Theorem scafeq 19654

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		fnov 7282	. . 3 ⊢ ( · Fn (𝐾 × 𝐵) ↔ · = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
2	1	biimpi 218	. 2 ⊢ ( · Fn (𝐾 × 𝐵) → · = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
3		scaffval.b	. . 3 ⊢ 𝐵 = (Base‘𝑊)
4		scaffval.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
5		scaffval.k	. . 3 ⊢ 𝐾 = (Base‘𝐹)
6		scaffval.a	. . 3 ⊢ ∙ = ( ·sf ‘𝑊)
7		scaffval.s	. . 3 ⊢ · = ( ·𝑠 ‘𝑊)
8	3, 4, 5, 6, 7	scaffval 19652	. 2 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦))
9	2, 8	syl6reqr 2875	1 ⊢ ( · Fn (𝐾 × 𝐵) → ∙ = · )

Syntax hints:   → wi 4   = wceq 1537   × cxp 5553   Fn wfn 6350  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158  Basecbs 16483  Scalarcsca 16568   ·_𝑠 cvsca 16569   ·_sf cscaf 19635

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-scaf 19637

This theorem is referenced by: (None)