Hilbert Space Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  HSE Home  >  Th. List  >  hvmulassi Structured version   Visualization version   GIF version

Theorem hvmulassi 27791
 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 27752 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
51, 2, 3, 4mp3an 1421 1 ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1480   ∈ wcel 1987  (class class class)co 6615  ℂcc 9894   · cmul 9901   ℋchil 27664   ·ℎ csm 27666 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-hvmulass 27752 This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1038 This theorem is referenced by:  hvmul2negi  27793  hvnegdii  27807  normlem0  27854  lnophmlem2  28764
 Copyright terms: Public domain W3C validator