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Theorem madjusmdetlem4 28730
Description: Lemma for madjusmdet 28731. (Contributed by Thierry Arnoux, 22-Aug-2020.)
Hypotheses
Ref Expression
madjusmdet.b  |-  B  =  ( Base `  A
)
madjusmdet.a  |-  A  =  ( ( 1 ... N ) Mat  R )
madjusmdet.d  |-  D  =  ( ( 1 ... N ) maDet  R )
madjusmdet.k  |-  K  =  ( ( 1 ... N ) maAdju  R )
madjusmdet.t  |-  .x.  =  ( .r `  R )
madjusmdet.z  |-  Z  =  ( ZRHom `  R
)
madjusmdet.e  |-  E  =  ( ( 1 ... ( N  -  1 ) ) maDet  R )
madjusmdet.n  |-  ( ph  ->  N  e.  NN )
madjusmdet.r  |-  ( ph  ->  R  e.  CRing )
madjusmdet.i  |-  ( ph  ->  I  e.  ( 1 ... N ) )
madjusmdet.j  |-  ( ph  ->  J  e.  ( 1 ... N ) )
madjusmdet.m  |-  ( ph  ->  M  e.  B )
madjusmdetlem2.p  |-  P  =  ( i  e.  ( 1 ... N ) 
|->  if ( i  =  1 ,  I ,  if ( i  <_  I ,  ( i  -  1 ) ,  i ) ) )
madjusmdetlem2.s  |-  S  =  ( i  e.  ( 1 ... N ) 
|->  if ( i  =  1 ,  N ,  if ( i  <_  N ,  ( i  - 
1 ) ,  i ) ) )
madjusmdetlem4.q  |-  Q  =  ( j  e.  ( 1 ... N ) 
|->  if ( j  =  1 ,  J ,  if ( j  <_  J ,  ( j  - 
1 ) ,  j ) ) )
madjusmdetlem4.t  |-  T  =  ( j  e.  ( 1 ... N ) 
|->  if ( j  =  1 ,  N ,  if ( j  <_  N ,  ( j  - 
1 ) ,  j ) ) )
Assertion
Ref Expression
madjusmdetlem4  |-  ( ph  ->  ( J ( K `
 M ) I )  =  ( ( Z `  ( -u
1 ^ ( I  +  J ) ) )  .x.  ( E `
 ( I (subMat1 `  M ) J ) ) ) )
Distinct variable groups:    B, i,
j    i, I, j    i, J, j    i, M, j   
i, N, j    P, i, j    Q, i, j    R, i, j    ph, i,
j    S, i, j    T, i, j
Allowed substitution hints:    A( i, j)    D( i, j)    .x. ( i, j)    E( i, j)    K( i, j)    Z( i, j)

Proof of Theorem madjusmdetlem4
Dummy variables  k 
l are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 madjusmdet.b . . 3  |-  B  =  ( Base `  A
)
2 madjusmdet.a . . 3  |-  A  =  ( ( 1 ... N ) Mat  R )
3 madjusmdet.d . . 3  |-  D  =  ( ( 1 ... N ) maDet  R )
4 madjusmdet.k . . 3  |-  K  =  ( ( 1 ... N ) maAdju  R )
5 madjusmdet.t . . 3  |-  .x.  =  ( .r `  R )
6 madjusmdet.z . . 3  |-  Z  =  ( ZRHom `  R
)
7 madjusmdet.e . . 3  |-  E  =  ( ( 1 ... ( N  -  1 ) ) maDet  R )
8 madjusmdet.n . . 3  |-  ( ph  ->  N  e.  NN )
9 madjusmdet.r . . 3  |-  ( ph  ->  R  e.  CRing )
10 madjusmdet.i . . 3  |-  ( ph  ->  I  e.  ( 1 ... N ) )
11 madjusmdet.j . . 3  |-  ( ph  ->  J  e.  ( 1 ... N ) )
12 madjusmdet.m . . 3  |-  ( ph  ->  M  e.  B )
13 eqid 2471 . . 3  |-  ( Base `  ( SymGrp `  ( 1 ... N ) ) )  =  ( Base `  ( SymGrp `
 ( 1 ... N ) ) )
14 eqid 2471 . . 3  |-  (pmSgn `  ( 1 ... N
) )  =  (pmSgn `  ( 1 ... N
) )
15 eqid 2471 . . 3  |-  ( I ( ( ( 1 ... N ) minMatR1  R
) `  M ) J )  =  ( I ( ( ( 1 ... N ) minMatR1  R ) `  M
) J )
16 fveq2 5879 . . . . 5  |-  ( k  =  i  ->  (
( P  o.  `' S ) `  k
)  =  ( ( P  o.  `' S
) `  i )
)
