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Theorem marrepval 19042
Description: Third substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.)
Hypotheses
Ref Expression
marrepfval.a  |-  A  =  ( N Mat  R )
marrepfval.b  |-  B  =  ( Base `  A
)
marrepfval.q  |-  Q  =  ( N matRRep  R )
marrepfval.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
marrepval  |-  ( ( ( M  e.  B  /\  S  e.  ( Base `  R ) )  /\  ( K  e.  N  /\  L  e.  N ) )  -> 
( K ( M Q S ) L )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  S ,  .0.  ) ,  ( i M j ) ) ) )
Distinct variable groups:    i, N, j    R, i, j    i, M, j    S, i, j   
i, K, j    i, L, j
Allowed substitution hints:    A( i, j)    B( i, j)    Q( i, j)    .0. ( i, j)

Proof of Theorem marrepval
Dummy variables  k 
l are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 marrepfval.a . . . 4  |-  A  =  ( N Mat  R )
2 marrepfval.b . . . 4  |-  B  =  ( Base `  A
)
3 marrepfval.q . . . 4  |-  Q  =  ( N matRRep  R )
4 marrepfval.z . . . 4  |-  .0.  =  ( 0g `  R )
51, 2, 3, 4marrepval0 19041 . . 3  |-  ( ( M  e.  B  /\  S  e.  ( Base `  R ) )  -> 
( M Q S )  =  ( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  S ,  .0.  ) ,  ( i M j ) ) ) ) )
65adantr 465 . 2  |-  ( ( ( M  e.  B  /\  S  e.  ( Base `  R ) )  /\  ( K  e.  N  /\  L  e.  N ) )  -> 
( M Q S )  =  ( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  S ,  .0.  ) ,  ( i M j ) ) ) ) )
7 simprl 756 . . 3  |-  ( ( ( M  e.  B  /\  S  e.  ( Base `  R ) )  /\  ( K  e.  N  /\  L  e.  N ) )  ->  K  e.  N )
8 simplrr 762 . . 3  |-  ( ( ( ( M  e.  B  /\  S  e.  ( Base `  R
) )  /\  ( K  e.  N  /\  L  e.  N )
)  /\  k  =  K )  ->  L  e.  N )
91, 2matrcl 18892 . . . . . . 7  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
109simpld 459 . . . . . 6  |-  ( M  e.  B  ->  N  e.  Fin )
1110, 10jca 532 . . . . 5  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  N  e.  Fin ) )
1211ad3antrrr 729 . . . 4  |-  ( ( ( ( M  e.  B  /\  S  e.  ( Base `  R
) )  /\  ( K  e.  N  /\  L  e.  N )
)  /\  ( k  =  K  /\  l  =  L ) )  -> 
( N  e.  Fin  /\  N  e.  Fin )
)
13 mpt2exga 6861 . . . 4  |-  ( ( N  e.  Fin  /\  N  e.  Fin )  ->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  S ,  .0.  ) ,  ( i M j ) ) )  e.  _V )
1412, 13syl 16 . . 3  |-  ( ( ( ( M  e.  B  /\  S  e.  ( Base `  R
) )  /\  ( K  e.  N  /\  L  e.  N )
)  /\  ( k  =  K  /\  l  =  L ) )  -> 
( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  S ,  .0.  ) ,  ( i M j ) ) )  e.  _V )
15 eqeq2 2458 . . . . . . 7  |-  ( k  =  K  ->  (
i  =  k  <->  i  =  K ) )
1615adantr 465 . . . . . 6  |-  ( ( k  =  K  /\  l  =  L )  ->  ( i  =  k  <-> 
i  =  K ) )
17 eqeq2 2458 . . . . . . . 8  |-  ( l  =  L  ->  (
j  =  l  <->  j  =  L ) )
1817ifbid 3948 . . . . . . 7  |-  ( l  =  L  ->  if ( j  =  l ,  S ,  .0.  )  =  if (
j  =  L ,  S ,  .0.  )
)
1918adantl 466 . . . . . 6  |-  ( ( k  =  K  /\  l  =  L )  ->  if ( j  =  l ,  S ,  .0.  )  =  if ( j  =  L ,  S ,  .0.  ) )
2016, 19ifbieq1d 3949 . . . . 5  |-  ( ( k  =  K  /\  l  =  L )  ->  if ( i  =  k ,  if ( j  =  l ,  S ,  .0.  ) ,  ( i M j ) )  =  if ( i  =  K ,  if ( j  =  L ,  S ,  .0.  ) ,  ( i M j ) ) )
2120mpt2eq3dv 6348 . . . 4  |-  ( ( k  =  K  /\  l  =  L )  ->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  S ,  .0.  ) ,  ( i M j ) ) )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  S ,  .0.  ) ,  ( i M j ) ) ) )
2221adantl 466 . . 3  |-  ( ( ( ( M  e.  B  /\  S  e.  ( Base `  R
) )  /\  ( K  e.  N  /\  L  e.  N )
)  /\  ( k  =  K  /\  l  =  L ) )  -> 
( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  S ,  .0.  ) ,  ( i M j ) ) )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  S ,  .0.  ) ,  ( i M j ) ) ) )
237, 8, 14, 22ovmpt2dv2 6421 . 2  |-  ( ( ( M  e.  B  /\  S  e.  ( Base `  R ) )  /\  ( K  e.  N  /\  L  e.  N ) )  -> 
( ( M Q S )  =  ( k  e.  N , 
l  e.  N  |->  ( i  e.  N , 
j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  S ,  .0.  ) ,  ( i M j ) ) ) )  -> 
( K ( M Q S ) L )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  S ,  .0.  ) ,  ( i M j ) ) ) ) )
246, 23mpd 15 1  |-  ( ( ( M  e.  B  /\  S  e.  ( Base `  R ) )  /\  ( K  e.  N  /\  L  e.  N ) )  -> 
( K ( M Q S ) L )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  S ,  .0.  ) ,  ( i M j ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   _Vcvv 3095   ifcif 3926   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   Fincfn 7518   Basecbs 14614   0gc0g 14819   Mat cmat 18887   matRRep cmarrep 19036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-slot 14618  df-base 14619  df-mat 18888  df-marrep 19038
This theorem is referenced by:  marrepeval  19043  marrepcl  19044  1marepvmarrepid  19055  smadiadetglem1  19151  smadiadetglem2  19152
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