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Theorem cvlatl 33630
 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.
StepHypRef Expression
1 eqid 2610 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2610 . . 3 (le‘𝐾) = (le‘𝐾)
3 eqid 2610 . . 3 (join‘𝐾) = (join‘𝐾)
4 eqid 2610 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
51, 2, 3, 4iscvlat 33628 . 2 (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)∀𝑥 ∈ (Base‘𝐾)((¬ 𝑝(le‘𝐾)𝑥𝑝(le‘𝐾)(𝑥(join‘𝐾)𝑞)) → 𝑞(le‘𝐾)(𝑥(join‘𝐾)𝑝))))
65simplbi 475 1 (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  Atomscatm 33568  AtLatcal 33569  CvLatclc 33570 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-cvlat 33627 This theorem is referenced by:  cvllat  33631  cvlexch3  33637  cvlexch4N  33638  cvlatexchb1  33639  cvlcvr1  33644  cvlcvrp  33645  cvlatcvr1  33646  cvlsupr2  33648  hlatl  33665
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