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Theorem minmar1val 18572
Description: Third 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
minmar1val  |-  ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N )  ->  ( K ( Q `
 M ) L )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) )
Distinct variable groups:    i, N, j    R, i, j    i, M, j    i, K, j   
i, L, j
Allowed substitution hints:    A( i, j)    B( i, j)    Q( i, j)    .1. ( i, j)    .0. ( i, j)

Proof of Theorem minmar1val
Dummy variables  k 
l are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 minmar1fval.a . . . 4  |-  A  =  ( N Mat  R )
2 minmar1fval.b . . . 4  |-  B  =  ( Base `  A
)
3 minmar1fval.q . . . 4  |-  Q  =  ( N minMatR1  R )
4 minmar1fval.o . . . 4  |-  .1.  =  ( 1r `  R )
5 minmar1fval.z . . . 4  |-  .0.  =  ( 0g `  R )
61, 2, 3, 4, 5minmar1val0 18571 . . 3  |-  ( M  e.  B  ->  ( Q `  M )  =  ( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) ) )
763ad2ant1 1009 . 2  |-  ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N )  ->  ( Q `  M
)  =  ( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) ) )
8 simp2 989 . . 3  |-  ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N )  ->  K  e.  N )
9 simpl3 993 . . 3  |-  ( ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N
)  /\  k  =  K )  ->  L  e.  N )
101, 2matrcl 18423 . . . . . . . 8  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
1110simpld 459 . . . . . . 7  |-  ( M  e.  B  ->  N  e.  Fin )
1211, 11jca 532 . . . . . 6  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  N  e.  Fin ) )
13123ad2ant1 1009 . . . . 5  |-  ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N )  ->  ( N  e.  Fin  /\  N  e.  Fin )
)
1413adantr 465 . . . 4  |-  ( ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N
)  /\  ( k  =  K  /\  l  =  L ) )  -> 
( N  e.  Fin  /\  N  e.  Fin )
)
15 mpt2exga 6751 . . . 4  |-  ( ( N  e.  Fin  /\  N  e.  Fin )  ->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i M j ) ) )  e.  _V )
1614, 15syl 16 . . 3  |-  ( ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N
)  /\  ( k  =  K  /\  l  =  L ) )  -> 
( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i M j ) ) )  e.  _V )
17 eqeq2 2466 . . . . . . 7  |-  ( k  =  K  ->  (
i  =  k  <->  i  =  K ) )
1817adantr 465 . . . . . 6  |-  ( ( k  =  K  /\  l  =  L )  ->  ( i  =  k  <-> 
i  =  K ) )
19 eqeq2 2466 . . . . . . . 8  |-  ( l  =  L  ->  (
j  =  l  <->  j  =  L ) )
2019ifbid 3911 . . . . . . 7  |-  ( l  =  L  ->  if ( j  =  l ,  .1.  ,  .0.  )  =  if (
j  =  L ,  .1.  ,  .0.  ) )
2120adantl 466 . . . . . 6  |-  ( ( k  =  K  /\  l  =  L )  ->  if ( j  =  l ,  .1.  ,  .0.  )  =  if ( j  =  L ,  .1.  ,  .0.  ) )
2218, 21ifbieq1d 3912 . . . . 5  |-  ( ( k  =  K  /\  l  =  L )  ->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i M j ) )  =  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) )
2322mpt2eq3dv 6253 . . . 4  |-  ( ( k  =  K  /\  l  =  L )  ->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i M j ) ) )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) )
2423adantl 466 . . 3  |-  ( ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N
)  /\  ( k  =  K  /\  l  =  L ) )  -> 
( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i M j ) ) )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) )
258, 9, 16, 24ovmpt2dv2 6326 . 2  |-  ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N )  ->  ( ( Q `  M )  =  ( k  e.  N , 
l  e.  N  |->  ( i  e.  N , 
j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) )  -> 
( K ( Q `
 M ) L )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) ) )
267, 25mpd 15 1  |-  ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N )  ->  ( K ( Q `
 M ) L )  =  ( 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:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3070   ifcif 3891   ` cfv 5518  (class class class)co 6192    |-> cmpt2 6194   Fincfn 7412   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  ax-un 6474
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-pw 3962  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-1st 6679  df-2nd 6680  df-slot 14282  df-base 14283  df-mat 18393  df-minmar1 18559
This theorem is referenced by:  minmar1eval  18573  maducoevalmin1  18576  smadiadet  18594  smadiadetglem2  18596
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