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Theorem cvlatl 39306
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 2734 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2734 . . 3 (le‘𝐾) = (le‘𝐾)
3 eqid 2734 . . 3 (join‘𝐾) = (join‘𝐾)
4 eqid 2734 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
51, 2, 3, 4iscvlat 39304 . 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 497 1 (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2105  wral 3058   class class class wbr 5147  cfv 6562  (class class class)co 7430  Basecbs 17244  lecple 17304  joincjn 18368  Atomscatm 39244  AtLatcal 39245  CvLatclc 39246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-ov 7433  df-cvlat 39303
This theorem is referenced by:  cvllat  39307  cvlexch3  39313  cvlexch4N  39314  cvlatexchb1  39315  cvlcvr1  39320  cvlcvrp  39321  cvlatcvr1  39322  cvlsupr2  39324  hlatl  39341
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