Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
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 28707 | . . 3 ⊢ 0ℎ ∈ ℋ | |
2 | ax-hvcom 28705 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴)) | |
3 | 1, 2 | mpan2 687 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴)) |
4 | ax-hvaddid 28708 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | |
5 | 3, 4 | eqtr3d 2855 | 1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 (class class class)co 7145 ℋchba 28623 +ℎ cva 28624 0ℎc0v 28628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-9 2115 ax-ext 2790 ax-hvcom 28705 ax-hv0cl 28707 ax-hvaddid 28708 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-cleq 2811 |
This theorem is referenced by: hv2neg 28732 hvaddid2i 28733 hvaddsub4 28782 hilablo 28864 hilid 28865 shunssi 29072 spanunsni 29283 5oalem2 29359 3oalem2 29367 |
Copyright terms: Public domain | W3C validator |