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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 29365 | . . 3 ⊢ 0ℎ ∈ ℋ | |
2 | ax-hvcom 29363 | . . 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 29366 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | |
5 | 3, 4 | eqtr3d 2780 | 1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 (class class class)co 7275 ℋchba 29281 +ℎ cva 29282 0ℎc0v 29286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-9 2116 ax-ext 2709 ax-hvcom 29363 ax-hv0cl 29365 ax-hvaddid 29366 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1783 df-cleq 2730 |
This theorem is referenced by: hv2neg 29390 hvaddid2i 29391 hvaddsub4 29440 hilablo 29522 hilid 29523 shunssi 29730 spanunsni 29941 5oalem2 30017 3oalem2 30025 |
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