Theorem hlcvl 40232

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 40229	. 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
2	1	simp3d 1162	1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  CLatccla 18586  OMLcoml 40048  CvLatclc 40138  HLchlt 40223

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-ov 7416  df-hlat 40224

This theorem is used by:  hlatl  40233  hlexch1  40255  hlexch2  40256  hlexchb1  40257  hlexchb2  40258  hlsupr2  40260  hlexch3  40264  hlexch4N  40265  hlatexchb1  40266  hlatexchb2  40267  hlatexch1  40268  hlatexch2  40269  llnexchb2lem  40741  4atexlemkc  40931  4atex  40949  4atex3  40954  cdleme02N  41095  cdleme0ex2N  41097  cdleme0moN  41098  cdleme0nex  41163  cdleme20zN  41174  cdleme19a  41176  cdleme19d  41179  cdleme21a  41198  cdleme21b  41199  cdleme21c  41200  cdleme21ct  41202  cdleme22f  41219  cdleme22f2  41220  cdleme22g  41221  cdlemf1  41434