Theorem cvlatl 39695

Description: An atomic lattice with the covering property is an atomic lattice. (Contributed by NM, 5-Nov-2012.)

Assertion

Ref	Expression
cvlatl	⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)

Proof of Theorem cvlatl

Dummy variables 𝑞 𝑝 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2737	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2737	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2737	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5	1, 2, 3, 4	iscvlat 39693	. 2 ⊢ (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)∀𝑥 ∈ (Base‘𝐾)((¬ 𝑝(le‘𝐾)𝑥 ∧ 𝑝(le‘𝐾)(𝑥(join‘𝐾)𝑞)) → 𝑞(le‘𝐾)(𝑥(join‘𝐾)𝑝))))
6	5	simplbi 496	1 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2114  ∀wral 3052   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  lecple 17196  joincjn 18246  Atomscatm 39633  AtLatcal 39634  CvLatclc 39635

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-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-cvlat 39692

This theorem is referenced by:  cvllat  39696  cvlexch3  39702  cvlexch4N  39703  cvlatexchb1  39704  cvlcvr1  39709  cvlcvrp  39710  cvlatcvr1  39711  cvlsupr2  39713  hlatl  39730