_195_minus__equation__iff,axiom,
! [A2: complex,B3: complex] :
( ( ( uminus1482373934393186551omplex @ A2 )
= B3 )
= ( ( uminus1482373934393186551omplex @ B3 )
= A2 ) ) ).
% minus_equation_iff
thf(fact_196_neg__equal__iff__equal,axiom,
! [A2: complex,B3: complex] :
( ( ( uminus1482373934393186551omplex @ A2 )
= ( uminus1482373934393186551omplex @ B3 ) )
= ( A2 = B3 ) ) ).
% neg_equal_iff_equal
thf(fact_197_fixed__carrier__mat_Osum__mat__carrier,axiom,
! [Fc_mats2: set_mat_complex,DimR2: nat,DimC2: nat,I: set_mat_nat,A: mat_nat > mat_complex] :
( ( linear8738132868031958293omplex @ Fc_mats2 @ DimR2 @ DimC2 )
=> ( ! [I2: mat_nat] :
( ( member_mat_nat @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ Fc_mats2 ) )
=> ( member_mat_complex @ ( linear4360147844572819692omplex @ DimR2 @ DimC2 @ A @ I ) @ ( carrier_mat_complex @ DimR2 @ DimC2 ) ) ) ) ).
% fixed_carrier_mat.sum_mat_carrier
thf(fact_198_fixed__carrier__mat_Osum__mat__carrier,axiom,
! [Fc_mats2: set_mat_complex,DimR2: nat,DimC2: nat,I: set_nat,A: nat > mat_complex] :
( ( linear8738132868031958293omplex @ Fc_mats2 @ DimR2 @ DimC2 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ Fc_mats2 ) )
=> ( member_mat_complex @ ( linear8108877306658443851omplex @ DimR2 @ DimC2 @ A @ I ) @ ( carrier_mat_complex @ DimR2 @ DimC2 ) ) ) ) ).
% fixed_carrier_mat.sum_mat_carrier
thf(fact_199_fixed__carrier__mat_Osum__mat__carrier,axiom,
! [Fc_mats2: set_mat_complex,DimR2: nat,DimC2: nat,I: set_set_complex,A: set_complex > mat_complex] :
( ( linear8738132868031958293omplex @ Fc_mats2 @ DimR2 @ DimC2 )
=> ( ! [I2: set_complex] :
( ( member_set_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ Fc_mats2 ) )
=> ( member_mat_complex @ ( linear7804121962625877471omplex @ DimR2 @ DimC2 @ A @ I ) @ ( carrier_mat_complex @ DimR2 @ DimC2 ) ) ) ) ).
% fixed_carrier_mat.sum_mat_carrier
thf(fact_200_fixed__carrier__mat_Osum__mat__carrier,axiom,
! [Fc_mats2: set_mat_complex,DimR2: nat,DimC2: nat,I: set_set_mat_complex,A: set_mat_complex > mat_complex] :
( ( linear8738132868031958293omplex @ Fc_mats2 @ DimR2 @ DimC2 )
=> ( ! [I2: set_mat_complex] :
( ( member3612512168372279472omplex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ Fc_mats2 ) )
=> ( member_mat_complex @ ( linear5733585793752561962omplex @ DimR2 @ DimC2 @ A @ I ) @ ( carrier_mat_complex @ DimR2 @ DimC2 ) ) ) ) ).
% fixed_carrier_mat.sum_mat_carrier
thf(fact_201_fixed__carrier__mat_Osum__mat__carrier,axiom,
! [Fc_mats2: set_mat_complex,DimR2: nat,DimC2: nat,I: set_mat_complex,A: mat_complex > mat_complex] :
( ( linear8738132868031958293omplex @ Fc_mats2 @ DimR2 @ DimC2 )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ Fc_mats2 ) )
=> ( member_mat_complex @ ( linear1795808462385993418omplex @ DimR2 @ DimC2 @ A @ I ) @ ( carrier_mat_complex @ DimR2 @ DimC2 ) ) ) ) ).
% fixed_carrier_mat.sum_mat_carrier
thf(fact_202_fixed__carrier__mat_Osum__mat__carrier,axiom,
! [Fc_mats2: set_mat_complex,DimR2: nat,DimC2: nat,I: set_complex,A: complex > mat_complex] :
( ( linear8738132868031958293omplex @ Fc_mats2 @ DimR2 @ DimC2 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ Fc_mats2 ) )
=> ( member_mat_complex @ ( linear8664352376190006057omplex @ DimR2 @ DimC2 @ A @ I ) @ ( carrier_mat_complex @ DimR2 @ DimC2 ) ) ) ) ).
% fixed_carrier_mat.sum_mat_carrier
thf(fact_203_fixed__carrier__mat_Osum__mat__empty,axiom,
! [Fc_mats2: set_mat_complex,DimR2: nat,DimC2: nat,A: nat > mat_complex] :
( ( linear8738132868031958293omplex @ Fc_mats2 @ DimR2 @ DimC2 )
=> ( ( linear8108877306658443851omplex @ DimR2 @ DimC2 @ A @ bot_bot_set_nat )
= ( zero_mat_complex @ DimR2 @ DimC2 ) ) ) ).
% fixed_carrier_mat.sum_mat_empty
thf(fact_204_fixed__carrier__mat_Osum__mat__empty,axiom,
! [Fc_mats2: set_mat_complex,DimR2: nat,DimC2: nat,A: mat_complex > mat_complex] :
( ( linear8738132868031958293omplex @ Fc_mats2 @ DimR2 @ DimC2 )
=> ( ( linear1795808462385993418omplex @ DimR2 @ DimC2 @ A @ bot_bo7165004461764951667omplex )
= ( zero_mat_complex @ DimR2 @ DimC2 ) ) ) ).
% fixed_carrier_mat.sum_mat_empty
thf(fact_205_fixed__carrier__mat_Osum__mat__empty,axiom,
! [Fc_mats2: set_mat_complex,DimR2: nat,DimC2: nat,A: complex > mat_complex] :
( ( linear8738132868031958293omplex @ Fc_mats2 @ DimR2 @ DimC2 )
=> ( ( linear8664352376190006057omplex @ DimR2 @ DimC2 @ A @ bot_bot_set_complex )
= ( zero_mat_complex @ DimR2 @ DimC2 ) ) ) ).
% fixed_carrier_mat.sum_mat_empty
thf(fact_206_arithmetic__simps_I63_J,axiom,
! [A2: complex] :
( ( times_times_complex @ A2 @ zero_zero_complex )
= zero_zero_complex ) ).
% arithmetic_simps(63)
thf(fact_207_arithmetic__simps_I63_J,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% arithmetic_simps(63)
thf(fact_208_arithmetic__simps_I62_J,axiom,
! [A2: complex] :
( ( times_times_complex @ zero_zero_complex @ A2 )
= zero_zero_complex ) ).
% arithmetic_simps(62)
thf(fact_209_arithmetic__simps_I62_J,axiom,
! [A2: nat] :
( ( times_times_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% arithmetic_simps(62)
thf(fact_210_mult__not__zero,axiom,
! [A2: complex,B3: complex] :
( ( ( times_times_complex @ A2 @ B3 )
!= zero_zero_complex )
=> ( ( A2 != zero_zero_complex )
& ( B3 != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_211_mult__not__zero,axiom,
! [A2: nat,B3: nat] :
( ( ( times_times_nat @ A2 @ B3 )
!= zero_zero_nat )
=> ( ( A2 != zero_zero_nat )
& ( B3 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_212_divisors__zero,axiom,
! [A2: complex,B3: complex] :
( ( ( times_times_complex @ A2 @ B3 )
= zero_zero_complex )
=> ( ( A2 = zero_zero_complex )
| ( B3 = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_213_divisors__zero,axiom,
! [A2: nat,B3: nat] :
( ( ( times_times_nat @ A2 @ B3 )
= zero_zero_nat )
=> ( ( A2 = zero_zero_nat )
| ( B3 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_214_mult__eq__0__iff,axiom,
! [A2: complex,B3: complex] :
( ( ( times_times_complex @ A2 @ B3 )
= zero_zero_complex )
= ( ( A2 = zero_zero_complex )
| ( B3 = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_215_mult__eq__0__iff,axiom,
! [A2: nat,B3: nat] :
( ( ( times_times_nat @ A2 @ B3 )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
| ( B3 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_216_no__zero__divisors,axiom,
! [A2: complex,B3: complex] :
( ( A2 != zero_zero_complex )
=> ( ( B3 != zero_zero_complex )
=> ( ( times_times_complex @ A2 @ B3 )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_217_no__zero__divisors,axiom,
! [A2: nat,B3: nat] :
( ( A2 != zero_zero_nat )
=> ( ( B3 != zero_zero_nat )
=> ( ( times_times_nat @ A2 @ B3 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_218_mult__cancel__left,axiom,
! [C: complex,A2: complex,B3: complex] :
( ( ( times_times_complex @ C @ A2 )
= ( times_times_complex @ C @ B3 ) )
= ( ( C = zero_zero_complex )
| ( A2 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_219_mult__cancel__left,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B3 ) )
= ( ( C = zero_zero_nat )
| ( A2 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_220_mult__left__cancel,axiom,
! [C: complex,A2: complex,B3: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ C @ A2 )
= ( times_times_complex @ C @ B3 ) )
= ( A2 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_221_mult__left__cancel,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B3 ) )
= ( A2 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_222_mult__cancel__right,axiom,
! [A2: complex,C: complex,B3: complex] :
( ( ( times_times_complex @ A2 @ C )
= ( times_times_complex @ B3 @ C ) )
= ( ( C = zero_zero_complex )
| ( A2 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_223_mult__cancel__right,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B3 @ C ) )
= ( ( C = zero_zero_nat )
| ( A2 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_224_mult__right__cancel,axiom,
! [C: complex,A2: complex,B3: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ A2 @ C )
= ( times_times_complex @ B3 @ C ) )
= ( A2 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_225_mult__right__cancel,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B3 @ C ) )
= ( A2 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_226_zero__order_I5_J,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% zero_order(5)
thf(fact_227_zero__order_I4_J,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_order(4)
thf(fact_228_zero__order_I3_J,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% zero_order(3)
thf(fact_229_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_230_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_231_pth__7_I2_J,axiom,
! [X: complex] :
( ( plus_plus_complex @ X @ zero_zero_complex )
= X ) ).
% pth_7(2)
thf(fact_232_pth__7_I1_J,axiom,
! [X: complex] :
( ( plus_plus_complex @ zero_zero_complex @ X )
= X ) ).
% pth_7(1)
thf(fact_233_arithmetic__simps_I50_J,axiom,
! [A2: set_complex] :
( ( plus_p7052360327008956141omplex @ A2 @ zero_z6614145512433583213omplex )
= A2 ) ).
% arithmetic_simps(50)
thf(fact_234_arithmetic__simps_I50_J,axiom,
! [A2: set_nat] :
( ( plus_plus_set_nat @ A2 @ zero_zero_set_nat )
= A2 ) ).
% arithmetic_simps(50)
thf(fact_235_arithmetic__simps_I50_J,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% arithmetic_simps(50)
thf(fact_236_arithmetic__simps_I50_J,axiom,
! [A2: complex] :
( ( plus_plus_complex @ A2 @ zero_zero_complex )
= A2 ) ).
% arithmetic_simps(50)
thf(fact_237_arithmetic__simps_I49_J,axiom,
! [A2: set_complex] :
( ( plus_p7052360327008956141omplex @ zero_z6614145512433583213omplex @ A2 )
= A2 ) ).
% arithmetic_simps(49)
thf(fact_238_arithmetic__simps_I49_J,axiom,
! [A2: set_nat] :
( ( plus_plus_set_nat @ zero_zero_set_nat @ A2 )
= A2 ) ).
% arithmetic_simps(49)
thf(fact_239_arithmetic__simps_I49_J,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% arithmetic_simps(49)
thf(fact_240_arithmetic__simps_I49_J,axiom,
! [A2: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A2 )
= A2 ) ).
% arithmetic_simps(49)
thf(fact_241_group__cancel_Orule0,axiom,
! [A2: set_complex] :
( A2
= ( plus_p7052360327008956141omplex @ A2 @ zero_z6614145512433583213omplex ) ) ).
% group_cancel.rule0
thf(fact_242_group__cancel_Orule0,axiom,
! [A2: set_nat] :
( A2
= ( plus_plus_set_nat @ A2 @ zero_zero_set_nat ) ) ).
% group_cancel.rule0
thf(fact_243_group__cancel_Orule0,axiom,
! [A2: nat] :
( A2
= ( plus_plus_nat @ A2 @ zero_zero_nat ) ) ).
% group_cancel.rule0
thf(fact_244_group__cancel_Orule0,axiom,
! [A2: complex] :
( A2
= ( plus_plus_complex @ A2 @ zero_zero_complex ) ) ).
% group_cancel.rule0
thf(fact_245_comm__monoid__add__class_Oadd__0,axiom,
! [A2: set_complex] :
( ( plus_p7052360327008956141omplex @ zero_z6614145512433583213omplex @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_246_comm__monoid__add__class_Oadd__0,axiom,
! [A2: set_nat] :
( ( plus_plus_set_nat @ zero_zero_set_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_247_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_248_comm__monoid__add__class_Oadd__0,axiom,
! [A2: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_249_add_Ogroup__left__neutral,axiom,
! [A2: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_250_add__cancel__left__left,axiom,
! [B3: nat,A2: nat] :
( ( ( plus_plus_nat @ B3 @ A2 )
= A2 )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_251_add__cancel__left__left,axiom,
! [B3: complex,A2: complex] :
( ( ( plus_plus_complex @ B3 @ A2 )
= A2 )
= ( B3 = zero_zero_complex ) ) ).
% add_cancel_left_left
thf(fact_252_add__cancel__left__right,axiom,
! [A2: nat,B3: nat] :
( ( ( plus_plus_nat @ A2 @ B3 )
= A2 )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_253_add__cancel__left__right,axiom,
! [A2: complex,B3: complex] :
( ( ( plus_plus_complex @ A2 @ B3 )
= A2 )
= ( B3 = zero_zero_complex ) ) ).
% add_cancel_left_right
thf(fact_254_add__cancel__right__left,axiom,
! [A2: nat,B3: nat] :
( ( A2
= ( plus_plus_nat @ B3 @ A2 ) )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_255_add__cancel__right__left,axiom,
! [A2: complex,B3: complex] :
( ( A2
= ( plus_plus_complex @ B3 @ A2 ) )
= ( B3 = zero_zero_complex ) ) ).
% add_cancel_right_left
thf(fact_256_add__cancel__right__right,axiom,
! [A2: nat,B3: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B3 ) )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_257_add__cancel__right__right,axiom,
! [A2: complex,B3: complex] :
( ( A2
= ( plus_plus_complex @ A2 @ B3 ) )
= ( B3 = zero_zero_complex ) ) ).
% add_cancel_right_right
thf(fact_258_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_259_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y2 ) )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_260_arithmetic__simps_I57_J,axiom,
! [A2: complex] :
( ( minus_minus_complex @ A2 @ zero_zero_complex )
= A2 ) ).
% arithmetic_simps(57)
thf(fact_261_diff__self,axiom,
! [A2: complex] :
( ( minus_minus_complex @ A2 @ A2 )
= zero_zero_complex ) ).
% diff_self
thf(fact_262_right__minus__eq,axiom,
! [A2: complex,B3: complex] :
( ( ( minus_minus_complex @ A2 @ B3 )
= zero_zero_complex )
= ( A2 = B3 ) ) ).
% right_minus_eq
thf(fact_263_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_264_diff__zero,axiom,
! [A2: complex] :
( ( minus_minus_complex @ A2 @ zero_zero_complex )
= A2 ) ).
% diff_zero
thf(fact_265_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_266_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: complex] :
( ( minus_minus_complex @ A2 @ A2 )
= zero_zero_complex ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_267_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_268_Rings_Oring__distribs_I2_J,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A2 @ B3 ) @ C )
= ( plus_plus_complex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B3 @ C ) ) ) ).
% Rings.ring_distribs(2)
thf(fact_269_Rings_Oring__distribs_I2_J,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B3 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ).
% Rings.ring_distribs(2)
thf(fact_270_Rings_Oring__distribs_I1_J,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( times_times_complex @ A2 @ ( plus_plus_complex @ B3 @ C ) )
= ( plus_plus_complex @ ( times_times_complex @ A2 @ B3 ) @ ( times_times_complex @ A2 @ C ) ) ) ).
% Rings.ring_distribs(1)
thf(fact_271_Rings_Oring__distribs_I1_J,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( times_times_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ B3 ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% Rings.ring_distribs(1)
thf(fact_272_ring__class_Oring__distribs_I2_J,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A2 @ B3 ) @ C )
= ( plus_plus_complex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B3 @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_273_ring__class_Oring__distribs_I1_J,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( times_times_complex @ A2 @ ( plus_plus_complex @ B3 @ C ) )
= ( plus_plus_complex @ ( times_times_complex @ A2 @ B3 ) @ ( times_times_complex @ A2 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_274_comm__semiring__class_Odistrib,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A2 @ B3 ) @ C )
= ( plus_plus_complex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B3 @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_275_comm__semiring__class_Odistrib,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B3 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_276_combine__common__factor,axiom,
! [A2: complex,E: complex,B3: complex,C: complex] :
( ( plus_plus_complex @ ( times_times_complex @ A2 @ E ) @ ( plus_plus_complex @ ( times_times_complex @ B3 @ E ) @ C ) )
= ( plus_plus_complex @ ( times_times_complex @ ( plus_plus_complex @ A2 @ B3 ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_277_combine__common__factor,axiom,
! [A2: nat,E: nat,B3: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A2 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B3 @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B3 ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_278_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_complex @ I3 @ J )
& ( ord_less_complex @ K @ L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_279_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_280_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( I3 = J )
& ( ord_less_complex @ K @ L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_281_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( I3 = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_282_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_complex @ I3 @ J )
& ( K = L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_283_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_284_add__strict__mono,axiom,
! [A2: complex,B3: complex,C: complex,D2: complex] :
( ( ord_less_complex @ A2 @ B3 )
=> ( ( ord_less_complex @ C @ D2 )
=> ( ord_less_complex @ ( plus_plus_complex @ A2 @ C ) @ ( plus_plus_complex @ B3 @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_285_add__strict__mono,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_286_add__less__cancel__left,axiom,
! [C: complex,A2: complex,B3: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ C @ A2 ) @ ( plus_plus_complex @ C @ B3 ) )
= ( ord_less_complex @ A2 @ B3 ) ) ).
% add_less_cancel_left
thf(fact_287_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
= ( ord_less_nat @ A2 @ B3 ) ) ).
% add_less_cancel_left
thf(fact_288_add__strict__left__mono,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( ord_less_complex @ A2 @ B3 )
=> ( ord_less_complex @ ( plus_plus_complex @ C @ A2 ) @ ( plus_plus_complex @ C @ B3 ) ) ) ).
% add_strict_left_mono
thf(fact_289_add__strict__left__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).
% add_strict_left_mono
thf(fact_290_add__less__cancel__right,axiom,
! [A2: complex,C: complex,B3: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ A2 @ C ) @ ( plus_plus_complex @ B3 @ C ) )
= ( ord_less_complex @ A2 @ B3 ) ) ).
% add_less_cancel_right
thf(fact_291_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
= ( ord_less_nat @ A2 @ B3 ) ) ).
% add_less_cancel_right
thf(fact_292_add__strict__right__mono,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( ord_less_complex @ A2 @ B3 )
=> ( ord_less_complex @ ( plus_plus_complex @ A2 @ C ) @ ( plus_plus_complex @ B3 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_293_add__strict__right__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_294_add__less__imp__less__left,axiom,
! [C: complex,A2: complex,B3: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ C @ A2 ) @ ( plus_plus_complex @ C @ B3 ) )
=> ( ord_less_complex @ A2 @ B3 ) ) ).
% add_less_imp_less_left
thf(fact_295_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
=> ( ord_less_nat @ A2 @ B3 ) ) ).
% add_less_imp_less_left
thf(fact_296_add__less__imp__less__right,axiom,
! [A2: complex,C: complex,B3: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ A2 @ C ) @ ( plus_plus_complex @ B3 @ C ) )
=> ( ord_less_complex @ A2 @ B3 ) ) ).
% add_less_imp_less_right
thf(fact_297_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
=> ( ord_less_nat @ A2 @ B3 ) ) ).
% add_less_imp_less_right
thf(fact_298_arithmetic__simps_I51_J,axiom,
( ( uminus1482373934393186551omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% arithmetic_simps(51)
thf(fact_299_neg__0__equal__iff__equal,axiom,
! [A2: complex] :
( ( zero_zero_complex
= ( uminus1482373934393186551omplex @ A2 ) )
= ( zero_zero_complex = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_300_neg__equal__0__iff__equal,axiom,
! [A2: complex] :
( ( ( uminus1482373934393186551omplex @ A2 )
= zero_zero_complex )
= ( A2 = zero_zero_complex ) ) ).
% neg_equal_0_iff_equal
thf(fact_301_Rings_Oring__distribs_I4_J,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( times_times_complex @ A2 @ ( minus_minus_complex @ B3 @ C ) )
= ( minus_minus_complex @ ( times_times_complex @ A2 @ B3 ) @ ( times_times_complex @ A2 @ C ) ) ) ).
% Rings.ring_distribs(4)
thf(fact_302_Rings_Oring__distribs_I3_J,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( times_times_complex @ ( minus_minus_complex @ A2 @ B3 ) @ C )
= ( minus_minus_complex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B3 @ C ) ) ) ).
% Rings.ring_distribs(3)
thf(fact_303_left__diff__distrib_H,axiom,
! [B3: complex,C: complex,A2: complex] :
( ( times_times_complex @ ( minus_minus_complex @ B3 @ C ) @ A2 )
= ( minus_minus_complex @ ( times_times_complex @ B3 @ A2 ) @ ( times_times_complex @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_304_left__diff__distrib_H,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B3 @ C ) @ A2 )
= ( minus_minus_nat @ ( times_times_nat @ B3 @ A2 ) @ ( times_times_nat @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_305_right__diff__distrib_H,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( times_times_complex @ A2 @ ( minus_minus_complex @ B3 @ C ) )
= ( minus_minus_complex @ ( times_times_complex @ A2 @ B3 ) @ ( times_times_complex @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_306_right__diff__distrib_H,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( times_times_nat @ A2 @ ( minus_minus_nat @ B3 @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A2 @ B3 ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_307_diff__strict__mono,axiom,
! [A2: complex,B3: complex,D2: complex,C: complex] :
( ( ord_less_complex @ A2 @ B3 )
=> ( ( ord_less_complex @ D2 @ C )
=> ( ord_less_complex @ ( minus_minus_complex @ A2 @ C ) @ ( minus_minus_complex @ B3 @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_308_diff__eq__diff__less,axiom,
! [A2: complex,B3: complex,C: complex,D2: complex] :
( ( ( minus_minus_complex @ A2 @ B3 )
= ( minus_minus_complex @ C @ D2 ) )
=> ( ( ord_less_complex @ A2 @ B3 )
= ( ord_less_complex @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_309_diff__strict__left__mono,axiom,
! [B3: complex,A2: complex,C: complex] :
( ( ord_less_complex @ B3 @ A2 )
=> ( ord_less_complex @ ( minus_minus_complex @ C @ A2 ) @ ( minus_minus_complex @ C @ B3 ) ) ) ).
% diff_strict_left_mono
thf(fact_310_diff__strict__right__mono,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( ord_less_complex @ A2 @ B3 )
=> ( ord_less_complex @ ( minus_minus_complex @ A2 @ C ) @ ( minus_minus_complex @ B3 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_311_group__cancel_Osub1,axiom,
! [A: complex,K: complex,A2: complex,B3: complex] :
( ( A
= ( plus_plus_complex @ K @ A2 ) )
=> ( ( minus_minus_complex @ A @ B3 )
= ( plus_plus_complex @ K @ ( minus_minus_complex @ A2 @ B3 ) ) ) ) ).
% group_cancel.sub1
thf(fact_312_diff__eq__eq,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( ( minus_minus_complex @ A2 @ B3 )
= C )
= ( A2
= ( plus_plus_complex @ C @ B3 ) ) ) ).
% diff_eq_eq
thf(fact_313_eq__diff__eq,axiom,
! [A2: complex,C: complex,B3: complex] :
( ( A2
= ( minus_minus_complex @ C @ B3 ) )
= ( ( plus_plus_complex @ A2 @ B3 )
= C ) ) ).
% eq_diff_eq
thf(fact_314_add__diff__eq,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( plus_plus_complex @ A2 @ ( minus_minus_complex @ B3 @ C ) )
= ( minus_minus_complex @ ( plus_plus_complex @ A2 @ B3 ) @ C ) ) ).
% add_diff_eq
thf(fact_315_diff__diff__eq2,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( minus_minus_complex @ A2 @ ( minus_minus_complex @ B3 @ C ) )
= ( minus_minus_complex @ ( plus_plus_complex @ A2 @ C ) @ B3 ) ) ).
% diff_diff_eq2
thf(fact_316_diff__add__eq,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( plus_plus_complex @ ( minus_minus_complex @ A2 @ B3 ) @ C )
= ( minus_minus_complex @ ( plus_plus_complex @ A2 @ C ) @ B3 ) ) ).
% diff_add_eq
thf(fact_317_add__diff__cancel,axiom,
! [A2: complex,B3: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A2 @ B3 ) @ B3 )
= A2 ) ).
% add_diff_cancel
thf(fact_318_diff__add__cancel,axiom,
! [A2: complex,B3: complex] :
( ( plus_plus_complex @ ( minus_minus_complex @ A2 @ B3 ) @ B3 )
= A2 ) ).
% diff_add_cancel
thf(fact_319_diff__add__eq__diff__diff__swap,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( minus_minus_complex @ A2 @ ( plus_plus_complex @ B3 @ C ) )
= ( minus_minus_complex @ ( minus_minus_complex @ A2 @ C ) @ B3 ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_320_diff__diff__add,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( minus_minus_complex @ ( minus_minus_complex @ A2 @ B3 ) @ C )
= ( minus_minus_complex @ A2 @ ( plus_plus_complex @ B3 @ C ) ) ) ).
% diff_diff_add
thf(fact_321_diff__diff__add,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B3 ) @ C )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% diff_diff_add
thf(fact_322_add__implies__diff,axiom,
! [C: complex,B3: complex,A2: complex] :
( ( ( plus_plus_complex @ C @ B3 )
= A2 )
=> ( C
= ( minus_minus_complex @ A2 @ B3 ) ) ) ).
% add_implies_diff
thf(fact_323_add__implies__diff,axiom,
! [C: nat,B3: nat,A2: nat] :
( ( ( plus_plus_nat @ C @ B3 )
= A2 )
=> ( C
= ( minus_minus_nat @ A2 @ B3 ) ) ) ).
% add_implies_diff
thf(fact_324_add__diff__cancel__left,axiom,
! [C: complex,A2: complex,B3: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ C @ A2 ) @ ( plus_plus_complex @ C @ B3 ) )
= ( minus_minus_complex @ A2 @ B3 ) ) ).
% add_diff_cancel_left
thf(fact_325_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
= ( minus_minus_nat @ A2 @ B3 ) ) ).
% add_diff_cancel_left
thf(fact_326_add__diff__cancel__left_H,axiom,
! [A2: complex,B3: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A2 @ B3 ) @ A2 )
= B3 ) ).
% add_diff_cancel_left'
thf(fact_327_add__diff__cancel__left_H,axiom,
! [A2: nat,B3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B3 ) @ A2 )
= B3 ) ).
% add_diff_cancel_left'
thf(fact_328_add__diff__cancel__right,axiom,
! [A2: complex,C: complex,B3: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A2 @ C ) @ ( plus_plus_complex @ B3 @ C ) )
= ( minus_minus_complex @ A2 @ B3 ) ) ).
% add_diff_cancel_right
thf(fact_329_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
= ( minus_minus_nat @ A2 @ B3 ) ) ).
% add_diff_cancel_right
thf(fact_330_add__diff__cancel__right_H,axiom,
! [A2: complex,B3: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A2 @ B3 ) @ B3 )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_331_add__diff__cancel__right_H,axiom,
! [A2: nat,B3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B3 ) @ B3 )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_332_square__eq__iff,axiom,
! [A2: complex,B3: complex] :
( ( ( times_times_complex @ A2 @ A2 )
= ( times_times_complex @ B3 @ B3 ) )
= ( ( A2 = B3 )
| ( A2
= ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).
% square_eq_iff
thf(fact_333_minus__mult__left,axiom,
! [A2: complex,B3: complex] :
( ( uminus1482373934393186551omplex @ ( times_times_complex @ A2 @ B3 ) )
= ( times_times_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B3 ) ) ).
