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Theorem hvmulassi 28420
Description: Scalar multiplication associative 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
hvmulassi ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))

Proof of Theorem hvmulassi
StepHypRef Expression
1 hvmulcom.1 . 2 𝐴 ∈ ℂ
2 hvmulcom.2 . 2 𝐵 ∈ ℂ
3 hvmulcom.3 . 2 𝐶 ∈ ℋ
4 ax-hvmulass 28381 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
51, 2, 3, 4mp3an 1586 1 ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  wcel 2157  (class class class)co 6876  cc 10220   · cmul 10227  chba 28293   · csm 28295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-hvmulass 28381
This theorem depends on definitions:  df-bi 199  df-an 386  df-3an 1110
This theorem is referenced by:  hvmul2negi  28422  hvnegdii  28436  normlem0  28483  lnophmlem2  29393
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