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Theorem hvaddid2 28008
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 27988 . . 3 0 ∈ ℋ
2 ax-hvcom 27986 . . 3 ((𝐴 ∈ ℋ ∧ 0 ∈ ℋ) → (𝐴 + 0) = (0 + 𝐴))
31, 2mpan2 707 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = (0 + 𝐴))
4 ax-hvaddid 27989 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)
53, 4eqtr3d 2687 1 (𝐴 ∈ ℋ → (0 + 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  (class class class)co 6690  chil 27904   + cva 27905  0c0v 27909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-ext 2631  ax-hvcom 27986  ax-hv0cl 27988  ax-hvaddid 27989
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-cleq 2644
This theorem is referenced by:  hv2neg  28013  hvaddid2i  28014  hvaddsub4  28063  hilablo  28145  hilid  28146  shunssi  28355  spanunsni  28566  5oalem2  28642  3oalem2  28650
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