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Theorem submafval 18954
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 6290 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  ( N Mat  R
) )
3 submafval.a . . . . . . . 8  |-  A  =  ( N Mat  R )
42, 3syl6eqr 2502 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  A )
54fveq2d 5860 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  ( Base `  A
) )
6 submafval.b . . . . . 6  |-  B  =  ( Base `  A
)
75, 6syl6eqr 2502 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  B )
8 simpl 457 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  n  =  N )
9 difeq1 3600 . . . . . . . 8  |-  ( n  =  N  ->  (
n  \  { k } )  =  ( N  \  { k } ) )
109adantr 465 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n  \  {
k } )  =  ( N  \  {
k } ) )
11 difeq1 3600 . . . . . . . 8  |-  ( n  =  N  ->  (
n  \  { l } )  =  ( N  \  { l } ) )
1211adantr 465 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n  \  {
l } )  =  ( N  \  {
l } ) )
13 eqidd 2444 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( i m j )  =  ( i m j ) )
1410, 12, 13mpt2eq123dv 6344 . . . . . 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 6344 . . . . 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 4515 . . . 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 18952 . . . 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 5866 . . . . . 6  |-  ( Base `  A )  e.  _V
196, 18eqeltri 2527 . . . . 5  |-  B  e. 
_V
2019mptex 6128 . . . 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 6418 . . 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 ) ) ) ) )
2217mpt2ndm0 6501 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N subMat  R )  =  (/) )
23 mpt0 5698 . . . . 5  |-  ( m  e.  (/)  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  ( N  \  {
k } ) ,  j  e.  ( N 
\  { l } )  |->  ( i m j ) ) ) )  =  (/)
2422, 23syl6eqr 2502 . . . 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 ) ) ) ) )
253fveq2i 5859 . . . . . . 7  |-  ( Base `  A )  =  (
Base `  ( N Mat  R ) )
266, 25eqtri 2472 . . . . . 6  |-  B  =  ( Base `  ( N Mat  R ) )
27 matbas0pc 18784 . . . . . 6  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )
2826, 27syl5eq 2496 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
2928mpteq1d 4518 . . . 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 ) ) ) ) )
3024, 29eqtr4d 2487 . . 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 ) ) ) ) )
3121, 30pm2.61i 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 ) ) ) )
321, 31eqtri 2472 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 1383    e. wcel 1804   _Vcvv 3095    \ cdif 3458   (/)c0 3770   {csn 4014    |-> cmpt 4495   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   Basecbs 14509   Mat cmat 18782   subMat csubma 18951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-slot 14513  df-base 14514  df-mat 18783  df-subma 18952
This theorem is referenced by:  submaval0  18955
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