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Theorem hvaddlid 30770
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 30750 . . 3 0 ∈ ℋ
2 ax-hvcom 30748 . . 3 ((𝐴 ∈ ℋ ∧ 0 ∈ ℋ) → (𝐴 + 0) = (0 + 𝐴))
31, 2mpan2 688 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = (0 + 𝐴))
4 ax-hvaddid 30751 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)
53, 4eqtr3d 2766 1 (𝐴 ∈ ℋ → (0 + 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  (class class class)co 7402  chba 30666   + cva 30667  0c0v 30671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-ext 2695  ax-hvcom 30748  ax-hv0cl 30750  ax-hvaddid 30751
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-cleq 2716
This theorem is referenced by:  hv2neg  30775  hvaddlidi  30776  hvaddsub4  30825  hilablo  30907  hilid  30908  shunssi  31115  spanunsni  31326  5oalem2  31402  3oalem2  31410
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