% minus_mult_left
thf(fact_334_minus__mult__minus,axiom,
! [A2: complex,B3: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( uminus1482373934393186551omplex @ B3 ) )
= ( times_times_complex @ A2 @ B3 ) ) ).
% minus_mult_minus
thf(fact_335_minus__mult__right,axiom,
! [A2: complex,B3: complex] :
( ( uminus1482373934393186551omplex @ ( times_times_complex @ A2 @ B3 ) )
= ( times_times_complex @ A2 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ).
% minus_mult_right
thf(fact_336_minus__mult__commute,axiom,
! [A2: complex,B3: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B3 )
= ( times_times_complex @ A2 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ).
% minus_mult_commute
thf(fact_337_less__minus__iff,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_complex @ A2 @ ( uminus1482373934393186551omplex @ B3 ) )
= ( ord_less_complex @ B3 @ ( uminus1482373934393186551omplex @ A2 ) ) ) ).
% less_minus_iff
thf(fact_338_minus__less__iff,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B3 )
= ( ord_less_complex @ ( uminus1482373934393186551omplex @ B3 ) @ A2 ) ) ).
% minus_less_iff
thf(fact_339_neg__less__iff__less,axiom,
! [B3: complex,A2: complex] :
( ( ord_less_complex @ ( uminus1482373934393186551omplex @ B3 ) @ ( uminus1482373934393186551omplex @ A2 ) )
= ( ord_less_complex @ A2 @ B3 ) ) ).
% neg_less_iff_less
thf(fact_340_group__cancel_Oneg1,axiom,
! [A: complex,K: complex,A2: complex] :
( ( A
= ( plus_plus_complex @ K @ A2 ) )
=> ( ( uminus1482373934393186551omplex @ A )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A2 ) ) ) ) ).
% group_cancel.neg1
thf(fact_341_minus__add,axiom,
! [A2: complex,B3: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A2 @ B3 ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B3 ) @ ( uminus1482373934393186551omplex @ A2 ) ) ) ).
% minus_add
thf(fact_342_add__minus__cancel,axiom,
! [A2: complex,B3: complex] :
( ( plus_plus_complex @ A2 @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B3 ) )
= B3 ) ).
% add_minus_cancel
thf(fact_343_minus__add__cancel,axiom,
! [A2: complex,B3: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( plus_plus_complex @ A2 @ B3 ) )
= B3 ) ).
% minus_add_cancel
thf(fact_344_minus__add__distrib,axiom,
! [A2: complex,B3: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A2 @ B3 ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( uminus1482373934393186551omplex @ B3 ) ) ) ).
% minus_add_distrib
thf(fact_345_minus__diff__eq,axiom,
! [A2: complex,B3: complex] :
( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A2 @ B3 ) )
= ( minus_minus_complex @ B3 @ A2 ) ) ).
% minus_diff_eq
thf(fact_346_minus__diff__commute,axiom,
! [B3: complex,A2: complex] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B3 ) @ A2 )
= ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B3 ) ) ).
% minus_diff_commute
thf(fact_347_mult__sign__intros_I7_J,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(7)
thf(fact_348_mult__sign__intros_I6_J,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(6)
thf(fact_349_mult__sign__intros_I5_J,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B3 ) ) ) ) ).
% mult_sign_intros(5)
thf(fact_350_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B3 @ A2 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_351_zero__less__mult__pos,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% zero_less_mult_pos
thf(fact_352_zero__less__mult__pos2,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B3 @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% zero_less_mult_pos2
thf(fact_353_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_354_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_355_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_356_add__sign__intros_I6_J,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_complex @ A2 @ zero_zero_complex )
=> ( ( ord_less_complex @ B3 @ zero_zero_complex )
=> ( ord_less_complex @ ( plus_plus_complex @ A2 @ B3 ) @ zero_zero_complex ) ) ) ).
% add_sign_intros(6)
thf(fact_357_add__sign__intros_I6_J,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_sign_intros(6)
thf(fact_358_add__sign__intros_I2_J,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_complex @ zero_zero_complex @ A2 )
=> ( ( ord_less_complex @ zero_zero_complex @ B3 )
=> ( ord_less_complex @ zero_zero_complex @ ( plus_plus_complex @ A2 @ B3 ) ) ) ) ).
% add_sign_intros(2)
thf(fact_359_add__sign__intros_I2_J,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% add_sign_intros(2)
thf(fact_360_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ~ ! [C3: nat] :
( ( B3
= ( plus_plus_nat @ A2 @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_361_pos__add__strict,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( ord_less_complex @ zero_zero_complex @ A2 )
=> ( ( ord_less_complex @ B3 @ C )
=> ( ord_less_complex @ B3 @ ( plus_plus_complex @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_362_pos__add__strict,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_363_add__less__same__cancel1,axiom,
! [B3: complex,A2: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ B3 @ A2 ) @ B3 )
= ( ord_less_complex @ A2 @ zero_zero_complex ) ) ).
% add_less_same_cancel1
thf(fact_364_add__less__same__cancel1,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B3 @ A2 ) @ B3 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_365_add__less__same__cancel2,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ A2 @ B3 ) @ B3 )
= ( ord_less_complex @ A2 @ zero_zero_complex ) ) ).
% add_less_same_cancel2
thf(fact_366_add__less__same__cancel2,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ B3 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_367_less__add__same__cancel1,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_complex @ A2 @ ( plus_plus_complex @ A2 @ B3 ) )
= ( ord_less_complex @ zero_zero_complex @ B3 ) ) ).
% less_add_same_cancel1
thf(fact_368_less__add__same__cancel1,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B3 ) )
= ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% less_add_same_cancel1
thf(fact_369_less__add__same__cancel2,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_complex @ A2 @ ( plus_plus_complex @ B3 @ A2 ) )
= ( ord_less_complex @ zero_zero_complex @ B3 ) ) ).
% less_add_same_cancel2
thf(fact_370_less__add__same__cancel2,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B3 @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% less_add_same_cancel2
thf(fact_371_diff__gt__0__iff__gt,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_complex @ zero_zero_complex @ ( minus_minus_complex @ A2 @ B3 ) )
= ( ord_less_complex @ B3 @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_372_diff__less__0__iff__less,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_complex @ ( minus_minus_complex @ A2 @ B3 ) @ zero_zero_complex )
= ( ord_less_complex @ A2 @ B3 ) ) ).
% diff_less_0_iff_less
thf(fact_373_diff__add__zero,axiom,
! [A2: nat,B3: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B3 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_374_neg__0__less__iff__less,axiom,
! [A2: complex] :
( ( ord_less_complex @ zero_zero_complex @ ( uminus1482373934393186551omplex @ A2 ) )
= ( ord_less_complex @ A2 @ zero_zero_complex ) ) ).
% neg_0_less_iff_less
thf(fact_375_neg__less__0__iff__less,axiom,
! [A2: complex] :
( ( ord_less_complex @ ( uminus1482373934393186551omplex @ A2 ) @ zero_zero_complex )
= ( ord_less_complex @ zero_zero_complex @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_376_left__minus,axiom,
! [A2: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ A2 )
= zero_zero_complex ) ).
% left_minus
thf(fact_377_right__minus,axiom,
! [A2: complex] :
( ( plus_plus_complex @ A2 @ ( uminus1482373934393186551omplex @ A2 ) )
= zero_zero_complex ) ).
% right_minus
thf(fact_378_add__eq__0__iff,axiom,
! [A2: complex,B3: complex] :
( ( ( plus_plus_complex @ A2 @ B3 )
= zero_zero_complex )
= ( B3
= ( uminus1482373934393186551omplex @ A2 ) ) ) ).
% add_eq_0_iff
thf(fact_379_minus__unique,axiom,
! [A2: complex,B3: complex] :
( ( ( plus_plus_complex @ A2 @ B3 )
= zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ A2 )
= B3 ) ) ).
% minus_unique
thf(fact_380_add__eq__0__iff2,axiom,
! [A2: complex,B3: complex] :
( ( ( plus_plus_complex @ A2 @ B3 )
= zero_zero_complex )
= ( A2
= ( uminus1482373934393186551omplex @ B3 ) ) ) ).
% add_eq_0_iff2
thf(fact_381_ab__group__add__class_Oab__left__minus,axiom,
! [A2: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ A2 )
= zero_zero_complex ) ).
% ab_group_add_class.ab_left_minus
thf(fact_382_neg__eq__iff__add__eq__0,axiom,
! [A2: complex,B3: complex] :
( ( ( uminus1482373934393186551omplex @ A2 )
= B3 )
= ( ( plus_plus_complex @ A2 @ B3 )
= zero_zero_complex ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_383_eq__add__iff1,axiom,
! [A2: complex,E: complex,C: complex,B3: complex,D2: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ A2 @ E ) @ C )
= ( plus_plus_complex @ ( times_times_complex @ B3 @ E ) @ D2 ) )
= ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A2 @ B3 ) @ E ) @ C )
= D2 ) ) ).
% eq_add_iff1
thf(fact_384_eq__add__iff2,axiom,
! [A2: complex,E: complex,C: complex,B3: complex,D2: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ A2 @ E ) @ C )
= ( plus_plus_complex @ ( times_times_complex @ B3 @ E ) @ D2 ) )
= ( C
= ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B3 @ A2 ) @ E ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_385_square__diff__square__factored,axiom,
! [X: complex,Y2: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ ( times_times_complex @ Y2 @ Y2 ) )
= ( times_times_complex @ ( plus_plus_complex @ X @ Y2 ) @ ( minus_minus_complex @ X @ Y2 ) ) ) ).
% square_diff_square_factored
thf(fact_386_diff__less__eq,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( ord_less_complex @ ( minus_minus_complex @ A2 @ B3 ) @ C )
= ( ord_less_complex @ A2 @ ( plus_plus_complex @ C @ B3 ) ) ) ).
% diff_less_eq
thf(fact_387_less__diff__eq,axiom,
! [A2: complex,C: complex,B3: complex] :
( ( ord_less_complex @ A2 @ ( minus_minus_complex @ C @ B3 ) )
= ( ord_less_complex @ ( plus_plus_complex @ A2 @ B3 ) @ C ) ) ).
% less_diff_eq
thf(fact_388_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: nat,B3: nat] :
( ~ ( ord_less_nat @ A2 @ B3 )
=> ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A2 @ B3 ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_389_arithmetic__simps_I56_J,axiom,
! [A2: complex] :
( ( minus_minus_complex @ zero_zero_complex @ A2 )
= ( uminus1482373934393186551omplex @ A2 ) ) ).
% arithmetic_simps(56)
thf(fact_390_group__cancel_Osub2,axiom,
! [B2: complex,K: complex,B3: complex,A2: complex] :
( ( B2
= ( plus_plus_complex @ K @ B3 ) )
=> ( ( minus_minus_complex @ A2 @ B2 )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A2 @ B3 ) ) ) ) ).
% group_cancel.sub2
thf(fact_391_diff__minus__eq__add,axiom,
! [A2: complex,B3: complex] :
( ( minus_minus_complex @ A2 @ ( uminus1482373934393186551omplex @ B3 ) )
= ( plus_plus_complex @ A2 @ B3 ) ) ).
% diff_minus_eq_add
thf(fact_392_local_Osum__mat__def,axiom,
! [A: complex > mat_complex,I: set_complex] :
( ( linear8664352376190006057omplex @ n @ n @ A @ I )
= ( group_4310869077125956188omplex @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ A @ I ) ) ).
% local.sum_mat_def
thf(fact_393_local_Osum__mat__def,axiom,
! [A: mat_complex > mat_complex,I: set_mat_complex] :
( ( linear1795808462385993418omplex @ n @ n @ A @ I )
= ( group_1588376139278055545omplex @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ A @ I ) ) ).
% local.sum_mat_def
thf(fact_394_sum__mat__left__ortho__zero,axiom,
! [I: set_mat_nat,A: mat_nat > mat_complex,B2: mat_complex] :
( ( finite9067146157826639218at_nat @ I )
=> ( ! [I2: mat_nat] :
( ( member_mat_nat @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( ! [I2: mat_nat] :
( ( member_mat_nat @ I2 @ I )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ B2 )
= ( zero_mat_complex @ n @ n ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear4360147844572819692omplex @ n @ n @ A @ I ) @ B2 )
= ( zero_mat_complex @ n @ n ) ) ) ) ) ) ).
% sum_mat_left_ortho_zero
thf(fact_395_sum__mat__left__ortho__zero,axiom,
! [I: set_set_complex,A: set_complex > mat_complex,B2: mat_complex] :
( ( finite6551019134538273531omplex @ I )
=> ( ! [I2: set_complex] :
( ( member_set_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( ! [I2: set_complex] :
( ( member_set_complex @ I2 @ I )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ B2 )
= ( zero_mat_complex @ n @ n ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear7804121962625877471omplex @ n @ n @ A @ I ) @ B2 )
= ( zero_mat_complex @ n @ n ) ) ) ) ) ) ).
% sum_mat_left_ortho_zero
thf(fact_396_sum__mat__left__ortho__zero,axiom,
! [I: set_set_mat_complex,A: set_mat_complex > mat_complex,B2: mat_complex] :
( ( finite1349200545324696496omplex @ I )
=> ( ! [I2: set_mat_complex] :
( ( member3612512168372279472omplex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( ! [I2: set_mat_complex] :
( ( member3612512168372279472omplex @ I2 @ I )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ B2 )
= ( zero_mat_complex @ n @ n ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear5733585793752561962omplex @ n @ n @ A @ I ) @ B2 )
= ( zero_mat_complex @ n @ n ) ) ) ) ) ) ).
% sum_mat_left_ortho_zero
thf(fact_397_sum__mat__left__ortho__zero,axiom,
! [I: set_nat,A: nat > mat_complex,B2: mat_complex] :
( ( finite_finite_nat @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ B2 )
= ( zero_mat_complex @ n @ n ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear8108877306658443851omplex @ n @ n @ A @ I ) @ B2 )
= ( zero_mat_complex @ n @ n ) ) ) ) ) ) ).
% sum_mat_left_ortho_zero
thf(fact_398_sum__mat__left__ortho__zero,axiom,
! [I: set_mat_complex,A: mat_complex > mat_complex,B2: mat_complex] :
( ( finite7047982916621727056omplex @ I )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ B2 )
= ( zero_mat_complex @ n @ n ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear1795808462385993418omplex @ n @ n @ A @ I ) @ B2 )
= ( zero_mat_complex @ n @ n ) ) ) ) ) ) ).
% sum_mat_left_ortho_zero
thf(fact_399_sum__mat__left__ortho__zero,axiom,
! [I: set_complex,A: complex > mat_complex,B2: mat_complex] :
( ( finite3207457112153483333omplex @ I )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ B2 )
= ( zero_mat_complex @ n @ n ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear8664352376190006057omplex @ n @ n @ A @ I ) @ B2 )
= ( zero_mat_complex @ n @ n ) ) ) ) ) ) ).
% sum_mat_left_ortho_zero
thf(fact_400_sum__mat__ortho__one,axiom,
! [I: set_mat_nat,J: mat_nat,B2: mat_complex,A: mat_nat > mat_complex] :
( ( finite9067146157826639218at_nat @ I )
=> ( ( member_mat_nat @ J @ I )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( ! [I2: mat_nat] :
( ( member_mat_nat @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: mat_nat] :
( ( member_mat_nat @ I2 @ I )
=> ( ( I2 != J )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ B2 )
= ( zero_mat_complex @ n @ n ) ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear4360147844572819692omplex @ n @ n @ A @ I ) @ B2 )
= ( times_8009071140041733218omplex @ ( A @ J ) @ B2 ) ) ) ) ) ) ) ).
% sum_mat_ortho_one
thf(fact_401_sum__mat__ortho__one,axiom,
! [I: set_set_complex,J: set_complex,B2: mat_complex,A: set_complex > mat_complex] :
( ( finite6551019134538273531omplex @ I )
=> ( ( member_set_complex @ J @ I )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( ! [I2: set_complex] :
( ( member_set_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: set_complex] :
( ( member_set_complex @ I2 @ I )
=> ( ( I2 != J )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ B2 )
= ( zero_mat_complex @ n @ n ) ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear7804121962625877471omplex @ n @ n @ A @ I ) @ B2 )
= ( times_8009071140041733218omplex @ ( A @ J ) @ B2 ) ) ) ) ) ) ) ).
% sum_mat_ortho_one
thf(fact_402_sum__mat__ortho__one,axiom,
! [I: set_set_mat_complex,J: set_mat_complex,B2: mat_complex,A: set_mat_complex > mat_complex] :
( ( finite1349200545324696496omplex @ I )
=> ( ( member3612512168372279472omplex @ J @ I )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( ! [I2: set_mat_complex] :
( ( member3612512168372279472omplex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: set_mat_complex] :
( ( member3612512168372279472omplex @ I2 @ I )
=> ( ( I2 != J )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ B2 )
= ( zero_mat_complex @ n @ n ) ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear5733585793752561962omplex @ n @ n @ A @ I ) @ B2 )
= ( times_8009071140041733218omplex @ ( A @ J ) @ B2 ) ) ) ) ) ) ) ).
% sum_mat_ortho_one
thf(fact_403_sum__mat__ortho__one,axiom,
! [I: set_nat,J: nat,B2: mat_complex,A: nat > mat_complex] :
( ( finite_finite_nat @ I )
=> ( ( member_nat @ J @ I )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ( I2 != J )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ B2 )
= ( zero_mat_complex @ n @ n ) ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear8108877306658443851omplex @ n @ n @ A @ I ) @ B2 )
= ( times_8009071140041733218omplex @ ( A @ J ) @ B2 ) ) ) ) ) ) ) ).
% sum_mat_ortho_one
thf(fact_404_sum__mat__ortho__one,axiom,
! [I: set_mat_complex,J: mat_complex,B2: mat_complex,A: mat_complex > mat_complex] :
( ( finite7047982916621727056omplex @ I )
=> ( ( member_mat_complex @ J @ I )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( ( I2 != J )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ B2 )
= ( zero_mat_complex @ n @ n ) ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear1795808462385993418omplex @ n @ n @ A @ I ) @ B2 )
= ( times_8009071140041733218omplex @ ( A @ J ) @ B2 ) ) ) ) ) ) ) ).
% sum_mat_ortho_one
thf(fact_405_sum__mat__ortho__one,axiom,
! [I: set_complex,J: complex,B2: mat_complex,A: complex > mat_complex] :
( ( finite3207457112153483333omplex @ I )
=> ( ( member_complex @ J @ I )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( ( I2 != J )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ B2 )
= ( zero_mat_complex @ n @ n ) ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear8664352376190006057omplex @ n @ n @ A @ I ) @ B2 )
= ( times_8009071140041733218omplex @ ( A @ J ) @ B2 ) ) ) ) ) ) ) ).
% sum_mat_ortho_one
thf(fact_406_sum__mat__ortho__proj,axiom,
! [I: set_mat_nat,J: mat_nat,A: mat_nat > mat_complex] :
( ( finite9067146157826639218at_nat @ I )
=> ( ( member_mat_nat @ J @ I )
=> ( ( ( times_8009071140041733218omplex @ ( A @ J ) @ ( A @ J ) )
= ( A @ J ) )
=> ( ! [I2: mat_nat] :
( ( member_mat_nat @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: mat_nat] :
( ( member_mat_nat @ I2 @ I )
=> ( ( I2 != J )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ ( A @ J ) )
= ( zero_mat_complex @ n @ n ) ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear4360147844572819692omplex @ n @ n @ A @ I ) @ ( A @ J ) )
= ( A @ J ) ) ) ) ) ) ) ).
% sum_mat_ortho_proj
thf(fact_407_sum__mat__ortho__proj,axiom,
! [I: set_set_complex,J: set_complex,A: set_complex > mat_complex] :
( ( finite6551019134538273531omplex @ I )
=> ( ( member_set_complex @ J @ I )
=> ( ( ( times_8009071140041733218omplex @ ( A @ J ) @ ( A @ J ) )
= ( A @ J ) )
=> ( ! [I2: set_complex] :
( ( member_set_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: set_complex] :
( ( member_set_complex @ I2 @ I )
=> ( ( I2 != J )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ ( A @ J ) )
= ( zero_mat_complex @ n @ n ) ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear7804121962625877471omplex @ n @ n @ A @ I ) @ ( A @ J ) )
= ( A @ J ) ) ) ) ) ) ) ).
% sum_mat_ortho_proj
thf(fact_408_sum__mat__ortho__proj,axiom,
! [I: set_set_mat_complex,J: set_mat_complex,A: set_mat_complex > mat_complex] :
( ( finite1349200545324696496omplex @ I )
=> ( ( member3612512168372279472omplex @ J @ I )
=> ( ( ( times_8009071140041733218omplex @ ( A @ J ) @ ( A @ J ) )
= ( A @ J ) )
=> ( ! [I2: set_mat_complex] :
( ( member3612512168372279472omplex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: set_mat_complex] :
( ( member3612512168372279472omplex @ I2 @ I )
=> ( ( I2 != J )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ ( A @ J ) )
= ( zero_mat_complex @ n @ n ) ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear5733585793752561962omplex @ n @ n @ A @ I ) @ ( A @ J ) )
= ( A @ J ) ) ) ) ) ) ) ).
% sum_mat_ortho_proj
thf(fact_409_sum__mat__ortho__proj,axiom,
! [I: set_nat,J: nat,A: nat > mat_complex] :
( ( finite_finite_nat @ I )
=> ( ( member_nat @ J @ I )
=> ( ( ( times_8009071140041733218omplex @ ( A @ J ) @ ( A @ J ) )
= ( A @ J ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ( I2 != J )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ ( A @ J ) )
= ( zero_mat_complex @ n @ n ) ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear8108877306658443851omplex @ n @ n @ A @ I ) @ ( A @ J ) )
= ( A @ J ) ) ) ) ) ) ) ).
% sum_mat_ortho_proj
thf(fact_410_sum__mat__ortho__proj,axiom,
! [I: set_mat_complex,J: mat_complex,A: mat_complex > mat_complex] :
( ( finite7047982916621727056omplex @ I )
=> ( ( member_mat_complex @ J @ I )
=> ( ( ( times_8009071140041733218omplex @ ( A @ J ) @ ( A @ J ) )
= ( A @ J ) )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( ( I2 != J )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ ( A @ J ) )
= ( zero_mat_complex @ n @ n ) ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear1795808462385993418omplex @ n @ n @ A @ I ) @ ( A @ J ) )
= ( A @ J ) ) ) ) ) ) ) ).
% sum_mat_ortho_proj
thf(fact_411_sum__mat__ortho__proj,axiom,
! [I: set_complex,J: complex,A: complex > mat_complex] :
( ( finite3207457112153483333omplex @ I )
=> ( ( member_complex @ J @ I )
=> ( ( ( times_8009071140041733218omplex @ ( A @ J ) @ ( A @ J ) )
= ( A @ J ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( ( I2 != J )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ ( A @ J ) )
= ( zero_mat_complex @ n @ n ) ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear8664352376190006057omplex @ n @ n @ A @ I ) @ ( A @ J ) )
= ( A @ J ) ) ) ) ) ) ) ).
% sum_mat_ortho_proj
thf(fact_412_sum__mat__ortho__square,axiom,
! [I: set_mat_nat,A: mat_nat > mat_complex] :
( ( finite9067146157826639218at_nat @ I )
=> ( ! [I2: mat_nat] :
( ( member_mat_nat @ I2 @ I )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ ( A @ I2 ) )
= ( A @ I2 ) ) )
=> ( ! [I2: mat_nat] :
( ( member_mat_nat @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: mat_nat,J2: mat_nat] :
( ( member_mat_nat @ I2 @ I )
=> ( ( member_mat_nat @ J2 @ I )
=> ( ( I2 != J2 )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ ( A @ J2 ) )
= ( zero_mat_complex @ n @ n ) ) ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear4360147844572819692omplex @ n @ n @ A @ I ) @ ( linear4360147844572819692omplex @ n @ n @ A @ I ) )
= ( linear4360147844572819692omplex @ n @ n @ A @ I ) ) ) ) ) ) ).
% sum_mat_ortho_square
thf(fact_413_sum__mat__ortho__square,axiom,
! [I: set_set_complex,A: set_complex > mat_complex] :
( ( finite6551019134538273531omplex @ I )
=> ( ! [I2: set_complex] :
( ( member_set_complex @ I2 @ I )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ ( A @ I2 ) )
= ( A @ I2 ) ) )
=> ( ! [I2: set_complex] :
( ( member_set_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: set_complex,J2: set_complex] :
( ( member_set_complex @ I2 @ I )
=> ( ( member_set_complex @ J2 @ I )
=> ( ( I2 != J2 )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ ( A @ J2 ) )
= ( zero_mat_complex @ n @ n ) ) ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear7804121962625877471omplex @ n @ n @ A @ I ) @ ( linear7804121962625877471omplex @ n @ n @ A @ I ) )
= ( linear7804121962625877471omplex @ n @ n @ A @ I ) ) ) ) ) ) ).
% sum_mat_ortho_square
thf(fact_414_sum__mat__ortho__square,axiom,
! [I: set_set_mat_complex,A: set_mat_complex > mat_complex] :
( ( finite1349200545324696496omplex @ I )
=> ( ! [I2: set_mat_complex] :
( ( member3612512168372279472omplex @ I2 @ I )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ ( A @ I2 ) )
= ( A @ I2 ) ) )
=> ( ! [I2: set_mat_complex] :
( ( member3612512168372279472omplex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: set_mat_complex,J2: set_mat_complex] :
( ( member3612512168372279472omplex @ I2 @ I )
=> ( ( member3612512168372279472omplex @ J2 @ I )
=> ( ( I2 != J2 )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ ( A @ J2 ) )
= ( zero_mat_complex @ n @ n ) ) ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear5733585793752561962omplex @ n @ n @ A @ I ) @ ( linear5733585793752561962omplex @ n @ n @ A @ I ) )
= ( linear5733585793752561962omplex @ n @ n @ A @ I ) ) ) ) ) ) ).
% sum_mat_ortho_square
thf(fact_415_sum__mat__ortho__square,axiom,
! [I: set_nat,A: nat > mat_complex] :
( ( finite_finite_nat @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ ( A @ I2 ) )
= ( A @ I2 ) ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: nat,J2: nat] :
( ( member_nat @ I2 @ I )
=> ( ( member_nat @ J2 @ I )
=> ( ( I2 != J2 )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ ( A @ J2 ) )
= ( zero_mat_complex @ n @ n ) ) ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear8108877306658443851omplex @ n @ n @ A @ I ) @ ( linear8108877306658443851omplex @ n @ n @ A @ I ) )
= ( linear8108877306658443851omplex @ n @ n @ A @ I ) ) ) ) ) ) ).
% sum_mat_ortho_square
thf(fact_416_sum__mat__ortho__square,axiom,
! [I: set_mat_complex,A: mat_complex > mat_complex] :
( ( finite7047982916621727056omplex @ I )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ ( A @ I2 ) )
= ( A @ I2 ) ) )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: mat_complex,J2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( ( member_mat_complex @ J2 @ I )
=> ( ( I2 != J2 )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ ( A @ J2 ) )
= ( zero_mat_complex @ n @ n ) ) ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear1795808462385993418omplex @ n @ n @ A @ I ) @ ( linear1795808462385993418omplex @ n @ n @ A @ I ) )
= ( linear1795808462385993418omplex @ n @ n @ A @ I ) ) ) ) ) ) ).
% sum_mat_ortho_square
thf(fact_417_sum__mat__ortho__square,axiom,
! [I: set_complex,A: complex > mat_complex] :
( ( finite3207457112153483333omplex @ I )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ ( A @ I2 ) )
= ( A @ I2 ) ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: complex,J2: complex] :
( ( member_complex @ I2 @ I )
=> ( ( member_complex @ J2 @ I )
=> ( ( I2 != J2 )
=> ( ( times_8009071140041733218omplex @ ( A @ I2 ) @ ( A @ J2 ) )
= ( zero_mat_complex @ n @ n ) ) ) ) )
=> ( ( times_8009071140041733218omplex @ ( linear8664352376190006057omplex @ n @ n @ A @ I ) @ ( linear8664352376190006057omplex @ n @ n @ A @ I ) )
= ( linear8664352376190006057omplex @ n @ n @ A @ I ) ) ) ) ) ) ).
% sum_mat_ortho_square
thf(fact_418_sum__mat__right__ortho__zero,axiom,
! [I: set_mat_nat,A: mat_nat > mat_complex,B2: mat_complex] :
( ( finite9067146157826639218at_nat @ I )
=> ( ! [I2: mat_nat] :
( ( member_mat_nat @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( ! [I2: mat_nat] :
( ( member_mat_nat @ I2 @ I )
=> ( ( times_8009071140041733218omplex @ B2 @ ( A @ I2 ) )
= ( zero_mat_complex @ n @ n ) ) )
=> ( ( times_8009071140041733218omplex @ B2 @ ( linear4360147844572819692omplex @ n @ n @ A @ I ) )
= ( zero_mat_complex @ n @ n ) ) ) ) ) ) ).
