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Theorem scafeq 17668
Description: If the scalar multiplication operation is already a function, the functionalization of it is equal to the original operation. (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  =  ( .sf `  W
)
scaffval.s  |-  .x.  =  ( .s `  W )
Assertion
Ref Expression
scafeq  |-  (  .x.  Fn  ( K  X.  B
)  ->  .xb  =  .x.  )

Proof of Theorem scafeq
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnov 6331 . . 3  |-  (  .x.  Fn  ( K  X.  B
)  <->  .x.  =  ( x  e.  K ,  y  e.  B  |->  ( x 
.x.  y ) ) )
21biimpi 194 . 2  |-  (  .x.  Fn  ( K  X.  B
)  ->  .x.  =  ( x  e.  K , 
y  e.  B  |->  ( x  .x.  y ) ) )
3 scaffval.b . . 3  |-  B  =  ( Base `  W
)
4 scaffval.f . . 3  |-  F  =  (Scalar `  W )
5 scaffval.k . . 3  |-  K  =  ( Base `  F
)
6 scaffval.a . . 3  |-  .xb  =  ( .sf `  W
)
7 scaffval.s . . 3  |-  .x.  =  ( .s `  W )
83, 4, 5, 6, 7scaffval 17666 . 2  |-  .xb  =  ( x  e.  K ,  y  e.  B  |->  ( x  .x.  y
) )
92, 8syl6reqr 2456 1  |-  (  .x.  Fn  ( K  X.  B
)  ->  .xb  =  .x.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    X. cxp 4928    Fn wfn 5508   ` cfv 5513  (class class class)co 6218    |-> cmpt2 6220   Basecbs 14657  Scalarcsca 14728   .scvsca 14729   .sfcscaf 17649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-fv 5521  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-1st 6721  df-2nd 6722  df-slot 14661  df-base 14662  df-scaf 17651
This theorem is referenced by: (None)
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