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Mathbox for Norm Megill |
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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 2799 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2799 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | eqid 2799 | . . 3 ⊢ (lt‘𝐾) = (lt‘𝐾) | |
4 | eqid 2799 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
5 | eqid 2799 | . . 3 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
6 | eqid 2799 | . . 3 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
7 | eqid 2799 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ishlat1 35373 | . 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 492 | 1 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∧ w3a 1108 ∈ wcel 2157 ≠ wne 2971 ∀wral 3089 ∃wrex 3090 class class class wbr 4843 ‘cfv 6101 (class class class)co 6878 Basecbs 16184 lecple 16274 ltcplt 17256 joincjn 17259 0.cp0 17352 1.cp1 17353 CLatccla 17422 OMLcoml 35196 Atomscatm 35284 CvLatclc 35286 HLchlt 35371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-iota 6064 df-fv 6109 df-ov 6881 df-hlat 35372 |
This theorem is referenced by: hloml 35378 hlclat 35379 hlcvl 35380 cvr1 35431 cvrp 35437 atcvr1 35438 atcvr2 35439 |
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