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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlcvl | Structured version Visualization version GIF version |
Description: A Hilbert lattice is an atomic lattice with the covering property. (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
hlcvl | ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlomcmcv 38765 | . 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat)) | |
2 | 1 | simp3d 1142 | 1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 CLatccla 18481 OMLcoml 38584 CvLatclc 38674 HLchlt 38759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-iota 6494 df-fv 6550 df-ov 7417 df-hlat 38760 |
This theorem is referenced by: hlatl 38769 hlexch1 38792 hlexch2 38793 hlexchb1 38794 hlexchb2 38795 hlsupr2 38797 hlexch3 38801 hlexch4N 38802 hlatexchb1 38803 hlatexchb2 38804 hlatexch1 38805 hlatexch2 38806 llnexchb2lem 39278 4atexlemkc 39468 4atex 39486 4atex3 39491 cdleme02N 39632 cdleme0ex2N 39634 cdleme0moN 39635 cdleme0nex 39700 cdleme20zN 39711 cdleme19a 39713 cdleme19d 39716 cdleme21a 39735 cdleme21b 39736 cdleme21c 39737 cdleme21ct 39739 cdleme22f 39756 cdleme22f2 39757 cdleme22g 39758 cdlemf1 39971 |
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