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Theorem cvlatl 34438
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 2621 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2621 . . 3 (le‘𝐾) = (le‘𝐾)
3 eqid 2621 . . 3 (join‘𝐾) = (join‘𝐾)
4 eqid 2621 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
51, 2, 3, 4iscvlat 34436 . 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 476 1 (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wcel 1989  wral 2911   class class class wbr 4651  cfv 5886  (class class class)co 6647  Basecbs 15851  lecple 15942  joincjn 16938  Atomscatm 34376  AtLatcal 34377  CvLatclc 34378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-iota 5849  df-fv 5894  df-ov 6650  df-cvlat 34435
This theorem is referenced by:  cvllat  34439  cvlexch3  34445  cvlexch4N  34446  cvlatexchb1  34447  cvlcvr1  34452  cvlcvrp  34453  cvlatcvr1  34454  cvlsupr2  34456  hlatl  34473
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