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Theorem hvaddid2 29385
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 29365 . . 3 0 ∈ ℋ
2 ax-hvcom 29363 . . 3 ((𝐴 ∈ ℋ ∧ 0 ∈ ℋ) → (𝐴 + 0) = (0 + 𝐴))
31, 2mpan2 688 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = (0 + 𝐴))
4 ax-hvaddid 29366 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)
53, 4eqtr3d 2780 1 (𝐴 ∈ ℋ → (0 + 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  (class class class)co 7275  chba 29281   + cva 29282  0c0v 29286
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 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2709  ax-hvcom 29363  ax-hv0cl 29365  ax-hvaddid 29366
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-cleq 2730
This theorem is referenced by:  hv2neg  29390  hvaddid2i  29391  hvaddsub4  29440  hilablo  29522  hilid  29523  shunssi  29730  spanunsni  29941  5oalem2  30017  3oalem2  30025
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