1716oveq1d 6323 . . . 4  |-  ( k  =  i  ->  (
( ( P  o.  `' S ) `  k
) ( I ( ( ( 1 ... N ) minMatR1  R ) `
 M ) J ) ( ( Q  o.  `' T ) `
 l ) )  =  ( ( ( P  o.  `' S
) `  i )
( I ( ( ( 1 ... N
) minMatR1  R ) `  M
) J ) ( ( Q  o.  `' T ) `  l
) ) )
18 fveq2 5879 . . . . 5  |-  ( l  =  j  ->  (
( Q  o.  `' T ) `  l
)  =  ( ( Q  o.  `' T
) `  j )
)
1918oveq2d 6324 . . . 4  |-  ( l  =  j  ->  (
( ( P  o.  `' S ) `  i
) ( I ( ( ( 1 ... N ) minMatR1  R ) `
 M ) J ) ( ( Q  o.  `' T ) `
 l ) )  =  ( ( ( P  o.  `' S
) `  i )
( I ( ( ( 1 ... N
) minMatR1  R ) `  M
) J ) ( ( Q  o.  `' T ) `  j
) ) )
2017, 19cbvmpt2v 6390 . . 3  |-  ( k  e.  ( 1 ... N ) ,  l  e.  ( 1 ... N )  |->  ( ( ( P  o.  `' S ) `  k
) ( I ( ( ( 1 ... N ) minMatR1  R ) `
 M ) J ) ( ( Q  o.  `' T ) `
 l ) ) )  =  ( i  e.  ( 1 ... N ) ,  j  e.  ( 1 ... N )  |->  ( ( ( P  o.  `' S ) `  i
) ( I ( ( ( 1 ... N ) minMatR1  R ) `
 M ) J ) ( ( Q  o.  `' T ) `
 j ) ) )
21 eqid 2471 . . . . . 6  |-  ( 1 ... N )  =  ( 1 ... N
)
22 madjusmdetlem2.p . . . . . 6  |-  P  =  ( i  e.  ( 1 ... N ) 
|->  if ( i  =  1 ,  I ,  if ( i  <_  I ,  ( i  -  1 ) ,  i ) ) )
23 eqid 2471 . . . . . 6  |-  ( SymGrp `  ( 1 ... N
) )  =  (
SymGrp `  ( 1 ... N ) )
2421, 22, 23, 13fzto1st 28690 . . . . 5  |-  ( I  e.  ( 1 ... N )  ->  P  e.  ( Base `  ( SymGrp `
 ( 1 ... N ) ) ) )
2510, 24syl 17 . . . 4  |-  ( ph  ->  P  e.  ( Base `  ( SymGrp `  ( 1 ... N ) ) ) )
26 nnuz 11218 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
278, 26syl6eleq 2559 . . . . . . . 8  |-  ( ph  ->  N  e.  ( ZZ>= ` 
1 ) )
28 eluzfz2 11833 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  1
)  ->  N  e.  ( 1 ... N
) )
2927, 28syl 17 . . . . . . 7  |-  ( ph  ->  N  e.  ( 1 ... N ) )
30 madjusmdetlem2.s . . . . . . . 8  |-  S  =  ( i  e.  ( 1 ... N ) 
|->  if ( i  =  1 ,  N ,  if ( i  <_  N ,  ( i  - 
1 ) ,  i ) ) )
3121, 30, 23, 13fzto1st 28690 . . . . . . 7  |-  ( N  e.  ( 1 ... N )  ->  S  e.  ( Base `  ( SymGrp `
 ( 1 ... N ) ) ) )
3229, 31syl 17 . . . . . 6  |-  ( ph  ->  S  e.  ( Base `  ( SymGrp `  ( 1 ... N ) ) ) )
33 eqid 2471 . . . . . . 7  |-  ( invg `  ( SymGrp `  ( 1 ... N
) ) )  =  ( invg `  ( SymGrp `  ( 1 ... N ) ) )
3423, 13, 33symginv 17121 . . . . . 6  |-  ( S  e.  ( Base `  ( SymGrp `
 ( 1 ... N ) ) )  ->  ( ( invg `  ( SymGrp `  ( 1 ... N
) ) ) `  S )  =  `' S )
3532, 34syl 17 . . . . 5  |-  ( ph  ->  ( ( invg `  ( SymGrp `  ( 1 ... N ) ) ) `
 S )  =  `' S )
36 fzfid 12224 . . . . . . 7  |-  ( ph  ->  ( 1 ... N
)  e.  Fin )
3723symggrp 17119 . . . . . . 7  |-  ( ( 1 ... N )  e.  Fin  ->  ( SymGrp `
 ( 1 ... N ) )  e. 
Grp )
3836, 37syl 17 . . . . . 6  |-  ( ph  ->  ( SymGrp `  ( 1 ... N ) )  e. 
Grp )
3913, 33grpinvcl 16789 . . . . . 6  |-  ( ( ( SymGrp `  ( 1 ... N ) )  e. 
