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Theorem lmodscaf 18708
 Description: The scalar multiplication operation is 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)
Assertion
Ref Expression
lmodscaf (𝑊 ∈ LMod → :(𝐾 × 𝐵)⟶𝐵)

Proof of Theorem lmodscaf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 scaffval.b . . . . 5 𝐵 = (Base‘𝑊)
2 scaffval.f . . . . 5 𝐹 = (Scalar‘𝑊)
3 eqid 2610 . . . . 5 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 scaffval.k . . . . 5 𝐾 = (Base‘𝐹)
51, 2, 3, 4lmodvscl 18703 . . . 4 ((𝑊 ∈ LMod ∧ 𝑥𝐾𝑦𝐵) → (𝑥( ·𝑠𝑊)𝑦) ∈ 𝐵)
653expb 1258 . . 3 ((𝑊 ∈ LMod ∧ (𝑥𝐾𝑦𝐵)) → (𝑥( ·𝑠𝑊)𝑦) ∈ 𝐵)
76ralrimivva 2954 . 2 (𝑊 ∈ LMod → ∀𝑥𝐾𝑦𝐵 (𝑥( ·𝑠𝑊)𝑦) ∈ 𝐵)
8 scaffval.a . . . 4 = ( ·sf𝑊)
91, 2, 4, 8, 3scaffval 18704 . . 3 = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥( ·𝑠𝑊)𝑦))
109fmpt2 7126 . 2 (∀𝑥𝐾𝑦𝐵 (𝑥( ·𝑠𝑊)𝑦) ∈ 𝐵 :(𝐾 × 𝐵)⟶𝐵)
117, 10sylib 207 1 (𝑊 ∈ LMod → :(𝐾 × 𝐵)⟶𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896   × cxp 5036  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Scalarcsca 15771   ·𝑠 cvsca 15772  LModclmod 18686   ·sf cscaf 18687 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-slot 15699  df-base 15700  df-lmod 18688  df-scaf 18689 This theorem is referenced by:  lmodfopnelem1  18722  nlmvscn  22301  cvsi  22738
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