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Theorem lmodscaf 20273
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 2738 . . . . 5 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 scaffval.k . . . . 5 𝐾 = (Base‘𝐹)
51, 2, 3, 4lmodvscl 20268 . . . 4 ((𝑊 ∈ LMod ∧ 𝑥𝐾𝑦𝐵) → (𝑥( ·𝑠𝑊)𝑦) ∈ 𝐵)
653expb 1121 . . 3 ((𝑊 ∈ LMod ∧ (𝑥𝐾𝑦𝐵)) → (𝑥( ·𝑠𝑊)𝑦) ∈ 𝐵)
76ralrimivva 3196 . 2 (𝑊 ∈ LMod → ∀𝑥𝐾𝑦𝐵 (𝑥( ·𝑠𝑊)𝑦) ∈ 𝐵)
8 scaffval.a . . . 4 = ( ·sf𝑊)
91, 2, 4, 8, 3scaffval 20269 . . 3 = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥( ·𝑠𝑊)𝑦))
109fmpo 7989 . 2 (∀𝑥𝐾𝑦𝐵 (𝑥( ·𝑠𝑊)𝑦) ∈ 𝐵 :(𝐾 × 𝐵)⟶𝐵)
117, 10sylib 217 1 (𝑊 ∈ LMod → :(𝐾 × 𝐵)⟶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wral 3063   × cxp 5629  wf 6488  cfv 6492  (class class class)co 7350  Basecbs 17019  Scalarcsca 17072   ·𝑠 cvsca 17073  LModclmod 20251   ·sf cscaf 20252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7353  df-oprab 7354  df-mpo 7355  df-1st 7912  df-2nd 7913  df-lmod 20253  df-scaf 20254
This theorem is referenced by:  lmodfopnelem1  20287  nlmvscn  23979  cvsi  24421
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