Theorem scafeq 20636

Description: If the scalar multiplication operation is already a function, the functionalization of it is equal to the original operation. (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)
scaffval.s	⊢ · = ( ·𝑠 ‘𝑊)

Assertion

Ref	Expression
scafeq	⊢ ( · Fn (𝐾 × 𝐵) → ∙ = · )

Proof of Theorem scafeq

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		scaffval.s	. . 3 ⊢ · = ( ·𝑠 ‘𝑊)
6	1, 2, 3, 4, 5	scaffval 20634	. 2 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦))
7		fnov 7542	. . 3 ⊢ ( · Fn (𝐾 × 𝐵) ↔ · = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
8	7	biimpi 215	. 2 ⊢ ( · Fn (𝐾 × 𝐵) → · = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
9	6, 8	eqtr4id 2789	1 ⊢ ( · Fn (𝐾 × 𝐵) → ∙ = · )

Syntax hints:   → wi 4   = wceq 1539   × cxp 5673   Fn wfn 6537  ‘cfv 6542  (class class class)co 7411   ∈ cmpo 7413  Basecbs 17148  Scalarcsca 17204   ·_𝑠 cvsca 17205   ·_sf cscaf 20615

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 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-scaf 20617

This theorem is referenced by: (None)