% sum_mat_right_ortho_zero
thf(fact_419_sum__mat__right__ortho__zero,axiom,
! [I: set_set_complex,A: set_complex > mat_complex,B2: mat_complex] :
( ( finite6551019134538273531omplex @ I )
=> ( ! [I2: set_complex] :
( ( member_set_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( ! [I2: set_complex] :
( ( member_set_complex @ I2 @ I )
=> ( ( times_8009071140041733218omplex @ B2 @ ( A @ I2 ) )
= ( zero_mat_complex @ n @ n ) ) )
=> ( ( times_8009071140041733218omplex @ B2 @ ( linear7804121962625877471omplex @ n @ n @ A @ I ) )
= ( zero_mat_complex @ n @ n ) ) ) ) ) ) ).
% sum_mat_right_ortho_zero
thf(fact_420_sum__mat__right__ortho__zero,axiom,
! [I: set_set_mat_complex,A: set_mat_complex > mat_complex,B2: mat_complex] :
( ( finite1349200545324696496omplex @ I )
=> ( ! [I2: set_mat_complex] :
( ( member3612512168372279472omplex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( ! [I2: set_mat_complex] :
( ( member3612512168372279472omplex @ I2 @ I )
=> ( ( times_8009071140041733218omplex @ B2 @ ( A @ I2 ) )
= ( zero_mat_complex @ n @ n ) ) )
=> ( ( times_8009071140041733218omplex @ B2 @ ( linear5733585793752561962omplex @ n @ n @ A @ I ) )
= ( zero_mat_complex @ n @ n ) ) ) ) ) ) ).
% sum_mat_right_ortho_zero
thf(fact_421_sum__mat__right__ortho__zero,axiom,
! [I: set_nat,A: nat > mat_complex,B2: mat_complex] :
( ( finite_finite_nat @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ( times_8009071140041733218omplex @ B2 @ ( A @ I2 ) )
= ( zero_mat_complex @ n @ n ) ) )
=> ( ( times_8009071140041733218omplex @ B2 @ ( linear8108877306658443851omplex @ n @ n @ A @ I ) )
= ( zero_mat_complex @ n @ n ) ) ) ) ) ) ).
% sum_mat_right_ortho_zero
thf(fact_422_sum__mat__right__ortho__zero,axiom,
! [I: set_mat_complex,A: mat_complex > mat_complex,B2: mat_complex] :
( ( finite7047982916621727056omplex @ I )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( ( times_8009071140041733218omplex @ B2 @ ( A @ I2 ) )
= ( zero_mat_complex @ n @ n ) ) )
=> ( ( times_8009071140041733218omplex @ B2 @ ( linear1795808462385993418omplex @ n @ n @ A @ I ) )
= ( zero_mat_complex @ n @ n ) ) ) ) ) ) ).
% sum_mat_right_ortho_zero
thf(fact_423_sum__mat__right__ortho__zero,axiom,
! [I: set_complex,A: complex > mat_complex,B2: mat_complex] :
( ( finite3207457112153483333omplex @ I )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ( member_mat_complex @ B2 @ fc )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( ( times_8009071140041733218omplex @ B2 @ ( A @ I2 ) )
= ( zero_mat_complex @ n @ n ) ) )
=> ( ( times_8009071140041733218omplex @ B2 @ ( linear8664352376190006057omplex @ n @ n @ A @ I ) )
= ( zero_mat_complex @ n @ n ) ) ) ) ) ) ).
% sum_mat_right_ortho_zero
thf(fact_424_sum__mat__hermitian,axiom,
! [I: set_nat,A: nat > mat_complex] :
( ( finite_finite_nat @ I )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ I )
=> ( comple8306762464034002205omplex @ ( A @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ I )
=> ( member_mat_complex @ ( A @ X3 ) @ fc ) )
=> ( comple8306762464034002205omplex @ ( linear8108877306658443851omplex @ n @ n @ A @ I ) ) ) ) ) ).
% sum_mat_hermitian
thf(fact_425_sum__mat__hermitian,axiom,
! [I: set_complex,A: complex > mat_complex] :
( ( finite3207457112153483333omplex @ I )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ I )
=> ( comple8306762464034002205omplex @ ( A @ X3 ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ I )
=> ( member_mat_complex @ ( A @ X3 ) @ fc ) )
=> ( comple8306762464034002205omplex @ ( linear8664352376190006057omplex @ n @ n @ A @ I ) ) ) ) ) ).
% sum_mat_hermitian
thf(fact_426_sum__mat__hermitian,axiom,
! [I: set_mat_complex,A: mat_complex > mat_complex] :
( ( finite7047982916621727056omplex @ I )
=> ( ! [X3: mat_complex] :
( ( member_mat_complex @ X3 @ I )
=> ( comple8306762464034002205omplex @ ( A @ X3 ) ) )
=> ( ! [X3: mat_complex] :
( ( member_mat_complex @ X3 @ I )
=> ( member_mat_complex @ ( A @ X3 ) @ fc ) )
=> ( comple8306762464034002205omplex @ ( linear1795808462385993418omplex @ n @ n @ A @ I ) ) ) ) ) ).
% sum_mat_hermitian
thf(fact_427_class__ring_Oring__simprules_I14_J,axiom,
( minus_minus_complex
= ( ^ [X2: complex,Y: complex] : ( plus_plus_complex @ X2 @ ( uminus1482373934393186551omplex @ Y ) ) ) ) ).
% class_ring.ring_simprules(14)
thf(fact_428_sum__with__infinite,axiom,
! [A: set_nat,G: nat > mat_complex] :
( ~ ( finite_finite_nat @ A )
=> ( ( group_3997526426246263166ex_nat @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ G @ A )
= ( zero_mat_complex @ n @ n ) ) ) ).
% sum_with_infinite
thf(fact_429_sum__with__infinite,axiom,
! [A: set_complex,G: complex > mat_complex] :
( ~ ( finite3207457112153483333omplex @ A )
=> ( ( group_4310869077125956188omplex @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ G @ A )
= ( zero_mat_complex @ n @ n ) ) ) ).
% sum_with_infinite
thf(fact_430_sum__with__infinite,axiom,
! [A: set_mat_complex,G: mat_complex > mat_complex] :
( ~ ( finite7047982916621727056omplex @ A )
=> ( ( group_1588376139278055545omplex @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ G @ A )
= ( zero_mat_complex @ n @ n ) ) ) ).
% sum_with_infinite
thf(fact_431_sum__mat__cong,axiom,
! [I: set_mat_nat,A: mat_nat > mat_complex,B2: mat_nat > mat_complex] :
( ( finite9067146157826639218at_nat @ I )
=> ( ! [I2: mat_nat] :
( ( member_mat_nat @ I2 @ I )
=> ( ( A @ I2 )
= ( B2 @ I2 ) ) )
=> ( ! [I2: mat_nat] :
( ( member_mat_nat @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: mat_nat] :
( ( member_mat_nat @ I2 @ I )
=> ( member_mat_complex @ ( B2 @ I2 ) @ fc ) )
=> ( ( linear4360147844572819692omplex @ n @ n @ A @ I )
= ( linear4360147844572819692omplex @ n @ n @ B2 @ I ) ) ) ) ) ) ).
% sum_mat_cong
thf(fact_432_sum__mat__cong,axiom,
! [I: set_set_complex,A: set_complex > mat_complex,B2: set_complex > mat_complex] :
( ( finite6551019134538273531omplex @ I )
=> ( ! [I2: set_complex] :
( ( member_set_complex @ I2 @ I )
=> ( ( A @ I2 )
= ( B2 @ I2 ) ) )
=> ( ! [I2: set_complex] :
( ( member_set_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: set_complex] :
( ( member_set_complex @ I2 @ I )
=> ( member_mat_complex @ ( B2 @ I2 ) @ fc ) )
=> ( ( linear7804121962625877471omplex @ n @ n @ A @ I )
= ( linear7804121962625877471omplex @ n @ n @ B2 @ I ) ) ) ) ) ) ).
% sum_mat_cong
thf(fact_433_sum__mat__cong,axiom,
! [I: set_set_mat_complex,A: set_mat_complex > mat_complex,B2: set_mat_complex > mat_complex] :
( ( finite1349200545324696496omplex @ I )
=> ( ! [I2: set_mat_complex] :
( ( member3612512168372279472omplex @ I2 @ I )
=> ( ( A @ I2 )
= ( B2 @ I2 ) ) )
=> ( ! [I2: set_mat_complex] :
( ( member3612512168372279472omplex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: set_mat_complex] :
( ( member3612512168372279472omplex @ I2 @ I )
=> ( member_mat_complex @ ( B2 @ I2 ) @ fc ) )
=> ( ( linear5733585793752561962omplex @ n @ n @ A @ I )
= ( linear5733585793752561962omplex @ n @ n @ B2 @ I ) ) ) ) ) ) ).
% sum_mat_cong
thf(fact_434_sum__mat__cong,axiom,
! [I: set_nat,A: nat > mat_complex,B2: nat > mat_complex] :
( ( finite_finite_nat @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ( A @ I2 )
= ( B2 @ I2 ) ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_mat_complex @ ( B2 @ I2 ) @ fc ) )
=> ( ( linear8108877306658443851omplex @ n @ n @ A @ I )
= ( linear8108877306658443851omplex @ n @ n @ B2 @ I ) ) ) ) ) ) ).
% sum_mat_cong
thf(fact_435_sum__mat__cong,axiom,
! [I: set_mat_complex,A: mat_complex > mat_complex,B2: mat_complex > mat_complex] :
( ( finite7047982916621727056omplex @ I )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( ( A @ I2 )
= ( B2 @ I2 ) ) )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( member_mat_complex @ ( B2 @ I2 ) @ fc ) )
=> ( ( linear1795808462385993418omplex @ n @ n @ A @ I )
= ( linear1795808462385993418omplex @ n @ n @ B2 @ I ) ) ) ) ) ) ).
% sum_mat_cong
thf(fact_436_sum__mat__cong,axiom,
! [I: set_complex,A: complex > mat_complex,B2: complex > mat_complex] :
( ( finite3207457112153483333omplex @ I )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( ( A @ I2 )
= ( B2 @ I2 ) ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( member_mat_complex @ ( B2 @ I2 ) @ fc ) )
=> ( ( linear8664352376190006057omplex @ n @ n @ A @ I )
= ( linear8664352376190006057omplex @ n @ n @ B2 @ I ) ) ) ) ) ) ).
% sum_mat_cong
thf(fact_437_sum__with__cong_H,axiom,
! [I: set_set_mat_complex,A: set_mat_complex > mat_complex,B2: set_mat_complex > mat_complex] :
( ( finite1349200545324696496omplex @ I )
=> ( ! [I2: set_mat_complex] :
( ( member3612512168372279472omplex @ I2 @ I )
=> ( ( A @ I2 )
= ( B2 @ I2 ) ) )
=> ( ! [I2: set_mat_complex] :
( ( member3612512168372279472omplex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: set_mat_complex] :
( ( member3612512168372279472omplex @ I2 @ I )
=> ( member_mat_complex @ ( B2 @ I2 ) @ fc ) )
=> ( ( group_4914184569911763673omplex @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ A @ I )
= ( group_4914184569911763673omplex @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ B2 @ I ) ) ) ) ) ) ).
% sum_with_cong'
thf(fact_438_sum__with__cong_H,axiom,
! [I: set_nat,A: nat > mat_complex,B2: nat > mat_complex] :
( ( finite_finite_nat @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ( A @ I2 )
= ( B2 @ I2 ) ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_mat_complex @ ( B2 @ I2 ) @ fc ) )
=> ( ( group_3997526426246263166ex_nat @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ A @ I )
= ( group_3997526426246263166ex_nat @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ B2 @ I ) ) ) ) ) ) ).
% sum_with_cong'
thf(fact_439_sum__with__cong_H,axiom,
! [I: set_mat_complex,A: mat_complex > mat_complex,B2: mat_complex > mat_complex] :
( ( finite7047982916621727056omplex @ I )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( ( A @ I2 )
= ( B2 @ I2 ) ) )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( member_mat_complex @ ( B2 @ I2 ) @ fc ) )
=> ( ( group_1588376139278055545omplex @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ A @ I )
= ( group_1588376139278055545omplex @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ B2 @ I ) ) ) ) ) ) ).
% sum_with_cong'
thf(fact_440_sum__with__cong_H,axiom,
! [I: set_complex,A: complex > mat_complex,B2: complex > mat_complex] :
( ( finite3207457112153483333omplex @ I )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( ( A @ I2 )
= ( B2 @ I2 ) ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( member_mat_complex @ ( B2 @ I2 ) @ fc ) )
=> ( ( group_4310869077125956188omplex @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ A @ I )
= ( group_4310869077125956188omplex @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ B2 @ I ) ) ) ) ) ) ).
% sum_with_cong'
thf(fact_441_comm__monoid__add__on__with_Osum__with__cong_H,axiom,
! [S: set_complex,Pls: complex > complex > complex,Z: complex,I: set_mat_complex,A: mat_complex > complex,B2: mat_complex > complex] :
( ( group_2796108508354279923omplex @ S @ Pls @ Z )
=> ( ( finite7047982916621727056omplex @ I )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( ( A @ I2 )
= ( B2 @ I2 ) ) )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( member_complex @ ( A @ I2 ) @ S ) )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( member_complex @ ( B2 @ I2 ) @ S ) )
=> ( ( group_6233491804913373438omplex @ Pls @ Z @ A @ I )
= ( group_6233491804913373438omplex @ Pls @ Z @ B2 @ I ) ) ) ) ) ) ) ).
% comm_monoid_add_on_with.sum_with_cong'
thf(fact_442_comm__monoid__add__on__with_Osum__with__cong_H,axiom,
! [S: set_complex,Pls: complex > complex > complex,Z: complex,I: set_complex,A: complex > complex,B2: complex > complex] :
( ( group_2796108508354279923omplex @ S @ Pls @ Z )
=> ( ( finite3207457112153483333omplex @ I )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( ( A @ I2 )
= ( B2 @ I2 ) ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( member_complex @ ( A @ I2 ) @ S ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( member_complex @ ( B2 @ I2 ) @ S ) )
=> ( ( group_4775205164212326935omplex @ Pls @ Z @ A @ I )
= ( group_4775205164212326935omplex @ Pls @ Z @ B2 @ I ) ) ) ) ) ) ) ).
% comm_monoid_add_on_with.sum_with_cong'
thf(fact_443_comm__monoid__add__on__with_Osum__with__cong_H,axiom,
! [S: set_mat_complex,Pls: mat_complex > mat_complex > mat_complex,Z: mat_complex,I: set_mat_complex,A: mat_complex > mat_complex,B2: mat_complex > mat_complex] :
( ( group_5394922976599784994omplex @ S @ Pls @ Z )
=> ( ( finite7047982916621727056omplex @ I )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( ( A @ I2 )
= ( B2 @ I2 ) ) )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ S ) )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( member_mat_complex @ ( B2 @ I2 ) @ S ) )
=> ( ( group_1588376139278055545omplex @ Pls @ Z @ A @ I )
= ( group_1588376139278055545omplex @ Pls @ Z @ B2 @ I ) ) ) ) ) ) ) ).
% comm_monoid_add_on_with.sum_with_cong'
thf(fact_444_comm__monoid__add__on__with_Osum__with__cong_H,axiom,
! [S: set_mat_complex,Pls: mat_complex > mat_complex > mat_complex,Z: mat_complex,I: set_complex,A: complex > mat_complex,B2: complex > mat_complex] :
( ( group_5394922976599784994omplex @ S @ Pls @ Z )
=> ( ( finite3207457112153483333omplex @ I )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( ( A @ I2 )
= ( B2 @ I2 ) ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ S ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( member_mat_complex @ ( B2 @ I2 ) @ S ) )
=> ( ( group_4310869077125956188omplex @ Pls @ Z @ A @ I )
= ( group_4310869077125956188omplex @ Pls @ Z @ B2 @ I ) ) ) ) ) ) ) ).
% comm_monoid_add_on_with.sum_with_cong'
thf(fact_445_class__cring_Ofactors__equal,axiom,
! [A2: complex,B3: complex,C: complex,D2: complex] :
( ( A2 = B3 )
=> ( ( C = D2 )
=> ( ( times_times_complex @ A2 @ C )
= ( times_times_complex @ B3 @ D2 ) ) ) ) ).
% class_cring.factors_equal
thf(fact_446_class__semiring_Oadd_Ofactors__equal,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( A2 = B3 )
=> ( ( C = D2 )
=> ( ( plus_plus_nat @ A2 @ C )
= ( plus_plus_nat @ B3 @ D2 ) ) ) ) ).
% class_semiring.add.factors_equal
thf(fact_447_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_448_class__ring_Ominus__zero,axiom,
( ( uminus1482373934393186551omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% class_ring.minus_zero
thf(fact_449_class__cring_Ocring__simprules_I22_J,axiom,
( ( uminus1482373934393186551omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% class_cring.cring_simprules(22)
thf(fact_450_is__num__normalize_I8_J,axiom,
! [A2: complex,B3: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A2 @ B3 ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B3 ) @ ( uminus1482373934393186551omplex @ A2 ) ) ) ).
% is_num_normalize(8)
thf(fact_451_sum__mono__neutral__cong__left,axiom,
! [H: mat_complex > mat_complex,G: mat_complex > mat_complex,T: set_mat_complex,Sa: set_mat_complex] :
( ! [X3: mat_complex] : ( member_mat_complex @ ( H @ X3 ) @ fc )
=> ( ! [X3: mat_complex] : ( member_mat_complex @ ( G @ X3 ) @ fc )
=> ( ( finite7047982916621727056omplex @ T )
=> ( ( ord_le3632134057777142183omplex @ Sa @ T )
=> ( ! [X3: mat_complex] :
( ( member_mat_complex @ X3 @ ( minus_8760755521168068590omplex @ T @ Sa ) )
=> ( ( H @ X3 )
= ( zero_mat_complex @ n @ n ) ) )
=> ( ! [X3: mat_complex] :
( ( member_mat_complex @ X3 @ Sa )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( group_1588376139278055545omplex @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ G @ Sa )
= ( group_1588376139278055545omplex @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ H @ T ) ) ) ) ) ) ) ) ).
% sum_mono_neutral_cong_left
thf(fact_452_sum__mono__neutral__cong__left,axiom,
! [H: complex > mat_complex,G: complex > mat_complex,T: set_complex,Sa: set_complex] :
( ! [X3: complex] : ( member_mat_complex @ ( H @ X3 ) @ fc )
=> ( ! [X3: complex] : ( member_mat_complex @ ( G @ X3 ) @ fc )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ( ord_le211207098394363844omplex @ Sa @ T )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T @ Sa ) )
=> ( ( H @ X3 )
= ( zero_mat_complex @ n @ n ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ Sa )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( group_4310869077125956188omplex @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ G @ Sa )
= ( group_4310869077125956188omplex @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ H @ T ) ) ) ) ) ) ) ) ).
% sum_mono_neutral_cong_left
thf(fact_453_sum__mat__positive,axiom,
! [I: set_mat_complex,A: mat_complex > mat_complex] :
( ( finite7047982916621727056omplex @ I )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( complex_positive @ ( A @ I2 ) ) )
=> ( ! [I2: mat_complex] :
( ( member_mat_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( complex_positive @ ( linear1795808462385993418omplex @ n @ n @ A @ I ) ) ) ) ) ).
% sum_mat_positive
thf(fact_454_sum__mat__positive,axiom,
! [I: set_complex,A: complex > mat_complex] :
( ( finite3207457112153483333omplex @ I )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( complex_positive @ ( A @ I2 ) ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I )
=> ( member_mat_complex @ ( A @ I2 ) @ fc ) )
=> ( complex_positive @ ( linear8664352376190006057omplex @ n @ n @ A @ I ) ) ) ) ) ).
% sum_mat_positive
thf(fact_455_ex__min__if__finite,axiom,
! [S: set_mat_complex] :
( ( finite7047982916621727056omplex @ S )
=> ( ( S != bot_bo7165004461764951667omplex )
=> ? [X3: mat_complex] :
( ( member_mat_complex @ X3 @ S )
& ~ ? [Xa: mat_complex] :
( ( member_mat_complex @ Xa @ S )
& ( ord_less_mat_complex @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_456_ex__min__if__finite,axiom,
! [S: set_complex] :
( ( finite3207457112153483333omplex @ S )
=> ( ( S != bot_bot_set_complex )
=> ? [X3: complex] :
( ( member_complex @ X3 @ S )
& ~ ? [Xa: complex] :
( ( member_complex @ Xa @ S )
& ( ord_less_complex @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_457_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_458_infinite__growing,axiom,
! [X4: set_nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X4 )
& ( ord_less_nat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X4 ) ) ) ).
% infinite_growing
thf(fact_459_semiring__norm_I113_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% semiring_norm(113)
thf(fact_460_zero__order_I2_J,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% zero_order(2)
thf(fact_461_zero__order_I1_J,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_order(1)
thf(fact_462_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_463_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_eq_complex @ I3 @ J )
& ( K = L ) )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_464_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( I3 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_465_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( I3 = J )
& ( ord_less_eq_complex @ K @ L ) )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_466_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_467_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_eq_complex @ I3 @ J )
& ( ord_less_eq_complex @ K @ L ) )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_468_add__mono,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D2 ) ) ) ) ).
% add_mono
thf(fact_469_add__mono,axiom,
! [A2: complex,B3: complex,C: complex,D2: complex] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ( ord_less_eq_complex @ C @ D2 )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A2 @ C ) @ ( plus_plus_complex @ B3 @ D2 ) ) ) ) ).
% add_mono
thf(fact_470_add__left__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).
% add_left_mono
thf(fact_471_add__left__mono,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ C @ A2 ) @ ( plus_plus_complex @ C @ B3 ) ) ) ).
% add_left_mono
thf(fact_472_less__eqE,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ~ ! [C3: nat] :
( B3
!= ( plus_plus_nat @ A2 @ C3 ) ) ) ).
% less_eqE
thf(fact_473_add__right__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add_right_mono
thf(fact_474_add__right__mono,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A2 @ C ) @ ( plus_plus_complex @ B3 @ C ) ) ) ).
% add_right_mono
thf(fact_475_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B4: nat] :
? [C4: nat] :
( B4
= ( plus_plus_nat @ A3 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_476_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
= ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% add_le_cancel_left
thf(fact_477_add__le__cancel__left,axiom,
! [C: complex,A2: complex,B3: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ C @ A2 ) @ ( plus_plus_complex @ C @ B3 ) )
= ( ord_less_eq_complex @ A2 @ B3 ) ) ).
% add_le_cancel_left
thf(fact_478_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% add_le_imp_le_left
thf(fact_479_add__le__imp__le__left,axiom,
! [C: complex,A2: complex,B3: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ C @ A2 ) @ ( plus_plus_complex @ C @ B3 ) )
=> ( ord_less_eq_complex @ A2 @ B3 ) ) ).
% add_le_imp_le_left
thf(fact_480_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
= ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% add_le_cancel_right
thf(fact_481_add__le__cancel__right,axiom,
! [A2: complex,C: complex,B3: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ A2 @ C ) @ ( plus_plus_complex @ B3 @ C ) )
= ( ord_less_eq_complex @ A2 @ B3 ) ) ).
% add_le_cancel_right
thf(fact_482_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% add_le_imp_le_right
thf(fact_483_add__le__imp__le__right,axiom,
! [A2: complex,C: complex,B3: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ A2 @ C ) @ ( plus_plus_complex @ B3 @ C ) )
=> ( ord_less_eq_complex @ A2 @ B3 ) ) ).
% add_le_imp_le_right
thf(fact_484_diff__mono,axiom,
! [A2: complex,B3: complex,D2: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ( ord_less_eq_complex @ D2 @ C )
=> ( ord_less_eq_complex @ ( minus_minus_complex @ A2 @ C ) @ ( minus_minus_complex @ B3 @ D2 ) ) ) ) ).
% diff_mono
thf(fact_485_diff__left__mono,axiom,
! [B3: complex,A2: complex,C: complex] :
( ( ord_less_eq_complex @ B3 @ A2 )
=> ( ord_less_eq_complex @ ( minus_minus_complex @ C @ A2 ) @ ( minus_minus_complex @ C @ B3 ) ) ) ).
% diff_left_mono
thf(fact_486_diff__right__mono,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ord_less_eq_complex @ ( minus_minus_complex @ A2 @ C ) @ ( minus_minus_complex @ B3 @ C ) ) ) ).
% diff_right_mono
thf(fact_487_diff__eq__diff__less__eq,axiom,
! [A2: complex,B3: complex,C: complex,D2: complex] :
( ( ( minus_minus_complex @ A2 @ B3 )
= ( minus_minus_complex @ C @ D2 ) )
=> ( ( ord_less_eq_complex @ A2 @ B3 )
= ( ord_less_eq_complex @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_488_le__minus__iff,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_eq_complex @ A2 @ ( uminus1482373934393186551omplex @ B3 ) )
= ( ord_less_eq_complex @ B3 @ ( uminus1482373934393186551omplex @ A2 ) ) ) ).
% le_minus_iff
thf(fact_489_minus__le__iff,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_eq_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B3 )
= ( ord_less_eq_complex @ ( uminus1482373934393186551omplex @ B3 ) @ A2 ) ) ).
% minus_le_iff
thf(fact_490_le__imp__neg__le,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ord_less_eq_complex @ ( uminus1482373934393186551omplex @ B3 ) @ ( uminus1482373934393186551omplex @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_491_neg__le__iff__le,axiom,
! [B3: complex,A2: complex] :
( ( ord_less_eq_complex @ ( uminus1482373934393186551omplex @ B3 ) @ ( uminus1482373934393186551omplex @ A2 ) )
= ( ord_less_eq_complex @ A2 @ B3 ) ) ).
% neg_le_iff_le
thf(fact_492_positive__is__hermitian,axiom,
! [A: mat_complex] :
( ( complex_positive @ A )
=> ( comple8306762464034002205omplex @ A ) ) ).
% positive_is_hermitian
thf(fact_493_Complex__Matrix_Opositive__zero,axiom,
! [N: nat] : ( complex_positive @ ( zero_mat_complex @ N @ N ) ) ).
% Complex_Matrix.positive_zero
thf(fact_494_mult__sign__intros_I4_J,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_eq_complex @ A2 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ B3 @ zero_zero_complex )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A2 @ B3 ) ) ) ) ).
% mult_sign_intros(4)
thf(fact_495_mult__sign__intros_I3_J,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(3)
thf(fact_496_mult__sign__intros_I3_J,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_eq_complex @ A2 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B3 )
=> ( ord_less_eq_complex @ ( times_times_complex @ A2 @ B3 ) @ zero_zero_complex ) ) ) ).
% mult_sign_intros(3)
thf(fact_497_mult__sign__intros_I2_J,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(2)
thf(fact_498_mult__sign__intros_I2_J,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A2 )
=> ( ( ord_less_eq_complex @ B3 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ A2 @ B3 ) @ zero_zero_complex ) ) ) ).
% mult_sign_intros(2)
thf(fact_499_mult__sign__intros_I1_J,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B3 ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_500_mult__sign__intros_I1_J,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B3 )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A2 @ B3 ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_501_mult__mono,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_502_mult__mono,axiom,
! [A2: complex,B3: complex,C: complex,D2: complex] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ( ord_less_eq_complex @ C @ D2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ord_less_eq_complex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B3 @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_503_mult__mono_H,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_504_mult__mono_H,axiom,
! [A2: complex,B3: complex,C: complex,D2: complex] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ( ord_less_eq_complex @ C @ D2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ A2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ord_less_eq_complex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B3 @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_505_split__mult__pos__le,axiom,
! [A2: complex,B3: complex] :
( ( ( ( ord_less_eq_complex @ zero_zero_complex @ A2 )
& ( ord_less_eq_complex @ zero_zero_complex @ B3 ) )
| ( ( ord_less_eq_complex @ A2 @ zero_zero_complex )
& ( ord_less_eq_complex @ B3 @ zero_zero_complex ) ) )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A2 @ B3 ) ) ) ).
% split_mult_pos_le
thf(fact_506_mult__left__mono__neg,axiom,
! [B3: complex,A2: complex,C: complex] :
( ( ord_less_eq_complex @ B3 @ A2 )
=> ( ( ord_less_eq_complex @ C @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ C @ A2 ) @ ( times_times_complex @ C @ B3 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_507_mult__left__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% mult_left_mono
thf(fact_508_mult__left__mono,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ord_less_eq_complex @ ( times_times_complex @ C @ A2 ) @ ( times_times_complex @ C @ B3 ) ) ) ) ).
