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| Mirrors > Home > HSE Home > Th. List > hvaddlid | Structured version Visualization version GIF version | ||
| Description: Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| hvaddlid | ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ax-hv0cl 31023 | . . 3 ⊢ 0ℎ ∈ ℋ | |
| 2 | ax-hvcom 31021 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴)) | |
| 3 | 1, 2 | mpan2 691 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴)) | 
| 4 | ax-hvaddid 31024 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | |
| 5 | 3, 4 | eqtr3d 2778 | 1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 (class class class)co 7432 ℋchba 30939 +ℎ cva 30940 0ℎc0v 30944 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-9 2117 ax-ext 2707 ax-hvcom 31021 ax-hv0cl 31023 ax-hvaddid 31024 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-cleq 2728 | 
| This theorem is referenced by: hv2neg 31048 hvaddlidi 31049 hvaddsub4 31098 hilablo 31180 hilid 31181 shunssi 31388 spanunsni 31599 5oalem2 31675 3oalem2 31683 | 
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