hvaddlidi - Hilbert Space Explorer

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Theorem hvaddlidi 31513

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	⊢ (0_ℎ +_ℎ 𝐴) = 𝐴

**Proof of Theorem hvaddlidi**
Step	Hyp	Ref	Expression
1		hvaddlid.1	. 2 ⊢ 𝐴 ∈ ℋ
2		hvaddlid 31507	. 2 ⊢ (𝐴 ∈ ℋ → (0_ℎ +_ℎ 𝐴) = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ (0_ℎ +_ℎ 𝐴) = 𝐴

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 ∈ wcel 2145 (class class class)co 7414 ℋchba 31403 +_ℎ cva 31404 0_ℎc0v 31408

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-9 2155 ax-ext 2732 ax-hvcom 31485 ax-hv0cl 31487 ax-hvaddid 31488

This proof depends on definitions: df-bi 210 df-an 402 df-ex 1813 df-cleq 2752

This theorem is used by: hvsubeq0i 31547 hvaddcani 31549 hsn0elch 31732 hhssnv 31748 shscli 31801