Theorem scafeq 20143

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 20141	. 2 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦))
7		fnov 7405	. . 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 2797	1 ⊢ ( · Fn (𝐾 × 𝐵) → ∙ = · )

Syntax hints:   → wi 4   = wceq 1539   × cxp 5587   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  Basecbs 16912  Scalarcsca 16965   ·_𝑠 cvsca 16966   ·_sf cscaf 20124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-scaf 20126

This theorem is referenced by: (None)