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Theorem hlcvl 38768
Description: A Hilbert lattice is an atomic lattice with the covering property. (Contributed by NM, 5-Nov-2012.)
Assertion
Ref Expression
hlcvl (𝐾 ∈ HL → 𝐾 ∈ CvLat)

Proof of Theorem hlcvl
StepHypRef Expression
1 hlomcmcv 38765 . 2 (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
21simp3d 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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