Theorem hvmulassi 30928

Description: Scalar multiplication associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hvmulcom.1	⊢ 𝐴 ∈ ℂ
hvmulcom.2	⊢ 𝐵 ∈ ℂ
hvmulcom.3	⊢ 𝐶 ∈ ℋ

Assertion

Ref	Expression
hvmulassi	⊢ ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶))

Proof of Theorem hvmulassi

Step	Hyp	Ref	Expression
1		hvmulcom.1	. 2 ⊢ 𝐴 ∈ ℂ
2		hvmulcom.2	. 2 ⊢ 𝐵 ∈ ℂ
3		hvmulcom.3	. 2 ⊢ 𝐶 ∈ ℋ
4		ax-hvmulass 30889	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)))
5	1, 2, 3, 4	mp3an 1457	1 ⊢ ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶))

Syntax hints:   = wceq 1533   ∈ wcel 2098  (class class class)co 7419  ℂcc 11138   · cmul 11145   ℋchba 30801   ·_ℎ csm 30803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-hvmulass 30889

This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086

This theorem is referenced by:  hvmul2negi  30930  hvnegdii  30944  normlem0  30991  lnophmlem2  31899