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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 39358 | . 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat)) | |
| 2 | 1 | simp3d 1144 | 1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 CLatccla 18544 OMLcoml 39177 CvLatclc 39267 HLchlt 39352 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-hlat 39353 | 
| This theorem is referenced by: hlatl 39362 hlexch1 39385 hlexch2 39386 hlexchb1 39387 hlexchb2 39388 hlsupr2 39390 hlexch3 39394 hlexch4N 39395 hlatexchb1 39396 hlatexchb2 39397 hlatexch1 39398 hlatexch2 39399 llnexchb2lem 39871 4atexlemkc 40061 4atex 40079 4atex3 40084 cdleme02N 40225 cdleme0ex2N 40227 cdleme0moN 40228 cdleme0nex 40293 cdleme20zN 40304 cdleme19a 40306 cdleme19d 40309 cdleme21a 40328 cdleme21b 40329 cdleme21c 40330 cdleme21ct 40332 cdleme22f 40349 cdleme22f2 40350 cdleme22g 40351 cdlemf1 40564 | 
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