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Theorem hvaddid2 28716
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 28696 . . 3 0 ∈ ℋ
2 ax-hvcom 28694 . . 3 ((𝐴 ∈ ℋ ∧ 0 ∈ ℋ) → (𝐴 + 0) = (0 + 𝐴))
31, 2mpan2 687 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = (0 + 𝐴))
4 ax-hvaddid 28697 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)
53, 4eqtr3d 2862 1 (𝐴 ∈ ℋ → (0 + 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  (class class class)co 7151  chba 28612   + cva 28613  0c0v 28617
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 1904  ax-6 1963  ax-7 2008  ax-9 2117  ax-ext 2797  ax-hvcom 28694  ax-hv0cl 28696  ax-hvaddid 28697
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-cleq 2818
This theorem is referenced by:  hv2neg  28721  hvaddid2i  28722  hvaddsub4  28771  hilablo  28853  hilid  28854  shunssi  29061  spanunsni  29272  5oalem2  29348  3oalem2  29356
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