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

Theorem hvmulcomi 28244
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 28240 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
51, 2, 3, 4mp3an 1572 1 (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wcel 2145  (class class class)co 6793  cc 10136  chil 28116   · csm 28118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-mulcom 10202  ax-hvmulass 28204
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator