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Theorem mulvfv 26999
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 26996 . . . 4  |-  ( ( A  e.  E  /\  B  e.  D )  ->  ( A . v B )  =  ( x  e.  RR  |->  ( A  x.  ( B `
 x ) ) ) )
21fveq1d 5607 . . 3  |-  ( ( A  e.  E  /\  B  e.  D )  ->  ( ( A . v B ) `  C )  =  ( ( x  e.  RR  |->  ( A  x.  ( B `  x )
) ) `  C
) )
3 fveq2 5605 . . . . 5  |-  ( x  =  C  ->  ( B `  x )  =  ( B `  C ) )
43oveq2d 5958 . . . 4  |-  ( x  =  C  ->  ( A  x.  ( B `  x ) )  =  ( A  x.  ( B `  C )
) )
5 eqid 2358 . . . 4  |-  ( x  e.  RR  |->  ( A  x.  ( B `  x ) ) )  =  ( x  e.  RR  |->  ( A  x.  ( B `  x ) ) )
6 ovex 5967 . . . 4  |-  ( A  x.  ( B `  C ) )  e. 
_V
74, 5, 6fvmpt 5682 . . 3  |-  ( C  e.  RR  ->  (
( x  e.  RR  |->  ( A  x.  ( B `  x )
) ) `  C
)  =  ( A  x.  ( B `  C ) ) )
82, 7sylan9eq 2410 . 2  |-  ( ( ( A  e.  E  /\  B  e.  D
)  /\  C  e.  RR )  ->  ( ( A . v B ) `  C )  =  ( A  x.  ( B `  C ) ) )
983impa 1146 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 358    /\ w3a 934    = wceq 1642    e. wcel 1710    e. cmpt 4156   ` cfv 5334  (class class class)co 5942   RRcr 8823    x. cmul 8829   . vctimesr 26987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pr 4293  ax-cnex 8880  ax-resscn 8881
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-mulv 26993
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