Theorem hvaddlid 31119

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 31099	. . 3 ⊢ 0ℎ ∈ ℋ
2		ax-hvcom 31097	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
3	1, 2	mpan2 697	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
4		ax-hvaddid 31100	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
5	3, 4	eqtr3d 2777	1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  (class class class)co 7363   ℋchba 31015   +_ℎ cva 31016  0_ℎc0v 31020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-ext 2712  ax-hvcom 31097  ax-hv0cl 31099  ax-hvaddid 31100

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-cleq 2732

This theorem is referenced by:  hv2neg  31124  hvaddlidi  31125  hvaddsub4  31174  hilablo  31256  hilid  31257  shunssi  31464  spanunsni  31675  5oalem2  31751  3oalem2  31759