Theorem hvaddlid 31404

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 31384	. . 3 ⊢ 0ℎ ∈ ℋ
2		ax-hvcom 31382	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
3	1, 2	mpan2 704	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
4		ax-hvaddid 31385	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
5	3, 4	eqtr3d 2802	1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  (class class class)co 7416   ℋchba 31300   +_ℎ cva 31301  0_ℎc0v 31305

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-ext 2737  ax-hvcom 31382  ax-hv0cl 31384  ax-hvaddid 31385

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2757

This theorem is used by:  hv2neg  31409  hvaddlidi  31410  hvaddsub4  31459  hilablo  31541  hilid  31542  shunssi  31749  spanunsni  31960  5oalem2  32036  3oalem2  32044