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Theorem marepvval0 18832
Description: Second 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
marepvval0  |-  ( ( 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 ) ) ) ) )
Distinct variable groups:    i, N, j, k    R, i, j, k    C, i, j, k   
i, M, j, k
Allowed substitution hints:    A( i, j, k)    B( i, j, k)    Q( i, j, k)    V( i, j, k)

Proof of Theorem marepvval0
Dummy variables  m  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 marepvfval.a . . . . . 6  |-  A  =  ( N Mat  R )
2 marepvfval.b . . . . . 6  |-  B  =  ( Base `  A
)
31, 2matrcl 18678 . . . . 5  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
43simpld 459 . . . 4  |-  ( M  e.  B  ->  N  e.  Fin )
54adantr 465 . . 3  |-  ( ( M  e.  B  /\  C  e.  V )  ->  N  e.  Fin )
6 mptexg 6128 . . 3  |-  ( N  e.  Fin  ->  (
k  e.  N  |->  ( i  e.  N , 
j  e.  N  |->  if ( j  =  k ,  ( C `  i ) ,  ( i M j ) ) ) )  e. 
_V )
75, 6syl 16 . 2  |-  ( ( 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 )
8 fveq1 5863 . . . . . . 7  |-  ( c  =  C  ->  (
c `  i )  =  ( C `  i ) )
98adantl 466 . . . . . 6  |-  ( ( m  =  M  /\  c  =  C )  ->  ( c `  i
)  =  ( C `
 i ) )
10 oveq 6288 . . . . . . 7  |-  ( m  =  M  ->  (
i m j )  =  ( i M j ) )
1110adantr 465 . . . . . 6  |-  ( ( m  =  M  /\  c  =  C )  ->  ( i m j )  =  ( i M j ) )
129, 11ifeq12d 3959 . . . . 5  |-  ( ( m  =  M  /\  c  =  C )  ->  if ( j  =  k ,  ( c `
 i ) ,  ( i m j ) )  =  if ( j  =  k ,  ( C `  i ) ,  ( i M j ) ) )
1312mpt2eq3dv 6345 . . . 4  |-  ( ( m  =  M  /\  c  =  C )  ->  ( 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 ) ) ) )
1413mpteq2dv 4534 . . 3  |-  ( ( m  =  M  /\  c  =  C )  ->  ( 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 ) ) ) ) )
15 marepvfval.q . . . 4  |-  Q  =  ( N matRepV  R )
16 marepvfval.v . . . 4  |-  V  =  ( ( Base `  R
)  ^m  N )
171, 2, 15, 16marepvfval 18831 . . 3  |-  Q  =  ( m  e.  B ,  c  e.  V  |->  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( c `
 i ) ,  ( i m j ) ) ) ) )
1814, 17ovmpt2ga 6414 . 2  |-  ( ( 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 )  -> 
( M Q C )  =  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( C `  i
) ,  ( i M j ) ) ) ) )
197, 18mpd3an3 1325 1  |-  ( ( 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 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = 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 14483   Mat cmat 18673   matRepV cmatrepV 18823
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 14487  df-base 14488  df-mat 18674  df-marepv 18825
This theorem is referenced by:  marepvval  18833
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