% mult_left_mono
thf(fact_509_mult__right__mono__neg,axiom,
! [B3: complex,A2: complex,C: complex] :
( ( ord_less_eq_complex @ B3 @ A2 )
=> ( ( ord_less_eq_complex @ C @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B3 @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_510_mult__right__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_511_mult__right__mono,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ord_less_eq_complex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B3 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_512_split__mult__neg__le,axiom,
! [A2: nat,B3: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
& ( ord_less_eq_nat @ B3 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_513_split__mult__neg__le,axiom,
! [A2: complex,B3: complex] :
( ( ( ( ord_less_eq_complex @ zero_zero_complex @ A2 )
& ( ord_less_eq_complex @ B3 @ zero_zero_complex ) )
| ( ( ord_less_eq_complex @ A2 @ zero_zero_complex )
& ( ord_less_eq_complex @ zero_zero_complex @ B3 ) ) )
=> ( ord_less_eq_complex @ ( times_times_complex @ A2 @ B3 ) @ zero_zero_complex ) ) ).
% split_mult_neg_le
thf(fact_514_mult__nonneg__nonpos2,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B3 @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_515_mult__nonneg__nonpos2,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A2 )
=> ( ( ord_less_eq_complex @ B3 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ B3 @ A2 ) @ zero_zero_complex ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_516_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_517_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ord_less_eq_complex @ ( times_times_complex @ C @ A2 ) @ ( times_times_complex @ C @ B3 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_518_zero__compare__simps_I3_J,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% zero_compare_simps(3)
thf(fact_519_zero__compare__simps_I3_J,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A2 )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ord_less_eq_complex @ B3 @ ( plus_plus_complex @ A2 @ C ) ) ) ) ).
% zero_compare_simps(3)
thf(fact_520_add__sign__intros_I8_J,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_sign_intros(8)
thf(fact_521_add__sign__intros_I8_J,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_eq_complex @ A2 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ B3 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A2 @ B3 ) @ zero_zero_complex ) ) ) ).
% add_sign_intros(8)
thf(fact_522_add__sign__intros_I4_J,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% add_sign_intros(4)
thf(fact_523_add__sign__intros_I4_J,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B3 )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( plus_plus_complex @ A2 @ B3 ) ) ) ) ).
% add_sign_intros(4)
thf(fact_524_add__decreasing,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B3 ) ) ) ).
% add_decreasing
thf(fact_525_add__decreasing,axiom,
! [A2: complex,C: complex,B3: complex] :
( ( ord_less_eq_complex @ A2 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ C @ B3 )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A2 @ C ) @ B3 ) ) ) ).
% add_decreasing
thf(fact_526_add__decreasing2,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B3 ) ) ) ).
% add_decreasing2
thf(fact_527_add__decreasing2,axiom,
! [C: complex,A2: complex,B3: complex] :
( ( ord_less_eq_complex @ C @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A2 @ C ) @ B3 ) ) ) ).
% add_decreasing2
thf(fact_528_add__increasing2,axiom,
! [C: nat,B3: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_529_add__increasing2,axiom,
! [C: complex,B3: complex,A2: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ( ord_less_eq_complex @ B3 @ A2 )
=> ( ord_less_eq_complex @ B3 @ ( plus_plus_complex @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_530_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_531_add__nonneg__eq__0__iff,axiom,
! [X: complex,Y2: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ X )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ Y2 )
=> ( ( ( plus_plus_complex @ X @ Y2 )
= zero_zero_complex )
= ( ( X = zero_zero_complex )
& ( Y2 = zero_zero_complex ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_532_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_533_add__nonpos__eq__0__iff,axiom,
! [X: complex,Y2: complex] :
( ( ord_less_eq_complex @ X @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ Y2 @ zero_zero_complex )
=> ( ( ( plus_plus_complex @ X @ Y2 )
= zero_zero_complex )
= ( ( X = zero_zero_complex )
& ( Y2 = zero_zero_complex ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_534_add__le__same__cancel1,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B3 @ A2 ) @ B3 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_535_add__le__same__cancel1,axiom,
! [B3: complex,A2: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ B3 @ A2 ) @ B3 )
= ( ord_less_eq_complex @ A2 @ zero_zero_complex ) ) ).
% add_le_same_cancel1
thf(fact_536_add__le__same__cancel2,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B3 ) @ B3 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_537_add__le__same__cancel2,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ A2 @ B3 ) @ B3 )
= ( ord_less_eq_complex @ A2 @ zero_zero_complex ) ) ).
% add_le_same_cancel2
thf(fact_538_le__add__same__cancel1,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B3 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% le_add_same_cancel1
thf(fact_539_le__add__same__cancel1,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_eq_complex @ A2 @ ( plus_plus_complex @ A2 @ B3 ) )
= ( ord_less_eq_complex @ zero_zero_complex @ B3 ) ) ).
% le_add_same_cancel1
thf(fact_540_le__add__same__cancel2,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B3 @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% le_add_same_cancel2
thf(fact_541_le__add__same__cancel2,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_eq_complex @ A2 @ ( plus_plus_complex @ B3 @ A2 ) )
= ( ord_less_eq_complex @ zero_zero_complex @ B3 ) ) ).
% le_add_same_cancel2
thf(fact_542_diff__ge__0__iff__ge,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ ( minus_minus_complex @ A2 @ B3 ) )
= ( ord_less_eq_complex @ B3 @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_543_diff__le__0__iff__le,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_eq_complex @ ( minus_minus_complex @ A2 @ B3 ) @ zero_zero_complex )
= ( ord_less_eq_complex @ A2 @ B3 ) ) ).
% diff_le_0_iff_le
thf(fact_544_add__mono__thms__linordered__field_I4_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_545_add__mono__thms__linordered__field_I4_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_eq_complex @ I3 @ J )
& ( ord_less_complex @ K @ L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_546_add__mono__thms__linordered__field_I3_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_547_add__mono__thms__linordered__field_I3_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_complex @ I3 @ J )
& ( ord_less_eq_complex @ K @ L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_548_add__le__less__mono,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_549_add__le__less__mono,axiom,
! [A2: complex,B3: complex,C: complex,D2: complex] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ( ord_less_complex @ C @ D2 )
=> ( ord_less_complex @ ( plus_plus_complex @ A2 @ C ) @ ( plus_plus_complex @ B3 @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_550_add__less__le__mono,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_551_add__less__le__mono,axiom,
! [A2: complex,B3: complex,C: complex,D2: complex] :
( ( ord_less_complex @ A2 @ B3 )
=> ( ( ord_less_eq_complex @ C @ D2 )
=> ( ord_less_complex @ ( plus_plus_complex @ A2 @ C ) @ ( plus_plus_complex @ B3 @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_552_neg__0__le__iff__le,axiom,
! [A2: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ ( uminus1482373934393186551omplex @ A2 ) )
= ( ord_less_eq_complex @ A2 @ zero_zero_complex ) ) ).
% neg_0_le_iff_le
thf(fact_553_neg__le__0__iff__le,axiom,
! [A2: complex] :
( ( ord_less_eq_complex @ ( uminus1482373934393186551omplex @ A2 ) @ zero_zero_complex )
= ( ord_less_eq_complex @ zero_zero_complex @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_554_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( minus_minus_nat @ B3 @ A2 )
= C )
= ( B3
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_555_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B3 @ A2 ) )
= B3 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_556_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_557_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_558_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_559_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B3 ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_560_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B3 ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_561_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_562_le__add__diff,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_563_add__le__add__imp__diff__le,axiom,
! [I3: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_564_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A2 ) @ A2 )
= B3 ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_565_le__add__diff__inverse2,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B3 ) @ B3 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_566_le__add__diff__inverse,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A2 @ B3 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_567_add__le__imp__le__diff,axiom,
! [I3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ N )
=> ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_568_le__diff__eq,axiom,
! [A2: complex,C: complex,B3: complex] :
( ( ord_less_eq_complex @ A2 @ ( minus_minus_complex @ C @ B3 ) )
= ( ord_less_eq_complex @ ( plus_plus_complex @ A2 @ B3 ) @ C ) ) ).
% le_diff_eq
thf(fact_569_diff__le__eq,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( ord_less_eq_complex @ ( minus_minus_complex @ A2 @ B3 ) @ C )
= ( ord_less_eq_complex @ A2 @ ( plus_plus_complex @ C @ B3 ) ) ) ).
% diff_le_eq
thf(fact_570_ab__group__add__on__with_Osum__mono__neutral__cong__left,axiom,
! [S: set_complex,Pls: complex > complex > complex,Z: complex,Mns: complex > complex > complex,Um: complex > complex,H: mat_complex > complex,G: mat_complex > complex,T: set_mat_complex,Sa: set_mat_complex] :
( ( group_3842241455253984271omplex @ S @ Pls @ Z @ Mns @ Um )
=> ( ! [X3: mat_complex] : ( member_complex @ ( H @ X3 ) @ S )
=> ( ! [X3: mat_complex] : ( member_complex @ ( G @ X3 ) @ S )
=> ( ( finite7047982916621727056omplex @ T )
=> ( ( ord_le3632134057777142183omplex @ Sa @ T )
=> ( ! [X3: mat_complex] :
( ( member_mat_complex @ X3 @ ( minus_8760755521168068590omplex @ T @ Sa ) )
=> ( ( H @ X3 )
= Z ) )
=> ( ! [X3: mat_complex] :
( ( member_mat_complex @ X3 @ Sa )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( group_6233491804913373438omplex @ Pls @ Z @ G @ Sa )
= ( group_6233491804913373438omplex @ Pls @ Z @ H @ T ) ) ) ) ) ) ) ) ) ).
% ab_group_add_on_with.sum_mono_neutral_cong_left
thf(fact_571_ab__group__add__on__with_Osum__mono__neutral__cong__left,axiom,
! [S: set_complex,Pls: complex > complex > complex,Z: complex,Mns: complex > complex > complex,Um: complex > complex,H: complex > complex,G: complex > complex,T: set_complex,Sa: set_complex] :
( ( group_3842241455253984271omplex @ S @ Pls @ Z @ Mns @ Um )
=> ( ! [X3: complex] : ( member_complex @ ( H @ X3 ) @ S )
=> ( ! [X3: complex] : ( member_complex @ ( G @ X3 ) @ S )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ( ord_le211207098394363844omplex @ Sa @ T )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T @ Sa ) )
=> ( ( H @ X3 )
= Z ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ Sa )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( group_4775205164212326935omplex @ Pls @ Z @ G @ Sa )
= ( group_4775205164212326935omplex @ Pls @ Z @ H @ T ) ) ) ) ) ) ) ) ) ).
% ab_group_add_on_with.sum_mono_neutral_cong_left
thf(fact_572_ab__group__add__on__with_Osum__mono__neutral__cong__left,axiom,
! [S: set_mat_complex,Pls: mat_complex > mat_complex > mat_complex,Z: mat_complex,Mns: mat_complex > mat_complex > mat_complex,Um: mat_complex > mat_complex,H: mat_complex > mat_complex,G: mat_complex > mat_complex,T: set_mat_complex,Sa: set_mat_complex] :
( ( group_7083379013581629702omplex @ S @ Pls @ Z @ Mns @ Um )
=> ( ! [X3: mat_complex] : ( member_mat_complex @ ( H @ X3 ) @ S )
=> ( ! [X3: mat_complex] : ( member_mat_complex @ ( G @ X3 ) @ S )
=> ( ( finite7047982916621727056omplex @ T )
=> ( ( ord_le3632134057777142183omplex @ Sa @ T )
=> ( ! [X3: mat_complex] :
( ( member_mat_complex @ X3 @ ( minus_8760755521168068590omplex @ T @ Sa ) )
=> ( ( H @ X3 )
= Z ) )
=> ( ! [X3: mat_complex] :
( ( member_mat_complex @ X3 @ Sa )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( group_1588376139278055545omplex @ Pls @ Z @ G @ Sa )
= ( group_1588376139278055545omplex @ Pls @ Z @ H @ T ) ) ) ) ) ) ) ) ) ).
% ab_group_add_on_with.sum_mono_neutral_cong_left
thf(fact_573_ab__group__add__on__with_Osum__mono__neutral__cong__left,axiom,
! [S: set_mat_complex,Pls: mat_complex > mat_complex > mat_complex,Z: mat_complex,Mns: mat_complex > mat_complex > mat_complex,Um: mat_complex > mat_complex,H: complex > mat_complex,G: complex > mat_complex,T: set_complex,Sa: set_complex] :
( ( group_7083379013581629702omplex @ S @ Pls @ Z @ Mns @ Um )
=> ( ! [X3: complex] : ( member_mat_complex @ ( H @ X3 ) @ S )
=> ( ! [X3: complex] : ( member_mat_complex @ ( G @ X3 ) @ S )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ( ord_le211207098394363844omplex @ Sa @ T )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T @ Sa ) )
=> ( ( H @ X3 )
= Z ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ Sa )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( group_4310869077125956188omplex @ Pls @ Z @ G @ Sa )
= ( group_4310869077125956188omplex @ Pls @ Z @ H @ T ) ) ) ) ) ) ) ) ) ).
% ab_group_add_on_with.sum_mono_neutral_cong_left
thf(fact_574_Complex__Matrix_Opositive__add,axiom,
! [A: mat_complex,B2: mat_complex,N: nat] :
( ( complex_positive @ A )
=> ( ( complex_positive @ B2 )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( complex_positive @ ( plus_p8323303612493835998omplex @ A @ B2 ) ) ) ) ) ) ).
% Complex_Matrix.positive_add
thf(fact_575_mult__left__less__imp__less,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A2 @ B3 ) ) ) ).
% mult_left_less_imp_less
thf(fact_576_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_577_mult__right__less__imp__less,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A2 @ B3 ) ) ) ).
% mult_right_less_imp_less
thf(fact_578_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_579_mult__left__le__imp__le,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).
% mult_left_le_imp_le
thf(fact_580_mult__right__le__imp__le,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).
% mult_right_le_imp_le
thf(fact_581_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_582_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_583_zero__compare__simps_I2_J,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% zero_compare_simps(2)
thf(fact_584_zero__compare__simps_I2_J,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A2 )
=> ( ( ord_less_complex @ B3 @ C )
=> ( ord_less_complex @ B3 @ ( plus_plus_complex @ A2 @ C ) ) ) ) ).
% zero_compare_simps(2)
thf(fact_585_zero__compare__simps_I1_J,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% zero_compare_simps(1)
thf(fact_586_zero__compare__simps_I1_J,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( ord_less_complex @ zero_zero_complex @ A2 )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ord_less_complex @ B3 @ ( plus_plus_complex @ A2 @ C ) ) ) ) ).
% zero_compare_simps(1)
thf(fact_587_add__sign__intros_I7_J,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_sign_intros(7)
thf(fact_588_add__sign__intros_I7_J,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_eq_complex @ A2 @ zero_zero_complex )
=> ( ( ord_less_complex @ B3 @ zero_zero_complex )
=> ( ord_less_complex @ ( plus_plus_complex @ A2 @ B3 ) @ zero_zero_complex ) ) ) ).
% add_sign_intros(7)
thf(fact_589_add__sign__intros_I5_J,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_sign_intros(5)
thf(fact_590_add__sign__intros_I5_J,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_complex @ A2 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ B3 @ zero_zero_complex )
=> ( ord_less_complex @ ( plus_plus_complex @ A2 @ B3 ) @ zero_zero_complex ) ) ) ).
% add_sign_intros(5)
thf(fact_591_add__sign__intros_I3_J,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% add_sign_intros(3)
thf(fact_592_add__sign__intros_I3_J,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A2 )
=> ( ( ord_less_complex @ zero_zero_complex @ B3 )
=> ( ord_less_complex @ zero_zero_complex @ ( plus_plus_complex @ A2 @ B3 ) ) ) ) ).
% add_sign_intros(3)
thf(fact_593_add__sign__intros_I1_J,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% add_sign_intros(1)
thf(fact_594_add__sign__intros_I1_J,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_complex @ zero_zero_complex @ A2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B3 )
=> ( ord_less_complex @ zero_zero_complex @ ( plus_plus_complex @ A2 @ B3 ) ) ) ) ).
% add_sign_intros(1)
thf(fact_595_ordered__ring__class_Ole__add__iff1,axiom,
! [A2: complex,E: complex,C: complex,B3: complex,D2: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ ( times_times_complex @ A2 @ E ) @ C ) @ ( plus_plus_complex @ ( times_times_complex @ B3 @ E ) @ D2 ) )
= ( ord_less_eq_complex @ ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A2 @ B3 ) @ E ) @ C ) @ D2 ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_596_ordered__ring__class_Ole__add__iff2,axiom,
! [A2: complex,E: complex,C: complex,B3: complex,D2: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ ( times_times_complex @ A2 @ E ) @ C ) @ ( plus_plus_complex @ ( times_times_complex @ B3 @ E ) @ D2 ) )
= ( ord_less_eq_complex @ C @ ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B3 @ A2 ) @ E ) @ D2 ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_597_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_598_comm__monoid__add__on__with_Oadd__zero,axiom,
! [S: set_complex,Pls: complex > complex > complex,Z: complex,A2: complex] :
( ( group_2796108508354279923omplex @ S @ Pls @ Z )
=> ( ( member_complex @ A2 @ S )
=> ( ( Pls @ Z @ A2 )
= A2 ) ) ) ).
% comm_monoid_add_on_with.add_zero
thf(fact_599_comm__monoid__add__on__with_Oadd__zero,axiom,
! [S: set_mat_complex,Pls: mat_complex > mat_complex > mat_complex,Z: mat_complex,A2: mat_complex] :
( ( group_5394922976599784994omplex @ S @ Pls @ Z )
=> ( ( member_mat_complex @ A2 @ S )
=> ( ( Pls @ Z @ A2 )
= A2 ) ) ) ).
% comm_monoid_add_on_with.add_zero
thf(fact_600_comm__monoid__add__on__with_Ozero__mem,axiom,
! [S: set_complex,Pls: complex > complex > complex,Z: complex] :
( ( group_2796108508354279923omplex @ S @ Pls @ Z )
=> ( member_complex @ Z @ S ) ) ).
% comm_monoid_add_on_with.zero_mem
thf(fact_601_comm__monoid__add__on__with_Ozero__mem,axiom,
! [S: set_mat_complex,Pls: mat_complex > mat_complex > mat_complex,Z: mat_complex] :
( ( group_5394922976599784994omplex @ S @ Pls @ Z )
=> ( member_mat_complex @ Z @ S ) ) ).
% comm_monoid_add_on_with.zero_mem
thf(fact_602_ab__group__add__on__with_Ouminus__mem,axiom,
! [S: set_complex,Pls: complex > complex > complex,Z: complex,Mns: complex > complex > complex,Um: complex > complex,A2: complex] :
( ( group_3842241455253984271omplex @ S @ Pls @ Z @ Mns @ Um )
=> ( ( member_complex @ A2 @ S )
=> ( member_complex @ ( Um @ A2 ) @ S ) ) ) ).
% ab_group_add_on_with.uminus_mem
thf(fact_603_ab__group__add__on__with_Ouminus__mem,axiom,
! [S: set_mat_complex,Pls: mat_complex > mat_complex > mat_complex,Z: mat_complex,Mns: mat_complex > mat_complex > mat_complex,Um: mat_complex > mat_complex,A2: mat_complex] :
( ( group_7083379013581629702omplex @ S @ Pls @ Z @ Mns @ Um )
=> ( ( member_mat_complex @ A2 @ S )
=> ( member_mat_complex @ ( Um @ A2 ) @ S ) ) ) ).
% ab_group_add_on_with.uminus_mem
thf(fact_604_ab__group__add__on__with_Oab__left__minus,axiom,
! [S: set_complex,Pls: complex > complex > complex,Z: complex,Mns: complex > complex > complex,Um: complex > complex,A2: complex] :
( ( group_3842241455253984271omplex @ S @ Pls @ Z @ Mns @ Um )
=> ( ( member_complex @ A2 @ S )
=> ( ( Pls @ ( Um @ A2 ) @ A2 )
= Z ) ) ) ).
% ab_group_add_on_with.ab_left_minus
thf(fact_605_ab__group__add__on__with_Oab__left__minus,axiom,
! [S: set_mat_complex,Pls: mat_complex > mat_complex > mat_complex,Z: mat_complex,Mns: mat_complex > mat_complex > mat_complex,Um: mat_complex > mat_complex,A2: mat_complex] :
( ( group_7083379013581629702omplex @ S @ Pls @ Z @ Mns @ Um )
=> ( ( member_mat_complex @ A2 @ S )
=> ( ( Pls @ ( Um @ A2 ) @ A2 )
= Z ) ) ) ).
% ab_group_add_on_with.ab_left_minus
thf(fact_606_ab__group__add__on__with_Oab__diff__conv__add__uminus,axiom,
! [S: set_complex,Pls: complex > complex > complex,Z: complex,Mns: complex > complex > complex,Um: complex > complex,A2: complex,B3: complex] :
( ( group_3842241455253984271omplex @ S @ Pls @ Z @ Mns @ Um )
=> ( ( member_complex @ A2 @ S )
=> ( ( member_complex @ B3 @ S )
=> ( ( Mns @ A2 @ B3 )
= ( Pls @ A2 @ ( Um @ B3 ) ) ) ) ) ) ).
% ab_group_add_on_with.ab_diff_conv_add_uminus
thf(fact_607_ab__group__add__on__with_Oab__diff__conv__add__uminus,axiom,
! [S: set_mat_complex,Pls: mat_complex > mat_complex > mat_complex,Z: mat_complex,Mns: mat_complex > mat_complex > mat_complex,Um: mat_complex > mat_complex,A2: mat_complex,B3: mat_complex] :
( ( group_7083379013581629702omplex @ S @ Pls @ Z @ Mns @ Um )
=> ( ( member_mat_complex @ A2 @ S )
=> ( ( member_mat_complex @ B3 @ S )
=> ( ( Mns @ A2 @ B3 )
= ( Pls @ A2 @ ( Um @ B3 ) ) ) ) ) ) ).
% ab_group_add_on_with.ab_diff_conv_add_uminus
thf(fact_608_ab__semigroup__add__on__with_Oadd__commute,axiom,
! [S: set_complex,Pls: complex > complex > complex,A2: complex,B3: complex] :
( ( group_4393537770075021309omplex @ S @ Pls )
=> ( ( member_complex @ A2 @ S )
=> ( ( member_complex @ B3 @ S )
=> ( ( Pls @ A2 @ B3 )
= ( Pls @ B3 @ A2 ) ) ) ) ) ).
% ab_semigroup_add_on_with.add_commute
thf(fact_609_ab__semigroup__add__on__with_Oadd__commute,axiom,
! [S: set_mat_complex,Pls: mat_complex > mat_complex > mat_complex,A2: mat_complex,B3: mat_complex] :
( ( group_8296285978151397272omplex @ S @ Pls )
=> ( ( member_mat_complex @ A2 @ S )
=> ( ( member_mat_complex @ B3 @ S )
=> ( ( Pls @ A2 @ B3 )
= ( Pls @ B3 @ A2 ) ) ) ) ) ).
% ab_semigroup_add_on_with.add_commute
thf(fact_610_semigroup__add__on__with_Ointro,axiom,
! [S: set_complex,Pls: complex > complex > complex] :
( ! [A4: complex,B5: complex,C3: complex] :
( ( member_complex @ A4 @ S )
=> ( ( member_complex @ B5 @ S )
=> ( ( member_complex @ C3 @ S )
=> ( ( Pls @ ( Pls @ A4 @ B5 ) @ C3 )
= ( Pls @ A4 @ ( Pls @ B5 @ C3 ) ) ) ) ) )
=> ( ! [A4: complex,B5: complex] :
( ( member_complex @ A4 @ S )
=> ( ( member_complex @ B5 @ S )
=> ( member_complex @ ( Pls @ A4 @ B5 ) @ S ) ) )
=> ( group_6437017389065004156omplex @ S @ Pls ) ) ) ).
% semigroup_add_on_with.intro
thf(fact_611_semigroup__add__on__with_Ointro,axiom,
! [S: set_mat_complex,Pls: mat_complex > mat_complex > mat_complex] :
( ! [A4: mat_complex,B5: mat_complex,C3: mat_complex] :
( ( member_mat_complex @ A4 @ S )
=> ( ( member_mat_complex @ B5 @ S )
=> ( ( member_mat_complex @ C3 @ S )
=> ( ( Pls @ ( Pls @ A4 @ B5 ) @ C3 )
= ( Pls @ A4 @ ( Pls @ B5 @ C3 ) ) ) ) ) )
=> ( ! [A4: mat_complex,B5: mat_complex] :
( ( member_mat_complex @ A4 @ S )
=> ( ( member_mat_complex @ B5 @ S )
=> ( member_mat_complex @ ( Pls @ A4 @ B5 ) @ S ) ) )
=> ( group_6724803037680873049omplex @ S @ Pls ) ) ) ).
% semigroup_add_on_with.intro
thf(fact_612_semigroup__add__on__with_Oadd__mem,axiom,
! [S: set_complex,Pls: complex > complex > complex,A2: complex,B3: complex] :
( ( group_6437017389065004156omplex @ S @ Pls )
=> ( ( member_complex @ A2 @ S )
=> ( ( member_complex @ B3 @ S )
=> ( member_complex @ ( Pls @ A2 @ B3 ) @ S ) ) ) ) ).
% semigroup_add_on_with.add_mem
thf(fact_613_semigroup__add__on__with_Oadd__mem,axiom,
! [S: set_mat_complex,Pls: mat_complex > mat_complex > mat_complex,A2: mat_complex,B3: mat_complex] :
( ( group_6724803037680873049omplex @ S @ Pls )
=> ( ( member_mat_complex @ A2 @ S )
=> ( ( member_mat_complex @ B3 @ S )
=> ( member_mat_complex @ ( Pls @ A2 @ B3 ) @ S ) ) ) ) ).
% semigroup_add_on_with.add_mem
thf(fact_614_semigroup__add__on__with_Oadd__assoc,axiom,
! [S: set_complex,Pls: complex > complex > complex,A2: complex,B3: complex,C: complex] :
( ( group_6437017389065004156omplex @ S @ Pls )
=> ( ( member_complex @ A2 @ S )
=> ( ( member_complex @ B3 @ S )
=> ( ( member_complex @ C @ S )
=> ( ( Pls @ ( Pls @ A2 @ B3 ) @ C )
= ( Pls @ A2 @ ( Pls @ B3 @ C ) ) ) ) ) ) ) ).
% semigroup_add_on_with.add_assoc
thf(fact_615_semigroup__add__on__with_Oadd__assoc,axiom,
! [S: set_mat_complex,Pls: mat_complex > mat_complex > mat_complex,A2: mat_complex,B3: mat_complex,C: mat_complex] :
( ( group_6724803037680873049omplex @ S @ Pls )
=> ( ( member_mat_complex @ A2 @ S )
=> ( ( member_mat_complex @ B3 @ S )
=> ( ( member_mat_complex @ C @ S )
=> ( ( Pls @ ( Pls @ A2 @ B3 ) @ C )
= ( Pls @ A2 @ ( Pls @ B3 @ C ) ) ) ) ) ) ) ).
% semigroup_add_on_with.add_assoc
thf(fact_616_semigroup__add__on__with__def,axiom,
( group_6437017389065004156omplex
= ( ^ [S2: set_complex,Pls2: complex > complex > complex] :
( ! [A3: complex,B4: complex,C4: complex] :
( ( member_complex @ A3 @ S2 )
=> ( ( member_complex @ B4 @ S2 )
=> ( ( member_complex @ C4 @ S2 )
=> ( ( Pls2 @ ( Pls2 @ A3 @ B4 ) @ C4 )
= ( Pls2 @ A3 @ ( Pls2 @ B4 @ C4 ) ) ) ) ) )
& ! [A3: complex,B4: complex] :
( ( member_complex @ A3 @ S2 )
=> ( ( member_complex @ B4 @ S2 )
=> ( member_complex @ ( Pls2 @ A3 @ B4 ) @ S2 ) ) ) ) ) ) ).
% semigroup_add_on_with_def
thf(fact_617_semigroup__add__on__with__def,axiom,
( group_6724803037680873049omplex
= ( ^ [S2: set_mat_complex,Pls2: mat_complex > mat_complex > mat_complex] :
( ! [A3: mat_complex,B4: mat_complex,C4: mat_complex] :
( ( member_mat_complex @ A3 @ S2 )
=> ( ( member_mat_complex @ B4 @ S2 )
=> ( ( member_mat_complex @ C4 @ S2 )
=> ( ( Pls2 @ ( Pls2 @ A3 @ B4 ) @ C4 )
= ( Pls2 @ A3 @ ( Pls2 @ B4 @ C4 ) ) ) ) ) )
& ! [A3: mat_complex,B4: mat_complex] :
( ( member_mat_complex @ A3 @ S2 )
=> ( ( member_mat_complex @ B4 @ S2 )
=> ( member_mat_complex @ ( Pls2 @ A3 @ B4 ) @ S2 ) ) ) ) ) ) ).
