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Theorem scafval 20896
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 = ( ·sf𝑊)
scaffval.s · = ( ·𝑠𝑊)
Assertion
Ref Expression
scafval ((𝑋𝐾𝑌𝐵) → (𝑋 𝑌) = (𝑋 · 𝑌))

Proof of Theorem scafval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 7440 . 2 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥 · 𝑦) = (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 = ( ·sf𝑊)
6 scaffval.s . . 3 · = ( ·𝑠𝑊)
72, 3, 4, 5, 6scaffval 20895 . 2 = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
8 ovex 7464 . 2 (𝑋 · 𝑌) ∈ V
91, 7, 8ovmpoa 7588 1 ((𝑋𝐾𝑌𝐵) → (𝑋 𝑌) = (𝑋 · 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  Scalarcsca 17301   ·𝑠 cvsca 17302   ·sf cscaf 20876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-scaf 20878
This theorem is referenced by:  lmodfopne  20915  cnmpt1vsca  24218  cnmpt2vsca  24219  nlmvscn  24724
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