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Mirrors > Home > HSE Home > Th. List > hvaddid2 | Structured version Visualization version GIF version |
Description: Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvaddid2 | ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hv0cl 27244 | . . 3 ⊢ 0ℎ ∈ ℋ | |
2 | ax-hvcom 27242 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴)) | |
3 | 1, 2 | mpan2 703 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴)) |
4 | ax-hvaddid 27245 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | |
5 | 3, 4 | eqtr3d 2646 | 1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 (class class class)co 6549 ℋchil 27160 +ℎ cva 27161 0ℎc0v 27165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-ext 2590 ax-hvcom 27242 ax-hv0cl 27244 ax-hvaddid 27245 |
This theorem depends on definitions: df-bi 196 df-an 385 df-cleq 2603 |
This theorem is referenced by: hv2neg 27269 hvaddid2i 27270 hvaddsub4 27319 hilablo 27401 hilid 27402 shunssi 27611 spanunsni 27822 5oalem2 27898 3oalem2 27906 |
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