Theorem hvaddid2 28220

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 28200	. . 3 ⊢ 0ℎ ∈ ℋ
2		ax-hvcom 28198	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
3	1, 2	mpan2 671	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
4		ax-hvaddid 28201	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
5	3, 4	eqtr3d 2807	1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1631   ∈ wcel 2145  (class class class)co 6793   ℋchil 28116   +_ℎ cva 28117  0_ℎc0v 28121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-ext 2751  ax-hvcom 28198  ax-hv0cl 28200  ax-hvaddid 28201

This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-cleq 2764

This theorem is referenced by:  hv2neg  28225  hvaddid2i  28226  hvaddsub4  28275  hilablo  28357  hilid  28358  shunssi  28567  spanunsni  28778  5oalem2  28854  3oalem2  28862