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Theorem mulvfv 27676
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 27673 . . . 4  |-  ( ( A  e.  E  /\  B  e.  D )  ->  ( A . v B )  =  ( x  e.  RR  |->  ( A  x.  ( B `
 x ) ) ) )
21fveq1d 5527 . . 3  |-  ( ( A  e.  E  /\  B  e.  D )  ->  ( ( A . v B ) `  C )  =  ( ( x  e.  RR  |->  ( A  x.  ( B `  x )
) ) `  C
) )
3 fveq2 5525 . . . . 5  |-  ( x  =  C  ->  ( B `  x )  =  ( B `  C ) )
43oveq2d 5874 . . . 4  |-  ( x  =  C  ->  ( A  x.  ( B `  x ) )  =  ( A  x.  ( B `  C )
) )
5 eqid 2283 . . . 4  |-  ( x  e.  RR  |->  ( A  x.  ( B `  x ) ) )  =  ( x  e.  RR  |->  ( A  x.  ( B `  x ) ) )
6 ovex 5883 . . . 4  |-  ( A  x.  ( B `  C ) )  e. 
_V
74, 5, 6fvmpt 5602 . . 3  |-  ( C  e.  RR  ->  (
( x  e.  RR  |->  ( A  x.  ( B `  x )
) ) `  C
)  =  ( A  x.  ( B `  C ) ) )
82, 7sylan9eq 2335 . 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 1623    e. wcel 1684    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   RRcr 8736    x. cmul 8742   . vctimesr 27664
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-mulv 27670
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