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Theorem hvaddid2 28405
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 28385 . . 3 0 ∈ ℋ
2 ax-hvcom 28383 . . 3 ((𝐴 ∈ ℋ ∧ 0 ∈ ℋ) → (𝐴 + 0) = (0 + 𝐴))
31, 2mpan2 683 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = (0 + 𝐴))
4 ax-hvaddid 28386 . 2 (𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)
53, 4eqtr3d 2835 1 (𝐴 ∈ ℋ → (0 + 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  (class class class)co 6878  chba 28301   + cva 28302  0c0v 28306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-ext 2777  ax-hvcom 28383  ax-hv0cl 28385  ax-hvaddid 28386
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-cleq 2792
This theorem is referenced by:  hv2neg  28410  hvaddid2i  28411  hvaddsub4  28460  hilablo  28542  hilid  28543  shunssi  28752  spanunsni  28963  5oalem2  29039  3oalem2  29047
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