Theorem hvmulcom 29405

Description: Scalar multiplication commutative law. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)

Assertion

Ref	Expression
hvmulcom	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶)))

Proof of Theorem hvmulcom

Step	Hyp	Ref	Expression
1		mulcom 10957	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
2	1	oveq1d 7290	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = ((𝐵 · 𝐴) ·ℎ 𝐶))
3	2	3adant3 1131	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = ((𝐵 · 𝐴) ·ℎ 𝐶))
4		ax-hvmulass 29369	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)))
5		ax-hvmulass 29369	. . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐵 · 𝐴) ·ℎ 𝐶) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶)))
6	5	3com12 1122	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐵 · 𝐴) ·ℎ 𝐶) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶)))
7	3, 4, 6	3eqtr3d 2786	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  (class class class)co 7275  ℂcc 10869   · cmul 10876   ℋchba 29281   ·_ℎ csm 29283

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-ext 2709  ax-mulcom 10935  ax-hvmulass 29369

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-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278

This theorem is referenced by:  hvmulcomi  29409  hvsubdistr1  29411  lnopmi  30362