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Theorem marrepeval 19232
Description: An entry of a matrix with a replaced row. (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
marrepeval  |-  ( ( ( M  e.  B  /\  S  e.  ( Base `  R ) )  /\  ( K  e.  N  /\  L  e.  N )  /\  (
I  e.  N  /\  J  e.  N )
)  ->  ( I
( K ( M Q S ) L ) J )  =  if ( I  =  K ,  if ( J  =  L ,  S ,  .0.  ) ,  ( I M J ) ) )

Proof of Theorem marrepeval
Dummy variables  i 
j 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, 4marrepval 19231 . . 3  |-  ( ( ( 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 ) ) ) )
653adant3 1014 . 2  |-  ( ( ( M  e.  B  /\  S  e.  ( Base `  R ) )  /\  ( K  e.  N  /\  L  e.  N )  /\  (
I  e.  N  /\  J  e.  N )
)  ->  ( K
( M Q S ) L )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  S ,  .0.  ) ,  ( i M j ) ) ) )
7 simp3l 1022 . . 3  |-  ( ( ( M  e.  B  /\  S  e.  ( Base `  R ) )  /\  ( K  e.  N  /\  L  e.  N )  /\  (
I  e.  N  /\  J  e.  N )
)  ->  I  e.  N )
8 simpl3r 1050 . . 3  |-  ( ( ( ( M  e.  B  /\  S  e.  ( Base `  R
) )  /\  ( K  e.  N  /\  L  e.  N )  /\  ( I  e.  N  /\  J  e.  N
) )  /\  i  =  I )  ->  J  e.  N )
9 fvex 5858 . . . . . . . . 9  |-  ( 0g
`  R )  e. 
_V
104, 9eqeltri 2538 . . . . . . . 8  |-  .0.  e.  _V
11 ifexg 3998 . . . . . . . 8  |-  ( ( S  e.  ( Base `  R )  /\  .0.  e.  _V )  ->  if ( j  =  L ,  S ,  .0.  )  e.  _V )
1210, 11mpan2 669 . . . . . . 7  |-  ( S  e.  ( Base `  R
)  ->  if (
j  =  L ,  S ,  .0.  )  e.  _V )
13 ovex 6298 . . . . . . . 8  |-  ( i M j )  e. 
_V
1413a1i 11 . . . . . . 7  |-  ( S  e.  ( Base `  R
)  ->  ( i M j )  e. 
_V )
1512, 14ifcld 3972 . . . . . 6  |-  ( S  e.  ( Base `  R
)  ->  if (
i  =  K ,  if ( j  =  L ,  S ,  .0.  ) ,  ( i M j ) )  e.  _V )
1615adantl 464 . . . . 5  |-  ( ( M  e.  B  /\  S  e.  ( Base `  R ) )  ->  if ( i  =  K ,  if ( j  =  L ,  S ,  .0.  ) ,  ( i M j ) )  e.  _V )
17163ad2ant1 1015 . . . 4  |-  ( ( ( 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 )
1817adantr 463 . . 3  |-  ( ( ( ( M  e.  B  /\  S  e.  ( Base `  R
) )  /\  ( K  e.  N  /\  L  e.  N )  /\  ( I  e.  N  /\  J  e.  N
) )  /\  (
i  =  I  /\  j  =  J )
)  ->  if (
i  =  K ,  if ( j  =  L ,  S ,  .0.  ) ,  ( i M j ) )  e.  _V )
19 eqeq1 2458 . . . . . 6  |-  ( i  =  I  ->  (
i  =  K  <->  I  =  K ) )
2019adantr 463 . . . . 5  |-  ( ( i  =  I  /\  j  =  J )  ->  ( i  =  K  <-> 
I  =  K ) )
21 eqeq1 2458 . . . . . . 7  |-  ( j  =  J  ->  (
j  =  L  <->  J  =  L ) )
2221ifbid 3951 . . . . . 6  |-  ( j  =  J  ->  if ( j  =  L ,  S ,  .0.  )  =  if ( J  =  L ,  S ,  .0.  )
)
2322adantl 464 . . . . 5  |-  ( ( i  =  I  /\  j  =  J )  ->  if ( j  =  L ,  S ,  .0.  )  =  if ( J  =  L ,  S ,  .0.  )
)
24 oveq12 6279 . . . . 5  |-  ( ( i  =  I  /\  j  =  J )  ->  ( i M j )  =  ( I M J ) )
2520, 23, 24ifbieq12d 3956 . . . 4  |-  ( ( i  =  I  /\  j  =  J )  ->  if ( i  =  K ,  if ( j  =  L ,  S ,  .0.  ) ,  ( i M j ) )  =  if ( I  =  K ,  if ( J  =  L ,  S ,  .0.  ) ,  ( I M J ) ) )
2625adantl 464 . . 3  |-  ( ( ( ( M  e.  B  /\  S  e.  ( Base `  R
) )  /\  ( K  e.  N  /\  L  e.  N )  /\  ( I  e.  N  /\  J  e.  N
) )  /\  (
i  =  I  /\  j  =  J )
)  ->  if (
i  =  K ,  if ( j  =  L ,  S ,  .0.  ) ,  ( i M j ) )  =  if ( I  =  K ,  if ( J  =  L ,  S ,  .0.  ) ,  ( I M J ) ) )
277, 8, 18, 26ovmpt2dv2 6409 . 2  |-  ( ( ( M  e.  B  /\  S  e.  ( Base `  R ) )  /\  ( K  e.  N  /\  L  e.  N )  /\  (
I  e.  N  /\  J  e.  N )
)  ->  ( ( K ( M Q S ) L )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  S ,  .0.  ) ,  ( i M j ) ) )  ->  ( I
( K ( M Q S ) L ) J )  =  if ( I  =  K ,  if ( J  =  L ,  S ,  .0.  ) ,  ( I M J ) ) ) )
286, 27mpd 15 1  |-  ( ( ( M  e.  B  /\  S  e.  ( Base `  R ) )  /\  ( K  e.  N  /\  L  e.  N )  /\  (
I  e.  N  /\  J  e.  N )
)  ->  ( I
( K ( M Q S ) L ) J )  =  if ( I  =  K ,  if ( J  =  L ,  S ,  .0.  ) ,  ( I M J ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106   ifcif 3929   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   Basecbs 14716   0gc0g 14929   Mat cmat 19076   matRRep cmarrep 19225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-slot 14720  df-base 14721  df-mat 19077  df-marrep 19227
This theorem is referenced by: (None)
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