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Theorem lmodscaf 20882
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 2737 . . . . 5 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 scaffval.k . . . . 5 𝐾 = (Base‘𝐹)
51, 2, 3, 4lmodvscl 20876 . . . 4 ((𝑊 ∈ LMod ∧ 𝑥𝐾𝑦𝐵) → (𝑥( ·𝑠𝑊)𝑦) ∈ 𝐵)
653expb 1121 . . 3 ((𝑊 ∈ LMod ∧ (𝑥𝐾𝑦𝐵)) → (𝑥( ·𝑠𝑊)𝑦) ∈ 𝐵)
76ralrimivva 3202 . 2 (𝑊 ∈ LMod → ∀𝑥𝐾𝑦𝐵 (𝑥( ·𝑠𝑊)𝑦) ∈ 𝐵)
8 scaffval.a . . . 4 = ( ·sf𝑊)
91, 2, 4, 8, 3scaffval 20878 . . 3 = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥( ·𝑠𝑊)𝑦))
109fmpo 8093 . 2 (∀𝑥𝐾𝑦𝐵 (𝑥( ·𝑠𝑊)𝑦) ∈ 𝐵 :(𝐾 × 𝐵)⟶𝐵)
117, 10sylib 218 1 (𝑊 ∈ LMod → :(𝐾 × 𝐵)⟶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wral 3061   × cxp 5683  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  Scalarcsca 17300   ·𝑠 cvsca 17301  LModclmod 20858   ·sf cscaf 20859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-lmod 20860  df-scaf 20861
This theorem is referenced by:  lmodfopnelem1  20896  nlmvscn  24708  cvsi  25163
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