% semigroup_add_on_with_def
thf(fact_618_semigroup__add__on__with__Ball__def,axiom,
( group_6437017389065004156omplex
= ( ^ [S2: set_complex,Pls2: complex > complex > complex] :
( ! [X2: complex] :
( ( member_complex @ X2 @ S2 )
=> ! [Y: complex] :
( ( member_complex @ Y @ S2 )
=> ! [Z2: complex] :
( ( member_complex @ Z2 @ S2 )
=> ( ( Pls2 @ ( Pls2 @ X2 @ Y ) @ Z2 )
= ( Pls2 @ X2 @ ( Pls2 @ Y @ Z2 ) ) ) ) ) )
& ! [X2: complex] :
( ( member_complex @ X2 @ S2 )
=> ! [Y: complex] :
( ( member_complex @ Y @ S2 )
=> ( member_complex @ ( Pls2 @ X2 @ Y ) @ S2 ) ) ) ) ) ) ).
% semigroup_add_on_with_Ball_def
thf(fact_619_semigroup__add__on__with__Ball__def,axiom,
( group_6724803037680873049omplex
= ( ^ [S2: set_mat_complex,Pls2: mat_complex > mat_complex > mat_complex] :
( ! [X2: mat_complex] :
( ( member_mat_complex @ X2 @ S2 )
=> ! [Y: mat_complex] :
( ( member_mat_complex @ Y @ S2 )
=> ! [Z2: mat_complex] :
( ( member_mat_complex @ Z2 @ S2 )
=> ( ( Pls2 @ ( Pls2 @ X2 @ Y ) @ Z2 )
= ( Pls2 @ X2 @ ( Pls2 @ Y @ Z2 ) ) ) ) ) )
& ! [X2: mat_complex] :
( ( member_mat_complex @ X2 @ S2 )
=> ! [Y: mat_complex] :
( ( member_mat_complex @ Y @ S2 )
=> ( member_mat_complex @ ( Pls2 @ X2 @ Y ) @ S2 ) ) ) ) ) ) ).
% semigroup_add_on_with_Ball_def
thf(fact_620_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_621_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_622_nat__mult__less__cancel__disj,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 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_623_comm__monoid__add__on__with_Ocarrier__ne,axiom,
! [S: set_complex,Pls: complex > complex > complex,Z: complex] :
( ( group_2796108508354279923omplex @ S @ Pls @ Z )
=> ( S != bot_bot_set_complex ) ) ).
% comm_monoid_add_on_with.carrier_ne
thf(fact_624_comm__monoid__add__on__with_Ocarrier__ne,axiom,
! [S: set_mat_complex,Pls: mat_complex > mat_complex > mat_complex,Z: mat_complex] :
( ( group_5394922976599784994omplex @ S @ Pls @ Z )
=> ( S != bot_bo7165004461764951667omplex ) ) ).
% comm_monoid_add_on_with.carrier_ne
thf(fact_625_ab__group__add__on__with_Oaxioms_I1_J,axiom,
! [S: set_mat_complex,Pls: mat_complex > mat_complex > mat_complex,Z: mat_complex,Mns: mat_complex > mat_complex > mat_complex,Um: mat_complex > mat_complex] :
( ( group_7083379013581629702omplex @ S @ Pls @ Z @ Mns @ Um )
=> ( group_5394922976599784994omplex @ S @ Pls @ Z ) ) ).
% ab_group_add_on_with.axioms(1)
thf(fact_626_ab__group__add__on__with__Ball__def,axiom,
( group_3842241455253984271omplex
= ( ^ [S2: set_complex,Pls2: complex > complex > complex,Z2: complex,Mns2: complex > complex > complex,Um2: complex > complex] :
( ( group_2796108508354279923omplex @ S2 @ Pls2 @ Z2 )
& ! [X2: complex] :
( ( member_complex @ X2 @ S2 )
=> ( ( Pls2 @ ( Um2 @ X2 ) @ X2 )
= Z2 ) )
& ! [X2: complex] :
( ( member_complex @ X2 @ S2 )
=> ! [Y: complex] :
( ( member_complex @ Y @ S2 )
=> ( ( Mns2 @ X2 @ Y )
= ( Pls2 @ X2 @ ( Um2 @ Y ) ) ) ) )
& ! [X2: complex] :
( ( member_complex @ X2 @ S2 )
=> ( member_complex @ ( Um2 @ X2 ) @ S2 ) ) ) ) ) ).
% ab_group_add_on_with_Ball_def
thf(fact_627_ab__group__add__on__with__Ball__def,axiom,
( group_7083379013581629702omplex
= ( ^ [S2: set_mat_complex,Pls2: mat_complex > mat_complex > mat_complex,Z2: mat_complex,Mns2: mat_complex > mat_complex > mat_complex,Um2: mat_complex > mat_complex] :
( ( group_5394922976599784994omplex @ S2 @ Pls2 @ Z2 )
& ! [X2: mat_complex] :
( ( member_mat_complex @ X2 @ S2 )
=> ( ( Pls2 @ ( Um2 @ X2 ) @ X2 )
= Z2 ) )
& ! [X2: mat_complex] :
( ( member_mat_complex @ X2 @ S2 )
=> ! [Y: mat_complex] :
( ( member_mat_complex @ Y @ S2 )
=> ( ( Mns2 @ X2 @ Y )
= ( Pls2 @ X2 @ ( Um2 @ Y ) ) ) ) )
& ! [X2: mat_complex] :
( ( member_mat_complex @ X2 @ S2 )
=> ( member_mat_complex @ ( Um2 @ X2 ) @ S2 ) ) ) ) ) ).
% ab_group_add_on_with_Ball_def
thf(fact_628_comm__monoid__add__on__with_Oaxioms_I1_J,axiom,
! [S: set_mat_complex,Pls: mat_complex > mat_complex > mat_complex,Z: mat_complex] :
( ( group_5394922976599784994omplex @ S @ Pls @ Z )
=> ( group_8296285978151397272omplex @ S @ Pls ) ) ).
% comm_monoid_add_on_with.axioms(1)
thf(fact_629_comm__monoid__add__on__with__Ball__def,axiom,
( group_2796108508354279923omplex
= ( ^ [S2: set_complex,Pls2: complex > complex > complex,Z2: complex] :
( ( group_4393537770075021309omplex @ S2 @ Pls2 )
& ! [X2: complex] :
( ( member_complex @ X2 @ S2 )
=> ( ( Pls2 @ Z2 @ X2 )
= X2 ) )
& ( member_complex @ Z2 @ S2 ) ) ) ) ).
% comm_monoid_add_on_with_Ball_def
thf(fact_630_comm__monoid__add__on__with__Ball__def,axiom,
( group_5394922976599784994omplex
= ( ^ [S2: set_mat_complex,Pls2: mat_complex > mat_complex > mat_complex,Z2: mat_complex] :
( ( group_8296285978151397272omplex @ S2 @ Pls2 )
& ! [X2: mat_complex] :
( ( member_mat_complex @ X2 @ S2 )
=> ( ( Pls2 @ Z2 @ X2 )
= X2 ) )
& ( member_mat_complex @ Z2 @ S2 ) ) ) ) ).
% comm_monoid_add_on_with_Ball_def
thf(fact_631_ab__semigroup__add__on__with_Oaxioms_I1_J,axiom,
! [S: set_mat_complex,Pls: mat_complex > mat_complex > mat_complex] :
( ( group_8296285978151397272omplex @ S @ Pls )
=> ( group_6724803037680873049omplex @ S @ Pls ) ) ).
% ab_semigroup_add_on_with.axioms(1)
thf(fact_632_ab__semigroup__add__on__with__Ball__def,axiom,
( group_8296285978151397272omplex
= ( ^ [S2: set_mat_complex,Pls2: mat_complex > mat_complex > mat_complex] :
( ( group_6724803037680873049omplex @ S2 @ Pls2 )
& ! [X2: mat_complex] :
( ( member_mat_complex @ X2 @ S2 )
=> ! [Y: mat_complex] :
( ( member_mat_complex @ Y @ S2 )
=> ( ( Pls2 @ X2 @ Y )
= ( Pls2 @ Y @ X2 ) ) ) ) ) ) ) ).
% ab_semigroup_add_on_with_Ball_def
thf(fact_633_nat__eq__add__iff1,axiom,
! [J: nat,I3: nat,U3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I3 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U3 ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U3 ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J ) @ U3 ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_634_nat__eq__add__iff2,axiom,
! [I3: nat,J: nat,U3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U3 ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U3 ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I3 ) @ U3 ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_635_nat__le__add__iff1,axiom,
! [J: nat,I3: nat,U3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I3 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U3 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U3 ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J ) @ U3 ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_636_nat__le__add__iff2,axiom,
! [I3: nat,J: nat,U3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U3 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U3 ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I3 ) @ U3 ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_637_nat__diff__add__eq1,axiom,
! [J: nat,I3: nat,U3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U3 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U3 ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J ) @ U3 ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_638_nat__diff__add__eq2,axiom,
! [I3: nat,J: nat,U3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U3 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U3 ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I3 ) @ U3 ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_639_left__add__mult__distrib,axiom,
! [I3: nat,U3: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I3 @ U3 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U3 ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I3 @ J ) @ U3 ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_640_nat__less__add__iff1,axiom,
! [J: nat,I3: nat,U3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I3 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U3 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U3 ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J ) @ U3 ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_641_nat__less__add__iff2,axiom,
! [I3: nat,J: nat,U3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U3 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U3 ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I3 ) @ U3 ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_642_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_643_nat__mult__le__cancel__disj,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 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_644_positive__smult,axiom,
! [A: mat_complex,N: nat,C: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_positive @ A )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( complex_positive @ ( smult_mat_complex @ C @ A ) ) ) ) ) ).
% positive_smult
thf(fact_645_sum__with__insert,axiom,
! [G: mat_complex > mat_complex,X: mat_complex,A: set_mat_complex] :
( ( member_mat_complex @ ( G @ X ) @ fc )
=> ( ( ord_le3632134057777142183omplex @ ( image_23760814813800901omplex @ G @ A ) @ fc )
=> ( ( finite7047982916621727056omplex @ A )
=> ( ~ ( member_mat_complex @ X @ A )
=> ( ( group_1588376139278055545omplex @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ G @ ( insert_mat_complex @ X @ A ) )
= ( plus_p8323303612493835998omplex @ ( G @ X ) @ ( group_1588376139278055545omplex @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ G @ A ) ) ) ) ) ) ) ).
% sum_with_insert
thf(fact_646_sum__with__insert,axiom,
! [G: complex > mat_complex,X: complex,A: set_complex] :
( ( member_mat_complex @ ( G @ X ) @ fc )
=> ( ( ord_le3632134057777142183omplex @ ( image_6107471045988054706omplex @ G @ A ) @ fc )
=> ( ( finite3207457112153483333omplex @ A )
=> ( ~ ( member_complex @ X @ A )
=> ( ( group_4310869077125956188omplex @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ G @ ( insert_complex @ X @ A ) )
= ( plus_p8323303612493835998omplex @ ( G @ X ) @ ( group_4310869077125956188omplex @ plus_p8323303612493835998omplex @ ( zero_mat_complex @ n @ n ) @ G @ A ) ) ) ) ) ) ) ).
% sum_with_insert
thf(fact_647_sum__mat__insert,axiom,
! [A: mat_complex > mat_complex,X: mat_complex,I: set_mat_complex] :
( ( member_mat_complex @ ( A @ X ) @ fc )
=> ( ( ord_le3632134057777142183omplex @ ( image_23760814813800901omplex @ A @ I ) @ fc )
=> ( ( finite7047982916621727056omplex @ I )
=> ( ~ ( member_mat_complex @ X @ I )
=> ( ( linear1795808462385993418omplex @ n @ n @ A @ ( insert_mat_complex @ X @ I ) )
= ( plus_p8323303612493835998omplex @ ( A @ X ) @ ( linear1795808462385993418omplex @ n @ n @ A @ I ) ) ) ) ) ) ) ).
% sum_mat_insert
thf(fact_648_sum__mat__insert,axiom,
! [A: complex > mat_complex,X: complex,I: set_complex] :
( ( member_mat_complex @ ( A @ X ) @ fc )
=> ( ( ord_le3632134057777142183omplex @ ( image_6107471045988054706omplex @ A @ I ) @ fc )
=> ( ( finite3207457112153483333omplex @ I )
=> ( ~ ( member_complex @ X @ I )
=> ( ( linear8664352376190006057omplex @ n @ n @ A @ ( insert_complex @ X @ I ) )
= ( plus_p8323303612493835998omplex @ ( A @ X ) @ ( linear8664352376190006057omplex @ n @ n @ A @ I ) ) ) ) ) ) ) ).
% sum_mat_insert
thf(fact_649_last__subset,axiom,
! [A: set_mat_complex,A2: mat_complex,B3: mat_complex] :
( ( ord_le3632134057777142183omplex @ A @ ( insert_mat_complex @ A2 @ ( insert_mat_complex @ B3 @ bot_bo7165004461764951667omplex ) ) )
=> ( ( A2 != B3 )
=> ( ( A
!= ( insert_mat_complex @ A2 @ ( insert_mat_complex @ B3 @ bot_bo7165004461764951667omplex ) ) )
=> ( ( A != bot_bo7165004461764951667omplex )
=> ( ( A
!= ( insert_mat_complex @ A2 @ bot_bo7165004461764951667omplex ) )
=> ( A
= ( insert_mat_complex @ B3 @ bot_bo7165004461764951667omplex ) ) ) ) ) ) ) ).
% last_subset
thf(fact_650_last__subset,axiom,
! [A: set_complex,A2: complex,B3: complex] :
( ( ord_le211207098394363844omplex @ A @ ( insert_complex @ A2 @ ( insert_complex @ B3 @ bot_bot_set_complex ) ) )
=> ( ( A2 != B3 )
=> ( ( A
!= ( insert_complex @ A2 @ ( insert_complex @ B3 @ bot_bot_set_complex ) ) )
=> ( ( A != bot_bot_set_complex )
=> ( ( A
!= ( insert_complex @ A2 @ bot_bot_set_complex ) )
=> ( A
= ( insert_complex @ B3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% last_subset
thf(fact_651_finite__ranking__induct,axiom,
! [S: set_mat_complex,P: set_mat_complex > $o,F: mat_complex > nat] :
( ( finite7047982916621727056omplex @ S )
=> ( ( P @ bot_bo7165004461764951667omplex )
=> ( ! [X3: mat_complex,S3: set_mat_complex] :
( ( finite7047982916621727056omplex @ S3 )
=> ( ! [Y3: mat_complex] :
( ( member_mat_complex @ Y3 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_mat_complex @ X3 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_652_finite__ranking__induct,axiom,
! [S: set_complex,P: set_complex > $o,F: complex > nat] :
( ( finite3207457112153483333omplex @ S )
=> ( ( P @ bot_bot_set_complex )
=> ( ! [X3: complex,S3: set_complex] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ! [Y3: complex] :
( ( member_complex @ Y3 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_complex @ X3 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_653_finite__linorder__max__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B5: nat,A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A5 )
=> ( ord_less_nat @ X5 @ B5 ) )
=> ( ( P @ A5 )
=> ( P @ ( insert_nat @ B5 @ A5 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_654_finite__linorder__min__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B5: nat,A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A5 )
=> ( ord_less_nat @ B5 @ X5 ) )
=> ( ( P @ A5 )
=> ( P @ ( insert_nat @ B5 @ A5 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_655_comm__monoid__add__on__with_Osum__with__insert,axiom,
! [S: set_complex,Pls: complex > complex > complex,Z: complex,G: mat_complex > complex,X: mat_complex,A: set_mat_complex] :
( ( group_2796108508354279923omplex @ S @ Pls @ Z )
=> ( ( member_complex @ ( G @ X ) @ S )
=> ( ( ord_le211207098394363844omplex @ ( image_4184848318200637456omplex @ G @ A ) @ S )
=> ( ( finite7047982916621727056omplex @ A )
=> ( ~ ( member_mat_complex @ X @ A )
=> ( ( group_6233491804913373438omplex @ Pls @ Z @ G @ ( insert_mat_complex @ X @ A ) )
= ( Pls @ ( G @ X ) @ ( group_6233491804913373438omplex @ Pls @ Z @ G @ A ) ) ) ) ) ) ) ) ).
% comm_monoid_add_on_with.sum_with_insert
thf(fact_656_comm__monoid__add__on__with_Osum__with__insert,axiom,
! [S: set_complex,Pls: complex > complex > complex,Z: complex,G: complex > complex,X: complex,A: set_complex] :
( ( group_2796108508354279923omplex @ S @ Pls @ Z )
=> ( ( member_complex @ ( G @ X ) @ S )
=> ( ( ord_le211207098394363844omplex @ ( image_1468599708987790691omplex @ G @ A ) @ S )
=> ( ( finite3207457112153483333omplex @ A )
=> ( ~ ( member_complex @ X @ A )
=> ( ( group_4775205164212326935omplex @ Pls @ Z @ G @ ( insert_complex @ X @ A ) )
= ( Pls @ ( G @ X ) @ ( group_4775205164212326935omplex @ Pls @ Z @ G @ A ) ) ) ) ) ) ) ) ).
% comm_monoid_add_on_with.sum_with_insert
thf(fact_657_comm__monoid__add__on__with_Osum__with__insert,axiom,
! [S: set_mat_complex,Pls: mat_complex > mat_complex > mat_complex,Z: mat_complex,G: mat_complex > mat_complex,X: mat_complex,A: set_mat_complex] :
( ( group_5394922976599784994omplex @ S @ Pls @ Z )
=> ( ( member_mat_complex @ ( G @ X ) @ S )
=> ( ( ord_le3632134057777142183omplex @ ( image_23760814813800901omplex @ G @ A ) @ S )
=> ( ( finite7047982916621727056omplex @ A )
=> ( ~ ( member_mat_complex @ X @ A )
=> ( ( group_1588376139278055545omplex @ Pls @ Z @ G @ ( insert_mat_complex @ X @ A ) )
= ( Pls @ ( G @ X ) @ ( group_1588376139278055545omplex @ Pls @ Z @ G @ A ) ) ) ) ) ) ) ) ).
% comm_monoid_add_on_with.sum_with_insert
thf(fact_658_comm__monoid__add__on__with_Osum__with__insert,axiom,
! [S: set_mat_complex,Pls: mat_complex > mat_complex > mat_complex,Z: mat_complex,G: complex > mat_complex,X: complex,A: set_complex] :
( ( group_5394922976599784994omplex @ S @ Pls @ Z )
=> ( ( member_mat_complex @ ( G @ X ) @ S )
=> ( ( ord_le3632134057777142183omplex @ ( image_6107471045988054706omplex @ G @ A ) @ S )
=> ( ( finite3207457112153483333omplex @ A )
=> ( ~ ( member_complex @ X @ A )
=> ( ( group_4310869077125956188omplex @ Pls @ Z @ G @ ( insert_complex @ X @ A ) )
= ( Pls @ ( G @ X ) @ ( group_4310869077125956188omplex @ Pls @ Z @ G @ A ) ) ) ) ) ) ) ) ).
% comm_monoid_add_on_with.sum_with_insert
thf(fact_659_fixed__carrier__mat_Osum__mat__insert,axiom,
! [Fc_mats2: set_mat_complex,DimR2: nat,DimC2: nat,A: mat_complex > mat_complex,X: mat_complex,I: set_mat_complex] :
( ( linear8738132868031958293omplex @ Fc_mats2 @ DimR2 @ DimC2 )
=> ( ( member_mat_complex @ ( A @ X ) @ Fc_mats2 )
=> ( ( ord_le3632134057777142183omplex @ ( image_23760814813800901omplex @ A @ I ) @ Fc_mats2 )
=> ( ( finite7047982916621727056omplex @ I )
=> ( ~ ( member_mat_complex @ X @ I )
=> ( ( linear1795808462385993418omplex @ DimR2 @ DimC2 @ A @ ( insert_mat_complex @ X @ I ) )
= ( plus_p8323303612493835998omplex @ ( A @ X ) @ ( linear1795808462385993418omplex @ DimR2 @ DimC2 @ A @ I ) ) ) ) ) ) ) ) ).
% fixed_carrier_mat.sum_mat_insert
thf(fact_660_fixed__carrier__mat_Osum__mat__insert,axiom,
! [Fc_mats2: set_mat_complex,DimR2: nat,DimC2: nat,A: complex > mat_complex,X: complex,I: set_complex] :
( ( linear8738132868031958293omplex @ Fc_mats2 @ DimR2 @ DimC2 )
=> ( ( member_mat_complex @ ( A @ X ) @ Fc_mats2 )
=> ( ( ord_le3632134057777142183omplex @ ( image_6107471045988054706omplex @ A @ I ) @ Fc_mats2 )
=> ( ( finite3207457112153483333omplex @ I )
=> ( ~ ( member_complex @ X @ I )
=> ( ( linear8664352376190006057omplex @ DimR2 @ DimC2 @ A @ ( insert_complex @ X @ I ) )
= ( plus_p8323303612493835998omplex @ ( A @ X ) @ ( linear8664352376190006057omplex @ DimR2 @ DimC2 @ A @ I ) ) ) ) ) ) ) ) ).
% fixed_carrier_mat.sum_mat_insert
thf(fact_661_remove__induct,axiom,
! [P: set_mat_complex > $o,B2: set_mat_complex] :
( ( P @ bot_bo7165004461764951667omplex )
=> ( ( ~ ( finite7047982916621727056omplex @ B2 )
=> ( P @ B2 ) )
=> ( ! [A5: set_mat_complex] :
( ( finite7047982916621727056omplex @ A5 )
=> ( ( A5 != bot_bo7165004461764951667omplex )
=> ( ( ord_le3632134057777142183omplex @ A5 @ B2 )
=> ( ! [X5: mat_complex] :
( ( member_mat_complex @ X5 @ A5 )
=> ( P @ ( minus_8760755521168068590omplex @ A5 @ ( insert_mat_complex @ X5 @ bot_bo7165004461764951667omplex ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_662_remove__induct,axiom,
! [P: set_complex > $o,B2: set_complex] :
( ( P @ bot_bot_set_complex )
=> ( ( ~ ( finite3207457112153483333omplex @ B2 )
=> ( P @ B2 ) )
=> ( ! [A5: set_complex] :
( ( finite3207457112153483333omplex @ A5 )
=> ( ( A5 != bot_bot_set_complex )
=> ( ( ord_le211207098394363844omplex @ A5 @ B2 )
=> ( ! [X5: complex] :
( ( member_complex @ X5 @ A5 )
=> ( P @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X5 @ bot_bot_set_complex ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_663_finite__remove__induct,axiom,
! [B2: set_mat_complex,P: set_mat_complex > $o] :
( ( finite7047982916621727056omplex @ B2 )
=> ( ( P @ bot_bo7165004461764951667omplex )
=> ( ! [A5: set_mat_complex] :
( ( finite7047982916621727056omplex @ A5 )
=> ( ( A5 != bot_bo7165004461764951667omplex )
=> ( ( ord_le3632134057777142183omplex @ A5 @ B2 )
=> ( ! [X5: mat_complex] :
( ( member_mat_complex @ X5 @ A5 )
=> ( P @ ( minus_8760755521168068590omplex @ A5 @ ( insert_mat_complex @ X5 @ bot_bo7165004461764951667omplex ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_664_finite__remove__induct,axiom,
! [B2: set_complex,P: set_complex > $o] :
( ( finite3207457112153483333omplex @ B2 )
=> ( ( P @ bot_bot_set_complex )
=> ( ! [A5: set_complex] :
( ( finite3207457112153483333omplex @ A5 )
=> ( ( A5 != bot_bot_set_complex )
=> ( ( ord_le211207098394363844omplex @ A5 @ B2 )
=> ( ! [X5: complex] :
( ( member_complex @ X5 @ A5 )
=> ( P @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X5 @ bot_bot_set_complex ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_665_less__diff__conv2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I3 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_666_linorder__neqE__nat,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_667_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_668_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_669_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_670_less__not__refl3,axiom,
! [S4: nat,T2: nat] :
( ( ord_less_nat @ S4 @ T2 )
=> ( S4 != T2 ) ) ).
% less_not_refl3
thf(fact_671_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_672_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_673_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_674_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B3 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_675_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_676_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_677_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_678_le__trans,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_679_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_680_diff__commute,axiom,
! [I3: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J ) ) ).
% diff_commute
thf(fact_681_finite__has__maximal2,axiom,
! [A: set_mat_complex,A2: mat_complex] :
( ( finite7047982916621727056omplex @ A )
=> ( ( member_mat_complex @ A2 @ A )
=> ? [X3: mat_complex] :
( ( member_mat_complex @ X3 @ A )
& ( ord_le1403324449407493959omplex @ A2 @ X3 )
& ! [Xa: mat_complex] :
( ( member_mat_complex @ Xa @ A )
=> ( ( ord_le1403324449407493959omplex @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_682_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ A2 @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_683_finite__has__maximal2,axiom,
! [A: set_complex,A2: complex] :
( ( finite3207457112153483333omplex @ A )
=> ( ( member_complex @ A2 @ A )
=> ? [X3: complex] :
( ( member_complex @ X3 @ A )
& ( ord_less_eq_complex @ A2 @ X3 )
& ! [Xa: complex] :
( ( member_complex @ Xa @ A )
=> ( ( ord_less_eq_complex @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_684_finite__has__minimal2,axiom,
! [A: set_mat_complex,A2: mat_complex] :
( ( finite7047982916621727056omplex @ A )
=> ( ( member_mat_complex @ A2 @ A )
=> ? [X3: mat_complex] :
( ( member_mat_complex @ X3 @ A )
& ( ord_le1403324449407493959omplex @ X3 @ A2 )
& ! [Xa: mat_complex] :
( ( member_mat_complex @ Xa @ A )
=> ( ( ord_le1403324449407493959omplex @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_685_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_686_finite__has__minimal2,axiom,
! [A: set_complex,A2: complex] :
( ( finite3207457112153483333omplex @ A )
=> ( ( member_complex @ A2 @ A )
=> ? [X3: complex] :
( ( member_complex @ X3 @ A )
& ( ord_less_eq_complex @ X3 @ A2 )
& ! [Xa: complex] :
( ( member_complex @ Xa @ A )
=> ( ( ord_less_eq_complex @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_687_infinite__imp__nonempty,axiom,
! [S: set_mat_complex] :
( ~ ( finite7047982916621727056omplex @ S )
=> ( S != bot_bo7165004461764951667omplex ) ) ).
% infinite_imp_nonempty
thf(fact_688_infinite__imp__nonempty,axiom,
! [S: set_complex] :
( ~ ( finite3207457112153483333omplex @ S )
=> ( S != bot_bot_set_complex ) ) ).
% infinite_imp_nonempty
thf(fact_689_finite_OemptyI,axiom,
finite7047982916621727056omplex @ bot_bo7165004461764951667omplex ).
% finite.emptyI
thf(fact_690_finite_OemptyI,axiom,
finite3207457112153483333omplex @ bot_bot_set_complex ).
% finite.emptyI
thf(fact_691_finite_Ointros_I2_J,axiom,
! [A: set_complex,A2: complex] :
( ( finite3207457112153483333omplex @ A )
=> ( finite3207457112153483333omplex @ ( insert_complex @ A2 @ A ) ) ) ).
% finite.intros(2)
thf(fact_692_finite__insert,axiom,
! [A2: complex,A: set_complex] :
( ( finite3207457112153483333omplex @ ( insert_complex @ A2 @ A ) )
= ( finite3207457112153483333omplex @ A ) ) ).
% finite_insert
thf(fact_693_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_694_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_695_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_696_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_697_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_698_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_699_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_700_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_701_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_702_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_703_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_704_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_705_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_706_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_707_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_708_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_709_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_710_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_711_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_712_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_713_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_714_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_715_plus__nat_Osimps_I1_J,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.simps(1)
thf(fact_716_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_717_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_718_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_719_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_720_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_721_trans__less__add2,axiom,
! [I3: nat,J: nat,M: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_722_trans__less__add1,axiom,
! [I3: nat,J: nat,M: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_723_add__less__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_724_not__add__less2,axiom,
! [J: nat,I3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I3 ) @ I3 ) ).
% not_add_less2
thf(fact_725_not__add__less1,axiom,
! [I3: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J ) @ I3 ) ).
% not_add_less1
thf(fact_726_add__less__mono,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_727_add__lessD1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J ) @ K )
=> ( ord_less_nat @ I3 @ K ) ) ).
