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Theorem submafval 18365
Description: First substitution for a submatrix. (Contributed by AV, 28-Dec-2018.)
Hypotheses
Ref Expression
submafval.a  |-  A  =  ( N Mat  R )
submafval.q  |-  Q  =  ( N subMat  R )
submafval.b  |-  B  =  ( Base `  A
)
Assertion
Ref Expression
submafval  |-  Q  =  ( m  e.  B  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  ( N  \  { k } ) ,  j  e.  ( N  \  { l } ) 
|->  ( i m j ) ) ) )
Distinct variable groups:    B, m    i, N, j, k, l, m    R, i, j, k, l, m
Allowed substitution hints:    A( i, j, k, m, l)    B( i, j, k, l)    Q( i, j, k, m, l)

Proof of Theorem submafval
Dummy variables  n  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 submafval.q . 2  |-  Q  =  ( N subMat  R )
2 oveq12 6095 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  ( N Mat  R
) )
3 submafval.a . . . . . . . 8  |-  A  =  ( N Mat  R )
42, 3syl6eqr 2488 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  A )
54fveq2d 5690 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  ( Base `  A
) )
6 submafval.b . . . . . 6  |-  B  =  ( Base `  A
)
75, 6syl6eqr 2488 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  B )
8 simpl 457 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  n  =  N )
9 difeq1 3462 . . . . . . . 8  |-  ( n  =  N  ->  (
n  \  { k } )  =  ( N  \  { k } ) )
109adantr 465 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n  \  {
k } )  =  ( N  \  {
k } ) )
11 difeq1 3462 . . . . . . . 8  |-  ( n  =  N  ->  (
n  \  { l } )  =  ( N  \  { l } ) )
1211adantr 465 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n  \  {
l } )  =  ( N  \  {
l } ) )
13 eqidd 2439 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( i m j )  =  ( i m j ) )
1410, 12, 13mpt2eq123dv 6143 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( i  e.  ( n  \  { k } ) ,  j  e.  ( n  \  { l } ) 
|->  ( i m j ) )  =  ( i  e.  ( N 
\  { k } ) ,  j  e.  ( N  \  {
l } )  |->  ( i m j ) ) )
158, 8, 14mpt2eq123dv 6143 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  ->  ( k  e.  n ,  l  e.  n  |->  ( i  e.  ( n  \  { k } ) ,  j  e.  ( n  \  { l } ) 
|->  ( i m j ) ) )  =  ( k  e.  N ,  l  e.  N  |->  ( i  e.  ( N  \  { k } ) ,  j  e.  ( N  \  { l } ) 
|->  ( i m j ) ) ) )
167, 15mpteq12dv 4365 . . . 4  |-  ( ( n  =  N  /\  r  =  R )  ->  ( m  e.  (
Base `  ( n Mat  r ) )  |->  ( k  e.  n ,  l  e.  n  |->  ( i  e.  ( n 
\  { k } ) ,  j  e.  ( n  \  {
l } )  |->  ( i m j ) ) ) )  =  ( m  e.  B  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  ( N  \  { k } ) ,  j  e.  ( N  \  { l } ) 
|->  ( i m j ) ) ) ) )
17 df-subma 18363 . . . 4  |- subMat  =  ( n  e.  _V , 
r  e.  _V  |->  ( m  e.  ( Base `  ( n Mat  r ) )  |->  ( k  e.  n ,  l  e.  n  |->  ( i  e.  ( n  \  {
k } ) ,  j  e.  ( n 
\  { l } )  |->  ( i m j ) ) ) ) )
18 fvex 5696 . . . . . 6  |-  ( Base `  A )  e.  _V
196, 18eqeltri 2508 . . . . 5  |-  B  e. 
_V
2019mptex 5943 . . . 4  |-  ( m  e.  B  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  ( N  \  { k } ) ,  j  e.  ( N  \  { l } )  |->  ( i m j ) ) ) )  e.  _V
2116, 17, 20ovmpt2a 6216 . . 3  |-  ( ( N  e.  _V  /\  R  e.  _V )  ->  ( N subMat  R )  =  ( m  e.  B  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  ( N  \  {
k } ) ,  j  e.  ( N 
\  { l } )  |->  ( i m j ) ) ) ) )
2217reldmmpt2 6196 . . . . . 6  |-  Rel  dom subMat
2322ovprc 6113 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N subMat  R )  =  (/) )
24 mpt0 5533 . . . . 5  |-  ( m  e.  (/)  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  ( N  \  {
k } ) ,  j  e.  ( N 
\  { l } )  |->  ( i m j ) ) ) )  =  (/)
2523, 24syl6eqr 2488 . . . 4  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N subMat  R )  =  ( m  e.  (/)  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  ( N  \  {
k } ) ,  j  e.  ( N 
\  { l } )  |->  ( i m j ) ) ) ) )
263fveq2i 5689 . . . . . . 7  |-  ( Base `  A )  =  (
Base `  ( N Mat  R ) )
276, 26eqtri 2458 . . . . . 6  |-  B  =  ( Base `  ( N Mat  R ) )
28 matbas0pc 18261 . . . . . 6  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )
2927, 28syl5eq 2482 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
3029mpteq1d 4368 . . . 4  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( m  e.  B  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  ( N  \  { k } ) ,  j  e.  ( N  \  { l } ) 
|->  ( i m j ) ) ) )  =  ( m  e.  (/)  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  ( N  \  {
k } ) ,  j  e.  ( N 
\  { l } )  |->  ( i m j ) ) ) ) )
3125, 30eqtr4d 2473 . . 3  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N subMat  R )  =  ( m  e.  B  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  ( N  \  {
k } ) ,  j  e.  ( N 
\  { l } )  |->  ( i m j ) ) ) ) )
3221, 31pm2.61i 164 . 2  |-  ( N subMat  R )  =  ( m  e.  B  |->  ( k  e.  N , 
l  e.  N  |->  ( i  e.  ( N 
\  { k } ) ,  j  e.  ( N  \  {
l } )  |->  ( i m j ) ) ) )
331, 32eqtri 2458 1  |-  Q  =  ( m  e.  B  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  ( N  \  { k } ) ,  j  e.  ( N  \  { l } ) 
|->  ( i m j ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2967    \ cdif 3320   (/)c0 3632   {csn 3872    e. cmpt 4345   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   Basecbs 14166   Mat cmat 18255   subMat csubma 18362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-slot 14170  df-base 14171  df-mat 18257  df-subma 18363
This theorem is referenced by:  submaval0  18366
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