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Theorem submafval 18841
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 6284 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  ( N Mat  R
) )
3 submafval.a . . . . . . . 8  |-  A  =  ( N Mat  R )
42, 3syl6eqr 2519 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  A )
54fveq2d 5861 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  ( Base `  A
) )
6 submafval.b . . . . . 6  |-  B  =  ( Base `  A
)
75, 6syl6eqr 2519 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  B )
8 simpl 457 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  n  =  N )
9 difeq1 3608 . . . . . . . 8  |-  ( n  =  N  ->  (
n  \  { k } )  =  ( N  \  { k } ) )
109adantr 465 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n  \  {
k } )  =  ( N  \  {
k } ) )
11 difeq1 3608 . . . . . . . 8  |-  ( n  =  N  ->  (
n  \  { l } )  =  ( N  \  { l } ) )
1211adantr 465 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n  \  {
l } )  =  ( N  \  {
l } ) )
13 eqidd 2461 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( i m j )  =  ( i m j ) )
1410, 12, 13mpt2eq123dv 6334 . . . . . 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 6334 . . . . 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 4518 . . . 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 18839 . . . 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 5867 . . . . . 6  |-  ( Base `  A )  e.  _V
196, 18eqeltri 2544 . . . . 5  |-  B  e. 
_V
2019mptex 6122 . . . 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 6408 . . 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 6388 . . . . . 6  |-  Rel  dom subMat
2322ovprc 6302 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N subMat  R )  =  (/) )
24 mpt0 5699 . . . . 5  |-  ( m  e.  (/)  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  ( N  \  {
k } ) ,  j  e.  ( N 
\  { l } )  |->  ( i m j ) ) ) )  =  (/)
2523, 24syl6eqr 2519 . . . 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 5860 . . . . . . 7  |-  ( Base `  A )  =  (
Base `  ( N Mat  R ) )
276, 26eqtri 2489 . . . . . 6  |-  B  =  ( Base `  ( N Mat  R ) )
28 matbas0pc 18671 . . . . . 6  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )
2927, 28syl5eq 2513 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
3029mpteq1d 4521 . . . 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 2504 . . 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 2489 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 1374    e. wcel 1762   _Vcvv 3106    \ cdif 3466   (/)c0 3778   {csn 4020    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   Basecbs 14479   Mat cmat 18669   subMat csubma 18838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-slot 14483  df-base 14484  df-mat 18670  df-subma 18839
This theorem is referenced by:  submaval0  18842
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