Theorem hvaddid2 29385

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 29365	. . 3 ⊢ 0ℎ ∈ ℋ
2		ax-hvcom 29363	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
3	1, 2	mpan2 688	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
4		ax-hvaddid 29366	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
5	3, 4	eqtr3d 2780	1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  (class class class)co 7275   ℋchba 29281   +_ℎ cva 29282  0_ℎc0v 29286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2709  ax-hvcom 29363  ax-hv0cl 29365  ax-hvaddid 29366

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-cleq 2730

This theorem is referenced by:  hv2neg  29390  hvaddid2i  29391  hvaddsub4  29440  hilablo  29522  hilid  29523  shunssi  29730  spanunsni  29941  5oalem2  30017  3oalem2  30025