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Theorem hvaddlid 30832
Description: Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
hvaddlid (𝐴 ∈ ℋ → (0 + 𝐴) = 𝐴)

Proof of Theorem hvaddlid
StepHypRef Expression
1 ax-hv0cl 30812 . . 3 0 ∈ ℋ
2 ax-hvcom 30810 . . 3 ((𝐴 ∈ ℋ ∧ 0 ∈ ℋ) → (𝐴 + 0) = (0 + 𝐴))
31, 2mpan2 690 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = (0 + 𝐴))
4 ax-hvaddid 30813 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)
53, 4eqtr3d 2770 1 (𝐴 ∈ ℋ → (0 + 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  (class class class)co 7420  chba 30728   + cva 30729  0c0v 30733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-ext 2699  ax-hvcom 30810  ax-hv0cl 30812  ax-hvaddid 30813
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-cleq 2720
This theorem is referenced by:  hv2neg  30837  hvaddlidi  30838  hvaddsub4  30887  hilablo  30969  hilid  30970  shunssi  31177  spanunsni  31388  5oalem2  31464  3oalem2  31472
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