| Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > HSE Home > Th. List > ela | Structured version Visualization version GIF version | ||
| Description: Atoms in a Hilbert lattice are the elements that cover the zero subspace. Definition of atom in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| ela | ⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ 0ℋ ⋖ℋ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | breq2 5147 | . 2 ⊢ (𝑥 = 𝐴 → (0ℋ ⋖ℋ 𝑥 ↔ 0ℋ ⋖ℋ 𝐴)) | |
| 2 | df-at 32357 | . 2 ⊢ HAtoms = {𝑥 ∈ Cℋ ∣ 0ℋ ⋖ℋ 𝑥} | |
| 3 | 1, 2 | elrab2 3695 | 1 ⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ 0ℋ ⋖ℋ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 Cℋ cch 30948 0ℋc0h 30954 ⋖ℋ ccv 30983 HAtomscat 30984 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-at 32357 |
| This theorem is referenced by: elat2 32359 elatcv0 32360 atcv0 32361 |
| Copyright terms: Public domain | W3C validator |