Theorem hvaddlidi 31009

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

Hypothesis

Ref	Expression
hvaddlid.1	⊢ 𝐴 ∈ ℋ

Assertion

Ref	Expression
hvaddlidi	⊢ (0ℎ +ℎ 𝐴) = 𝐴

Proof of Theorem hvaddlidi

Step	Hyp	Ref	Expression
1		hvaddlid.1	. 2 ⊢ 𝐴 ∈ ℋ
2		hvaddlid 31003	. 2 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ (0ℎ +ℎ 𝐴) = 𝐴

Syntax hints:   = wceq 1541   ∈ wcel 2111  (class class class)co 7346   ℋchba 30899   +_ℎ cva 30900  0_ℎc0v 30904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2121  ax-ext 2703  ax-hvcom 30981  ax-hv0cl 30983  ax-hvaddid 30984

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2723

This theorem is referenced by:  hvsubeq0i  31043  hvaddcani  31045  hsn0elch  31228  hhssnv  31244  shscli  31297