Theorem hvaddlid 30953

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 30933	. . 3 ⊢ 0ℎ ∈ ℋ
2		ax-hvcom 30931	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
3	1, 2	mpan2 689	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
4		ax-hvaddid 30934	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
5	3, 4	eqtr3d 2768	1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  (class class class)co 7416   ℋchba 30849   +_ℎ cva 30850  0_ℎc0v 30854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-ext 2697  ax-hvcom 30931  ax-hv0cl 30933  ax-hvaddid 30934

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1775  df-cleq 2718

This theorem is referenced by:  hv2neg  30958  hvaddlidi  30959  hvaddsub4  31008  hilablo  31090  hilid  31091  shunssi  31298  spanunsni  31509  5oalem2  31585  3oalem2  31593