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Theorem marrepval 18933
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 18932 . . 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 755 . . 3  |-  ( ( ( M  e.  B  /\  S  e.  ( Base `  R ) )  /\  ( K  e.  N  /\  L  e.  N ) )  ->  K  e.  N )
8 simplrr 760 . . 3  |-  ( ( ( ( M  e.  B  /\  S  e.  ( Base `  R
) )  /\  ( K  e.  N  /\  L  e.  N )
)  /\  k  =  K )  ->  L  e.  N )
91, 2matrcl 18783 . . . . . . 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 6871 . . . 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 2482 . . . . . . 7  |-  ( k  =  K  ->  (
i  =  k  <->  i  =  K ) )
1615adantr 465 . . . . . 6  |-  ( ( k  =  K  /\  l  =  L )  ->  ( i  =  k  <-> 
i  =  K ) )
17 eqeq2 2482 . . . . . . . 8  |-  ( l  =  L  ->  (
j  =  l  <->  j  =  L ) )
1817ifbid 3967 . . . . . . 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 3968 . . . . 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 6358 . . . 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 6431 . 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 1379    e. wcel 1767   _Vcvv 3118   ifcif 3945   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   Fincfn 7528   Basecbs 14507   0gc0g 14712   Mat cmat 18778   matRRep cmarrep 18927
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-slot 14511  df-base 14512  df-mat 18779  df-marrep 18929
This theorem is referenced by:  marrepeval  18934  marrepcl  18935  1marepvmarrepid  18946  smadiadetglem1  19042  smadiadetglem2  19043
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