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Theorem hvmulassi 28233
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 28194 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
51, 2, 3, 4mp3an 1573 1 ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wcel 2139  (class class class)co 6814  cc 10146   · cmul 10153  chil 28106   · csm 28108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-hvmulass 28194
This theorem depends on definitions:  df-bi 197  df-an 385  df-3an 1074
This theorem is referenced by:  hvmul2negi  28235  hvnegdii  28249  normlem0  28296  lnophmlem2  29206
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