Theorem hvaddlid 31043

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 31023	. . 3 ⊢ 0ℎ ∈ ℋ
2		ax-hvcom 31021	. . 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 31024	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
5	3, 4	eqtr3d 2778	1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  (class class class)co 7432   ℋchba 30939   +_ℎ cva 30940  0_ℎc0v 30944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2707  ax-hvcom 31021  ax-hv0cl 31023  ax-hvaddid 31024

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-cleq 2728

This theorem is referenced by:  hv2neg  31048  hvaddlidi  31049  hvaddsub4  31098  hilablo  31180  hilid  31181  shunssi  31388  spanunsni  31599  5oalem2  31675  3oalem2  31683