Grp  /\  S  e.  ( Base `  ( SymGrp `  ( 1 ... N
) ) ) )  ->  ( ( invg `  ( SymGrp `  ( 1 ... N
) ) ) `  S )  e.  (
Base `  ( SymGrp `  ( 1 ... N
) ) ) )
4038, 32, 39syl2anc 673 . . . . 5  |-  ( ph  ->  ( ( invg `  ( SymGrp `  ( 1 ... N ) ) ) `
 S )  e.  ( Base `  ( SymGrp `
 ( 1 ... N ) ) ) )
4135, 40eqeltrrd 2550 . . . 4  |-  ( ph  ->  `' S  e.  ( Base `  ( SymGrp `  (
1 ... N ) ) ) )
42 eqid 2471 . . . . . 6  |-  ( +g  `  ( SymGrp `  ( 1 ... N ) ) )  =  ( +g  `  ( SymGrp `
 ( 1 ... N ) ) )
4323, 13, 42symgov 17109 . . . . 5  |-  ( ( P  e.  ( Base `  ( SymGrp `  ( 1 ... N ) ) )  /\  `' S  e.  ( Base `  ( SymGrp `
 ( 1 ... N ) ) ) )  ->  ( P
( +g  `  ( SymGrp `  ( 1 ... N
) ) ) `' S )  =  ( P  o.  `' S
) )
4423, 13, 42symgcl 17110 . . . . 5  |-  ( ( P  e.  ( Base `  ( SymGrp `  ( 1 ... N ) ) )  /\  `' S  e.  ( Base `  ( SymGrp `
 ( 1 ... N ) ) ) )  ->  ( P
( +g  `  ( SymGrp `  ( 1 ... N
) ) ) `' S )  e.  (
Base `  ( SymGrp `  ( 1 ... N
) ) ) )
4543, 44eqeltrrd 2550 . . . 4  |-  ( ( P  e.  ( Base `  ( SymGrp `  ( 1 ... N ) ) )  /\  `' S  e.  ( Base `  ( SymGrp `
 ( 1 ... N ) ) ) )  ->  ( P  o.  `' S )  e.  (
Base `  ( SymGrp `  ( 1 ... N
) ) ) )
4625, 41, 45syl2anc 673 . . 3  |-  ( ph  ->  ( P  o.  `' S )  e.  (
Base `  ( SymGrp `  ( 1 ... N
) ) ) )
47 madjusmdetlem4.q . . . . . 6  |-  Q  =  ( j  e.  ( 1 ... N ) 
|->  if ( j  =  1 ,  J ,  if ( j  <_  J ,  ( j  - 
1 ) ,  j ) ) )
4821, 47, 23, 13fzto1st 28690 . . . . 5  |-  ( J  e.  ( 1 ... N )  ->  Q  e.  ( Base `  ( SymGrp `
 ( 1 ... N ) ) ) )
4911, 48syl 17 . . . 4  |-  ( ph  ->  Q  e.  ( Base `  ( SymGrp `  ( 1 ... N ) ) ) )
50 madjusmdetlem4.t . . . . . . . 8  |-  T  =  ( j  e.  ( 1 ... N ) 
|->  if ( j  =  1 ,  N ,  if ( j  <_  N ,  ( j  - 
1 ) ,  j ) ) )
5121, 50, 23, 13fzto1st 28690 . . . . . . 7  |-  ( N  e.  ( 1 ... N )  ->  T  e.  ( Base `  ( SymGrp `
 ( 1 ... N ) ) ) )
5229, 51syl 17 . . . . . 6  |-  ( ph  ->  T  e.  ( Base `  ( SymGrp `  ( 1 ... N ) ) ) )
5323, 13, 33symginv 17121 . . . . . 6  |-  ( T  e.  ( Base `  ( SymGrp `
 ( 1 ... N ) ) )  ->  ( ( invg `  ( SymGrp `  ( 1 ... N
) ) ) `  T )  =  `' T )
5452, 53syl 17 . . . . 5  |-  ( ph  ->  ( ( invg `  ( SymGrp `  ( 1 ... N ) ) ) `
 T )  =  `' T )
5513, 33grpinvcl 16789 . . . . . 6  |-  ( ( ( SymGrp `  ( 1 ... N ) )  e. 
Grp  /\  T  e.  ( Base `  ( SymGrp `  ( 1 ... N
) ) ) )  ->  ( ( invg `  ( SymGrp `  ( 1 ... N
) ) ) `  T )  e.  (
Base `  ( SymGrp `  ( 1 ... N
) ) ) )
5638, 52, 55syl2anc 673 . . . . 5  |-  ( ph  ->  ( ( invg `  ( SymGrp `  ( 1 ... N ) ) ) `
 T )  e.  ( Base `  ( SymGrp `
 ( 1 ... N ) ) ) )
5754, 56eqeltrrd 2550 . . . 4  |-  ( ph  ->  `' T  e.  ( Base `  ( SymGrp `  (
1 ... N ) ) ) )
5823, 13, 42symgov 17109 . . . . 5  |-  ( ( Q  e.  ( Base `  ( SymGrp `  ( 1 ... N ) ) )  /\  `' T  e.  ( Base `  ( SymGrp `
 ( 1 ... N ) ) ) )  ->  ( Q
( +g  `  ( SymGrp `  ( 1 ... N
) ) ) `' T )  =  ( Q  o.  `' T
) )
5923, 13, 42symgcl 17110 . . . . 5  |-  ( ( Q  e.  ( Base `  ( SymGrp `  ( 1 ... N ) ) )  /\  `' T  e.  ( Base `  ( SymGrp `
 ( 1 ... N ) ) ) )  ->  ( Q
( +g  `  ( SymGrp `  ( 1 ... N
) ) ) `' T )  e.  (
Base `  ( SymGrp `  ( 1 ... N
) ) ) )