% add_lessD1
thf(fact_728_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_729_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_730_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_731_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_732_times__nat_Osimps_I1_J,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% times_nat.simps(1)
thf(fact_733_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_734_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_735_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_736_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_737_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_738_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_739_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_740_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_741_trans__le__add2,axiom,
! [I3: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_742_trans__le__add1,axiom,
! [I3: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_743_add__le__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_744_add__le__mono,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_745_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_746_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_747_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_748_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_749_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_750_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_751_diff__diff__cancel,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_752_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_753_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B3 ) )
= ( ord_less_eq_nat @ B3 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_754_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_755_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_756_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_757_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_758_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_759_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_760_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_761_mult__le__mono,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_762_mult__le__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_763_mult__le__mono2,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_764_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_765_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_766_diff__diff__left,axiom,
! [I3: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K )
= ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_767_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_768_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_769_nat__distrib_I4_J,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 ) ) ) ).
% nat_distrib(4)
thf(fact_770_nat__distrib_I3_J,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 ) ) ) ).
% nat_distrib(3)
thf(fact_771_finite__has__minimal,axiom,
! [A: set_mat_complex] :
( ( finite7047982916621727056omplex @ A )
=> ( ( A != bot_bo7165004461764951667omplex )
=> ? [X3: mat_complex] :
( ( member_mat_complex @ X3 @ A )
& ! [Xa: mat_complex] :
( ( member_mat_complex @ Xa @ A )
=> ( ( ord_le1403324449407493959omplex @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_772_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_773_finite__has__minimal,axiom,
! [A: set_complex] :
( ( finite3207457112153483333omplex @ A )
=> ( ( A != bot_bot_set_complex )
=> ? [X3: complex] :
( ( member_complex @ X3 @ A )
& ! [Xa: complex] :
( ( member_complex @ Xa @ A )
=> ( ( ord_less_eq_complex @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_774_finite__has__maximal,axiom,
! [A: set_mat_complex] :
( ( finite7047982916621727056omplex @ A )
=> ( ( A != bot_bo7165004461764951667omplex )
=> ? [X3: mat_complex] :
( ( member_mat_complex @ X3 @ A )
& ! [Xa: mat_complex] :
( ( member_mat_complex @ Xa @ A )
=> ( ( ord_le1403324449407493959omplex @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_775_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_776_finite__has__maximal,axiom,
! [A: set_complex] :
( ( finite3207457112153483333omplex @ A )
=> ( ( A != bot_bot_set_complex )
=> ? [X3: complex] :
( ( member_complex @ X3 @ A )
& ! [Xa: complex] :
( ( member_complex @ Xa @ A )
=> ( ( ord_less_eq_complex @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_777_finite_Ocases,axiom,
! [A2: set_mat_complex] :
( ( finite7047982916621727056omplex @ A2 )
=> ( ( A2 != bot_bo7165004461764951667omplex )
=> ~ ! [A5: set_mat_complex] :
( ? [A4: mat_complex] :
( A2
= ( insert_mat_complex @ A4 @ A5 ) )
=> ~ ( finite7047982916621727056omplex @ A5 ) ) ) ) ).
% finite.cases
thf(fact_778_finite_Ocases,axiom,
! [A2: set_complex] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( A2 != bot_bot_set_complex )
=> ~ ! [A5: set_complex] :
( ? [A4: complex] :
( A2
= ( insert_complex @ A4 @ A5 ) )
=> ~ ( finite3207457112153483333omplex @ A5 ) ) ) ) ).
% finite.cases
thf(fact_779_finite_Osimps,axiom,
( finite7047982916621727056omplex
= ( ^ [A3: set_mat_complex] :
( ( A3 = bot_bo7165004461764951667omplex )
| ? [A6: set_mat_complex,B4: mat_complex] :
( ( A3
= ( insert_mat_complex @ B4 @ A6 ) )
& ( finite7047982916621727056omplex @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_780_finite_Osimps,axiom,
( finite3207457112153483333omplex
= ( ^ [A3: set_complex] :
( ( A3 = bot_bot_set_complex )
| ? [A6: set_complex,B4: complex] :
( ( A3
= ( insert_complex @ B4 @ A6 ) )
& ( finite3207457112153483333omplex @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_781_finite_Oinduct,axiom,
! [X: set_mat_complex,P: set_mat_complex > $o] :
( ( finite7047982916621727056omplex @ X )
=> ( ( P @ bot_bo7165004461764951667omplex )
=> ( ! [A5: set_mat_complex,A4: mat_complex] :
( ( finite7047982916621727056omplex @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( insert_mat_complex @ A4 @ A5 ) ) ) )
=> ( P @ X ) ) ) ) ).
% finite.induct
thf(fact_782_finite_Oinduct,axiom,
! [X: set_complex,P: set_complex > $o] :
( ( finite3207457112153483333omplex @ X )
=> ( ( P @ bot_bot_set_complex )
=> ( ! [A5: set_complex,A4: complex] :
( ( finite3207457112153483333omplex @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( insert_complex @ A4 @ A5 ) ) ) )
=> ( P @ X ) ) ) ) ).
% finite.induct
thf(fact_783_finite__induct,axiom,
! [F2: set_mat_complex,P: set_mat_complex > $o] :
( ( finite7047982916621727056omplex @ F2 )
=> ( ( P @ bot_bo7165004461764951667omplex )
=> ( ! [X3: mat_complex,F3: set_mat_complex] :
( ( finite7047982916621727056omplex @ F3 )
=> ( ~ ( member_mat_complex @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_mat_complex @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_784_finite__induct,axiom,
! [F2: set_complex,P: set_complex > $o] :
( ( finite3207457112153483333omplex @ F2 )
=> ( ( P @ bot_bot_set_complex )
=> ( ! [X3: complex,F3: set_complex] :
( ( finite3207457112153483333omplex @ F3 )
=> ( ~ ( member_complex @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_complex @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_785_finite__ne__induct,axiom,
! [F2: set_mat_complex,P: set_mat_complex > $o] :
( ( finite7047982916621727056omplex @ F2 )
=> ( ( F2 != bot_bo7165004461764951667omplex )
=> ( ! [X3: mat_complex] : ( P @ ( insert_mat_complex @ X3 @ bot_bo7165004461764951667omplex ) )
=> ( ! [X3: mat_complex,F3: set_mat_complex] :
( ( finite7047982916621727056omplex @ F3 )
=> ( ( F3 != bot_bo7165004461764951667omplex )
=> ( ~ ( member_mat_complex @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_mat_complex @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_786_finite__ne__induct,axiom,
! [F2: set_complex,P: set_complex > $o] :
( ( finite3207457112153483333omplex @ F2 )
=> ( ( F2 != bot_bot_set_complex )
=> ( ! [X3: complex] : ( P @ ( insert_complex @ X3 @ bot_bot_set_complex ) )
=> ( ! [X3: complex,F3: set_complex] :
( ( finite3207457112153483333omplex @ F3 )
=> ( ( F3 != bot_bot_set_complex )
=> ( ~ ( member_complex @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_complex @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_787_infinite__finite__induct,axiom,
! [P: set_mat_complex > $o,A: set_mat_complex] :
( ! [A5: set_mat_complex] :
( ~ ( finite7047982916621727056omplex @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bo7165004461764951667omplex )
=> ( ! [X3: mat_complex,F3: set_mat_complex] :
( ( finite7047982916621727056omplex @ F3 )
=> ( ~ ( member_mat_complex @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_mat_complex @ X3 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_788_infinite__finite__induct,axiom,
! [P: set_complex > $o,A: set_complex] :
( ! [A5: set_complex] :
( ~ ( finite3207457112153483333omplex @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bot_set_complex )
=> ( ! [X3: complex,F3: set_complex] :
( ( finite3207457112153483333omplex @ F3 )
=> ( ~ ( member_complex @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_complex @ X3 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_789_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_790_finite__Diff__insert,axiom,
! [A: set_complex,A2: complex,B2: set_complex] :
( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A @ ( insert_complex @ A2 @ B2 ) ) )
= ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_791_less__imp__add__positive,axiom,
! [I3: nat,J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I3 @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_792_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_793_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_794_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_795_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_796_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_797_mult__less__mono2,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_798_mult__less__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_799_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_800_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_801_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_802_diff__less__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B3 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_803_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_804_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_805_less__diff__conv,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_806_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_807_le__diff__conv,axiom,
! [J: nat,K: nat,I3: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I3 @ K ) ) ) ).
% le_diff_conv
thf(fact_808_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_809_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_810_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K )
= ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_811_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_812_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I3 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_813_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_814_Nat_Ole__imp__diff__is__add,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ( minus_minus_nat @ J @ I3 )
= K )
= ( J
= ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_815_finite__subset__induct,axiom,
! [F2: set_mat_complex,A: set_mat_complex,P: set_mat_complex > $o] :
( ( finite7047982916621727056omplex @ F2 )
=> ( ( ord_le3632134057777142183omplex @ F2 @ A )
=> ( ( P @ bot_bo7165004461764951667omplex )
=> ( ! [A4: mat_complex,F3: set_mat_complex] :
( ( finite7047982916621727056omplex @ F3 )
=> ( ( member_mat_complex @ A4 @ A )
=> ( ~ ( member_mat_complex @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_mat_complex @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_816_finite__subset__induct,axiom,
! [F2: set_complex,A: set_complex,P: set_complex > $o] :
( ( finite3207457112153483333omplex @ F2 )
=> ( ( ord_le211207098394363844omplex @ F2 @ A )
=> ( ( P @ bot_bot_set_complex )
=> ( ! [A4: complex,F3: set_complex] :
( ( finite3207457112153483333omplex @ F3 )
=> ( ( member_complex @ A4 @ A )
=> ( ~ ( member_complex @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_complex @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_817_finite__subset__induct_H,axiom,
! [F2: set_mat_complex,A: set_mat_complex,P: set_mat_complex > $o] :
( ( finite7047982916621727056omplex @ F2 )
=> ( ( ord_le3632134057777142183omplex @ F2 @ A )
=> ( ( P @ bot_bo7165004461764951667omplex )
=> ( ! [A4: mat_complex,F3: set_mat_complex] :
( ( finite7047982916621727056omplex @ F3 )
=> ( ( member_mat_complex @ A4 @ A )
=> ( ( ord_le3632134057777142183omplex @ F3 @ A )
=> ( ~ ( member_mat_complex @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_mat_complex @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_818_finite__subset__induct_H,axiom,
! [F2: set_complex,A: set_complex,P: set_complex > $o] :
( ( finite3207457112153483333omplex @ F2 )
=> ( ( ord_le211207098394363844omplex @ F2 @ A )
=> ( ( P @ bot_bot_set_complex )
=> ( ! [A4: complex,F3: set_complex] :
( ( finite3207457112153483333omplex @ F3 )
=> ( ( member_complex @ A4 @ A )
=> ( ( ord_le211207098394363844omplex @ F3 @ A )
=> ( ~ ( member_complex @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_complex @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_819_infinite__remove,axiom,
! [S: set_mat_complex,A2: mat_complex] :
( ~ ( finite7047982916621727056omplex @ S )
=> ~ ( finite7047982916621727056omplex @ ( minus_8760755521168068590omplex @ S @ ( insert_mat_complex @ A2 @ bot_bo7165004461764951667omplex ) ) ) ) ).
% infinite_remove
thf(fact_820_infinite__remove,axiom,
! [S: set_complex,A2: complex] :
( ~ ( finite3207457112153483333omplex @ S )
=> ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ).
% infinite_remove
thf(fact_821_infinite__coinduct,axiom,
! [X4: set_mat_complex > $o,A: set_mat_complex] :
( ( X4 @ A )
=> ( ! [A5: set_mat_complex] :
( ( X4 @ A5 )
=> ? [X5: mat_complex] :
( ( member_mat_complex @ X5 @ A5 )
& ( ( X4 @ ( minus_8760755521168068590omplex @ A5 @ ( insert_mat_complex @ X5 @ bot_bo7165004461764951667omplex ) ) )
| ~ ( finite7047982916621727056omplex @ ( minus_8760755521168068590omplex @ A5 @ ( insert_mat_complex @ X5 @ bot_bo7165004461764951667omplex ) ) ) ) ) )
=> ~ ( finite7047982916621727056omplex @ A ) ) ) ).
% infinite_coinduct
thf(fact_822_infinite__coinduct,axiom,
! [X4: set_complex > $o,A: set_complex] :
( ( X4 @ A )
=> ( ! [A5: set_complex] :
( ( X4 @ A5 )
=> ? [X5: complex] :
( ( member_complex @ X5 @ A5 )
& ( ( X4 @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X5 @ bot_bot_set_complex ) ) )
| ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X5 @ bot_bot_set_complex ) ) ) ) ) )
=> ~ ( finite3207457112153483333omplex @ A ) ) ) ).
% infinite_coinduct
thf(fact_823_finite__empty__induct,axiom,
! [A: set_mat_complex,P: set_mat_complex > $o] :
( ( finite7047982916621727056omplex @ A )
=> ( ( P @ A )
=> ( ! [A4: mat_complex,A5: set_mat_complex] :
( ( finite7047982916621727056omplex @ A5 )
=> ( ( member_mat_complex @ A4 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_8760755521168068590omplex @ A5 @ ( insert_mat_complex @ A4 @ bot_bo7165004461764951667omplex ) ) ) ) ) )
=> ( P @ bot_bo7165004461764951667omplex ) ) ) ) ).
% finite_empty_induct
thf(fact_824_finite__empty__induct,axiom,
! [A: set_complex,P: set_complex > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( P @ A )
=> ( ! [A4: complex,A5: set_complex] :
( ( finite3207457112153483333omplex @ A5 )
=> ( ( member_complex @ A4 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ A4 @ bot_bot_set_complex ) ) ) ) ) )
=> ( P @ bot_bot_set_complex ) ) ) ) ).
% finite_empty_induct
thf(fact_825_finite__induct__select,axiom,
! [S: set_mat_complex,P: set_mat_complex > $o] :
( ( finite7047982916621727056omplex @ S )
=> ( ( P @ bot_bo7165004461764951667omplex )
=> ( ! [T3: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X5: mat_complex] :
( ( member_mat_complex @ X5 @ ( minus_8760755521168068590omplex @ S @ T3 ) )
& ( P @ ( insert_mat_complex @ X5 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_826_finite__induct__select,axiom,
! [S: set_complex,P: set_complex > $o] :
( ( finite3207457112153483333omplex @ S )
=> ( ( P @ bot_bot_set_complex )
=> ( ! [T3: set_complex] :
( ( ord_less_set_complex @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X5: complex] :
( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ S @ T3 ) )
& ( P @ ( insert_complex @ X5 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_827_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_828_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B3: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B3 ) )
= ( ( ( ord_less_nat @ A2 @ B3 )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A2
= ( plus_plus_nat @ B3 @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_829_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B3: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B3 ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B3 )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A2
= ( plus_plus_nat @ B3 @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_830_subset__insert__iff,axiom,
! [A: set_mat_complex,X: mat_complex,B2: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ A @ ( insert_mat_complex @ X @ B2 ) )
= ( ( ( member_mat_complex @ X @ A )
=> ( ord_le3632134057777142183omplex @ ( minus_8760755521168068590omplex @ A @ ( insert_mat_complex @ X @ bot_bo7165004461764951667omplex ) ) @ B2 ) )
& ( ~ ( member_mat_complex @ X @ A )
=> ( ord_le3632134057777142183omplex @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_831_subset__insert__iff,axiom,
! [A: set_complex,X: complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ A @ ( insert_complex @ X @ B2 ) )
= ( ( ( member_complex @ X @ A )
=> ( ord_le211207098394363844omplex @ ( minus_811609699411566653omplex @ A @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ B2 ) )
& ( ~ ( member_complex @ X @ A )
=> ( ord_le211207098394363844omplex @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_832_Diff__single__insert,axiom,
! [A: set_mat_complex,X: mat_complex,B2: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ ( minus_8760755521168068590omplex @ A @ ( insert_mat_complex @ X @ bot_bo7165004461764951667omplex ) ) @ B2 )
=> ( ord_le3632134057777142183omplex @ A @ ( insert_mat_complex @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_833_Diff__single__insert,axiom,
! [A: set_complex,X: complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ ( minus_811609699411566653omplex @ A @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ B2 )
=> ( ord_le211207098394363844omplex @ A @ ( insert_complex @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_834_psubset__insert__iff,axiom,
! [A: set_mat_complex,X: mat_complex,B2: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ A @ ( insert_mat_complex @ X @ B2 ) )
= ( ( ( member_mat_complex @ X @ B2 )
=> ( ord_le5598786136212072115omplex @ A @ B2 ) )
& ( ~ ( member_mat_complex @ X @ B2 )
=> ( ( ( member_mat_complex @ X @ A )
=> ( ord_le5598786136212072115omplex @ ( minus_8760755521168068590omplex @ A @ ( insert_mat_complex @ X @ bot_bo7165004461764951667omplex ) ) @ B2 ) )
& ( ~ ( member_mat_complex @ X @ A )
=> ( ord_le3632134057777142183omplex @ A @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_835_psubset__insert__iff,axiom,
! [A: set_complex,X: complex,B2: set_complex] :
( ( ord_less_set_complex @ A @ ( insert_complex @ X @ B2 ) )
= ( ( ( member_complex @ X @ B2 )
=> ( ord_less_set_complex @ A @ B2 ) )
& ( ~ ( member_complex @ X @ B2 )
=> ( ( ( member_complex @ X @ A )
=> ( ord_less_set_complex @ ( minus_811609699411566653omplex @ A @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ B2 ) )
& ( ~ ( member_complex @ X @ A )
=> ( ord_le211207098394363844omplex @ A @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_836_basic__trans__rules_I26_J,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% basic_trans_rules(26)
thf(fact_837_basic__trans__rules_I26_J,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( A2 = B3 )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ord_less_eq_complex @ A2 @ C ) ) ) ).
% basic_trans_rules(26)
thf(fact_838_basic__trans__rules_I25_J,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% basic_trans_rules(25)
thf(fact_839_basic__trans__rules_I25_J,axiom,
! [A2: complex,B3: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_complex @ A2 @ C ) ) ) ).
% basic_trans_rules(25)
thf(fact_840_basic__trans__rules_I24_J,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% basic_trans_rules(24)
thf(fact_841_basic__trans__rules_I24_J,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ( ord_less_eq_complex @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% basic_trans_rules(24)
thf(fact_842_basic__trans__rules_I23_J,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% basic_trans_rules(23)
thf(fact_843_basic__trans__rules_I23_J,axiom,
! [X: complex,Y2: complex,Z: complex] :
( ( ord_less_eq_complex @ X @ Y2 )
=> ( ( ord_less_eq_complex @ Y2 @ Z )
=> ( ord_less_eq_complex @ X @ Z ) ) ) ).
% basic_trans_rules(23)
thf(fact_844_basic__trans__rules_I10_J,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_845_basic__trans__rules_I10_J,axiom,
! [A2: complex,F: nat > complex,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_complex @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_846_basic__trans__rules_I10_J,axiom,
! [A2: nat,F: complex > nat,B3: complex,C: complex] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ! [X3: complex,Y4: complex] :
( ( ord_less_eq_complex @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_847_basic__trans__rules_I10_J,axiom,
! [A2: complex,F: complex > complex,B3: complex,C: complex] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ! [X3: complex,Y4: complex] :
( ( ord_less_eq_complex @ X3 @ Y4 )
=> ( ord_less_eq_complex @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_848_basic__trans__rules_I9_J,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_849_basic__trans__rules_I9_J,axiom,
! [A2: nat,B3: nat,F: nat > complex,C: complex] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_complex @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_850_basic__trans__rules_I9_J,axiom,
! [A2: complex,B3: complex,F: complex > nat,C: nat] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: complex,Y4: complex] :
( ( ord_less_eq_complex @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_851_basic__trans__rules_I9_J,axiom,
! [A2: complex,B3: complex,F: complex > complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: complex,Y4: complex] :
( ( ord_less_eq_complex @ X3 @ Y4 )
=> ( ord_less_eq_complex @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_852_basic__trans__rules_I8_J,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_853_basic__trans__rules_I8_J,axiom,
! [A2: nat,F: complex > nat,B3: complex,C: complex] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ! [X3: complex,Y4: complex] :
( ( ord_less_eq_complex @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_854_basic__trans__rules_I8_J,axiom,
! [A2: complex,F: nat > complex,B3: nat,C: nat] :
( ( ord_less_eq_complex @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_complex @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_855_basic__trans__rules_I8_J,axiom,
! [A2: complex,F: complex > complex,B3: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ! [X3: complex,Y4: complex] :
( ( ord_less_eq_complex @ X3 @ Y4 )
=> ( ord_less_eq_complex @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_856_basic__trans__rules_I7_J,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_857_basic__trans__rules_I7_J,axiom,
! [A2: nat,B3: nat,F: nat > complex,C: complex] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_complex @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_complex @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_858_basic__trans__rules_I7_J,axiom,
! [A2: complex,B3: complex,F: complex > nat,C: nat] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: complex,Y4: complex] :
( ( ord_less_eq_complex @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_859_basic__trans__rules_I7_J,axiom,
! [A2: complex,B3: complex,F: complex > complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ( ord_less_eq_complex @ ( F @ B3 ) @ C )
=> ( ! [X3: complex,Y4: complex] :
( ( ord_less_eq_complex @ X3 @ Y4 )
=> ( ord_less_eq_complex @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_860_basic__trans__rules_I28_J,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( A2 = B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% basic_trans_rules(28)
thf(fact_861_basic__trans__rules_I27_J,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% basic_trans_rules(27)
thf(fact_862_basic__trans__rules_I20_J,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A2 ) ) ).
% basic_trans_rules(20)
thf(fact_863_basic__trans__rules_I19_J,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% basic_trans_rules(19)
thf(fact_864_basic__trans__rules_I12_J,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(12)
thf(fact_865_basic__trans__rules_I11_J,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(11)
thf(fact_866_basic__trans__rules_I2_J,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(2)
thf(fact_867_basic__trans__rules_I1_J,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(1)
thf(fact_868_Set_Obasic__monos_I7_J,axiom,
! [A: set_mat_complex,B2: set_mat_complex,X: mat_complex] :
( ( ord_le3632134057777142183omplex @ A @ B2 )
=> ( ( member_mat_complex @ X @ A )
=> ( member_mat_complex @ X @ B2 ) ) ) ).
% Set.basic_monos(7)
thf(fact_869_Set_Obasic__monos_I7_J,axiom,
! [A: set_complex,B2: set_complex,X: complex] :
( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ( member_complex @ X @ A )
=> ( member_complex @ X @ B2 ) ) ) ).
% Set.basic_monos(7)
thf(fact_870_basic__trans__rules_I31_J,axiom,
! [A: set_mat_complex,B2: set_mat_complex,C: mat_complex] :
( ( ord_le3632134057777142183omplex @ A @ B2 )
=> ( ( member_mat_complex @ C @ A )
=> ( member_mat_complex @ C @ B2 ) ) ) ).
% basic_trans_rules(31)
thf(fact_871_basic__trans__rules_I31_J,axiom,
! [A: set_complex,B2: set_complex,C: complex] :
( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ( member_complex @ C @ A )
=> ( member_complex @ C @ B2 ) ) ) ).
% basic_trans_rules(31)
thf(fact_872_subsetI,axiom,
! [A: set_mat_complex,B2: set_mat_complex] :
( ! [X3: mat_complex] :
( ( member_mat_complex @ X3 @ A )
=> ( member_mat_complex @ X3 @ B2 ) )
=> ( ord_le3632134057777142183omplex @ A @ B2 ) ) ).
% subsetI
thf(fact_873_subsetI,axiom,
! [A: set_complex,B2: set_complex] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( member_complex @ X3 @ B2 ) )
=> ( ord_le211207098394363844omplex @ A @ B2 ) ) ).
% subsetI
thf(fact_874_subset__eq,axiom,
( ord_le3632134057777142183omplex
= ( ^ [A6: set_mat_complex,B6: set_mat_complex] :
! [X2: mat_complex] :
( ( member_mat_complex @ X2 @ A6 )
=> ( member_mat_complex @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_875_subset__eq,axiom,
( ord_le211207098394363844omplex
= ( ^ [A6: set_complex,B6: set_complex] :
! [X2: complex] :
( ( member_complex @ X2 @ A6 )
=> ( member_complex @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_876_subset__iff,axiom,
( ord_le3632134057777142183omplex
= ( ^ [A6: set_mat_complex,B6: set_mat_complex] :
! [T4: mat_complex] :
( ( member_mat_complex @ T4 @ A6 )
=> ( member_mat_complex @ T4 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_877_subset__iff,axiom,
( ord_le211207098394363844omplex
= ( ^ [A6: set_complex,B6: set_complex] :
! [T4: complex] :
( ( member_complex @ T4 @ A6 )
=> ( member_complex @ T4 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_878_not__psubset__empty,axiom,
! [A: set_mat_complex] :
~ ( ord_le5598786136212072115omplex @ A @ bot_bo7165004461764951667omplex ) ).
% not_psubset_empty
thf(fact_879_not__psubset__empty,axiom,
! [A: set_complex] :
~ ( ord_less_set_complex @ A @ bot_bot_set_complex ) ).
% not_psubset_empty
thf(fact_880_emptyE,axiom,
! [A2: mat_complex] :
~ ( member_mat_complex @ A2 @ bot_bo7165004461764951667omplex ) ).
% emptyE
thf(fact_881_emptyE,axiom,
! [A2: complex] :
~ ( member_complex @ A2 @ bot_bot_set_complex ) ).
% emptyE
thf(fact_882_equals0D,axiom,
! [A: set_mat_complex,A2: mat_complex] :
( ( A = bot_bo7165004461764951667omplex )
=> ~ ( member_mat_complex @ A2 @ A ) ) ).
% equals0D
thf(fact_883_equals0D,axiom,
! [A: set_complex,A2: complex] :
( ( A = bot_bot_set_complex )
=> ~ ( member_complex @ A2 @ A ) ) ).
% equals0D
thf(fact_884_equals0I,axiom,
! [A: set_mat_complex] :
( ! [Y4: mat_complex] :
~ ( member_mat_complex @ Y4 @ A )
=> ( A = bot_bo7165004461764951667omplex ) ) ).
% equals0I
thf(fact_885_equals0I,axiom,
! [A: set_complex] :
( ! [Y4: complex] :
~ ( member_complex @ Y4 @ A )
=> ( A = bot_bot_set_complex ) ) ).
% equals0I
thf(fact_886_empty__iff,axiom,
! [C: mat_complex] :
~ ( member_mat_complex @ C @ bot_bo7165004461764951667omplex ) ).
% empty_iff
thf(fact_887_empty__iff,axiom,
! [C: complex] :
~ ( member_complex @ C @ bot_bot_set_complex ) ).
% empty_iff
thf(fact_888_ex__in__conv,axiom,
! [A: set_mat_complex] :
( ( ? [X2: mat_complex] : ( member_mat_complex @ X2 @ A ) )
= ( A != bot_bo7165004461764951667omplex ) ) ).
% ex_in_conv
thf(fact_889_ex__in__conv,axiom,
! [A: set_complex] :
( ( ? [X2: complex] : ( member_complex @ X2 @ A ) )
= ( A != bot_bot_set_complex ) ) ).
% ex_in_conv
thf(fact_890_all__not__in__conv,axiom,
! [A: set_mat_complex] :
( ( ! [X2: mat_complex] :
~ ( member_mat_complex @ X2 @ A ) )
= ( A = bot_bo7165004461764951667omplex ) ) ).
% all_not_in_conv
thf(fact_891_all__not__in__conv,axiom,
! [A: set_complex] :
( ( ! [X2: complex] :
~ ( member_complex @ X2 @ A ) )
= ( A = bot_bot_set_complex ) ) ).
% all_not_in_conv
thf(fact_892_Collect__empty__eq,axiom,
! [P: mat_complex > $o] :
( ( ( collect_mat_complex @ P )
= bot_bo7165004461764951667omplex )
= ( ! [X2: mat_complex] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_893_Collect__empty__eq,axiom,
! [P: complex > $o] :
( ( ( collect_complex @ P )
= bot_bot_set_complex )
= ( ! [X2: complex] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_894_empty__Collect__eq,axiom,
! [P: mat_complex > $o] :
( ( bot_bo7165004461764951667omplex
= ( collect_mat_complex @ P ) )
= ( ! [X2: mat_complex] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_895_empty__Collect__eq,axiom,
! [P: complex > $o] :
( ( bot_bot_set_complex
= ( collect_complex @ P ) )
= ( ! [X2: complex] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_896_DiffE,axiom,
! [C: mat_complex,A: set_mat_complex,B2: set_mat_complex] :
( ( member_mat_complex @ C @ ( minus_8760755521168068590omplex @ A @ B2 ) )
=> ~ ( ( member_mat_complex @ C @ A )
=> ( member_mat_complex @ C @ B2 ) ) ) ).
% DiffE
thf(fact_897_DiffE,axiom,
! [C: complex,A: set_complex,B2: set_complex] :
( ( member_complex @ C @ ( minus_811609699411566653omplex @ A @ B2 ) )
=> ~ ( ( member_complex @ C @ A )
=> ( member_complex @ C @ B2 ) ) ) ).
