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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 5151 | . 2 ⊢ (𝑥 = 𝐴 → (0ℋ ⋖ℋ 𝑥 ↔ 0ℋ ⋖ℋ 𝐴)) | |
2 | df-at 31569 | . 2 ⊢ HAtoms = {𝑥 ∈ Cℋ ∣ 0ℋ ⋖ℋ 𝑥} | |
3 | 1, 2 | elrab2 3685 | 1 ⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ 0ℋ ⋖ℋ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∈ wcel 2107 class class class wbr 5147 Cℋ cch 30160 0ℋc0h 30166 ⋖ℋ ccv 30195 HAtomscat 30196 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-at 31569 |
This theorem is referenced by: elat2 31571 elatcv0 31572 atcv0 31573 |
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