6058, 59eqeltrrd 2550 . . . 4  |-  ( ( Q  e.  ( Base `  ( SymGrp `  ( 1 ... N ) ) )  /\  `' T  e.  ( Base `  ( SymGrp `
 ( 1 ... N ) ) ) )  ->  ( Q  o.  `' T )  e.  (
Base `  ( SymGrp `  ( 1 ... N
) ) ) )
6149, 57, 60syl2anc 673 . . 3  |-  ( ph  ->  ( Q  o.  `' T )  e.  (
Base `  ( SymGrp `  ( 1 ... N
) ) ) )
6223, 13symgbasf1o 17102 . . . . . . 7  |-  ( S  e.  ( Base `  ( SymGrp `
 ( 1 ... N ) ) )  ->  S : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
6332, 62syl 17 . . . . . 6  |-  ( ph  ->  S : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
64 f1of1 5827 . . . . . 6  |-  ( S : ( 1 ... N ) -1-1-onto-> ( 1 ... N
)  ->  S :
( 1 ... N
) -1-1-> ( 1 ... N ) )
65 df-f1 5594 . . . . . . 7  |-  ( S : ( 1 ... N ) -1-1-> ( 1 ... N )  <->  ( S : ( 1 ... N ) --> ( 1 ... N )  /\  Fun  `' S ) )
6665simprbi 471 . . . . . 6  |-  ( S : ( 1 ... N ) -1-1-> ( 1 ... N )  ->  Fun  `' S )
6763, 64, 663syl 18 . . . . 5  |-  ( ph  ->  Fun  `' S )
68 f1ocnv 5840 . . . . . . 7  |-  ( S : ( 1 ... N ) -1-1-onto-> ( 1 ... N
)  ->  `' S : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )
69 f1odm 5832 . . . . . . 7  |-  ( `' S : ( 1 ... N ) -1-1-onto-> ( 1 ... N )  ->  dom  `' S  =  (
1 ... N ) )
7063, 68, 693syl 18 . . . . . 6  |-  ( ph  ->  dom  `' S  =  ( 1 ... N
) )
7129, 70eleqtrrd 2552 . . . . 5  |-  ( ph  ->  N  e.  dom  `' S )
72 fvco 5956 . . . . 5  |-  ( ( Fun  `' S  /\  N  e.  dom  `' S
)  ->  ( ( P  o.  `' S
) `  N )  =  ( P `  ( `' S `  N ) ) )
7367, 71, 72syl2anc 673 . . . 4  |-  ( ph  ->  ( ( P  o.  `' S ) `  N
)  =  ( P `
 ( `' S `  N ) ) )
7421, 30, 23, 13fzto1stinvn 28691 . . . . . 6  |-  ( N  e.  ( 1 ... N )  ->  ( `' S `  N )  =  1 )
7529, 74syl 17 . . . . 5  |-  ( ph  ->  ( `' S `  N )  =  1 )
7675fveq2d 5883 . . . 4  |-  ( ph  ->  ( P `  ( `' S `  N ) )  =  ( P `
 1 ) )
7721, 22fzto1stfv1 28688 . . . . 5  |-  ( I  e.  ( 1 ... N )  ->  ( P `  1 )  =  I )
7810, 77syl 17 . . . 4  |-  ( ph  ->  ( P `  1
)  =  I )
7973, 76, 783eqtrd 2509 . . 3  |-  ( ph  ->  ( ( P  o.  `' S ) `  N
)  =  I )
8023, 13symgbasf1o 17102 . . . . . . 7  |-  ( T  e.  ( Base `  ( SymGrp `
 ( 1 ... N ) ) )  ->  T : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
8152, 80syl 17 . . . . . 6  |-  ( ph  ->  T : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
82 f1of1 5827 . . . . . 6  |-  ( T : ( 1 ... N ) -1-1-onto-> ( 1 ... N
)  ->  T :
( 1 ... N
) -1-1-> ( 1 ... N ) )
83 df-f1 5594 . . . . . . 7  |-  ( T : ( 1 ... N ) -1-1-> ( 1 ... N )  <->  ( T : ( 1 ... N ) --> ( 1 ... N )  /\  Fun  `' T ) )
8483simprbi 471 . . . . . 6  |-  ( T : ( 1 ... N ) -1-1-> ( 1 ... N )  ->  Fun  `' T )
8581, 82, 843syl 18 . . . . 5  |-  ( ph  ->  Fun  `' T )
86 f1ocnv 5840 . . . . . . 7  |-  ( T : ( 1 ... N ) -1-1-onto-> ( 1 ... N
)  ->  `' T : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )
87 f1odm 5832 . . . . . . 7  |-  ( `' T : ( 1 ... N ) -1-1-onto-> ( 1 ... N )  ->  dom  `' T  =  (
1 ... N ) )
8881, 86, 873syl 18 . . . . . 6  |-  ( ph  ->  dom  `' T  =  ( 1 ... N
) )
8929, 88eleqtrrd 2552 . . . . 5  |-  ( ph  ->  N  e.  dom  `' T )
90 fvco 5956 . . . . 5  |-  ( ( Fun  `' T  /\  N  e.  dom  `' T
)  ->  ( ( Q  o.  `' T
) `  N )  =  ( Q `  ( `' T `  N ) ) )
