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Theorem marepvval 19361
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 19360 . . . 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 1017 . . 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 5851 . 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 999 . . 3  |-  ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N )  ->  K  e.  N )
91, 2matrcl 19206 . . . . . . 7  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
109simpld 457 . . . . . 6  |-  ( M  e.  B  ->  N  e.  Fin )
1110, 10jca 530 . . . . 5  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  N  e.  Fin ) )
12113ad2ant1 1018 . . . 4  |-  ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N )  ->  ( N  e.  Fin  /\  N  e.  Fin )
)
13 mpt2exga 6860 . . . 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 17 . . 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 2417 . . . . . 6  |-  ( k  =  K  ->  (
j  =  k  <->  j  =  K ) )
1615ifbid 3907 . . . . 5  |-  ( k  =  K  ->  if ( j  =  k ,  ( C `  i ) ,  ( i M j ) )  =  if ( j  =  K , 
( C `  i
) ,  ( i M j ) ) )
1716mpt2eq3dv 6344 . . . 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 2402 . . . 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 5930 . . 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 659 . 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 2443 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   _Vcvv 3059   ifcif 3885    |-> cmpt 4453   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280    ^m cmap 7457   Fincfn 7554   Basecbs 14841   Mat cmat 19201   matRepV cmatrepV 19351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-slot 14845  df-base 14846  df-mat 19202  df-marepv 19353
This theorem is referenced by:  marepveval  19362  marepvcl  19363  1marepvmarrepid  19369  cramerimplem2  19478
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