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Theorem minmar1fval 18915
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 6291 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  ( N Mat  R
) )
3 minmar1fval.a . . . . . . . 8  |-  A  =  ( N Mat  R )
42, 3syl6eqr 2526 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  A )
54fveq2d 5868 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  ( Base `  A
) )
6 minmar1fval.b . . . . . 6  |-  B  =  ( Base `  A
)
75, 6syl6eqr 2526 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  B )
8 simpl 457 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  n  =  N )
9 fveq2 5864 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
10 minmar1fval.o . . . . . . . . . . 11  |-  .1.  =  ( 1r `  R )
119, 10syl6eqr 2526 . . . . . . . . . 10  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
12 fveq2 5864 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
13 minmar1fval.z . . . . . . . . . . 11  |-  .0.  =  ( 0g `  R )
1412, 13syl6eqr 2526 . . . . . . . . . 10  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
1511, 14ifeq12d 3959 . . . . . . . . 9  |-  ( r  =  R  ->  if ( j  =  l ,  ( 1r `  r ) ,  ( 0g `  r ) )  =  if ( j  =  l ,  .1.  ,  .0.  )
)
1615ifeq1d 3957 . . . . . . . 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 6341 . . . . . 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 6341 . . . . 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 4525 . . . 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 18904 . . . 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 5874 . . . . . 6  |-  ( Base `  A )  e.  _V
236, 22eqeltri 2551 . . . . 5  |-  B  e. 
_V
2423mptex 6129 . . . 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 6415 . . 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 6498 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N minMatR1  R )  =  (/) )
27 mpt0 5706 . . . . 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 2526 . . . 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 5867 . . . . . . 7  |-  ( Base `  A )  =  (
Base `  ( N Mat  R ) )
306, 29eqtri 2496 . . . . . 6  |-  B  =  ( Base `  ( N Mat  R ) )
31 matbas0pc 18678 . . . . . 6  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )
3230, 31syl5eq 2520 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
3332mpteq1d 4528 . . . 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 2511 . . 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 2496 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 1379    e. wcel 1767   _Vcvv 3113   (/)c0 3785   ifcif 3939    |-> cmpt 4505   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   Basecbs 14486   0gc0g 14691   1rcur 16943   Mat cmat 18676   minMatR1 cminmar1 18902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-slot 14490  df-base 14491  df-mat 18677  df-minmar1 18904
This theorem is referenced by:  minmar1val0  18916
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