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Theorem scafeq 17092
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 6309 . . 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 17090 . 2  |-  .xb  =  ( x  e.  K ,  y  e.  B  |->  ( x  .x.  y
) )
92, 8syl6reqr 2514 1  |-  (  .x.  Fn  ( K  X.  B
)  ->  .xb  =  .x.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    X. cxp 4947    Fn wfn 5522   ` cfv 5527  (class class class)co 6201    |-> cmpt2 6203   Basecbs 14293  Scalarcsca 14361   .scvsca 14362   .sfcscaf 17073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-slot 14297  df-base 14298  df-scaf 17075
This theorem is referenced by: (None)
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