Theorem hvaddlid 30958

Description: Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)

Assertion

Ref	Expression
hvaddlid	⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)

Proof of Theorem hvaddlid

Step	Hyp	Ref	Expression
1		ax-hv0cl 30938	. . 3 ⊢ 0ℎ ∈ ℋ
2		ax-hvcom 30936	. . 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 30939	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
5	3, 4	eqtr3d 2767	1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  (class class class)co 7389   ℋchba 30854   +_ℎ cva 30855  0_ℎc0v 30859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2702  ax-hvcom 30936  ax-hv0cl 30938  ax-hvaddid 30939

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2722

This theorem is referenced by:  hv2neg  30963  hvaddlidi  30964  hvaddsub4  31013  hilablo  31095  hilid  31096  shunssi  31303  spanunsni  31514  5oalem2  31590  3oalem2  31598