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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 5051 | . 2 ⊢ (𝑥 = 𝐴 → (0ℋ ⋖ℋ 𝑥 ↔ 0ℋ ⋖ℋ 𝐴)) | |
2 | df-at 30094 | . 2 ⊢ HAtoms = {𝑥 ∈ Cℋ ∣ 0ℋ ⋖ℋ 𝑥} | |
3 | 1, 2 | elrab2 3669 | 1 ⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ 0ℋ ⋖ℋ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2114 class class class wbr 5047 Cℋ cch 28685 0ℋc0h 28691 ⋖ℋ ccv 28720 HAtomscat 28721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-rab 3142 df-v 3483 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-nul 4275 df-if 4449 df-sn 4549 df-pr 4551 df-op 4555 df-br 5048 df-at 30094 |
This theorem is referenced by: elat2 30096 elatcv0 30097 atcv0 30098 |
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