Theorem hvmulcom 31132

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 11115	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
2	1	oveq1d 7371	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = ((𝐵 · 𝐴) ·ℎ 𝐶))
3	2	3adant3 1138	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = ((𝐵 · 𝐴) ·ℎ 𝐶))
4		ax-hvmulass 31096	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)))
5		ax-hvmulass 31096	. . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐵 · 𝐴) ·ℎ 𝐶) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶)))
6	5	3com12 1129	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐵 · 𝐴) ·ℎ 𝐶) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶)))
7	3, 4, 6	3eqtr3d 2782	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  (class class class)co 7356  ℂcc 11027   · cmul 11034   ℋchba 31008   ·_ℎ csm 31010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-mulcom 11093  ax-hvmulass 31096

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-ov 7359

This theorem is referenced by:  hvmulcomi  31136  hvsubdistr1  31138  lnopmi  32089