% DiffE
thf(fact_898_DiffI,axiom,
! [C: mat_complex,A: set_mat_complex,B2: set_mat_complex] :
( ( member_mat_complex @ C @ A )
=> ( ~ ( member_mat_complex @ C @ B2 )
=> ( member_mat_complex @ C @ ( minus_8760755521168068590omplex @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_899_DiffI,axiom,
! [C: complex,A: set_complex,B2: set_complex] :
( ( member_complex @ C @ A )
=> ( ~ ( member_complex @ C @ B2 )
=> ( member_complex @ C @ ( minus_811609699411566653omplex @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_900_DiffD1,axiom,
! [C: mat_complex,A: set_mat_complex,B2: set_mat_complex] :
( ( member_mat_complex @ C @ ( minus_8760755521168068590omplex @ A @ B2 ) )
=> ( member_mat_complex @ C @ A ) ) ).
% DiffD1
thf(fact_901_DiffD1,axiom,
! [C: complex,A: set_complex,B2: set_complex] :
( ( member_complex @ C @ ( minus_811609699411566653omplex @ A @ B2 ) )
=> ( member_complex @ C @ A ) ) ).
% DiffD1
thf(fact_902_DiffD2,axiom,
! [C: mat_complex,A: set_mat_complex,B2: set_mat_complex] :
( ( member_mat_complex @ C @ ( minus_8760755521168068590omplex @ A @ B2 ) )
=> ~ ( member_mat_complex @ C @ B2 ) ) ).
% DiffD2
thf(fact_903_DiffD2,axiom,
! [C: complex,A: set_complex,B2: set_complex] :
( ( member_complex @ C @ ( minus_811609699411566653omplex @ A @ B2 ) )
=> ~ ( member_complex @ C @ B2 ) ) ).
% DiffD2
thf(fact_904_Diff__iff,axiom,
! [C: mat_complex,A: set_mat_complex,B2: set_mat_complex] :
( ( member_mat_complex @ C @ ( minus_8760755521168068590omplex @ A @ B2 ) )
= ( ( member_mat_complex @ C @ A )
& ~ ( member_mat_complex @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_905_Diff__iff,axiom,
! [C: complex,A: set_complex,B2: set_complex] :
( ( member_complex @ C @ ( minus_811609699411566653omplex @ A @ B2 ) )
= ( ( member_complex @ C @ A )
& ~ ( member_complex @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_906_psubset__imp__ex__mem,axiom,
! [A: set_mat_complex,B2: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ A @ B2 )
=> ? [B5: mat_complex] : ( member_mat_complex @ B5 @ ( minus_8760755521168068590omplex @ B2 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_907_psubset__imp__ex__mem,axiom,
! [A: set_complex,B2: set_complex] :
( ( ord_less_set_complex @ A @ B2 )
=> ? [B5: complex] : ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B2 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_908_basic__trans__rules_I3_J,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_909_basic__trans__rules_I3_J,axiom,
! [A2: nat,B3: nat,F: nat > complex,C: complex] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_complex @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_complex @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_910_basic__trans__rules_I3_J,axiom,
! [A2: complex,B3: complex,F: complex > nat,C: nat] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: complex,Y4: complex] :
( ( ord_less_eq_complex @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_911_basic__trans__rules_I3_J,axiom,
! [A2: complex,B3: complex,F: complex > complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ( ord_less_complex @ ( F @ B3 ) @ C )
=> ( ! [X3: complex,Y4: complex] :
( ( ord_less_eq_complex @ X3 @ Y4 )
=> ( ord_less_eq_complex @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_912_basic__trans__rules_I4_J,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(4)
thf(fact_913_basic__trans__rules_I4_J,axiom,
! [A2: complex,F: nat > complex,B3: nat,C: nat] :
( ( ord_less_eq_complex @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_complex @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(4)
thf(fact_914_basic__trans__rules_I5_J,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(5)
thf(fact_915_basic__trans__rules_I5_J,axiom,
! [A2: nat,B3: nat,F: nat > complex,C: complex] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_complex @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_complex @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(5)
thf(fact_916_basic__trans__rules_I6_J,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_917_basic__trans__rules_I6_J,axiom,
! [A2: complex,F: nat > complex,B3: nat,C: nat] :
( ( ord_less_complex @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_complex @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_918_basic__trans__rules_I6_J,axiom,
! [A2: nat,F: complex > nat,B3: complex,C: complex] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ! [X3: complex,Y4: complex] :
( ( ord_less_eq_complex @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_919_basic__trans__rules_I6_J,axiom,
! [A2: complex,F: complex > complex,B3: complex,C: complex] :
( ( ord_less_complex @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ! [X3: complex,Y4: complex] :
( ( ord_less_eq_complex @ X3 @ Y4 )
=> ( ord_less_eq_complex @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_920_basic__trans__rules_I17_J,axiom,
! [A2: nat,B3: nat] :
( ( A2 != B3 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ord_less_nat @ A2 @ B3 ) ) ) ).
% basic_trans_rules(17)
thf(fact_921_basic__trans__rules_I17_J,axiom,
! [A2: complex,B3: complex] :
( ( A2 != B3 )
=> ( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ord_less_complex @ A2 @ B3 ) ) ) ).
% basic_trans_rules(17)
thf(fact_922_basic__trans__rules_I18_J,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_nat @ A2 @ B3 ) ) ) ).
% basic_trans_rules(18)
thf(fact_923_basic__trans__rules_I18_J,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_eq_complex @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_complex @ A2 @ B3 ) ) ) ).
% basic_trans_rules(18)
thf(fact_924_basic__trans__rules_I21_J,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% basic_trans_rules(21)
thf(fact_925_basic__trans__rules_I21_J,axiom,
! [X: complex,Y2: complex,Z: complex] :
( ( ord_less_eq_complex @ X @ Y2 )
=> ( ( ord_less_complex @ Y2 @ Z )
=> ( ord_less_complex @ X @ Z ) ) ) ).
% basic_trans_rules(21)
thf(fact_926_basic__trans__rules_I22_J,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% basic_trans_rules(22)
thf(fact_927_basic__trans__rules_I22_J,axiom,
! [X: complex,Y2: complex,Z: complex] :
( ( ord_less_complex @ X @ Y2 )
=> ( ( ord_less_eq_complex @ Y2 @ Z )
=> ( ord_less_complex @ X @ Z ) ) ) ).
% basic_trans_rules(22)
thf(fact_928_image__subsetI,axiom,
! [A: set_mat_complex,F: mat_complex > mat_complex,B2: set_mat_complex] :
( ! [X3: mat_complex] :
( ( member_mat_complex @ X3 @ A )
=> ( member_mat_complex @ ( F @ X3 ) @ B2 ) )
=> ( ord_le3632134057777142183omplex @ ( image_23760814813800901omplex @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_929_image__subsetI,axiom,
! [A: set_mat_complex,F: mat_complex > complex,B2: set_complex] :
( ! [X3: mat_complex] :
( ( member_mat_complex @ X3 @ A )
=> ( member_complex @ ( F @ X3 ) @ B2 ) )
=> ( ord_le211207098394363844omplex @ ( image_4184848318200637456omplex @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_930_image__subsetI,axiom,
! [A: set_complex,F: complex > mat_complex,B2: set_mat_complex] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( member_mat_complex @ ( F @ X3 ) @ B2 ) )
=> ( ord_le3632134057777142183omplex @ ( image_6107471045988054706omplex @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_931_image__subsetI,axiom,
! [A: set_complex,F: complex > complex,B2: set_complex] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( member_complex @ ( F @ X3 ) @ B2 ) )
=> ( ord_le211207098394363844omplex @ ( image_1468599708987790691omplex @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_932_image__is__empty,axiom,
! [F: mat_complex > mat_complex,A: set_mat_complex] :
( ( ( image_23760814813800901omplex @ F @ A )
= bot_bo7165004461764951667omplex )
= ( A = bot_bo7165004461764951667omplex ) ) ).
% image_is_empty
thf(fact_933_image__is__empty,axiom,
! [F: complex > mat_complex,A: set_complex] :
( ( ( image_6107471045988054706omplex @ F @ A )
= bot_bo7165004461764951667omplex )
= ( A = bot_bot_set_complex ) ) ).
% image_is_empty
thf(fact_934_image__is__empty,axiom,
! [F: mat_complex > complex,A: set_mat_complex] :
( ( ( image_4184848318200637456omplex @ F @ A )
= bot_bot_set_complex )
= ( A = bot_bo7165004461764951667omplex ) ) ).
% image_is_empty
thf(fact_935_image__is__empty,axiom,
! [F: complex > complex,A: set_complex] :
( ( ( image_1468599708987790691omplex @ F @ A )
= bot_bot_set_complex )
= ( A = bot_bot_set_complex ) ) ).
% image_is_empty
thf(fact_936_empty__is__image,axiom,
! [F: mat_complex > mat_complex,A: set_mat_complex] :
( ( bot_bo7165004461764951667omplex
= ( image_23760814813800901omplex @ F @ A ) )
= ( A = bot_bo7165004461764951667omplex ) ) ).
% empty_is_image
thf(fact_937_empty__is__image,axiom,
! [F: complex > mat_complex,A: set_complex] :
( ( bot_bo7165004461764951667omplex
= ( image_6107471045988054706omplex @ F @ A ) )
= ( A = bot_bot_set_complex ) ) ).
% empty_is_image
thf(fact_938_empty__is__image,axiom,
! [F: mat_complex > complex,A: set_mat_complex] :
( ( bot_bot_set_complex
= ( image_4184848318200637456omplex @ F @ A ) )
= ( A = bot_bo7165004461764951667omplex ) ) ).
% empty_is_image
thf(fact_939_empty__is__image,axiom,
! [F: complex > complex,A: set_complex] :
( ( bot_bot_set_complex
= ( image_1468599708987790691omplex @ F @ A ) )
= ( A = bot_bot_set_complex ) ) ).
% empty_is_image
thf(fact_940_image__empty,axiom,
! [F: mat_complex > mat_complex] :
( ( image_23760814813800901omplex @ F @ bot_bo7165004461764951667omplex )
= bot_bo7165004461764951667omplex ) ).
% image_empty
thf(fact_941_image__empty,axiom,
! [F: mat_complex > complex] :
( ( image_4184848318200637456omplex @ F @ bot_bo7165004461764951667omplex )
= bot_bot_set_complex ) ).
% image_empty
thf(fact_942_image__empty,axiom,
! [F: complex > mat_complex] :
( ( image_6107471045988054706omplex @ F @ bot_bot_set_complex )
= bot_bo7165004461764951667omplex ) ).
% image_empty
thf(fact_943_image__empty,axiom,
! [F: complex > complex] :
( ( image_1468599708987790691omplex @ F @ bot_bot_set_complex )
= bot_bot_set_complex ) ).
% image_empty
thf(fact_944_empty__subsetI,axiom,
! [A: set_mat_complex] : ( ord_le3632134057777142183omplex @ bot_bo7165004461764951667omplex @ A ) ).
% empty_subsetI
thf(fact_945_empty__subsetI,axiom,
! [A: set_complex] : ( ord_le211207098394363844omplex @ bot_bot_set_complex @ A ) ).
% empty_subsetI
thf(fact_946_subset__empty,axiom,
! [A: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ A @ bot_bo7165004461764951667omplex )
= ( A = bot_bo7165004461764951667omplex ) ) ).
% subset_empty
thf(fact_947_subset__empty,axiom,
! [A: set_complex] :
( ( ord_le211207098394363844omplex @ A @ bot_bot_set_complex )
= ( A = bot_bot_set_complex ) ) ).
% subset_empty
thf(fact_948_subset__Compl__self__eq,axiom,
! [A: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ A @ ( uminus5815530220087396478omplex @ A ) )
= ( A = bot_bo7165004461764951667omplex ) ) ).
% subset_Compl_self_eq
thf(fact_949_subset__Compl__self__eq,axiom,
! [A: set_complex] :
( ( ord_le211207098394363844omplex @ A @ ( uminus8566677241136511917omplex @ A ) )
= ( A = bot_bot_set_complex ) ) ).
% subset_Compl_self_eq
thf(fact_950_Set_Oinsert__mono,axiom,
! [C2: set_complex,D: set_complex,A2: complex] :
( ( ord_le211207098394363844omplex @ C2 @ D )
=> ( ord_le211207098394363844omplex @ ( insert_complex @ A2 @ C2 ) @ ( insert_complex @ A2 @ D ) ) ) ).
% Set.insert_mono
thf(fact_951_insert__subset,axiom,
! [X: mat_complex,A: set_mat_complex,B2: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ ( insert_mat_complex @ X @ A ) @ B2 )
= ( ( member_mat_complex @ X @ B2 )
& ( ord_le3632134057777142183omplex @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_952_insert__subset,axiom,
! [X: complex,A: set_complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ ( insert_complex @ X @ A ) @ B2 )
= ( ( member_complex @ X @ B2 )
& ( ord_le211207098394363844omplex @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_953_subset__insert,axiom,
! [X: mat_complex,A: set_mat_complex,B2: set_mat_complex] :
( ~ ( member_mat_complex @ X @ A )
=> ( ( ord_le3632134057777142183omplex @ A @ ( insert_mat_complex @ X @ B2 ) )
= ( ord_le3632134057777142183omplex @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_954_subset__insert,axiom,
! [X: complex,A: set_complex,B2: set_complex] :
( ~ ( member_complex @ X @ A )
=> ( ( ord_le211207098394363844omplex @ A @ ( insert_complex @ X @ B2 ) )
= ( ord_le211207098394363844omplex @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_955_subset__insertI,axiom,
! [B2: set_complex,A2: complex] : ( ord_le211207098394363844omplex @ B2 @ ( insert_complex @ A2 @ B2 ) ) ).
% subset_insertI
thf(fact_956_subset__insertI2,axiom,
! [A: set_complex,B2: set_complex,B3: complex] :
( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ord_le211207098394363844omplex @ A @ ( insert_complex @ B3 @ B2 ) ) ) ).
% subset_insertI2
thf(fact_957_singleton__inject,axiom,
! [A2: mat_complex,B3: mat_complex] :
( ( ( insert_mat_complex @ A2 @ bot_bo7165004461764951667omplex )
= ( insert_mat_complex @ B3 @ bot_bo7165004461764951667omplex ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_958_singleton__inject,axiom,
! [A2: complex,B3: complex] :
( ( ( insert_complex @ A2 @ bot_bot_set_complex )
= ( insert_complex @ B3 @ bot_bot_set_complex ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_959_insert__not__empty,axiom,
! [A2: mat_complex,A: set_mat_complex] :
( ( insert_mat_complex @ A2 @ A )
!= bot_bo7165004461764951667omplex ) ).
% insert_not_empty
thf(fact_960_insert__not__empty,axiom,
! [A2: complex,A: set_complex] :
( ( insert_complex @ A2 @ A )
!= bot_bot_set_complex ) ).
% insert_not_empty
thf(fact_961_doubleton__eq__iff,axiom,
! [A2: mat_complex,B3: mat_complex,C: mat_complex,D2: mat_complex] :
( ( ( insert_mat_complex @ A2 @ ( insert_mat_complex @ B3 @ bot_bo7165004461764951667omplex ) )
= ( insert_mat_complex @ C @ ( insert_mat_complex @ D2 @ bot_bo7165004461764951667omplex ) ) )
= ( ( ( A2 = C )
& ( B3 = D2 ) )
| ( ( A2 = D2 )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_962_doubleton__eq__iff,axiom,
! [A2: complex,B3: complex,C: complex,D2: complex] :
( ( ( insert_complex @ A2 @ ( insert_complex @ B3 @ bot_bot_set_complex ) )
= ( insert_complex @ C @ ( insert_complex @ D2 @ bot_bot_set_complex ) ) )
= ( ( ( A2 = C )
& ( B3 = D2 ) )
| ( ( A2 = D2 )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_963_singleton__iff,axiom,
! [B3: mat_complex,A2: mat_complex] :
( ( member_mat_complex @ B3 @ ( insert_mat_complex @ A2 @ bot_bo7165004461764951667omplex ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_964_singleton__iff,axiom,
! [B3: complex,A2: complex] :
( ( member_complex @ B3 @ ( insert_complex @ A2 @ bot_bot_set_complex ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_965_singletonI,axiom,
! [A2: mat_complex] : ( member_mat_complex @ A2 @ ( insert_mat_complex @ A2 @ bot_bo7165004461764951667omplex ) ) ).
% singletonI
thf(fact_966_singletonI,axiom,
! [A2: complex] : ( member_complex @ A2 @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ).
% singletonI
thf(fact_967_singletonD,axiom,
! [B3: mat_complex,A2: mat_complex] :
( ( member_mat_complex @ B3 @ ( insert_mat_complex @ A2 @ bot_bo7165004461764951667omplex ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_968_singletonD,axiom,
! [B3: complex,A2: complex] :
( ( member_complex @ B3 @ ( insert_complex @ A2 @ bot_bot_set_complex ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_969_Diff__empty,axiom,
! [A: set_mat_complex] :
( ( minus_8760755521168068590omplex @ A @ bot_bo7165004461764951667omplex )
= A ) ).
% Diff_empty
thf(fact_970_Diff__empty,axiom,
! [A: set_complex] :
( ( minus_811609699411566653omplex @ A @ bot_bot_set_complex )
= A ) ).
% Diff_empty
thf(fact_971_empty__Diff,axiom,
! [A: set_mat_complex] :
( ( minus_8760755521168068590omplex @ bot_bo7165004461764951667omplex @ A )
= bot_bo7165004461764951667omplex ) ).
% empty_Diff
thf(fact_972_empty__Diff,axiom,
! [A: set_complex] :
( ( minus_811609699411566653omplex @ bot_bot_set_complex @ A )
= bot_bot_set_complex ) ).
% empty_Diff
thf(fact_973_Diff__cancel,axiom,
! [A: set_mat_complex] :
( ( minus_8760755521168068590omplex @ A @ A )
= bot_bo7165004461764951667omplex ) ).
% Diff_cancel
thf(fact_974_Diff__cancel,axiom,
! [A: set_complex] :
( ( minus_811609699411566653omplex @ A @ A )
= bot_bot_set_complex ) ).
% Diff_cancel
thf(fact_975_insert__Diff__if,axiom,
! [X: mat_complex,B2: set_mat_complex,A: set_mat_complex] :
( ( ( member_mat_complex @ X @ B2 )
=> ( ( minus_8760755521168068590omplex @ ( insert_mat_complex @ X @ A ) @ B2 )
= ( minus_8760755521168068590omplex @ A @ B2 ) ) )
& ( ~ ( member_mat_complex @ X @ B2 )
=> ( ( minus_8760755521168068590omplex @ ( insert_mat_complex @ X @ A ) @ B2 )
= ( insert_mat_complex @ X @ ( minus_8760755521168068590omplex @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_976_insert__Diff__if,axiom,
! [X: complex,B2: set_complex,A: set_complex] :
( ( ( member_complex @ X @ B2 )
=> ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A ) @ B2 )
= ( minus_811609699411566653omplex @ A @ B2 ) ) )
& ( ~ ( member_complex @ X @ B2 )
=> ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A ) @ B2 )
= ( insert_complex @ X @ ( minus_811609699411566653omplex @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_977_insert__Diff1,axiom,
! [X: mat_complex,B2: set_mat_complex,A: set_mat_complex] :
( ( member_mat_complex @ X @ B2 )
=> ( ( minus_8760755521168068590omplex @ ( insert_mat_complex @ X @ A ) @ B2 )
= ( minus_8760755521168068590omplex @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_978_insert__Diff1,axiom,
! [X: complex,B2: set_complex,A: set_complex] :
( ( member_complex @ X @ B2 )
=> ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A ) @ B2 )
= ( minus_811609699411566653omplex @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_979_Diff__insert0,axiom,
! [X: mat_complex,A: set_mat_complex,B2: set_mat_complex] :
( ~ ( member_mat_complex @ X @ A )
=> ( ( minus_8760755521168068590omplex @ A @ ( insert_mat_complex @ X @ B2 ) )
= ( minus_8760755521168068590omplex @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_980_Diff__insert0,axiom,
! [X: complex,A: set_complex,B2: set_complex] :
( ~ ( member_complex @ X @ A )
=> ( ( minus_811609699411566653omplex @ A @ ( insert_complex @ X @ B2 ) )
= ( minus_811609699411566653omplex @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_981_image__add__0,axiom,
! [S: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_982_image__add__0,axiom,
! [S: set_complex] :
( ( image_1468599708987790691omplex @ ( plus_plus_complex @ zero_zero_complex ) @ S )
= S ) ).
% image_add_0
thf(fact_983_subset__Compl__singleton,axiom,
! [A: set_mat_complex,B3: mat_complex] :
( ( ord_le3632134057777142183omplex @ A @ ( uminus5815530220087396478omplex @ ( insert_mat_complex @ B3 @ bot_bo7165004461764951667omplex ) ) )
= ( ~ ( member_mat_complex @ B3 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_984_subset__Compl__singleton,axiom,
! [A: set_complex,B3: complex] :
( ( ord_le211207098394363844omplex @ A @ ( uminus8566677241136511917omplex @ ( insert_complex @ B3 @ bot_bot_set_complex ) ) )
= ( ~ ( member_complex @ B3 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_985_singleton__insert__inj__eq_H,axiom,
! [A2: mat_complex,A: set_mat_complex,B3: mat_complex] :
( ( ( insert_mat_complex @ A2 @ A )
= ( insert_mat_complex @ B3 @ bot_bo7165004461764951667omplex ) )
= ( ( A2 = B3 )
& ( ord_le3632134057777142183omplex @ A @ ( insert_mat_complex @ B3 @ bot_bo7165004461764951667omplex ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_986_singleton__insert__inj__eq_H,axiom,
! [A2: complex,A: set_complex,B3: complex] :
( ( ( insert_complex @ A2 @ A )
= ( insert_complex @ B3 @ bot_bot_set_complex ) )
= ( ( A2 = B3 )
& ( ord_le211207098394363844omplex @ A @ ( insert_complex @ B3 @ bot_bot_set_complex ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_987_singleton__insert__inj__eq,axiom,
! [B3: mat_complex,A2: mat_complex,A: set_mat_complex] :
( ( ( insert_mat_complex @ B3 @ bot_bo7165004461764951667omplex )
= ( insert_mat_complex @ A2 @ A ) )
= ( ( A2 = B3 )
& ( ord_le3632134057777142183omplex @ A @ ( insert_mat_complex @ B3 @ bot_bo7165004461764951667omplex ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_988_singleton__insert__inj__eq,axiom,
! [B3: complex,A2: complex,A: set_complex] :
( ( ( insert_complex @ B3 @ bot_bot_set_complex )
= ( insert_complex @ A2 @ A ) )
= ( ( A2 = B3 )
& ( ord_le211207098394363844omplex @ A @ ( insert_complex @ B3 @ bot_bot_set_complex ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_989_subset__singleton__iff,axiom,
! [X4: set_mat_complex,A2: mat_complex] :
( ( ord_le3632134057777142183omplex @ X4 @ ( insert_mat_complex @ A2 @ bot_bo7165004461764951667omplex ) )
= ( ( X4 = bot_bo7165004461764951667omplex )
| ( X4
= ( insert_mat_complex @ A2 @ bot_bo7165004461764951667omplex ) ) ) ) ).
% subset_singleton_iff
thf(fact_990_subset__singleton__iff,axiom,
! [X4: set_complex,A2: complex] :
( ( ord_le211207098394363844omplex @ X4 @ ( insert_complex @ A2 @ bot_bot_set_complex ) )
= ( ( X4 = bot_bot_set_complex )
| ( X4
= ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ).
% subset_singleton_iff
thf(fact_991_subset__singletonD,axiom,
! [A: set_mat_complex,X: mat_complex] :
( ( ord_le3632134057777142183omplex @ A @ ( insert_mat_complex @ X @ bot_bo7165004461764951667omplex ) )
=> ( ( A = bot_bo7165004461764951667omplex )
| ( A
= ( insert_mat_complex @ X @ bot_bo7165004461764951667omplex ) ) ) ) ).
% subset_singletonD
thf(fact_992_subset__singletonD,axiom,
! [A: set_complex,X: complex] :
( ( ord_le211207098394363844omplex @ A @ ( insert_complex @ X @ bot_bot_set_complex ) )
=> ( ( A = bot_bot_set_complex )
| ( A
= ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ).
% subset_singletonD
thf(fact_993_Diff__eq__empty__iff,axiom,
! [A: set_mat_complex,B2: set_mat_complex] :
( ( ( minus_8760755521168068590omplex @ A @ B2 )
= bot_bo7165004461764951667omplex )
= ( ord_le3632134057777142183omplex @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_994_Diff__eq__empty__iff,axiom,
! [A: set_complex,B2: set_complex] :
( ( ( minus_811609699411566653omplex @ A @ B2 )
= bot_bot_set_complex )
= ( ord_le211207098394363844omplex @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_995_subset__Diff__insert,axiom,
! [A: set_mat_complex,B2: set_mat_complex,X: mat_complex,C2: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ A @ ( minus_8760755521168068590omplex @ B2 @ ( insert_mat_complex @ X @ C2 ) ) )
= ( ( ord_le3632134057777142183omplex @ A @ ( minus_8760755521168068590omplex @ B2 @ C2 ) )
& ~ ( member_mat_complex @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_996_subset__Diff__insert,axiom,
! [A: set_complex,B2: set_complex,X: complex,C2: set_complex] :
( ( ord_le211207098394363844omplex @ A @ ( minus_811609699411566653omplex @ B2 @ ( insert_complex @ X @ C2 ) ) )
= ( ( ord_le211207098394363844omplex @ A @ ( minus_811609699411566653omplex @ B2 @ C2 ) )
& ~ ( member_complex @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_997_in__image__insert__iff,axiom,
! [B2: set_set_mat_complex,X: mat_complex,A: set_mat_complex] :
( ! [C5: set_mat_complex] :
( ( member3612512168372279472omplex @ C5 @ B2 )
=> ~ ( member_mat_complex @ X @ C5 ) )
=> ( ( member3612512168372279472omplex @ A @ ( image_7857974177869717957omplex @ ( insert_mat_complex @ X ) @ B2 ) )
= ( ( member_mat_complex @ X @ A )
& ( member3612512168372279472omplex @ ( minus_8760755521168068590omplex @ A @ ( insert_mat_complex @ X @ bot_bo7165004461764951667omplex ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_998_in__image__insert__iff,axiom,
! [B2: set_set_complex,X: complex,A: set_complex] :
( ! [C5: set_complex] :
( ( member_set_complex @ C5 @ B2 )
=> ~ ( member_complex @ X @ C5 ) )
=> ( ( member_set_complex @ A @ ( image_7998606247489673935omplex @ ( insert_complex @ X ) @ B2 ) )
= ( ( member_complex @ X @ A )
& ( member_set_complex @ ( minus_811609699411566653omplex @ A @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_999_Diff__insert,axiom,
! [A: set_mat_complex,A2: mat_complex,B2: set_mat_complex] :
( ( minus_8760755521168068590omplex @ A @ ( insert_mat_complex @ A2 @ B2 ) )
= ( minus_8760755521168068590omplex @ ( minus_8760755521168068590omplex @ A @ B2 ) @ ( insert_mat_complex @ A2 @ bot_bo7165004461764951667omplex ) ) ) ).
% Diff_insert
thf(fact_1000_Diff__insert,axiom,
! [A: set_complex,A2: complex,B2: set_complex] :
( ( minus_811609699411566653omplex @ A @ ( insert_complex @ A2 @ B2 ) )
= ( minus_811609699411566653omplex @ ( minus_811609699411566653omplex @ A @ B2 ) @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ).
% Diff_insert
thf(fact_1001_insert__Diff,axiom,
! [A2: mat_complex,A: set_mat_complex] :
( ( member_mat_complex @ A2 @ A )
=> ( ( insert_mat_complex @ A2 @ ( minus_8760755521168068590omplex @ A @ ( insert_mat_complex @ A2 @ bot_bo7165004461764951667omplex ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1002_insert__Diff,axiom,
! [A2: complex,A: set_complex] :
( ( member_complex @ A2 @ A )
=> ( ( insert_complex @ A2 @ ( minus_811609699411566653omplex @ A @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1003_Compl__insert,axiom,
! [X: mat_complex,A: set_mat_complex] :
( ( uminus5815530220087396478omplex @ ( insert_mat_complex @ X @ A ) )
= ( minus_8760755521168068590omplex @ ( uminus5815530220087396478omplex @ A ) @ ( insert_mat_complex @ X @ bot_bo7165004461764951667omplex ) ) ) ).