9185, 89, 90syl2anc 673 . . . 4  |-  ( ph  ->  ( ( Q  o.  `' T ) `  N
)  =  ( Q `
 ( `' T `  N ) ) )
9221, 50, 23, 13fzto1stinvn 28691 . . . . . 6  |-  ( N  e.  ( 1 ... N )  ->  ( `' T `  N )  =  1 )
9329, 92syl 17 . . . . 5  |-  ( ph  ->  ( `' T `  N )  =  1 )
9493fveq2d 5883 . . . 4  |-  ( ph  ->  ( Q `  ( `' T `  N ) )  =  ( Q `
 1 ) )
9521, 47fzto1stfv1 28688 . . . . 5  |-  ( J  e.  ( 1 ... N )  ->  ( Q `  1 )  =  J )
9611, 95syl 17 . . . 4  |-  ( ph  ->  ( Q `  1
)  =  J )
9791, 94, 963eqtrd 2509 . . 3  |-  ( ph  ->  ( ( Q  o.  `' T ) `  N
)  =  J )
98 crngring 17869 . . . . . 6  |-  ( R  e.  CRing  ->  R  e.  Ring )
999, 98syl 17 . . . . 5  |-  ( ph  ->  R  e.  Ring )
1002, 1minmar1cl 19753 . . . . 5  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  ( I  e.  ( 1 ... N
)  /\  J  e.  ( 1 ... N
) ) )  -> 
( I ( ( ( 1 ... N
) minMatR1  R ) `  M
) J )  e.  B )
10199, 12, 10, 11, 100syl22anc 1293 . . . 4  |-  ( ph  ->  ( I ( ( ( 1 ... N
) minMatR1  R ) `  M
) J )  e.  B )
1021, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 22, 30, 47, 50, 20, 101madjusmdetlem3 28729 . . 3  |-  ( ph  ->  ( I (subMat1 `  (
I ( ( ( 1 ... N ) minMatR1  R ) `  M
) J ) ) J )  =  ( N (subMat1 `  (
k  e.  ( 1 ... N ) ,  l  e.  ( 1 ... N )  |->  ( ( ( P  o.  `' S ) `  k
) ( I ( ( ( 1 ... N ) minMatR1  R ) `
 M ) J ) ( ( Q  o.  `' T ) `
 l ) ) ) ) N ) )
1031, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 20, 46, 61, 79, 97, 102madjusmdetlem1 28727 . 2  |-  ( ph  ->  ( J ( K `
 M ) I )  =  ( ( Z `  ( ( (pmSgn `  ( 1 ... N ) ) `  ( P  o.  `' S ) )  x.  ( (pmSgn `  (
1 ... N ) ) `
 ( Q  o.  `' T ) ) ) )  .x.  ( E `
 ( I (subMat1 `  M ) J ) ) ) )
10423, 14, 13psgnco 19228 . . . . . . . 8  |-  ( ( ( 1 ... N
)  e.  Fin  /\  P  e.  ( Base `  ( SymGrp `  ( 1 ... N ) ) )  /\  `' S  e.  ( Base `  ( SymGrp `
 ( 1 ... N ) ) ) )  ->  ( (pmSgn `  ( 1 ... N
) ) `  ( P  o.  `' S
) )  =  ( ( (pmSgn `  (
1 ... N ) ) `
 P )  x.  ( (pmSgn `  (
1 ... N ) ) `
 `' S ) ) )
10536, 25, 41, 104syl3anc 1292 . . . . . . 7  |-  ( ph  ->  ( (pmSgn `  (
1 ... N ) ) `
 ( P  o.  `' S ) )  =  ( ( (pmSgn `  ( 1 ... N
) ) `  P
)  x.  ( (pmSgn `  ( 1 ... N
) ) `  `' S ) ) )
10621, 22, 23, 13, 14psgnfzto1st 28692 . . . . . . . . 9  |-  ( I  e.  ( 1 ... N )  ->  (
(pmSgn `  ( 1 ... N ) ) `  P )  =  (
-u 1 ^ (
I  +  1 ) ) )
10710, 106syl 17 . . . . . . . 8  |-  ( ph  ->  ( (pmSgn `  (
1 ... N ) ) `
 P )  =  ( -u 1 ^ ( I  +  1 ) ) )
10823, 14, 13psgninv 19227 . . . . . . . . . 10  |-  ( ( ( 1 ... N
)  e.  Fin  /\  S  e.  ( Base `  ( SymGrp `  ( 1 ... N ) ) ) )  ->  ( (pmSgn `  ( 1 ... N
) ) `  `' S )  =  ( (pmSgn `  ( 1 ... N ) ) `  S ) )
10936, 32, 108syl2anc 673 . . . . . . . . 9  |-  ( ph  ->  ( (pmSgn `  (
1 ... N ) ) `
 `' S )  =  ( (pmSgn `  ( 1 ... N
) ) `  S
) )
11021, 30, 23, 13, 14psgnfzto1st 28692 . . . . . . . . . 10  |-  ( N  e.  ( 1 ... N )  ->  (
(pmSgn `  ( 1 ... N ) ) `  S )  =  (
-u 1 ^ ( N  +  1 ) ) )
11129, 110syl 17 . . . . . . . . 9  |-  ( ph  ->  ( (pmSgn `  (
1 ... N ) ) `
 S )  =  ( -u 1 ^ ( N  +  1 ) ) )
112109, 111eqtrd 2505 . . . . . . . 8  |-  ( ph  ->  ( (pmSgn `  (
1 ... N ) ) `
 `' S )  =  ( -u 1 ^ ( N  + 
1 ) ) )
