Theorem hvaddlid 31356

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 31336	. . 3 ⊢ 0ℎ ∈ ℋ
2		ax-hvcom 31334	. . 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 31337	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
5	3, 4	eqtr3d 2800	1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  (class class class)co 7412   ℋchba 31252   +_ℎ cva 31253  0_ℎc0v 31257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-ext 2735  ax-hvcom 31334  ax-hv0cl 31336  ax-hvaddid 31337

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-cleq 2755

This theorem is referenced by:  hv2neg  31361  hvaddlidi  31362  hvaddsub4  31411  hilablo  31493  hilid  31494  shunssi  31701  spanunsni  31912  5oalem2  31988  3oalem2  31996