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Theorem marrepval0 18502
Description: Second 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
marrepval0  |-  ( ( 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 ) ) ) ) )
Distinct variable groups:    i, N, j, k, l    R, i, j, k, l    i, M, j, k, l    S, i, j, k, l
Allowed substitution hints:    A( i, j, k, l)    B( i, j, k, l)    Q( i, j, k, l)    .0. ( i, j, k, l)

Proof of Theorem marrepval0
Dummy variables  m  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 marrepfval.a . . . . . . 7  |-  A  =  ( N Mat  R )
2 marrepfval.b . . . . . . 7  |-  B  =  ( Base `  A
)
31, 2matrcl 18440 . . . . . 6  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
43simpld 459 . . . . 5  |-  ( M  e.  B  ->  N  e.  Fin )
54, 4jca 532 . . . 4  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  N  e.  Fin ) )
65adantr 465 . . 3  |-  ( ( M  e.  B  /\  S  e.  ( Base `  R ) )  -> 
( N  e.  Fin  /\  N  e.  Fin )
)
7 mpt2exga 6762 . . 3  |-  ( ( N  e.  Fin  /\  N  e.  Fin )  ->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  S ,  .0.  ) ,  ( i M j ) ) ) )  e.  _V )
86, 7syl 16 . 2  |-  ( ( M  e.  B  /\  S  e.  ( Base `  R ) )  -> 
( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  S ,  .0.  ) ,  ( i M j ) ) ) )  e.  _V )
9 ifeq1 3906 . . . . . . 7  |-  ( s  =  S  ->  if ( j  =  l ,  s ,  .0.  )  =  if (
j  =  l ,  S ,  .0.  )
)
109adantl 466 . . . . . 6  |-  ( ( m  =  M  /\  s  =  S )  ->  if ( j  =  l ,  s ,  .0.  )  =  if ( j  =  l ,  S ,  .0.  ) )
11 oveq 6209 . . . . . . 7  |-  ( m  =  M  ->  (
i m j )  =  ( i M j ) )
1211adantr 465 . . . . . 6  |-  ( ( m  =  M  /\  s  =  S )  ->  ( i m j )  =  ( i M j ) )
1310, 12ifeq12d 3920 . . . . 5  |-  ( ( m  =  M  /\  s  =  S )  ->  if ( i  =  k ,  if ( j  =  l ,  s ,  .0.  ) ,  ( i m j ) )  =  if ( i  =  k ,  if ( j  =  l ,  S ,  .0.  ) ,  ( i M j ) ) )
1413mpt2eq3dv 6264 . . . 4  |-  ( ( m  =  M  /\  s  =  S )  ->  ( 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 ) ) ) )
1514mpt2eq3dv 6264 . . 3  |-  ( ( m  =  M  /\  s  =  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  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  S ,  .0.  ) ,  ( i M j ) ) ) ) )
16 marrepfval.q . . . 4  |-  Q  =  ( N matRRep  R )
17 marrepfval.z . . . 4  |-  .0.  =  ( 0g `  R )
181, 2, 16, 17marrepfval 18501 . . 3  |-  Q  =  ( m  e.  B ,  s  e.  ( Base `  R )  |->  ( k  e.  N , 
l  e.  N  |->  ( i  e.  N , 
j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  s ,  .0.  ) ,  ( i m j ) ) ) ) )
1915, 18ovmpt2ga 6333 . 2  |-  ( ( M  e.  B  /\  S  e.  ( Base `  R )  /\  (
k  e.  N , 
l  e.  N  |->  ( i  e.  N , 
j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  S ,  .0.  ) ,  ( i M j ) ) ) )  e. 
_V )  ->  ( 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 ) ) ) ) )
208, 19mpd3an3 1316 1  |-  ( ( 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 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078   ifcif 3902   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   Fincfn 7423   Basecbs 14295   0gc0g 14500   Mat cmat 18408   matRRep cmarrep 18497
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-slot 14299  df-base 14300  df-mat 18410  df-marrep 18499
This theorem is referenced by:  marrepval  18503  minmar1marrep  18591
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