113107, 112oveq12d 6326 . . . . . . 7  |-  ( ph  ->  ( ( (pmSgn `  ( 1 ... N
) ) `  P
)  x.  ( (pmSgn `  ( 1 ... N
) ) `  `' S ) )  =  ( ( -u 1 ^ ( I  + 
1 ) )  x.  ( -u 1 ^ ( N  +  1 ) ) ) )
114105, 113eqtrd 2505 . . . . . 6  |-  ( ph  ->  ( (pmSgn `  (
1 ... N ) ) `
 ( P  o.  `' S ) )  =  ( ( -u 1 ^ ( I  + 
1 ) )  x.  ( -u 1 ^ ( N  +  1 ) ) ) )
11523, 14, 13psgnco 19228 . . . . . . . 8  |-  ( ( ( 1 ... N
)  e.  Fin  /\  Q  e.  ( Base `  ( SymGrp `  ( 1 ... N ) ) )  /\  `' T  e.  ( Base `  ( SymGrp `
 ( 1 ... N ) ) ) )  ->  ( (pmSgn `  ( 1 ... N
) ) `  ( Q  o.  `' T
) )  =  ( ( (pmSgn `  (
1 ... N ) ) `
 Q )  x.  ( (pmSgn `  (
1 ... N ) ) `
 `' T ) ) )
11636, 49, 57, 115syl3anc 1292 . . . . . . 7  |-  ( ph  ->  ( (pmSgn `  (
1 ... N ) ) `
 ( Q  o.  `' T ) )  =  ( ( (pmSgn `  ( 1 ... N
) ) `  Q
)  x.  ( (pmSgn `  ( 1 ... N
) ) `  `' T ) ) )
11721, 47, 23, 13, 14psgnfzto1st 28692 . . . . . . . . 9  |-  ( J  e.  ( 1 ... N )  ->  (
(pmSgn `  ( 1 ... N ) ) `  Q )  =  (
-u 1 ^ ( J  +  1 ) ) )
11811, 117syl 17 . . . . . . . 8  |-  ( ph  ->  ( (pmSgn `  (
1 ... N ) ) `
 Q )  =  ( -u 1 ^ ( J  +  1 ) ) )
11923, 14, 13psgninv 19227 . . . . . . . . . 10  |-  ( ( ( 1 ... N
)  e.  Fin  /\  T  e.  ( Base `  ( SymGrp `  ( 1 ... N ) ) ) )  ->  ( (pmSgn `  ( 1 ... N
) ) `  `' T )  =  ( (pmSgn `  ( 1 ... N ) ) `  T ) )
12036, 52, 119syl2anc 673 . . . . . . . . 9  |-  ( ph  ->  ( (pmSgn `  (
1 ... N ) ) `
 `' T )  =  ( (pmSgn `  ( 1 ... N
) ) `  T
) )
12121, 50, 23, 13, 14psgnfzto1st 28692 . . . . . . . . . 10  |-  ( N  e.  ( 1 ... N )  ->  (
(pmSgn `  ( 1 ... N ) ) `  T )  =  (
-u 1 ^ ( N  +  1 ) ) )
12229, 121syl 17 . . . . . . . . 9  |-  ( ph  ->  ( (pmSgn `  (
1 ... N ) ) `
 T )  =  ( -u 1 ^ ( N  +  1 ) ) )
123120, 122eqtrd 2505 . . . . . . . 8  |-  ( ph  ->  ( (pmSgn `  (
1 ... N ) ) `
 `' T )  =  ( -u 1 ^ ( N  + 
1 ) ) )
124118, 123oveq12d 6326 . . . . . . 7  |-  ( ph  ->  ( ( (pmSgn `  ( 1 ... N
) ) `  Q
)  x.  ( (pmSgn `  ( 1 ... N
) ) `  `' T ) )  =  ( ( -u 1 ^ ( J  + 
1 ) )  x.  ( -u 1 ^ ( N  +  1 ) ) ) )
125116, 124eqtrd 2505 . . . . . 6  |-  ( ph  ->  ( (pmSgn `  (
1 ... N ) ) `
 ( Q  o.  `' T ) )  =  ( ( -u 1 ^ ( J  + 
1 ) )  x.  ( -u 1 ^ ( N  +  1 ) ) ) )
126114, 125oveq12d 6326 . . . . 5  |-  ( ph  ->  ( ( (pmSgn `  ( 1 ... N
) ) `  ( P  o.  `' S
) )  x.  (
(pmSgn `  ( 1 ... N ) ) `  ( Q  o.  `' T ) ) )  =  ( ( (
-u 1 ^ (
I  +  1 ) )  x.  ( -u
1 ^ ( N  +  1 ) ) )  x.  ( (
-u 1 ^ ( J  +  1 ) )  x.  ( -u
1 ^ ( N  +  1 ) ) ) ) )
127 1cnd 9677 . . . . . . . . 9  |-  ( ph  ->  1  e.  CC )
128127negcld 9992 . . . . . . . 8  |-  ( ph  -> 
-u 1  e.  CC )
129 fz1ssnn 11856 . . . . . . . . . . 11  |-  ( 1 ... N )  C_  NN
130129, 10sseldi 3416 . . . . . . . . . 10  |-  ( ph  ->  I  e.  NN )
131130nnnn0d 10949 . . . . . . . . 9  |-  ( ph  ->  I  e.  NN0 )
132 1nn0 10909 . . . . . . . . . 10  |-  1  e.  NN0
133132a1i 11 . . . . . . . . 9  |-  ( ph  ->  1  e.  NN0 )
134131, 133nn0addcld 10953 . . . . . . . 8  |-  ( ph  ->  ( I  +  1 )  e.  NN0 )
135128, 134expcld 12454 . . . . . . 7  |-  ( ph  ->  ( -u 1 ^ ( I  +  1 ) )  e.  CC )
1368nnnn0d 10949 . . . . . . . . 9  |-  ( ph  ->  N  e.  NN0 )
137136, 133nn0addcld 10953 . . . . . . . 8  |-  ( ph  ->  ( N  +  1 )  e.  NN0 )
