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Theorem hvaddid2 29968
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 29948 . . 3 0 ∈ ℋ
2 ax-hvcom 29946 . . 3 ((𝐴 ∈ ℋ ∧ 0 ∈ ℋ) → (𝐴 + 0) = (0 + 𝐴))
31, 2mpan2 690 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = (0 + 𝐴))
4 ax-hvaddid 29949 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)
53, 4eqtr3d 2779 1 (𝐴 ∈ ℋ → (0 + 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  (class class class)co 7358  chba 29864   + cva 29865  0c0v 29869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-ext 2708  ax-hvcom 29946  ax-hv0cl 29948  ax-hvaddid 29949
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-cleq 2729
This theorem is referenced by:  hv2neg  29973  hvaddid2i  29974  hvaddsub4  30023  hilablo  30105  hilid  30106  shunssi  30313  spanunsni  30524  5oalem2  30600  3oalem2  30608
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