Theorem hvaddlid 30959

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 30939	. . 3 ⊢ 0ℎ ∈ ℋ
2		ax-hvcom 30937	. . 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 30940	. 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 7390   ℋchba 30855   +_ℎ cva 30856  0_ℎc0v 30860

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 30937  ax-hv0cl 30939  ax-hvaddid 30940

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

This theorem is referenced by:  hv2neg  30964  hvaddlidi  30965  hvaddsub4  31014  hilablo  31096  hilid  31097  shunssi  31304  spanunsni  31515  5oalem2  31591  3oalem2  31599