Theorem hlcvl 39805

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 39802	. 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
2	1	simp3d 1145	1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat)

Syntax hints:   → wi 4   ∈ wcel 2114  CLatccla 18464  OMLcoml 39621  CvLatclc 39711  HLchlt 39796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-hlat 39797

This theorem is referenced by:  hlatl  39806  hlexch1  39828  hlexch2  39829  hlexchb1  39830  hlexchb2  39831  hlsupr2  39833  hlexch3  39837  hlexch4N  39838  hlatexchb1  39839  hlatexchb2  39840  hlatexch1  39841  hlatexch2  39842  llnexchb2lem  40314  4atexlemkc  40504  4atex  40522  4atex3  40527  cdleme02N  40668  cdleme0ex2N  40670  cdleme0moN  40671  cdleme0nex  40736  cdleme20zN  40747  cdleme19a  40749  cdleme19d  40752  cdleme21a  40771  cdleme21b  40772  cdleme21c  40773  cdleme21ct  40775  cdleme22f  40792  cdleme22f2  40793  cdleme22g  40794  cdlemf1  41007