Theorem hvaddlidi 31117

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 31111	. 2 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ (0ℎ +ℎ 𝐴) = 𝐴

Syntax hints:   = wceq 1542   ∈ wcel 2114  (class class class)co 7368   ℋchba 31007   +_ℎ cva 31008  0_ℎc0v 31012

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 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709  ax-hvcom 31089  ax-hv0cl 31091  ax-hvaddid 31092

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729

This theorem is referenced by:  hvsubeq0i  31151  hvaddcani  31153  hsn0elch  31336  hhssnv  31352  shscli  31405