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Theorem hlcvl 35429
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 35426 . 2 (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
21simp3d 1178 1 (𝐾 ∈ HL → 𝐾 ∈ CvLat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2164  CLatccla 17467  OMLcoml 35245  CvLatclc 35335  HLchlt 35420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-iota 6090  df-fv 6135  df-ov 6913  df-hlat 35421
This theorem is referenced by:  hlatl  35430  hlexch1  35452  hlexch2  35453  hlexchb1  35454  hlexchb2  35455  hlsupr2  35457  hlexch3  35461  hlexch4N  35462  hlatexchb1  35463  hlatexchb2  35464  hlatexch1  35465  hlatexch2  35466  llnexchb2lem  35938  4atexlemkc  36128  4atex  36146  4atex3  36151  cdleme02N  36292  cdleme0ex2N  36294  cdleme0moN  36295  cdleme0nex  36360  cdleme20zN  36371  cdleme19a  36373  cdleme19d  36376  cdleme21a  36395  cdleme21b  36396  cdleme21c  36397  cdleme21ct  36399  cdleme22f  36416  cdleme22f2  36417  cdleme22g  36418  cdlemf1  36631
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