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Theorem marepveval 18832
Description: An entry of a matrix with a replaced column. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)
Hypotheses
Ref Expression
marepvfval.a  |-  A  =  ( N Mat  R )
marepvfval.b  |-  B  =  ( Base `  A
)
marepvfval.q  |-  Q  =  ( N matRepV  R )
marepvfval.v  |-  V  =  ( ( Base `  R
)  ^m  N )
Assertion
Ref Expression
marepveval  |-  ( ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N ) )  -> 
( I ( ( M Q C ) `
 K ) J )  =  if ( J  =  K , 
( C `  I
) ,  ( I M J ) ) )

Proof of Theorem marepveval
Dummy variables  i 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 marepvfval.a . . . 4  |-  A  =  ( N Mat  R )
2 marepvfval.b . . . 4  |-  B  =  ( Base `  A
)
3 marepvfval.q . . . 4  |-  Q  =  ( N matRepV  R )
4 marepvfval.v . . . 4  |-  V  =  ( ( Base `  R
)  ^m  N )
51, 2, 3, 4marepvval 18831 . . 3  |-  ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N )  ->  ( ( M Q C ) `  K
)  =  ( i  e.  N ,  j  e.  N  |->  if ( j  =  K , 
( C `  i
) ,  ( i M j ) ) ) )
65adantr 465 . 2  |-  ( ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N ) )  -> 
( ( M Q C ) `  K
)  =  ( i  e.  N ,  j  e.  N  |->  if ( j  =  K , 
( C `  i
) ,  ( i M j ) ) ) )
7 simprl 755 . . 3  |-  ( ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N ) )  ->  I  e.  N )
8 simplrr 760 . . 3  |-  ( ( ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  (
I  e.  N  /\  J  e.  N )
)  /\  i  =  I )  ->  J  e.  N )
9 fvex 5869 . . . . . 6  |-  ( C `
 i )  e. 
_V
109a1i 11 . . . . 5  |-  ( ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N ) )  -> 
( C `  i
)  e.  _V )
11 ovex 6302 . . . . . 6  |-  ( i M j )  e. 
_V
1211a1i 11 . . . . 5  |-  ( ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N ) )  -> 
( i M j )  e.  _V )
1310, 12ifcld 3977 . . . 4  |-  ( ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N ) )  ->  if ( j  =  K ,  ( C `  i ) ,  ( i M j ) )  e.  _V )
1413adantr 465 . . 3  |-  ( ( ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  (
I  e.  N  /\  J  e.  N )
)  /\  ( i  =  I  /\  j  =  J ) )  ->  if ( j  =  K ,  ( C `  i ) ,  ( i M j ) )  e.  _V )
15 eqeq1 2466 . . . . . 6  |-  ( j  =  J  ->  (
j  =  K  <->  J  =  K ) )
1615adantl 466 . . . . 5  |-  ( ( i  =  I  /\  j  =  J )  ->  ( j  =  K  <-> 
J  =  K ) )
17 fveq2 5859 . . . . . 6  |-  ( i  =  I  ->  ( C `  i )  =  ( C `  I ) )
1817adantr 465 . . . . 5  |-  ( ( i  =  I  /\  j  =  J )  ->  ( C `  i
)  =  ( C `
 I ) )
19 oveq12 6286 . . . . 5  |-  ( ( i  =  I  /\  j  =  J )  ->  ( i M j )  =  ( I M J ) )
2016, 18, 19ifbieq12d 3961 . . . 4  |-  ( ( i  =  I  /\  j  =  J )  ->  if ( j  =  K ,  ( C `
 i ) ,  ( i M j ) )  =  if ( J  =  K ,  ( C `  I ) ,  ( I M J ) ) )
2120adantl 466 . . 3  |-  ( ( ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  (
I  e.  N  /\  J  e.  N )
)  /\  ( i  =  I  /\  j  =  J ) )  ->  if ( j  =  K ,  ( C `  i ) ,  ( i M j ) )  =  if ( J  =  K , 
( C `  I
) ,  ( I M J ) ) )
227, 8, 14, 21ovmpt2dv2 6413 . 2  |-  ( ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N ) )  -> 
( ( ( M Q C ) `  K )  =  ( i  e.  N , 
j  e.  N  |->  if ( j  =  K ,  ( C `  i ) ,  ( i M j ) ) )  ->  (
I ( ( M Q C ) `  K ) J )  =  if ( J  =  K ,  ( C `  I ) ,  ( I M J ) ) ) )
236, 22mpd 15 1  |-  ( ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N ) )  -> 
( I ( ( M Q C ) `
 K ) J )  =  if ( J  =  K , 
( C `  I
) ,  ( I M J ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   _Vcvv 3108   ifcif 3934   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279    ^m cmap 7412   Basecbs 14481   Mat cmat 18671   matRepV cmatrepV 18821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-slot 14485  df-base 14486  df-mat 18672  df-marepv 18823
This theorem is referenced by:  ma1repveval  18835  1marepvsma1  18847
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