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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 4877 | . 2 ⊢ (𝑥 = 𝐴 → (0ℋ ⋖ℋ 𝑥 ↔ 0ℋ ⋖ℋ 𝐴)) | |
2 | df-at 29752 | . 2 ⊢ HAtoms = {𝑥 ∈ Cℋ ∣ 0ℋ ⋖ℋ 𝑥} | |
3 | 1, 2 | elrab2 3589 | 1 ⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ 0ℋ ⋖ℋ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 ∈ wcel 2166 class class class wbr 4873 Cℋ cch 28341 0ℋc0h 28347 ⋖ℋ ccv 28376 HAtomscat 28377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-at 29752 |
This theorem is referenced by: elat2 29754 elatcv0 29755 atcv0 29756 |
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