Theorem hvaddlid 31182

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 31162	. . 3 ⊢ 0ℎ ∈ ℋ
2		ax-hvcom 31160	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
3	1, 2	mpan2 701	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
4		ax-hvaddid 31163	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
5	3, 4	eqtr3d 2798	1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  (class class class)co 7390   ℋchba 31078   +_ℎ cva 31079  0_ℎc0v 31083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-9 2151  ax-ext 2733  ax-hvcom 31160  ax-hv0cl 31162  ax-hvaddid 31163

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-cleq 2753

This theorem is referenced by:  hv2neg  31187  hvaddlidi  31188  hvaddsub4  31237  hilablo  31319  hilid  31320  shunssi  31527  spanunsni  31738  5oalem2  31814  3oalem2  31822