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Theorem hvaddlid 31043
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 31023 . . 3 0 ∈ ℋ
2 ax-hvcom 31021 . . 3 ((𝐴 ∈ ℋ ∧ 0 ∈ ℋ) → (𝐴 + 0) = (0 + 𝐴))
31, 2mpan2 691 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = (0 + 𝐴))
4 ax-hvaddid 31024 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)
53, 4eqtr3d 2778 1 (𝐴 ∈ ℋ → (0 + 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  (class class class)co 7432  chba 30939   + cva 30940  0c0v 30944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2707  ax-hvcom 31021  ax-hv0cl 31023  ax-hvaddid 31024
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-cleq 2728
This theorem is referenced by:  hv2neg  31048  hvaddlidi  31049  hvaddsub4  31098  hilablo  31180  hilid  31181  shunssi  31388  spanunsni  31599  5oalem2  31675  3oalem2  31683
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