Theorem hvaddid2 27264

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 27244	. . 3 ⊢ 0ℎ ∈ ℋ
2		ax-hvcom 27242	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
3	1, 2	mpan2 703	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
4		ax-hvaddid 27245	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
5	3, 4	eqtr3d 2646	1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  (class class class)co 6549   ℋchil 27160   +_ℎ cva 27161  0_ℎc0v 27165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-ext 2590  ax-hvcom 27242  ax-hv0cl 27244  ax-hvaddid 27245

This theorem depends on definitions:  df-bi 196  df-an 385  df-cleq 2603

This theorem is referenced by:  hv2neg  27269  hvaddid2i  27270  hvaddsub4  27319  hilablo  27401  hilid  27402  shunssi  27611  spanunsni  27822  5oalem2  27898  3oalem2  27906