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Theorem hvaddid2 28958
Description: Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
hvaddid2 (𝐴 ∈ ℋ → (0 + 𝐴) = 𝐴)

Proof of Theorem hvaddid2
StepHypRef Expression
1 ax-hv0cl 28938 . . 3 0 ∈ ℋ
2 ax-hvcom 28936 . . 3 ((𝐴 ∈ ℋ ∧ 0 ∈ ℋ) → (𝐴 + 0) = (0 + 𝐴))
31, 2mpan2 691 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = (0 + 𝐴))
4 ax-hvaddid 28939 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)
53, 4eqtr3d 2775 1 (𝐴 ∈ ℋ → (0 + 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2113  (class class class)co 7171  chba 28854   + cva 28855  0c0v 28859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-9 2123  ax-ext 2710  ax-hvcom 28936  ax-hv0cl 28938  ax-hvaddid 28939
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-cleq 2730
This theorem is referenced by:  hv2neg  28963  hvaddid2i  28964  hvaddsub4  29013  hilablo  29095  hilid  29096  shunssi  29303  spanunsni  29514  5oalem2  29590  3oalem2  29598
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