138128, 137expcld 12454 . . . . . . 7  |-  ( ph  ->  ( -u 1 ^ ( N  +  1 ) )  e.  CC )
139129, 11sseldi 3416 . . . . . . . . . 10  |-  ( ph  ->  J  e.  NN )
140139nnnn0d 10949 . . . . . . . . 9  |-  ( ph  ->  J  e.  NN0 )
141140, 133nn0addcld 10953 . . . . . . . 8  |-  ( ph  ->  ( J  +  1 )  e.  NN0 )
142128, 141expcld 12454 . . . . . . 7  |-  ( ph  ->  ( -u 1 ^ ( J  +  1 ) )  e.  CC )
143135, 138, 142, 138mul4d 9863 . . . . . 6  |-  ( ph  ->  ( ( ( -u
1 ^ ( I  +  1 ) )  x.  ( -u 1 ^ ( N  + 
1 ) ) )  x.  ( ( -u
1 ^ ( J  +  1 ) )  x.  ( -u 1 ^ ( N  + 
1 ) ) ) )  =  ( ( ( -u 1 ^ ( I  +  1 ) )  x.  ( -u 1 ^ ( J  +  1 ) ) )  x.  ( (
-u 1 ^ ( N  +  1 ) )  x.  ( -u
1 ^ ( N  +  1 ) ) ) ) )
144128, 141, 134expaddd 12456 . . . . . . . 8  |-  ( ph  ->  ( -u 1 ^ ( ( I  + 
1 )  +  ( J  +  1 ) ) )  =  ( ( -u 1 ^ ( I  +  1 ) )  x.  ( -u 1 ^ ( J  +  1 ) ) ) )
145130nncnd 10647 . . . . . . . . . . . 12  |-  ( ph  ->  I  e.  CC )
146139nncnd 10647 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  CC )
147145, 127, 146, 127add4d 9878 . . . . . . . . . . 11  |-  ( ph  ->  ( ( I  + 
1 )  +  ( J  +  1 ) )  =  ( ( I  +  J )  +  ( 1  +  1 ) ) )
148 1p1e2 10745 . . . . . . . . . . . 12  |-  ( 1  +  1 )  =  2
149148oveq2i 6319 . . . . . . . . . . 11  |-  ( ( I  +  J )  +  ( 1  +  1 ) )  =  ( ( I  +  J )  +  2 )
150147, 149syl6eq 2521 . . . . . . . . . 10  |-  ( ph  ->  ( ( I  + 
1 )  +  ( J  +  1 ) )  =  ( ( I  +  J )  +  2 ) )
151150oveq2d 6324 . . . . . . . . 9  |-  ( ph  ->  ( -u 1 ^ ( ( I  + 
1 )  +  ( J  +  1 ) ) )  =  (
-u 1 ^ (
( I  +  J
)  +  2 ) ) )
152 2nn0 10910 . . . . . . . . . . . 12  |-  2  e.  NN0
153152a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  2  e.  NN0 )
154131, 140nn0addcld 10953 . . . . . . . . . . 11  |-  ( ph  ->  ( I  +  J
)  e.  NN0 )
155128, 153, 154expaddd 12456 . . . . . . . . . 10  |-  ( ph  ->  ( -u 1 ^ ( ( I  +  J )  +  2 ) )  =  ( ( -u 1 ^ ( I  +  J
) )  x.  ( -u 1 ^ 2 ) ) )
156 neg1sqe1 12408 . . . . . . . . . . 11  |-  ( -u
1 ^ 2 )  =  1
157156oveq2i 6319 . . . . . . . . . 10  |-  ( (
-u 1 ^ (
I  +  J ) )  x.  ( -u
1 ^ 2 ) )  =  ( (
-u 1 ^ (
I  +  J ) )  x.  1 )
158155, 157syl6eq 2521 . . . . . . . . 9  |-  ( ph  ->  ( -u 1 ^ ( ( I  +  J )  +  2 ) )  =  ( ( -u 1 ^ ( I  +  J
) )  x.  1 ) )
159128, 154expcld 12454 . . . . . . . . . 10  |-  ( ph  ->  ( -u 1 ^ ( I  +  J
) )  e.  CC )
160159mulid1d 9678 . . . . . . . . 9  |-  ( ph  ->  ( ( -u 1 ^ ( I  +  J ) )  x.  1 )  =  (
-u 1 ^ (
I  +  J ) ) )
161151, 158, 1603eqtrd 2509 . . . . . . . 8  |-  ( ph  ->  ( -u 1 ^ ( ( I  + 
1 )  +  ( J  +  1 ) ) )  =  (
-u 1 ^ (
I  +  J ) ) )
162144, 161eqtr3d 2507 . . . . . . 7  |-  ( ph  ->  ( ( -u 1 ^ ( I  + 
1 ) )  x.  ( -u 1 ^ ( J  +  1 ) ) )  =  ( -u 1 ^ ( I  +  J
) ) )
163137nn0zd 11061 . . . . . . . 8  |-  ( ph  ->  ( N  +  1 )  e.  ZZ )
164 m1expcl2 12332 . . . . . . . 8  |-  ( ( N  +  1 )  e.  ZZ  ->  ( -u 1 ^ ( N  +  1 ) )  e.  { -u 1 ,  1 } )
165 1neg1t1neg1 28400 . . . . . . . 8  |-  ( (
-u 1 ^ ( N  +  1 ) )  e.  { -u
1 ,  1 }  ->  ( ( -u
1 ^ ( N  +  1 ) )  x.  ( -u 1 ^ ( N  + 
1 ) ) )  =  1 )
166163, 164, 1653syl 18 . . . . . . 7  |-  ( ph  ->  ( ( -u 1 ^ ( N  + 
