Theorem hvaddid2 28806

Description: Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)

Assertion

Ref	Expression
hvaddid2	⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)

Proof of Theorem hvaddid2

Step	Hyp	Ref	Expression
1		ax-hv0cl 28786	. . 3 ⊢ 0ℎ ∈ ℋ
2		ax-hvcom 28784	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
3	1, 2	mpan2 690	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
4		ax-hvaddid 28787	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
5	3, 4	eqtr3d 2835	1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  (class class class)co 7135   ℋchba 28702   +_ℎ cva 28703  0_ℎc0v 28707

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 1911  ax-6 1970  ax-7 2015  ax-9 2121  ax-ext 2770  ax-hvcom 28784  ax-hv0cl 28786  ax-hvaddid 28787

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791

This theorem is referenced by:  hv2neg  28811  hvaddid2i  28812  hvaddsub4  28861  hilablo  28943  hilid  28944  shunssi  29151  spanunsni  29362  5oalem2  29438  3oalem2  29446