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Theorem mulvfv 27547
Description: Scalar multiplication at a value. (Contributed by Andrew Salmon, 27-Jan-2012.)
Assertion
Ref Expression
mulvfv  |-  ( ( A  e.  E  /\  B  e.  D  /\  C  e.  RR )  ->  ( ( A . v B ) `  C )  =  ( A  x.  ( B `
 C ) ) )

Proof of Theorem mulvfv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 mulvval 27544 . . . 4  |-  ( ( A  e.  E  /\  B  e.  D )  ->  ( A . v B )  =  ( x  e.  RR  |->  ( A  x.  ( B `
 x ) ) ) )
21fveq1d 5693 . . 3  |-  ( ( A  e.  E  /\  B  e.  D )  ->  ( ( A . v B ) `  C )  =  ( ( x  e.  RR  |->  ( A  x.  ( B `  x )
) ) `  C
) )
3 fveq2 5691 . . . . 5  |-  ( x  =  C  ->  ( B `  x )  =  ( B `  C ) )
43oveq2d 6060 . . . 4  |-  ( x  =  C  ->  ( A  x.  ( B `  x ) )  =  ( A  x.  ( B `  C )
) )
5 eqid 2408 . . . 4  |-  ( x  e.  RR  |->  ( A  x.  ( B `  x ) ) )  =  ( x  e.  RR  |->  ( A  x.  ( B `  x ) ) )
6 ovex 6069 . . . 4  |-  ( A  x.  ( B `  C ) )  e. 
_V
74, 5, 6fvmpt 5769 . . 3  |-  ( C  e.  RR  ->  (
( x  e.  RR  |->  ( A  x.  ( B `  x )
) ) `  C
)  =  ( A  x.  ( B `  C ) ) )
82, 7sylan9eq 2460 . 2  |-  ( ( ( A  e.  E  /\  B  e.  D
)  /\  C  e.  RR )  ->  ( ( A . v B ) `  C )  =  ( A  x.  ( B `  C ) ) )
983impa 1148 1  |-  ( ( A  e.  E  /\  B  e.  D  /\  C  e.  RR )  ->  ( ( A . v B ) `  C )  =  ( A  x.  ( B `
 C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    e. cmpt 4230   ` cfv 5417  (class class class)co 6044   RRcr 8949    x. cmul 8955   . vctimesr 27535
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-cnex 9006  ax-resscn 9007
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-mulv 27541
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