Theorem hvaddid2 28958

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 28938	. . 3 ⊢ 0ℎ ∈ ℋ
2		ax-hvcom 28936	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
3	1, 2	mpan2 691	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
4		ax-hvaddid 28939	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
5	3, 4	eqtr3d 2775	1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2113  (class class class)co 7171   ℋchba 28854   +_ℎ cva 28855  0_ℎc0v 28859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-9 2123  ax-ext 2710  ax-hvcom 28936  ax-hv0cl 28938  ax-hvaddid 28939

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-cleq 2730

This theorem is referenced by:  hv2neg  28963  hvaddid2i  28964  hvaddsub4  29013  hilablo  29095  hilid  29096  shunssi  29303  spanunsni  29514  5oalem2  29590  3oalem2  29598