Theorem hvmulcomi 31399

Description: Scalar multiplication commutative 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
hvmulcomi	⊢ (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶))

Proof of Theorem hvmulcomi

Step	Hyp	Ref	Expression
1		hvmulcom.1	. 2 ⊢ 𝐴 ∈ ℂ
2		hvmulcom.2	. 2 ⊢ 𝐵 ∈ ℂ
3		hvmulcom.3	. 2 ⊢ 𝐶 ∈ ℋ
4		hvmulcom 31395	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶)))
5	1, 2, 3, 4	mp3an 1490	1 ⊢ (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶))

Syntax hints:   = wceq 1570   ∈ wcel 2143  (class class class)co 7410  ℂcc 11093   ℋchba 31271   ·_ℎ csm 31273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-mulcom 11159  ax-hvmulass 31359

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413

This theorem is referenced by: (None)