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Theorem hvaddid2 28727
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 28707 . . 3 0 ∈ ℋ
2 ax-hvcom 28705 . . 3 ((𝐴 ∈ ℋ ∧ 0 ∈ ℋ) → (𝐴 + 0) = (0 + 𝐴))
31, 2mpan2 687 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = (0 + 𝐴))
4 ax-hvaddid 28708 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)
53, 4eqtr3d 2855 1 (𝐴 ∈ ℋ → (0 + 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  (class class class)co 7145  chba 28623   + cva 28624  0c0v 28628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-9 2115  ax-ext 2790  ax-hvcom 28705  ax-hv0cl 28707  ax-hvaddid 28708
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-cleq 2811
This theorem is referenced by:  hv2neg  28732  hvaddid2i  28733  hvaddsub4  28782  hilablo  28864  hilid  28865  shunssi  29072  spanunsni  29283  5oalem2  29359  3oalem2  29367
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