Theorem scaffn 20639

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
scaffn	⊢ ∙ Fn (𝐾 × 𝐵)

Proof of Theorem scaffn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		scaffval.b	. . 3 ⊢ 𝐵 = (Base‘𝑊)
2		scaffval.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3		scaffval.k	. . 3 ⊢ 𝐾 = (Base‘𝐹)
4		scaffval.a	. . 3 ⊢ ∙ = ( ·sf ‘𝑊)
5		eqid 2730	. . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
6	1, 2, 3, 4, 5	scaffval 20636	. 2 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦))
7		ovex 7446	. 2 ⊢ (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V
8	6, 7	fnmpoi 8060	1 ⊢ ∙ Fn (𝐾 × 𝐵)

Syntax hints:   = wceq 1539   × cxp 5675   Fn wfn 6539  ‘cfv 6544  (class class class)co 7413  Basecbs 17150  Scalarcsca 17206   ·_𝑠 cvsca 17207   ·_sf cscaf 20617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7979  df-2nd 7980  df-scaf 20619

This theorem is referenced by: (None)