1 ) )  x.  ( -u 1 ^ ( N  +  1 ) ) )  =  1 )
167162, 166oveq12d 6326 . . . . . 6  |-  ( ph  ->  ( ( ( -u
1 ^ ( I  +  1 ) )  x.  ( -u 1 ^ ( J  + 
1 ) ) )  x.  ( ( -u
1 ^ ( N  +  1 ) )  x.  ( -u 1 ^ ( N  + 
1 ) ) ) )  =  ( (
-u 1 ^ (
I  +  J ) )  x.  1 ) )
168143, 167, 1603eqtrd 2509 . . . . 5  |-  ( ph  ->  ( ( ( -u
1 ^ ( I  +  1 ) )  x.  ( -u 1 ^ ( N  + 
1 ) ) )  x.  ( ( -u
1 ^ ( J  +  1 ) )  x.  ( -u 1 ^ ( N  + 
1 ) ) ) )  =  ( -u
1 ^ ( I  +  J ) ) )
169126, 168eqtrd 2505 . . . 4  |-  ( ph  ->  ( ( (pmSgn `  ( 1 ... N
) ) `  ( P  o.  `' S
) )  x.  (
(pmSgn `  ( 1 ... N ) ) `  ( Q  o.  `' T ) ) )  =  ( -u 1 ^ ( I  +  J ) ) )
170169fveq2d 5883 . . 3  |-  ( ph  ->  ( Z `  (
( (pmSgn `  (
1 ... N ) ) `
 ( P  o.  `' S ) )  x.  ( (pmSgn `  (
1 ... N ) ) `
 ( Q  o.  `' T ) ) ) )  =  ( Z `
 ( -u 1 ^ ( I  +  J ) ) ) )
171170oveq1d 6323 . 2  |-  ( ph  ->  ( ( Z `  ( ( (pmSgn `  ( 1 ... N
) ) `  ( P  o.  `' S
) )  x.  (
(pmSgn `  ( 1 ... N ) ) `  ( Q  o.  `' T ) ) ) )  .x.  ( E `
 ( I (subMat1 `  M ) J ) ) )  =  ( ( Z `  ( -u 1 ^ ( I  +  J ) ) )  .x.  ( E `
 ( I (subMat1 `  M ) J ) ) ) )
172103, 171eqtrd 2505 1  |-  ( ph  ->  ( J ( K `
 M ) I )  =  ( ( Z `  ( -u
1 ^ ( I  +  J ) ) )  .x.  ( E `
 ( I (subMat1 `  M ) J ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   ifcif 3872   {cpr 3961   class class class wbr 4395    |-> cmpt 4454   `'ccnv 4838   dom cdm 4839    o. ccom 4843   Fun wfun 5583   -->wf 5585   -1-1->wf1 5586   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   Fincfn 7587   1c1 9558    + caddc 9560    x. cmul 9562    <_ cle 9694    - cmin 9880   -ucneg 9881   NNcn 10631   2c2 10681   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810   ^cexp 12310   Basecbs 15199   +g cplusg 15268   .rcmulr 15269   Grpcgrp 16747   invgcminusg 16748   SymGrpcsymg 17096  pmSgncpsgn 17208   Ringcrg 17858   CRingccrg 17859   ZRHomczrh 19148   Mat cmat 19509   maDet cmdat 19686   maAdju cmadu 19734   minMatR1 cminmar1 19735  subMat1csmat 28693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-xor 1431  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-ot 3968  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-sup 7974  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-word 12711  df-lsw 12712  df-concat 12713  df-s1 12714  df-substr 12715  df-splice 12716  df-reverse 12717  df-s2 13003  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-0g 15418  df-gsum 15419  df-prds 15424  df-pws 15426  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-grp 16751  df-minusg 16752  df-mulg 16754  df-subg 16892  df-ghm 16959  df-gim 17001  df-cntz 17049  df-oppg 17075  df-symg 17097  df-pmtr 17161  df-psgn 17210  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-cring 17861  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-dvr 17989  df-rnghom 18021  df-drng 18055  df-subrg 18084  df-sra 18473  df-rgmod 18474  df-cnfld 19048  df-zring 19117  df-zrh 19152  df-dsmm 19372  df-frlm 19387  df-mat 19510  df-marrep 19660  df-subma 19679  df-mdet 19687  df-madu 19736  df-minmar1 19737  df-smat 28694
This theorem is referenced by:  madjusmdet  28731
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