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Theorem hvmulcomi 28871
 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 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))

Proof of Theorem hvmulcomi
StepHypRef Expression
1 hvmulcom.1 . 2 𝐴 ∈ ℂ
2 hvmulcom.2 . 2 𝐵 ∈ ℂ
3 hvmulcom.3 . 2 𝐶 ∈ ℋ
4 hvmulcom 28867 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
51, 2, 3, 4mp3an 1458 1 (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  (class class class)co 7142  ℂcc 10539   ℋchba 28743   ·ℎ csm 28745 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-mulcom 10605  ax-hvmulass 28831 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3887  df-in 3889  df-ss 3899  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-br 5034  df-iota 6288  df-fv 6337  df-ov 7145 This theorem is referenced by: (None)
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