MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  marrepeval Structured version   Unicode version

Theorem marrepeval 18496
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 18495 . . 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 1008 . 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 1016 . . 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 1044 . . 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 5804 . . . . . . . . 9  |-  ( 0g
`  R )  e. 
_V
104, 9eqeltri 2536 . . . . . . . 8  |-  .0.  e.  _V
11 ifexg 3962 . . . . . . . 8  |-  ( ( S  e.  ( Base `  R )  /\  .0.  e.  _V )  ->  if ( j  =  L ,  S ,  .0.  )  e.  _V )
1210, 11mpan2 671 . . . . . . 7  |-  ( S  e.  ( Base `  R
)  ->  if (
j  =  L ,  S ,  .0.  )  e.  _V )
13 ovex 6220 . . . . . . . 8  |-  ( i M j )  e. 
_V
1413a1i 11 . . . . . . 7  |-  ( S  e.  ( Base `  R
)  ->  ( i M j )  e. 
_V )
1512, 14ifcld 3935 . . . . . 6  |-  ( S  e.  ( Base `  R
)  ->  if (
i  =  K ,  if ( j  =  L ,  S ,  .0.  ) ,  ( i M j ) )  e.  _V )
1615adantl 466 . . . . 5  |-  ( ( M  e.  B  /\  S  e.  ( Base `  R ) )  ->  if ( i  =  K ,  if ( j  =  L ,  S ,  .0.  ) ,  ( i M j ) )  e.  _V )
17163ad2ant1 1009 . . . 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 465 . . 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 2456 . . . . . 6  |-  ( i  =  I  ->  (
i  =  K  <->  I  =  K ) )
2019adantr 465 . . . . 5  |-  ( ( i  =  I  /\  j  =  J )  ->  ( i  =  K  <-> 
I  =  K ) )
21 eqeq1 2456 . . . . . . 7  |-  ( j  =  J  ->  (
j  =  L  <->  J  =  L ) )
2221ifbid 3914 . . . . . 6  |-  ( j  =  J  ->  if ( j  =  L ,  S ,  .0.  )  =  if ( J  =  L ,  S ,  .0.  )
)
2322adantl 466 . . . . 5  |-  ( ( i  =  I  /\  j  =  J )  ->  if ( j  =  L ,  S ,  .0.  )  =  if ( J  =  L ,  S ,  .0.  )
)
24 oveq12 6204 . . . . 5  |-  ( ( i  =  I  /\  j  =  J )  ->  ( i M j )  =  ( I M J ) )
2520, 23, 24ifbieq12d 3919 . . . 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 466 . . 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 6329 . 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3072   ifcif 3894   ` cfv 5521  (class class class)co 6195    |-> cmpt2 6197   Basecbs 14287   0gc0g 14492   Mat cmat 18400   matRRep cmarrep 18489
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683  df-slot 14291  df-base 14292  df-mat 18402  df-marrep 18491
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator