Theorem hvaddlid 30271

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 30251	. . 3 ⊢ 0ℎ ∈ ℋ
2		ax-hvcom 30249	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
3	1, 2	mpan2 689	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
4		ax-hvaddid 30252	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
5	3, 4	eqtr3d 2774	1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  (class class class)co 7408   ℋchba 30167   +_ℎ cva 30168  0_ℎc0v 30172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2703  ax-hvcom 30249  ax-hv0cl 30251  ax-hvaddid 30252

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-cleq 2724

This theorem is referenced by:  hv2neg  30276  hvaddlidi  30277  hvaddsub4  30326  hilablo  30408  hilid  30409  shunssi  30616  spanunsni  30827  5oalem2  30903  3oalem2  30911