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Theorem minmar1val 19442
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 19441 . . 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 1018 . 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 998 . . 3  |-  ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N )  ->  K  e.  N )
9 simpl3 1002 . . 3  |-  ( ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N
)  /\  k  =  K )  ->  L  e.  N )
101, 2matrcl 19206 . . . . . . . 8  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
1110simpld 457 . . . . . . 7  |-  ( M  e.  B  ->  N  e.  Fin )
1211, 11jca 530 . . . . . 6  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  N  e.  Fin ) )
13123ad2ant1 1018 . . . . 5  |-  ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N )  ->  ( N  e.  Fin  /\  N  e.  Fin )
)
1413adantr 463 . . . 4  |-  ( ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N
)  /\  ( k  =  K  /\  l  =  L ) )  -> 
( N  e.  Fin  /\  N  e.  Fin )
)
15 mpt2exga 6860 . . . 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 17 . . 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 2417 . . . . . . 7  |-  ( k  =  K  ->  (
i  =  k  <->  i  =  K ) )
1817adantr 463 . . . . . 6  |-  ( ( k  =  K  /\  l  =  L )  ->  ( i  =  k  <-> 
i  =  K ) )
19 eqeq2 2417 . . . . . . . 8  |-  ( l  =  L  ->  (
j  =  l  <->  j  =  L ) )
2019ifbid 3907 . . . . . . 7  |-  ( l  =  L  ->  if ( j  =  l ,  .1.  ,  .0.  )  =  if (
j  =  L ,  .1.  ,  .0.  ) )
2120adantl 464 . . . . . 6  |-  ( ( k  =  K  /\  l  =  L )  ->  if ( j  =  l ,  .1.  ,  .0.  )  =  if ( j  =  L ,  .1.  ,  .0.  ) )
2218, 21ifbieq1d 3908 . . . . 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 6344 . . . 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 464 . . 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 6417 . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   _Vcvv 3059   ifcif 3885   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280   Fincfn 7554   Basecbs 14841   0gc0g 15054   1rcur 17473   Mat cmat 19201   minMatR1 cminmar1 19427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-slot 14845  df-base 14846  df-mat 19202  df-minmar1 19429
This theorem is referenced by:  minmar1eval  19443  maducoevalmin1  19446  smadiadet  19464  smadiadetglem2  19466
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