Theorem lmodscaf 20060

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	⊢ ∙ = ( ·sf ‘𝑊)

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.

Step	Hyp	Ref	Expression
1		scaffval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
2		scaffval.f	. . . . 5 ⊢ 𝐹 = (Scalar‘𝑊)
3		eqid 2738	. . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		scaffval.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐹)
5	1, 2, 3, 4	lmodvscl 20055	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵)
6	5	3expb 1118	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵)
7	6	ralrimivva 3114	. 2 ⊢ (𝑊 ∈ LMod → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵)
8		scaffval.a	. . . 4 ⊢ ∙ = ( ·sf ‘𝑊)
9	1, 2, 4, 8, 3	scaffval 20056	. . 3 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦))
10	9	fmpo 7881	. 2 ⊢ (∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵 ↔ ∙ :(𝐾 × 𝐵)⟶𝐵)
11	7, 10	sylib 217	1 ⊢ (𝑊 ∈ LMod → ∙ :(𝐾 × 𝐵)⟶𝐵)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  ∀wral 3063   × cxp 5578  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Scalarcsca 16891   ·_𝑠 cvsca 16892  LModclmod 20038   ·_sf cscaf 20039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-lmod 20040  df-scaf 20041

This theorem is referenced by:  lmodfopnelem1  20074  nlmvscn  23757  cvsi  24199