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Theorem scafval 15662
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 5883 . 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 15661 . 2  |-  .xb  =  ( x  e.  K ,  y  e.  B  |->  ( x  .x.  y
) )
8 ovex 5899 . 2  |-  ( X 
.x.  Y )  e. 
_V
91, 7, 8ovmpt2a 5994 1  |-  ( ( X  e.  K  /\  Y  e.  B )  ->  ( X  .xb  Y
)  =  ( X 
.x.  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164  Scalarcsca 13227   .scvsca 13228   .s fcscaf 15644
This theorem is referenced by:  cnmpt1vsca  17892  cnmpt2vsca  17893  nlmvscn  18214
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-slot 13168  df-base 13169  df-scaf 15646
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