Theorem hvaddlid 30998

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 30978	. . 3 ⊢ 0ℎ ∈ ℋ
2		ax-hvcom 30976	. . 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 30979	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
5	3, 4	eqtr3d 2768	1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  (class class class)co 7346   ℋchba 30894   +_ℎ cva 30895  0_ℎc0v 30899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2121  ax-ext 2703  ax-hvcom 30976  ax-hv0cl 30978  ax-hvaddid 30979

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2723

This theorem is referenced by:  hv2neg  31003  hvaddlidi  31004  hvaddsub4  31053  hilablo  31135  hilid  31136  shunssi  31343  spanunsni  31554  5oalem2  31630  3oalem2  31638