Theorem hlcvl 39819

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 39816	. 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 18455  OMLcoml 39635  CvLatclc 39725  HLchlt 39810

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 2709

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 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6448  df-fv 6500  df-ov 7363  df-hlat 39811

This theorem is referenced by:  hlatl  39820  hlexch1  39842  hlexch2  39843  hlexchb1  39844  hlexchb2  39845  hlsupr2  39847  hlexch3  39851  hlexch4N  39852  hlatexchb1  39853  hlatexchb2  39854  hlatexch1  39855  hlatexch2  39856  llnexchb2lem  40328  4atexlemkc  40518  4atex  40536  4atex3  40541  cdleme02N  40682  cdleme0ex2N  40684  cdleme0moN  40685  cdleme0nex  40750  cdleme20zN  40761  cdleme19a  40763  cdleme19d  40766  cdleme21a  40785  cdleme21b  40786  cdleme21c  40787  cdleme21ct  40789  cdleme22f  40806  cdleme22f2  40807  cdleme22g  40808  cdlemf1  41021