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Theorem scafval 19098
 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 6800 . 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 19097 . 2 = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
8 ovex 6821 . 2 (𝑋 · 𝑌) ∈ V
91, 7, 8ovmpt2a 6936 1 ((𝑋𝐾𝑌𝐵) → (𝑋 𝑌) = (𝑋 · 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1629   ∈ wcel 2143  ‘cfv 6030  (class class class)co 6791  Basecbs 16070  Scalarcsca 16158   ·𝑠 cvsca 16159   ·sf cscaf 19080 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-sep 4911  ax-nul 4919  ax-pow 4970  ax-pr 5033  ax-un 7094 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-ral 3064  df-rex 3065  df-rab 3068  df-v 3350  df-sbc 3585  df-csb 3680  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4572  df-iun 4653  df-br 4784  df-opab 4844  df-mpt 4861  df-id 5156  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-rn 5259  df-res 5260  df-ima 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-1st 7313  df-2nd 7314  df-slot 16074  df-base 16076  df-scaf 19082 This theorem is referenced by:  lmodfopne  19117  cnmpt1vsca  22223  cnmpt2vsca  22224  nlmvscn  22717
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