% Compl_insert
thf(fact_1004_Compl__insert,axiom,
! [X: complex,A: set_complex] :
( ( uminus8566677241136511917omplex @ ( insert_complex @ X @ A ) )
= ( minus_811609699411566653omplex @ ( uminus8566677241136511917omplex @ A ) @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ).
% Compl_insert
thf(fact_1005_Diff__insert2,axiom,
! [A: set_mat_complex,A2: mat_complex,B2: set_mat_complex] :
( ( minus_8760755521168068590omplex @ A @ ( insert_mat_complex @ A2 @ B2 ) )
= ( minus_8760755521168068590omplex @ ( minus_8760755521168068590omplex @ A @ ( insert_mat_complex @ A2 @ bot_bo7165004461764951667omplex ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_1006_Diff__insert2,axiom,
! [A: set_complex,A2: complex,B2: set_complex] :
( ( minus_811609699411566653omplex @ A @ ( insert_complex @ A2 @ B2 ) )
= ( minus_811609699411566653omplex @ ( minus_811609699411566653omplex @ A @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_1007_Diff__insert__absorb,axiom,
! [X: mat_complex,A: set_mat_complex] :
( ~ ( member_mat_complex @ X @ A )
=> ( ( minus_8760755521168068590omplex @ ( insert_mat_complex @ X @ A ) @ ( insert_mat_complex @ X @ bot_bo7165004461764951667omplex ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1008_Diff__insert__absorb,axiom,
! [X: complex,A: set_complex] :
( ~ ( member_complex @ X @ A )
=> ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A ) @ ( insert_complex @ X @ bot_bot_set_complex ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1009_insert__Diff__single,axiom,
! [A2: mat_complex,A: set_mat_complex] :
( ( insert_mat_complex @ A2 @ ( minus_8760755521168068590omplex @ A @ ( insert_mat_complex @ A2 @ bot_bo7165004461764951667omplex ) ) )
= ( insert_mat_complex @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_1010_insert__Diff__single,axiom,
! [A2: complex,A: set_complex] :
( ( insert_complex @ A2 @ ( minus_811609699411566653omplex @ A @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
= ( insert_complex @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_1011_Diff__not__in,axiom,
! [A2: mat_complex,A: set_mat_complex] :
~ ( member_mat_complex @ A2 @ ( minus_8760755521168068590omplex @ A @ ( insert_mat_complex @ A2 @ bot_bo7165004461764951667omplex ) ) ) ).
% Diff_not_in
thf(fact_1012_Diff__not__in,axiom,
! [A2: complex,A: set_complex] :
~ ( member_complex @ A2 @ ( minus_811609699411566653omplex @ A @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ).
% Diff_not_in
thf(fact_1013_set__add__0__right,axiom,
! [A: set_nat] :
( ( plus_plus_set_nat @ A @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) )
= A ) ).
% set_add_0_right
thf(fact_1014_set__add__0__right,axiom,
! [A: set_complex] :
( ( plus_p7052360327008956141omplex @ A @ ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) )
= A ) ).
% set_add_0_right
thf(fact_1015_set__times__elim,axiom,
! [X: mat_complex,A: set_mat_complex,B2: set_mat_complex] :
( ( member_mat_complex @ X @ ( times_6731331324747250370omplex @ A @ B2 ) )
=> ~ ! [A4: mat_complex,B5: mat_complex] :
( ( X
= ( times_8009071140041733218omplex @ A4 @ B5 ) )
=> ( ( member_mat_complex @ A4 @ A )
=> ~ ( member_mat_complex @ B5 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_1016_set__times__elim,axiom,
! [X: complex,A: set_complex,B2: set_complex] :
( ( member_complex @ X @ ( times_6048082448287401577omplex @ A @ B2 ) )
=> ~ ! [A4: complex,B5: complex] :
( ( X
= ( times_times_complex @ A4 @ B5 ) )
=> ( ( member_complex @ A4 @ A )
=> ~ ( member_complex @ B5 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_1017_set__times__elim,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ X @ ( times_times_set_nat @ A @ B2 ) )
=> ~ ! [A4: nat,B5: nat] :
( ( X
= ( times_times_nat @ A4 @ B5 ) )
=> ( ( member_nat @ A4 @ A )
=> ~ ( member_nat @ B5 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_1018_set__times__intro,axiom,
! [A2: mat_complex,C2: set_mat_complex,B3: mat_complex,D: set_mat_complex] :
( ( member_mat_complex @ A2 @ C2 )
=> ( ( member_mat_complex @ B3 @ D )
=> ( member_mat_complex @ ( times_8009071140041733218omplex @ A2 @ B3 ) @ ( times_6731331324747250370omplex @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_1019_set__times__intro,axiom,
! [A2: complex,C2: set_complex,B3: complex,D: set_complex] :
( ( member_complex @ A2 @ C2 )
=> ( ( member_complex @ B3 @ D )
=> ( member_complex @ ( times_times_complex @ A2 @ B3 ) @ ( times_6048082448287401577omplex @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_1020_set__times__intro,axiom,
! [A2: nat,C2: set_nat,B3: nat,D: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( member_nat @ B3 @ D )
=> ( member_nat @ ( times_times_nat @ A2 @ B3 ) @ ( times_times_set_nat @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_1021_set__plus__elim,axiom,
! [X: complex,A: set_complex,B2: set_complex] :
( ( member_complex @ X @ ( plus_p7052360327008956141omplex @ A @ B2 ) )
=> ~ ! [A4: complex,B5: complex] :
( ( X
= ( plus_plus_complex @ A4 @ B5 ) )
=> ( ( member_complex @ A4 @ A )
=> ~ ( member_complex @ B5 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_1022_set__plus__elim,axiom,
! [X: mat_complex,A: set_mat_complex,B2: set_mat_complex] :
( ( member_mat_complex @ X @ ( plus_p4229080058245121342omplex @ A @ B2 ) )
=> ~ ! [A4: mat_complex,B5: mat_complex] :
( ( X
= ( plus_p8323303612493835998omplex @ A4 @ B5 ) )
=> ( ( member_mat_complex @ A4 @ A )
=> ~ ( member_mat_complex @ B5 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_1023_set__plus__elim,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ X @ ( plus_plus_set_nat @ A @ B2 ) )
=> ~ ! [A4: nat,B5: nat] :
( ( X
= ( plus_plus_nat @ A4 @ B5 ) )
=> ( ( member_nat @ A4 @ A )
=> ~ ( member_nat @ B5 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_1024_set__plus__intro,axiom,
! [A2: complex,C2: set_complex,B3: complex,D: set_complex] :
( ( member_complex @ A2 @ C2 )
=> ( ( member_complex @ B3 @ D )
=> ( member_complex @ ( plus_plus_complex @ A2 @ B3 ) @ ( plus_p7052360327008956141omplex @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_1025_set__plus__intro,axiom,
! [A2: mat_complex,C2: set_mat_complex,B3: mat_complex,D: set_mat_complex] :
( ( member_mat_complex @ A2 @ C2 )
=> ( ( member_mat_complex @ B3 @ D )
=> ( member_mat_complex @ ( plus_p8323303612493835998omplex @ A2 @ B3 ) @ ( plus_p4229080058245121342omplex @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_1026_set__plus__intro,axiom,
! [A2: nat,C2: set_nat,B3: nat,D: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( member_nat @ B3 @ D )
=> ( member_nat @ ( plus_plus_nat @ A2 @ B3 ) @ ( plus_plus_set_nat @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_1027_sumset__empty_I2_J,axiom,
! [A: set_mat_complex] :
( ( plus_p4229080058245121342omplex @ bot_bo7165004461764951667omplex @ A )
= bot_bo7165004461764951667omplex ) ).
% sumset_empty(2)
thf(fact_1028_sumset__empty_I2_J,axiom,
! [A: set_complex] :
( ( plus_p7052360327008956141omplex @ bot_bot_set_complex @ A )
= bot_bot_set_complex ) ).
% sumset_empty(2)
thf(fact_1029_sumset__empty_I1_J,axiom,
! [A: set_mat_complex] :
( ( plus_p4229080058245121342omplex @ A @ bot_bo7165004461764951667omplex )
= bot_bo7165004461764951667omplex ) ).
% sumset_empty(1)
thf(fact_1030_sumset__empty_I1_J,axiom,
! [A: set_complex] :
( ( plus_p7052360327008956141omplex @ A @ bot_bot_set_complex )
= bot_bot_set_complex ) ).
% sumset_empty(1)
thf(fact_1031_eq__add__iff,axiom,
! [X: complex,Y2: complex] :
( ( X
= ( plus_plus_complex @ X @ Y2 ) )
= ( Y2 = zero_zero_complex ) ) ).
% eq_add_iff
thf(fact_1032_set__zero__plus2,axiom,
! [A: set_nat,B2: set_nat] :
( ( member_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_set_nat @ B2 @ ( plus_plus_set_nat @ A @ B2 ) ) ) ).
% set_zero_plus2
thf(fact_1033_set__zero__plus2,axiom,
! [A: set_complex,B2: set_complex] :
( ( member_complex @ zero_zero_complex @ A )
=> ( ord_le211207098394363844omplex @ B2 @ ( plus_p7052360327008956141omplex @ A @ B2 ) ) ) ).
% set_zero_plus2
thf(fact_1034_set__zero,axiom,
( zero_zero_set_nat
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% set_zero
thf(fact_1035_set__zero,axiom,
( zero_z6614145512433583213omplex
= ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) ) ).
% set_zero
thf(fact_1036_set__add__0,axiom,
! [A: set_nat] :
( ( plus_plus_set_nat @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) @ A )
= A ) ).
% set_add_0
thf(fact_1037_set__add__0,axiom,
! [A: set_complex] :
( ( plus_p7052360327008956141omplex @ ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) @ A )
= A ) ).
% set_add_0
thf(fact_1038_verit__minus__simplify_I3_J,axiom,
! [B3: complex] :
( ( minus_minus_complex @ zero_zero_complex @ B3 )
= ( uminus1482373934393186551omplex @ B3 ) ) ).
% verit_minus_simplify(3)
thf(fact_1039_verit__la__disequality,axiom,
! [A2: nat,B3: nat] :
( ( A2 = B3 )
| ~ ( ord_less_eq_nat @ A2 @ B3 )
| ~ ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% verit_la_disequality
thf(fact_1040_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_1041_verit__comp__simplify1_I2_J,axiom,
! [A2: complex] : ( ord_less_eq_complex @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_1042_verit__eq__simplify_I6_J,axiom,
! [X: nat,Y2: nat] :
( ( X = Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% verit_eq_simplify(6)
thf(fact_1043_verit__eq__simplify_I6_J,axiom,
! [X: complex,Y2: complex] :
( ( X = Y2 )
=> ( ord_less_eq_complex @ X @ Y2 ) ) ).
% verit_eq_simplify(6)
thf(fact_1044_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_1045_verit__negate__coefficient_I3_J,axiom,
! [A2: complex,B3: complex] :
( ( A2 = B3 )
=> ( ( uminus1482373934393186551omplex @ A2 )
= ( uminus1482373934393186551omplex @ B3 ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_1046_verit__minus__simplify_I4_J,axiom,
! [B3: complex] :
( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B3 ) )
= B3 ) ).
% verit_minus_simplify(4)
thf(fact_1047_verit__comp__simplify_I3_J,axiom,
! [B7: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
= ( ord_less_nat @ A7 @ B7 ) ) ).
% verit_comp_simplify(3)
thf(fact_1048_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_1049_verit__sum__simplify,axiom,
! [A2: complex] :
( ( plus_plus_complex @ A2 @ zero_zero_complex )
= A2 ) ).
% verit_sum_simplify
thf(fact_1050_cross3__simps_I48_J,axiom,
! [A2: complex,X: complex,Y2: complex] :
( ( times_times_complex @ A2 @ ( plus_plus_complex @ X @ Y2 ) )
= ( plus_plus_complex @ ( times_times_complex @ A2 @ X ) @ ( times_times_complex @ A2 @ Y2 ) ) ) ).
% cross3_simps(48)
thf(fact_1051_cross3__simps_I49_J,axiom,
! [A2: complex,B3: complex,X: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A2 @ B3 ) @ X )
= ( plus_plus_complex @ ( times_times_complex @ A2 @ X ) @ ( times_times_complex @ B3 @ X ) ) ) ).
% cross3_simps(49)
thf(fact_1052_cross3__simps_I50_J,axiom,
! [A2: complex,B3: complex,X: complex] :
( ( times_times_complex @ ( minus_minus_complex @ A2 @ B3 ) @ X )
= ( minus_minus_complex @ ( times_times_complex @ A2 @ X ) @ ( times_times_complex @ B3 @ X ) ) ) ).
% cross3_simps(50)
thf(fact_1053_cross3__simps_I51_J,axiom,
! [A2: complex,X: complex,Y2: complex] :
( ( times_times_complex @ A2 @ ( minus_minus_complex @ X @ Y2 ) )
= ( minus_minus_complex @ ( times_times_complex @ A2 @ X ) @ ( times_times_complex @ A2 @ Y2 ) ) ) ).
% cross3_simps(51)
thf(fact_1054_verit__negate__coefficient_I2_J,axiom,
! [A2: complex,B3: complex] :
( ( ord_less_complex @ A2 @ B3 )
=> ( ord_less_complex @ ( uminus1482373934393186551omplex @ B3 ) @ ( uminus1482373934393186551omplex @ A2 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1055_mult__diff__mult,axiom,
! [X: complex,Y2: complex,A2: complex,B3: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X @ Y2 ) @ ( times_times_complex @ A2 @ B3 ) )
= ( plus_plus_complex @ ( times_times_complex @ X @ ( minus_minus_complex @ Y2 @ B3 ) ) @ ( times_times_complex @ ( minus_minus_complex @ X @ A2 ) @ B3 ) ) ) ).
% mult_diff_mult
thf(fact_1056_diff__shunt__var,axiom,
! [X: set_mat_complex,Y2: set_mat_complex] :
( ( ( minus_8760755521168068590omplex @ X @ Y2 )
= bot_bo7165004461764951667omplex )
= ( ord_le3632134057777142183omplex @ X @ Y2 ) ) ).
% diff_shunt_var
thf(fact_1057_diff__shunt__var,axiom,
! [X: set_complex,Y2: set_complex] :
( ( ( minus_811609699411566653omplex @ X @ Y2 )
= bot_bot_set_complex )
= ( ord_le211207098394363844omplex @ X @ Y2 ) ) ).
% diff_shunt_var
thf(fact_1058_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1059_bot__set__def,axiom,
( bot_bo7165004461764951667omplex
= ( collect_mat_complex @ bot_bo2514468519737825834plex_o ) ) ).
% bot_set_def
thf(fact_1060_bot__set__def,axiom,
( bot_bot_set_complex
= ( collect_complex @ bot_bot_complex_o ) ) ).
% bot_set_def
thf(fact_1061_minus__diff__minus,axiom,
! [A2: complex,B3: complex] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( uminus1482373934393186551omplex @ B3 ) )
= ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A2 @ B3 ) ) ) ).
% minus_diff_minus
thf(fact_1062_poly__cancel__eq__conv,axiom,
! [X: complex,A2: complex,Y2: complex,B3: complex] :
( ( X = zero_zero_complex )
=> ( ( A2 != zero_zero_complex )
=> ( ( Y2 = zero_zero_complex )
= ( ( minus_minus_complex @ ( times_times_complex @ A2 @ Y2 ) @ ( times_times_complex @ B3 @ X ) )
= zero_zero_complex ) ) ) ) ).
% poly_cancel_eq_conv
thf(fact_1063_mult__hom_Ohom__add__eq__zero,axiom,
! [X: complex,Y2: complex,C: complex] :
( ( ( plus_plus_complex @ X @ Y2 )
= zero_zero_complex )
=> ( ( plus_plus_complex @ ( times_times_complex @ C @ X ) @ ( times_times_complex @ C @ Y2 ) )
= zero_zero_complex ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_1064_mult__hom_Ohom__add__eq__zero,axiom,
! [X: nat,Y2: nat,C: nat] :
( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
=> ( ( plus_plus_nat @ ( times_times_nat @ C @ X ) @ ( times_times_nat @ C @ Y2 ) )
= zero_zero_nat ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_1065_mult__hom_Ohom__zero,axiom,
! [C: complex] :
( ( times_times_complex @ C @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_hom.hom_zero
thf(fact_1066_mult__hom_Ohom__zero,axiom,
! [C: nat] :
( ( times_times_nat @ C @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_1067_mult__hom_Ohom__add,axiom,
! [C: complex,X: complex,Y2: complex] :
( ( times_times_complex @ C @ ( plus_plus_complex @ X @ Y2 ) )
= ( plus_plus_complex @ ( times_times_complex @ C @ X ) @ ( times_times_complex @ C @ Y2 ) ) ) ).
% mult_hom.hom_add
thf(fact_1068_mult__hom_Ohom__add,axiom,
! [C: nat,X: nat,Y2: nat] :
( ( times_times_nat @ C @ ( plus_plus_nat @ X @ Y2 ) )
= ( plus_plus_nat @ ( times_times_nat @ C @ X ) @ ( times_times_nat @ C @ Y2 ) ) ) ).
% mult_hom.hom_add
thf(fact_1069_add__scale__eq__noteq,axiom,
! [R: complex,A2: complex,B3: complex,C: complex,D2: complex] :
( ( R != zero_zero_complex )
=> ( ( ( A2 = B3 )
& ( C != D2 ) )
=> ( ( plus_plus_complex @ A2 @ ( times_times_complex @ R @ C ) )
!= ( plus_plus_complex @ B3 @ ( times_times_complex @ R @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1070_add__scale__eq__noteq,axiom,
! [R: nat,A2: nat,B3: nat,C: nat,D2: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A2 = B3 )
& ( C != D2 ) )
=> ( ( plus_plus_nat @ A2 @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B3 @ ( times_times_nat @ R @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1071_add__0__iff,axiom,
! [B3: nat,A2: nat] :
( ( B3
= ( plus_plus_nat @ B3 @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_1072_add__0__iff,axiom,
! [B3: complex,A2: complex] :
( ( B3
= ( plus_plus_complex @ B3 @ A2 ) )
= ( A2 = zero_zero_complex ) ) ).
% add_0_iff
thf(fact_1073_crossproduct__eq,axiom,
! [W: complex,Y2: complex,X: complex,Z: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ W @ Y2 ) @ ( times_times_complex @ X @ Z ) )
= ( plus_plus_complex @ ( times_times_complex @ W @ Z ) @ ( times_times_complex @ X @ Y2 ) ) )
= ( ( W = X )
| ( Y2 = Z ) ) ) ).
% crossproduct_eq
thf(fact_1074_crossproduct__eq,axiom,
! [W: nat,Y2: nat,X: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y2 ) @ ( times_times_nat @ X @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y2 ) ) )
= ( ( W = X )
| ( Y2 = Z ) ) ) ).
% crossproduct_eq
thf(fact_1075_crossproduct__noteq,axiom,
! [A2: complex,B3: complex,C: complex,D2: complex] :
( ( ( A2 != B3 )
& ( C != D2 ) )
= ( ( plus_plus_complex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B3 @ D2 ) )
!= ( plus_plus_complex @ ( times_times_complex @ A2 @ D2 ) @ ( times_times_complex @ B3 @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_1076_crossproduct__noteq,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ( A2 != B3 )
& ( C != D2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D2 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A2 @ D2 ) @ ( times_times_nat @ B3 @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_1077_max__mix__density__carrier,axiom,
! [N: nat] : ( member_mat_complex @ ( projec8360710381328234318ensity @ N ) @ ( carrier_mat_complex @ N @ N ) ) ).
% max_mix_density_carrier
thf(fact_1078_hermitian__real__diag__decomp,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( comple8306762464034002205omplex @ A )
=> ~ ! [B: mat_complex,U: mat_complex] :
~ ( spectr5409772854192057952omplex @ A @ B @ U ) ) ) ) ).
% hermitian_real_diag_decomp
thf(fact_1079_real__diag__decomp__hermitian,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( spectr5409772854192057952omplex @ A @ B2 @ U2 )
=> ( comple8306762464034002205omplex @ A ) ) ).
% real_diag_decomp_hermitian
thf(fact_1080_hermitian__decomp__decomp_H,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( projec5943904436471448624omplex @ A @ B2 @ U2 )
=> ( spectr5409772854192057952omplex @ A @ B2 @ U2 ) ) ).
% hermitian_decomp_decomp'
thf(fact_1081_real__diag__decompD_I1_J,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( spectr5409772854192057952omplex @ A @ B2 @ U2 )
=> ( spectr532731689276696518omplex @ A @ B2 @ U2 ) ) ).
% real_diag_decompD(1)
thf(fact_1082_spectrum__ne,axiom,
! [A: mat_complex] :
( ( member_mat_complex @ A @ fc )
=> ( ( comple8306762464034002205omplex @ A )
=> ( ( projec527831343749723810omplex @ A )
!= bot_bot_set_complex ) ) ) ).
% spectrum_ne
thf(fact_1083_OP__def,axiom,
( v_OP____
= ( comple3482886691669134300omplex @ ( col_complex @ u @ i ) @ ( col_complex @ u @ j ) ) ) ).
% OP_def
thf(fact_1084_zero__col__col,axiom,
! [I3: nat,U2: mat_complex] :
( ( ord_less_nat @ I3 @ n )
=> ( ( linear2567731547100889994omplex @ n @ U2 @ I3 )
= ( col_complex @ U2 @ I3 ) ) ) ).
% zero_col_col
thf(fact_1085_eigen__projector__carrier,axiom,
! [A: mat_complex,A2: complex] :
( ( member_mat_complex @ A @ fc )
=> ( ( member_complex @ A2 @ ( projec527831343749723810omplex @ A ) )
=> ( ( comple8306762464034002205omplex @ A )
=> ( member_mat_complex @ ( projec1689266477789839993jector @ n @ n @ A @ A2 ) @ fc ) ) ) ) ).
% eigen_projector_carrier
thf(fact_1086_col__add,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,B2: mat_complex,J: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( ord_less_nat @ J @ Nc )
=> ( ( col_complex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ J )
= ( plus_p3079357308422357842omplex @ ( col_complex @ A @ J ) @ ( col_complex @ B2 @ J ) ) ) ) ) ) ).
% col_add
thf(fact_1087_spectrum__finite,axiom,
! [M5: mat_complex] : ( finite3207457112153483333omplex @ ( projec527831343749723810omplex @ M5 ) ) ).
% spectrum_finite
thf(fact_1088_rf,axiom,
! [I3: nat] :
( ( ord_less_nat @ I3 @ n )
=> ( member_mat_complex @ ( linear1949544614684794075omplex @ ( col_complex @ u @ I3 ) ) @ fc ) ) ).
% rf
thf(fact_1089__092_060open_062rank__1__proj_A_IMatrix_Ocol_AU_Aj_J_A_092_060in_062_Afc_092_060close_062,axiom,
member_mat_complex @ ( linear1949544614684794075omplex @ ( col_complex @ u @ j ) ) @ fc ).
% \<open>rank_1_proj (Matrix.col U j) \<in> fc\<close>
thf(fact_1090_rank__1__proj__def,axiom,
( linear1949544614684794075omplex
= ( ^ [V2: vec_complex] : ( comple3482886691669134300omplex @ V2 @ V2 ) ) ) ).
% rank_1_proj_def
thf(fact_1091_rank__1__proj__square__mat,axiom,
! [V: vec_complex] : ( square_mat_complex @ ( linear1949544614684794075omplex @ V ) ) ).
% rank_1_proj_square_mat
thf(fact_1092_rank__1__proj__hermitian,axiom,
! [V: vec_complex] : ( comple8306762464034002205omplex @ ( linear1949544614684794075omplex @ V ) ) ).
% rank_1_proj_hermitian
thf(fact_1093_rank__1__proj__unitary__ne,axiom,
! [A: mat_complex,J: nat,K: nat] :
( ( member_mat_complex @ A @ fc )
=> ( ( comple6660659447773130958omplex @ A )
=> ( ( ord_less_nat @ J @ n )
=> ( ( ord_less_nat @ K @ n )
=> ( ( J != K )
=> ( ( times_8009071140041733218omplex @ ( linear1949544614684794075omplex @ ( col_complex @ A @ J ) ) @ ( linear1949544614684794075omplex @ ( col_complex @ A @ K ) ) )
= ( zero_mat_complex @ n @ n ) ) ) ) ) ) ) ).
% rank_1_proj_unitary_ne
thf(fact_1094_rank__1__proj__unitary__eq,axiom,
! [A: mat_complex,J: nat] :
( ( member_mat_complex @ A @ fc )
=> ( ( comple6660659447773130958omplex @ A )
=> ( ( ord_less_nat @ J @ n )
=> ( ( times_8009071140041733218omplex @ ( linear1949544614684794075omplex @ ( col_complex @ A @ J ) ) @ ( linear1949544614684794075omplex @ ( col_complex @ A @ J ) ) )
= ( linear1949544614684794075omplex @ ( col_complex @ A @ J ) ) ) ) ) ) ).
% rank_1_proj_unitary_eq
thf(fact_1095_hermitian__decomp__unitary,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( projec5943904436471448624omplex @ A @ B2 @ U2 )
=> ( comple6660659447773130958omplex @ U2 ) ) ).
% hermitian_decomp_unitary
thf(fact_1096_unitary__diagD_I3_J,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( spectr532731689276696518omplex @ A @ B2 @ U2 )
=> ( comple6660659447773130958omplex @ U2 ) ) ).
% unitary_diagD(3)
thf(fact_1097_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_1098_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_1099_zero__col__def,axiom,
! [U2: mat_complex] :
( ( linear2567731547100889994omplex @ n @ U2 )
= ( ^ [I5: nat] : ( if_vec_complex @ ( ord_less_nat @ I5 @ n ) @ ( col_complex @ U2 @ I5 ) @ ( zero_vec_complex @ n ) ) ) ) ).
% zero_col_def
thf(fact_1100_hermitian__spectrum__real,axiom,
! [A: mat_complex,A2: complex] :
( ( member_mat_complex @ A @ fc )
=> ( ( comple8306762464034002205omplex @ A )
=> ( ( member_complex @ A2 @ ( projec527831343749723810omplex @ A ) )
=> ( member_complex @ A2 @ real_V2521375963428798218omplex ) ) ) ) ).
% hermitian_spectrum_real
thf(fact_1101_col__zero,axiom,
! [J: nat,Nc: nat,Nr: nat] :
( ( ord_less_nat @ J @ Nc )
=> ( ( col_complex @ ( zero_mat_complex @ Nr @ Nc ) @ J )
= ( zero_vec_complex @ Nr ) ) ) ).
% col_zero
thf(fact_1102_complex__is__real__iff__compare0,axiom,
! [X: complex] :
( ( member_complex @ X @ real_V2521375963428798218omplex )
= ( ( ord_less_eq_complex @ X @ zero_zero_complex )
| ( ord_less_eq_complex @ zero_zero_complex @ X ) ) ) ).
% complex_is_real_iff_compare0
thf(fact_1103_nonnegative__complex__is__real,axiom,
! [X: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ X )
=> ( member_complex @ X @ real_V2521375963428798218omplex ) ) ).
% nonnegative_complex_is_real
thf(fact_1104_sm,axiom,
! [I3: nat] :
( ( ord_less_nat @ I3 @ n )
=> ( member_mat_complex @ ( smult_mat_complex @ ( nth_complex @ ( diag_mat_complex @ b ) @ I3 ) @ ( linear1949544614684794075omplex @ ( col_complex @ u @ I3 ) ) ) @ fc ) ) ).
% sm
thf(fact_1105_unitary__hermitian__eigenvalues,axiom,
! [U2: mat_complex,N: nat,K: complex] :
( ( comple6660659447773130958omplex @ U2 )
=> ( ( comple8306762464034002205omplex @ U2 )
=> ( ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_complex @ K @ ( projec527831343749723810omplex @ U2 ) )
=> ( member_complex @ K @ ( insert_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( insert_complex @ one_one_complex @ bot_bot_set_complex ) ) ) ) ) ) ) ) ).
% unitary_hermitian_eigenvalues
thf(fact_1106_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= zero_zero_complex ) ).
% add_neg_numeral_special(7)
thf(fact_1107_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
= zero_zero_complex ) ).
% add_neg_numeral_special(8)
thf(fact_1108_one__neq__neg__one,axiom,
( one_one_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% one_neq_neg_one
% Helper facts (3)
thf(help_If_3_1_If_001t__Matrix__Ovec_It__Complex__Ocomplex_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Matrix__Ovec_It__Complex__Ocomplex_J_T,axiom,
! [X: vec_complex,Y2: vec_complex] :
( ( if_vec_complex @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Matrix__Ovec_It__Complex__Ocomplex_J_T,axiom,
! [X: vec_complex,Y2: vec_complex] :
( ( if_vec_complex @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_mat_complex @ u @ fc ).
%------------------------------------------------------------------------------