Theorem hlcvl 39341

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

Step	Hyp	Ref	Expression
1		hlomcmcv 39338	. 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
2	1	simp3d 1143	1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat)

Syntax hints:   → wi 4   ∈ wcel 2106  CLatccla 18556  OMLcoml 39157  CvLatclc 39247  HLchlt 39332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-hlat 39333

This theorem is referenced by:  hlatl  39342  hlexch1  39365  hlexch2  39366  hlexchb1  39367  hlexchb2  39368  hlsupr2  39370  hlexch3  39374  hlexch4N  39375  hlatexchb1  39376  hlatexchb2  39377  hlatexch1  39378  hlatexch2  39379  llnexchb2lem  39851  4atexlemkc  40041  4atex  40059  4atex3  40064  cdleme02N  40205  cdleme0ex2N  40207  cdleme0moN  40208  cdleme0nex  40273  cdleme20zN  40284  cdleme19a  40286  cdleme19d  40289  cdleme21a  40308  cdleme21b  40309  cdleme21c  40310  cdleme21ct  40312  cdleme22f  40329  cdleme22f2  40330  cdleme22g  40331  cdlemf1  40544