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Theorem scafval 15969
Description: The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
scaffval.b  |-  B  =  ( Base `  W
)
scaffval.f  |-  F  =  (Scalar `  W )
scaffval.k  |-  K  =  ( Base `  F
)
scaffval.a  |-  .xb  =  ( .s f `  W
)
scaffval.s  |-  .x.  =  ( .s `  W )
Assertion
Ref Expression
scafval  |-  ( ( X  e.  K  /\  Y  e.  B )  ->  ( X  .xb  Y
)  =  ( X 
.x.  Y ) )

Proof of Theorem scafval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 6090 . 2  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x  .x.  y
)  =  ( X 
.x.  Y ) )
2 scaffval.b . . 3  |-  B  =  ( Base `  W
)
3 scaffval.f . . 3  |-  F  =  (Scalar `  W )
4 scaffval.k . . 3  |-  K  =  ( Base `  F
)
5 scaffval.a . . 3  |-  .xb  =  ( .s f `  W
)
6 scaffval.s . . 3  |-  .x.  =  ( .s `  W )
72, 3, 4, 5, 6scaffval 15968 . 2  |-  .xb  =  ( x  e.  K ,  y  e.  B  |->  ( x  .x.  y
) )
8 ovex 6106 . 2  |-  ( X 
.x.  Y )  e. 
_V
91, 7, 8ovmpt2a 6204 1  |-  ( ( X  e.  K  /\  Y  e.  B )  ->  ( X  .xb  Y
)  =  ( X 
.x.  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   Basecbs 13469  Scalarcsca 13532   .scvsca 13533   .s fcscaf 15951
This theorem is referenced by:  cnmpt1vsca  18223  cnmpt2vsca  18224  nlmvscn  18723
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-slot 13473  df-base 13474  df-scaf 15953
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