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Theorem minmar1fval 18570
Description: First substitution for the definition of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)
Hypotheses
Ref Expression
minmar1fval.a  |-  A  =  ( N Mat  R )
minmar1fval.b  |-  B  =  ( Base `  A
)
minmar1fval.q  |-  Q  =  ( N minMatR1  R )
minmar1fval.o  |-  .1.  =  ( 1r `  R )
minmar1fval.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
minmar1fval  |-  Q  =  ( m  e.  B  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( 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)    .1. ( i, j, k, m, l)    .0. ( i, j, k, m, l)

Proof of Theorem minmar1fval
Dummy variables  n  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 minmar1fval.q . 2  |-  Q  =  ( N minMatR1  R )
2 oveq12 6201 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  ( N Mat  R
) )
3 minmar1fval.a . . . . . . . 8  |-  A  =  ( N Mat  R )
42, 3syl6eqr 2510 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  A )
54fveq2d 5795 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  ( Base `  A
) )
6 minmar1fval.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 fveq2 5791 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
10 minmar1fval.o . . . . . . . . . . 11  |-  .1.  =  ( 1r `  R )
119, 10syl6eqr 2510 . . . . . . . . . 10  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
12 fveq2 5791 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
13 minmar1fval.z . . . . . . . . . . 11  |-  .0.  =  ( 0g `  R )
1412, 13syl6eqr 2510 . . . . . . . . . 10  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
1511, 14ifeq12d 3909 . . . . . . . . 9  |-  ( r  =  R  ->  if ( j  =  l ,  ( 1r `  r ) ,  ( 0g `  r ) )  =  if ( j  =  l ,  .1.  ,  .0.  )
)
1615ifeq1d 3907 . . . . . . . 8  |-  ( r  =  R  ->  if ( i  =  k ,  if ( j  =  l ,  ( 1r `  r ) ,  ( 0g `  r ) ) ,  ( i m j ) )  =  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i m j ) ) )
1716adantl 466 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  if ( i  =  k ,  if ( j  =  l ,  ( 1r `  r
) ,  ( 0g
`  r ) ) ,  ( i m j ) )  =  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i m j ) ) )
188, 8, 17mpt2eq123dv 6249 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( i  e.  n ,  j  e.  n  |->  if ( i  =  k ,  if ( j  =  l ,  ( 1r `  r
) ,  ( 0g
`  r ) ) ,  ( i m j ) ) )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i
m j ) ) ) )
198, 8, 18mpt2eq123dv 6249 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  ->  ( k  e.  n ,  l  e.  n  |->  ( i  e.  n ,  j  e.  n  |->  if ( i  =  k ,  if ( j  =  l ,  ( 1r `  r
) ,  ( 0g
`  r ) ) ,  ( i m j ) ) ) )  =  ( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i m j ) ) ) ) )
207, 19mpteq12dv 4470 . . . 4  |-  ( ( n  =  N  /\  r  =  R )  ->  ( m  e.  (
Base `  ( n Mat  r ) )  |->  ( k  e.  n ,  l  e.  n  |->  ( i  e.  n ,  j  e.  n  |->  if ( i  =  k ,  if ( j  =  l ,  ( 1r `  r ) ,  ( 0g `  r ) ) ,  ( i m j ) ) ) ) )  =  ( m  e.  B  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i m j ) ) ) ) ) )
21 df-minmar1 18559 . . . 4  |- minMatR1  =  ( n  e.  _V , 
r  e.  _V  |->  ( m  e.  ( Base `  ( n Mat  r ) )  |->  ( k  e.  n ,  l  e.  n  |->  ( i  e.  n ,  j  e.  n  |->  if ( i  =  k ,  if ( j  =  l ,  ( 1r `  r ) ,  ( 0g `  r ) ) ,  ( i m j ) ) ) ) ) )
22 fvex 5801 . . . . . 6  |-  ( Base `  A )  e.  _V
236, 22eqeltri 2535 . . . . 5  |-  B  e. 
_V
2423mptex 6049 . . . 4  |-  ( m  e.  B  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i m j ) ) ) ) )  e. 
_V
2520, 21, 24ovmpt2a 6323 . . 3  |-  ( ( N  e.  _V  /\  R  e.  _V )  ->  ( N minMatR1  R )  =  ( m  e.  B  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i
m j ) ) ) ) ) )
2621mpt2ndm0 6841 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N minMatR1  R )  =  (/) )
27 mpt0 5638 . . . . 5  |-  ( m  e.  (/)  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i
m j ) ) ) ) )  =  (/)
2826, 27syl6eqr 2510 . . . 4  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N minMatR1  R )  =  ( m  e.  (/)  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i
m j ) ) ) ) ) )
293fveq2i 5794 . . . . . . 7  |-  ( Base `  A )  =  (
Base `  ( N Mat  R ) )
306, 29eqtri 2480 . . . . . 6  |-  B  =  ( Base `  ( N Mat  R ) )
31 matbas0pc 18397 . . . . . 6  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )
3230, 31syl5eq 2504 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
3332mpteq1d 4473 . . . 4  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( m  e.  B  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i m j ) ) ) ) )  =  ( m  e.  (/)  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i m j ) ) ) ) ) )
3428, 33eqtr4d 2495 . . 3  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N minMatR1  R )  =  ( m  e.  B  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i
m j ) ) ) ) ) )
3525, 34pm2.61i 164 . 2  |-  ( N minMatR1  R )  =  ( m  e.  B  |->  ( k  e.  N , 
l  e.  N  |->  ( i  e.  N , 
j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i m j ) ) ) ) )
361, 35eqtri 2480 1  |-  Q  =  ( m  e.  B  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i m j ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3070   (/)c0 3737   ifcif 3891    |-> cmpt 4450   ` cfv 5518  (class class class)co 6192    |-> cmpt2 6194   Basecbs 14278   0gc0g 14482   1rcur 16710   Mat cmat 18391   minMatR1 cminmar1 18557
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 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
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 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-slot 14282  df-base 14283  df-mat 18393  df-minmar1 18559
This theorem is referenced by:  minmar1val0  18571
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