Theorem hlcvl 33664

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 33661	. 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
2	1	simp3d 1068	1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat)

Syntax hints:   → wi 4   ∈ wcel 1977  CLatccla 16930  OMLcoml 33480  CvLatclc 33570  HLchlt 33655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-hlat 33656

This theorem is referenced by:  hlatl  33665  hlexch1  33686  hlexch2  33687  hlexchb1  33688  hlexchb2  33689  hlsupr2  33691  hlexch3  33695  hlexch4N  33696  hlatexchb1  33697  hlatexchb2  33698  hlatexch1  33699  hlatexch2  33700  llnexchb2lem  34172  4atexlemkc  34362  4atex  34380  4atex3  34385  cdleme02N  34527  cdleme0ex2N  34529  cdleme0moN  34530  cdleme0nex  34595  cdleme20zN  34606  cdleme20yOLD  34608  cdleme19a  34609  cdleme19d  34612  cdleme21a  34631  cdleme21b  34632  cdleme21c  34633  cdleme21ct  34635  cdleme22f  34652  cdleme22f2  34653  cdleme22g  34654  cdlemf1  34867