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Theorem scafval 18648
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 6533 . 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 18647 . 2 = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
8 ovex 6552 . 2 (𝑋 · 𝑌) ∈ V
91, 7, 8ovmpt2a 6664 1 ((𝑋𝐾𝑌𝐵) → (𝑋 𝑌) = (𝑋 · 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  cfv 5787  (class class class)co 6524  Basecbs 15638  Scalarcsca 15714   ·𝑠 cvsca 15715   ·sf cscaf 18630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-fv 5795  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-1st 7033  df-2nd 7034  df-slot 15642  df-base 15643  df-scaf 18632
This theorem is referenced by:  lmodfopne  18667  cnmpt1vsca  21746  cnmpt2vsca  21747  nlmvscn  22231
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