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Theorem marepvval 18498
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 18497 . . . 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 5794 . 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 18430 . . . . . . 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 6752 . . . 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 2466 . . . . . 6  |-  ( k  =  K  ->  (
j  =  k  <->  j  =  K ) )
1615ifbid 3912 . . . . 5  |-  ( k  =  K  ->  if ( j  =  k ,  ( C `  i ) ,  ( i M j ) )  =  if ( j  =  K , 
( C `  i
) ,  ( i M j ) ) )
1716mpt2eq3dv 6254 . . . 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 2451 . . . 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 5874 . . 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 2492 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 1370    e. wcel 1758   _Vcvv 3071   ifcif 3892    |-> cmpt 4451   ` cfv 5519  (class class class)co 6193    |-> cmpt2 6195    ^m cmap 7317   Fincfn 7413   Basecbs 14285   Mat cmat 18398   matRepV cmatrepV 18488
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 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-slot 14289  df-base 14290  df-mat 18400  df-marepv 18490
This theorem is referenced by:  marepveval  18499  marepvcl  18500  1marepvmarrepid  18506  cramerimplem2  18615
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