Theorem hvmulcomi 31118

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 31114	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶)))
5	1, 2, 3, 4	mp3an 1464	1 ⊢ (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶))

Syntax hints:   = wceq 1542   ∈ wcel 2114  (class class class)co 7367  ℂcc 11036   ℋchba 30990   ·_ℎ csm 30992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-mulcom 11102  ax-hvmulass 31078

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370

This theorem is referenced by: (None)