Theorem hlcvl 39315

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 39312	. 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
2	1	simp3d 1144	1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat)

Syntax hints:   → wi 4   ∈ wcel 2108  CLatccla 18568  OMLcoml 39131  CvLatclc 39221  HLchlt 39306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451  df-hlat 39307

This theorem is referenced by:  hlatl  39316  hlexch1  39339  hlexch2  39340  hlexchb1  39341  hlexchb2  39342  hlsupr2  39344  hlexch3  39348  hlexch4N  39349  hlatexchb1  39350  hlatexchb2  39351  hlatexch1  39352  hlatexch2  39353  llnexchb2lem  39825  4atexlemkc  40015  4atex  40033  4atex3  40038  cdleme02N  40179  cdleme0ex2N  40181  cdleme0moN  40182  cdleme0nex  40247  cdleme20zN  40258  cdleme19a  40260  cdleme19d  40263  cdleme21a  40282  cdleme21b  40283  cdleme21c  40284  cdleme21ct  40286  cdleme22f  40303  cdleme22f2  40304  cdleme22g  40305  cdlemf1  40518