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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 2818 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2818 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | eqid 2818 | . . 3 ⊢ (lt‘𝐾) = (lt‘𝐾) | |
4 | eqid 2818 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
5 | eqid 2818 | . . 3 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
6 | eqid 2818 | . . 3 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
7 | eqid 2818 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ishlat1 36368 | . 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 498 | 1 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ∃wrex 3136 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 lecple 16560 ltcplt 17539 joincjn 17542 0.cp0 17635 1.cp1 17636 CLatccla 17705 OMLcoml 36191 Atomscatm 36279 CvLatclc 36281 HLchlt 36366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-hlat 36367 |
This theorem is referenced by: hloml 36373 hlclat 36374 hlcvl 36375 cvr1 36426 cvrp 36432 atcvr1 36433 atcvr2 36434 |
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