Theorem lmodscaf 20983

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 2769	. . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		scaffval.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐹)
5	1, 2, 3, 4	lmodvscl 20977	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵)
6	5	3expb 1136	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵)
7	6	ralrimivva 3214	. 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 20979	. . 3 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦))
10	9	fmpo 8065	. 2 ⊢ (∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵 ↔ ∙ :(𝐾 × 𝐵)⟶𝐵)
11	7, 10	sylib 221	1 ⊢ (𝑊 ∈ LMod → ∙ :(𝐾 × 𝐵)⟶𝐵)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ∀wral 3085   × cxp 5660  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  Scalarcsca 17313   ·_𝑠 cvsca 17314  LModclmod 20959   ·_sf cscaf 20960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-lmod 20961  df-scaf 20962

This theorem is referenced by:  lmodfopnelem1  20997  nlmvscn  24813  cvsi  25258