Theorem hvaddlid 30985

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 30965	. . 3 ⊢ 0ℎ ∈ ℋ
2		ax-hvcom 30963	. . 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 30966	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
5	3, 4	eqtr3d 2766	1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  (class class class)co 7353   ℋchba 30881   +_ℎ cva 30882  0_ℎc0v 30886

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 2701  ax-hvcom 30963  ax-hv0cl 30965  ax-hvaddid 30966

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2721

This theorem is referenced by:  hv2neg  30990  hvaddlidi  30991  hvaddsub4  31040  hilablo  31122  hilid  31123  shunssi  31330  spanunsni  31541  5oalem2  31617  3oalem2  31625