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Theorem marepvval 18836
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 18835 . . . 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 1016 . . 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 5866 . 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 998 . . 3  |-  ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N )  ->  K  e.  N )
91, 2matrcl 18681 . . . . . . 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 1017 . . . 4  |-  ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N )  ->  ( N  e.  Fin  /\  N  e.  Fin )
)
13 mpt2exga 6856 . . . 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 2482 . . . . . 6  |-  ( k  =  K  ->  (
j  =  k  <->  j  =  K ) )
1615ifbid 3961 . . . . 5  |-  ( k  =  K  ->  if ( j  =  k ,  ( C `  i ) ,  ( i M j ) )  =  if ( j  =  K , 
( C `  i
) ,  ( i M j ) ) )
1716mpt2eq3dv 6345 . . . 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 2467 . . . 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 5946 . . 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 2508 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 973    = wceq 1379    e. wcel 1767   _Vcvv 3113   ifcif 3939    |-> cmpt 4505   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284    ^m cmap 7417   Fincfn 7513   Basecbs 14486   Mat cmat 18676   matRepV cmatrepV 18826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-slot 14490  df-base 14491  df-mat 18677  df-marepv 18828
This theorem is referenced by:  marepveval  18837  marepvcl  18838  1marepvmarrepid  18844  cramerimplem2  18953
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