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Theorem hvaddid2 28220
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 28200 . . 3 0 ∈ ℋ
2 ax-hvcom 28198 . . 3 ((𝐴 ∈ ℋ ∧ 0 ∈ ℋ) → (𝐴 + 0) = (0 + 𝐴))
31, 2mpan2 671 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = (0 + 𝐴))
4 ax-hvaddid 28201 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)
53, 4eqtr3d 2807 1 (𝐴 ∈ ℋ → (0 + 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  (class class class)co 6793  chil 28116   + cva 28117  0c0v 28121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-ext 2751  ax-hvcom 28198  ax-hv0cl 28200  ax-hvaddid 28201
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-cleq 2764
This theorem is referenced by:  hv2neg  28225  hvaddid2i  28226  hvaddsub4  28275  hilablo  28357  hilid  28358  shunssi  28567  spanunsni  28778  5oalem2  28854  3oalem2  28862
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