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Theorem hvmulcomi 31079
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 31075 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
51, 2, 3, 4mp3an 1461 1 (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2108  (class class class)co 7448  cc 11182  chba 30951   · csm 30953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-mulcom 11248  ax-hvmulass 31039
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451
This theorem is referenced by: (None)
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