Theorem lmodscaf 20812

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 2731	. . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		scaffval.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐹)
5	1, 2, 3, 4	lmodvscl 20806	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵)
6	5	3expb 1120	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵)
7	6	ralrimivva 3175	. 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 20808	. . 3 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦))
10	9	fmpo 7995	. 2 ⊢ (∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵 ↔ ∙ :(𝐾 × 𝐵)⟶𝐵)
11	7, 10	sylib 218	1 ⊢ (𝑊 ∈ LMod → ∙ :(𝐾 × 𝐵)⟶𝐵)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ∀wral 3047   × cxp 5609  ⟶wf 6472  ‘cfv 6476  (class class class)co 7341  Basecbs 17115  Scalarcsca 17159   ·_𝑠 cvsca 17160  LModclmod 20788   ·_sf cscaf 20789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-lmod 20790  df-scaf 20791

This theorem is referenced by:  lmodfopnelem1  20826  nlmvscn  24597  cvsi  25052