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Theorem marrepfval 19516
Description: First 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
marrepfval  |-  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 ) ) ) ) )
Distinct variable groups:    B, m, s    i, N, j, k, l, m, s    R, i, j, k, l, m, s
Allowed substitution hints:    A( i, j, k, m, s, l)    B( i, j, k, l)    Q( i, j, k, m, s, l)    .0. ( i,
j, k, m, s, l)

Proof of Theorem marrepfval
Dummy variables  n  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 marrepfval.q . 2  |-  Q  =  ( N matRRep  R )
2 marrepfval.b . . . . . 6  |-  B  =  ( Base `  A
)
3 fvex 5891 . . . . . 6  |-  ( Base `  A )  e.  _V
42, 3eqeltri 2513 . . . . 5  |-  B  e. 
_V
5 fvex 5891 . . . . . 6  |-  ( Base `  R )  e.  _V
65a1i 11 . . . . 5  |-  ( ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  R
)  e.  _V )
7 mpt2exga 6883 . . . . 5  |-  ( ( B  e.  _V  /\  ( Base `  R )  e.  _V )  ->  (
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 )
84, 6, 7sylancr 667 . . . 4  |-  ( ( N  e.  _V  /\  R  e.  _V )  ->  ( 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 oveq12 6314 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  ( N Mat  R
) )
109fveq2d 5885 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  ( Base `  ( N Mat  R ) ) )
11 marrepfval.a . . . . . . . . 9  |-  A  =  ( N Mat  R )
1211fveq2i 5884 . . . . . . . 8  |-  ( Base `  A )  =  (
Base `  ( N Mat  R ) )
132, 12eqtri 2458 . . . . . . 7  |-  B  =  ( Base `  ( N Mat  R ) )
1410, 13syl6eqr 2488 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  B )
15 fveq2 5881 . . . . . . 7  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
1615adantl 467 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  r
)  =  ( Base `  R ) )
17 simpl 458 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  n  =  N )
18 fveq2 5881 . . . . . . . . . . . 12  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
19 marrepfval.z . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  R )
2018, 19syl6eqr 2488 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
2120ifeq2d 3934 . . . . . . . . . 10  |-  ( r  =  R  ->  if ( j  =  l ,  s ,  ( 0g `  r ) )  =  if ( j  =  l ,  s ,  .0.  )
)
2221ifeq1d 3933 . . . . . . . . 9  |-  ( r  =  R  ->  if ( i  =  k ,  if ( j  =  l ,  s ,  ( 0g `  r ) ) ,  ( i m j ) )  =  if ( i  =  k ,  if ( j  =  l ,  s ,  .0.  ) ,  ( i m j ) ) )
2322adantl 467 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  if ( i  =  k ,  if ( j  =  l ,  s ,  ( 0g
`  r ) ) ,  ( i m j ) )  =  if ( i  =  k ,  if ( j  =  l ,  s ,  .0.  ) ,  ( i m j ) ) )
2417, 17, 23mpt2eq123dv 6367 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( i  e.  n ,  j  e.  n  |->  if ( i  =  k ,  if ( j  =  l ,  s ,  ( 0g
`  r ) ) ,  ( i m j ) ) )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  s ,  .0.  ) ,  ( i
m j ) ) ) )
2517, 17, 24mpt2eq123dv 6367 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( k  e.  n ,  l  e.  n  |->  ( i  e.  n ,  j  e.  n  |->  if ( i  =  k ,  if ( j  =  l ,  s ,  ( 0g
`  r ) ) ,  ( 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 ) ) ) ) )
2614, 16, 25mpt2eq123dv 6367 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  ->  ( m  e.  (
Base `  ( n Mat  r ) ) ,  s  e.  ( Base `  r )  |->  ( k  e.  n ,  l  e.  n  |->  ( i  e.  n ,  j  e.  n  |->  if ( i  =  k ,  if ( j  =  l ,  s ,  ( 0g `  r
) ) ,  ( i m j ) ) ) ) )  =  ( 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 ) ) ) ) ) )
27 df-marrep 19514 . . . . 5  |- matRRep  =  ( n  e.  _V , 
r  e.  _V  |->  ( m  e.  ( Base `  ( n Mat  r ) ) ,  s  e.  ( Base `  r
)  |->  ( k  e.  n ,  l  e.  n  |->  ( i  e.  n ,  j  e.  n  |->  if ( i  =  k ,  if ( j  =  l ,  s ,  ( 0g `  r ) ) ,  ( i m j ) ) ) ) ) )
2826, 27ovmpt2ga 6440 . . . 4  |-  ( ( N  e.  _V  /\  R  e.  _V  /\  (
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 )  -> 
( N matRRep  R )  =  ( 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 ) ) ) ) ) )
298, 28mpd3an3 1361 . . 3  |-  ( ( N  e.  _V  /\  R  e.  _V )  ->  ( N matRRep  R )  =  ( 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 ) ) ) ) ) )
3027mpt2ndm0 6524 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N matRRep  R )  =  (/) )
31 mpt20 6375 . . . . 5  |-  ( m  e.  (/) ,  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 ) ) ) ) )  =  (/)
3230, 31syl6eqr 2488 . . . 4  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N matRRep  R )  =  ( m  e.  (/) ,  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 ) ) ) ) ) )
33 matbas0pc 19365 . . . . . 6  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )
3413, 33syl5eq 2482 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
35 eqidd 2430 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  R
)  =  ( Base `  R ) )
36 mpt2eq12 6365 . . . . 5  |-  ( ( B  =  (/)  /\  ( Base `  R )  =  ( Base `  R
) )  ->  (
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 ) ) ) ) )  =  ( m  e.  (/) ,  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 ) ) ) ) ) )
3734, 35, 36syl2anc 665 . . . 4  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( 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 ) ) ) ) )  =  ( m  e.  (/) ,  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 ) ) ) ) ) )
3832, 37eqtr4d 2473 . . 3  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N matRRep  R )  =  ( 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 ) ) ) ) ) )
3929, 38pm2.61i 167 . 2  |-  ( N matRRep  R )  =  ( 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 ) ) ) ) )
401, 39eqtri 2458 1  |-  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 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087   (/)c0 3767   ifcif 3915   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   Basecbs 15084   0gc0g 15297   Mat cmat 19363   matRRep cmarrep 19512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-slot 15088  df-base 15089  df-mat 19364  df-marrep 19514
This theorem is referenced by:  marrepval0  19517
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