Theorem hlcvl 39361

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 39358	. 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
2	1	simp3d 1144	1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat)

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