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Theorem submafval 18510
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 6202 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  ( N Mat  R
) )
3 submafval.a . . . . . . . 8  |-  A  =  ( N Mat  R )
42, 3syl6eqr 2510 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  A )
54fveq2d 5796 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  ( Base `  A
) )
6 submafval.b . . . . . 6  |-  B  =  ( Base `  A
)
75, 6syl6eqr 2510 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  B )
8 simpl 457 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  n  =  N )
9 difeq1 3568 . . . . . . . 8  |-  ( n  =  N  ->  (
n  \  { k } )  =  ( N  \  { k } ) )
109adantr 465 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n  \  {
k } )  =  ( N  \  {
k } ) )
11 difeq1 3568 . . . . . . . 8  |-  ( n  =  N  ->  (
n  \  { l } )  =  ( N  \  { l } ) )
1211adantr 465 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n  \  {
l } )  =  ( N  \  {
l } ) )
13 eqidd 2452 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( i m j )  =  ( i m j ) )
1410, 12, 13mpt2eq123dv 6250 . . . . . 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 6250 . . . . 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 4471 . . . 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 18508 . . . 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 5802 . . . . . 6  |-  ( Base `  A )  e.  _V
196, 18eqeltri 2535 . . . . 5  |-  B  e. 
_V
2019mptex 6050 . . . 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 6324 . . 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 6304 . . . . . 6  |-  Rel  dom subMat
2322ovprc 6220 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N subMat  R )  =  (/) )
24 mpt0 5639 . . . . 5  |-  ( m  e.  (/)  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  ( N  \  {
k } ) ,  j  e.  ( N 
\  { l } )  |->  ( i m j ) ) ) )  =  (/)
2523, 24syl6eqr 2510 . . . 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 5795 . . . . . . 7  |-  ( Base `  A )  =  (
Base `  ( N Mat  R ) )
276, 26eqtri 2480 . . . . . 6  |-  B  =  ( Base `  ( N Mat  R ) )
28 matbas0pc 18404 . . . . . 6  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )
2927, 28syl5eq 2504 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
3029mpteq1d 4474 . . . 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 2495 . . 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 2480 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 1370    e. wcel 1758   _Vcvv 3071    \ cdif 3426   (/)c0 3738   {csn 3978    |-> cmpt 4451   ` cfv 5519  (class class class)co 6193    |-> cmpt2 6195   Basecbs 14285   Mat cmat 18398   subMat csubma 18507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-slot 14289  df-base 14290  df-mat 18400  df-subma 18508
This theorem is referenced by:  submaval0  18511
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