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

Theorem hvcomi 27860
Description: Commutation of vector addition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
hvaddcl.1 𝐴 ∈ ℋ
hvaddcl.2 𝐵 ∈ ℋ
Assertion
Ref Expression
hvcomi (𝐴 + 𝐵) = (𝐵 + 𝐴)

Proof of Theorem hvcomi
StepHypRef Expression
1 hvaddcl.1 . 2 𝐴 ∈ ℋ
2 hvaddcl.2 . 2 𝐵 ∈ ℋ
3 ax-hvcom 27842 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
41, 2, 3mp2an 708 1 (𝐴 + 𝐵) = (𝐵 + 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1482  wcel 1989  (class class class)co 6647  chil 27760   + cva 27761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-hvcom 27842
This theorem depends on definitions:  df-bi 197  df-an 386
This theorem is referenced by:  hvadd12i  27898  hvnegdii  27903  norm3difi  27988  normpar2i  27997  nonbooli  28494  lnophmlem2  28860
  Copyright terms: Public domain W3C validator