Theorem hlcvl 39352

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 39349	. 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
2	1	simp3d 1144	1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat)

Syntax hints:   → wi 4   ∈ wcel 2109  CLatccla 18457  OMLcoml 39168  CvLatclc 39258  HLchlt 39343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-hlat 39344

This theorem is referenced by:  hlatl  39353  hlexch1  39376  hlexch2  39377  hlexchb1  39378  hlexchb2  39379  hlsupr2  39381  hlexch3  39385  hlexch4N  39386  hlatexchb1  39387  hlatexchb2  39388  hlatexch1  39389  hlatexch2  39390  llnexchb2lem  39862  4atexlemkc  40052  4atex  40070  4atex3  40075  cdleme02N  40216  cdleme0ex2N  40218  cdleme0moN  40219  cdleme0nex  40284  cdleme20zN  40295  cdleme19a  40297  cdleme19d  40300  cdleme21a  40319  cdleme21b  40320  cdleme21c  40321  cdleme21ct  40323  cdleme22f  40340  cdleme22f2  40341  cdleme22g  40342  cdlemf1  40555