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 → (0ℎ +ℎ 𝐴) = 𝐴)

Proof of Theorem hvaddid2

Step	Hyp	Ref	Expression
1		ax-hv0cl 28385	. . 3 ⊢ 0ℎ ∈ ℋ
2		ax-hvcom 28383	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
3	1, 2	mpan2 683	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
4		ax-hvaddid 28386	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
5	3, 4	eqtr3d 2835	1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1653   ∈ wcel 2157  (class class class)co 6878   ℋchba 28301   +_ℎ cva 28302  0_ℎc0v 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