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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlomcmcv | Structured version Visualization version GIF version |
Description: A Hilbert lattice is orthomodular, complete, and has the covering (exchange) property. (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
hlomcmcv | ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2738 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | eqid 2738 | . . 3 ⊢ (lt‘𝐾) = (lt‘𝐾) | |
4 | eqid 2738 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
5 | eqid 2738 | . . 3 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
6 | eqid 2738 | . . 3 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
7 | eqid 2738 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ishlat1 37129 | . 2 ⊢ (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ (∀𝑥 ∈ (Atoms‘𝐾)∀𝑦 ∈ (Atoms‘𝐾)(𝑥 ≠ 𝑦 → ∃𝑧 ∈ (Atoms‘𝐾)(𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧(le‘𝐾)(𝑥(join‘𝐾)𝑦))) ∧ ∃𝑥 ∈ (Base‘𝐾)∃𝑦 ∈ (Base‘𝐾)∃𝑧 ∈ (Base‘𝐾)(((0.‘𝐾)(lt‘𝐾)𝑥 ∧ 𝑥(lt‘𝐾)𝑦) ∧ (𝑦(lt‘𝐾)𝑧 ∧ 𝑧(lt‘𝐾)(1.‘𝐾)))))) |
9 | 8 | simplbi 501 | 1 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2111 ≠ wne 2941 ∀wral 3062 ∃wrex 3063 class class class wbr 5067 ‘cfv 6397 (class class class)co 7231 Basecbs 16784 lecple 16833 ltcplt 17839 joincjn 17842 0.cp0 17953 1.cp1 17954 CLatccla 18028 OMLcoml 36952 Atomscatm 37040 CvLatclc 37042 HLchlt 37127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-ext 2709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3422 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-br 5068 df-iota 6355 df-fv 6405 df-ov 7234 df-hlat 37128 |
This theorem is referenced by: hloml 37134 hlclat 37135 hlcvl 37136 cvr1 37187 cvrp 37193 atcvr1 37194 atcvr2 37195 |
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