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Theorem marepvval 18353
Description: Third substitution for the definition of the function replacing a column of a matrix by a vector. (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
marepvval  |-  ( ( 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 ) ) ) )
Distinct variable groups:    i, N, j    R, i, j    C, i, j    i, M, j   
i, K, j
Allowed substitution hints:    A( i, j)    B( i, j)    Q( i, j)    V( i, j)

Proof of Theorem marepvval
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 marepvfval.a . . . . 5  |-  A  =  ( N Mat  R )
2 marepvfval.b . . . . 5  |-  B  =  ( Base `  A
)
3 marepvfval.q . . . . 5  |-  Q  =  ( N matRepV  R )
4 marepvfval.v . . . . 5  |-  V  =  ( ( Base `  R
)  ^m  N )
51, 2, 3, 4marepvval0 18352 . . . 4  |-  ( ( M  e.  B  /\  C  e.  V )  ->  ( M Q C )  =  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( C `  i
) ,  ( i M j ) ) ) ) )
653adant3 1008 . . 3  |-  ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N )  ->  ( M Q C )  =  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( C `  i
) ,  ( i M j ) ) ) ) )
76fveq1d 5688 . 2  |-  ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N )  ->  ( ( M Q C ) `  K
)  =  ( ( k  e.  N  |->  ( i  e.  N , 
j  e.  N  |->  if ( j  =  k ,  ( C `  i ) ,  ( i M j ) ) ) ) `  K ) )
8 simp3 990 . . 3  |-  ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N )  ->  K  e.  N )
91, 2matrcl 18287 . . . . . . 7  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
109simpld 459 . . . . . 6  |-  ( M  e.  B  ->  N  e.  Fin )
1110, 10jca 532 . . . . 5  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  N  e.  Fin ) )
12113ad2ant1 1009 . . . 4  |-  ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N )  ->  ( N  e.  Fin  /\  N  e.  Fin )
)
13 mpt2exga 6644 . . . 4  |-  ( ( N  e.  Fin  /\  N  e.  Fin )  ->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  K ,  ( C `
 i ) ,  ( i M j ) ) )  e. 
_V )
1412, 13syl 16 . . 3  |-  ( ( 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 )
15 eqeq2 2447 . . . . . 6  |-  ( k  =  K  ->  (
j  =  k  <->  j  =  K ) )
1615ifbid 3806 . . . . 5  |-  ( k  =  K  ->  if ( j  =  k ,  ( C `  i ) ,  ( i M j ) )  =  if ( j  =  K , 
( C `  i
) ,  ( i M j ) ) )
1716mpt2eq3dv 6147 . . . 4  |-  ( k  =  K  ->  (
i  e.  N , 
j  e.  N  |->  if ( j  =  k ,  ( C `  i ) ,  ( i M j ) ) )  =  ( i  e.  N , 
j  e.  N  |->  if ( j  =  K ,  ( C `  i ) ,  ( i M j ) ) ) )
18 eqid 2438 . . . 4  |-  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( C `  i
) ,  ( i M j ) ) ) )  =  ( k  e.  N  |->  ( i  e.  N , 
j  e.  N  |->  if ( j  =  k ,  ( C `  i ) ,  ( i M j ) ) ) )
1917, 18fvmptg 5767 . . 3  |-  ( ( K  e.  N  /\  ( i  e.  N ,  j  e.  N  |->  if ( j  =  K ,  ( C `
 i ) ,  ( i M j ) ) )  e. 
_V )  ->  (
( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( C `
 i ) ,  ( i M j ) ) ) ) `
 K )  =  ( i  e.  N ,  j  e.  N  |->  if ( j  =  K ,  ( C `
 i ) ,  ( i M j ) ) ) )
208, 14, 19syl2anc 661 . 2  |-  ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N )  ->  ( ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( C `  i ) ,  ( i M j ) ) ) ) `  K )  =  ( i  e.  N ,  j  e.  N  |->  if ( j  =  K ,  ( C `  i ) ,  ( i M j ) ) ) )
217, 20eqtrd 2470 1  |-  ( ( 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 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2967   ifcif 3786    e. cmpt 4345   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088    ^m cmap 7206   Fincfn 7302   Basecbs 14166   Mat cmat 18255   matRepV cmatrepV 18343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-slot 14170  df-base 14171  df-mat 18257  df-marepv 18345
This theorem is referenced by:  marepveval  18354  marepvcl  18355  1marepvmarrepid  18361  cramerimplem2  18465
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