Theorem scaffn 20845

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 2736	. . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
6	1, 2, 3, 4, 5	scaffval 20842	. 2 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦))
7		ovex 7443	. 2 ⊢ (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V
8	6, 7	fnmpoi 8074	1 ⊢ ∙ Fn (𝐾 × 𝐵)

Syntax hints:   = wceq 1540   × cxp 5657   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  Scalarcsca 17279   ·_𝑠 cvsca 17280   ·_sf cscaf 20823

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-scaf 20825

This theorem is referenced by: (None)