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Theorem lmodscaf 18651
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 2606 . . . . 5 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 scaffval.k . . . . 5 𝐾 = (Base‘𝐹)
51, 2, 3, 4lmodvscl 18646 . . . 4 ((𝑊 ∈ LMod ∧ 𝑥𝐾𝑦𝐵) → (𝑥( ·𝑠𝑊)𝑦) ∈ 𝐵)
653expb 1257 . . 3 ((𝑊 ∈ LMod ∧ (𝑥𝐾𝑦𝐵)) → (𝑥( ·𝑠𝑊)𝑦) ∈ 𝐵)
76ralrimivva 2950 . 2 (𝑊 ∈ LMod → ∀𝑥𝐾𝑦𝐵 (𝑥( ·𝑠𝑊)𝑦) ∈ 𝐵)
8 scaffval.a . . . 4 = ( ·sf𝑊)
91, 2, 4, 8, 3scaffval 18647 . . 3 = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥( ·𝑠𝑊)𝑦))
109fmpt2 7100 . 2 (∀𝑥𝐾𝑦𝐵 (𝑥( ·𝑠𝑊)𝑦) ∈ 𝐵 :(𝐾 × 𝐵)⟶𝐵)
117, 10sylib 206 1 (𝑊 ∈ LMod → :(𝐾 × 𝐵)⟶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  wral 2892   × cxp 5023  wf 5783  cfv 5787  (class class class)co 6524  Basecbs 15638  Scalarcsca 15714   ·𝑠 cvsca 15715  LModclmod 18629   ·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-lmod 18631  df-scaf 18632
This theorem is referenced by:  lmodfopnelem1  18665  nlmvscn  22231  cvsi  22664
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