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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 30750 | . . 3 ⊢ 0ℎ ∈ ℋ | |
2 | ax-hvcom 30748 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴)) | |
3 | 1, 2 | mpan2 688 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴)) |
4 | ax-hvaddid 30751 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | |
5 | 3, 4 | eqtr3d 2766 | 1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 (class class class)co 7402 ℋchba 30666 +ℎ cva 30667 0ℎc0v 30671 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-ext 2695 ax-hvcom 30748 ax-hv0cl 30750 ax-hvaddid 30751 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-cleq 2716 |
This theorem is referenced by: hv2neg 30775 hvaddlidi 30776 hvaddsub4 30825 hilablo 30907 hilid 30908 shunssi 31115 spanunsni 31326 5oalem2 31402 3oalem2 31410 |
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