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Theorem cvlatl 39326
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 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)
51, 2, 3, 4iscvlat 39324 . 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 2108  wral 3061   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  Atomscatm 39264  AtLatcal 39265  CvLatclc 39266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-cvlat 39323
This theorem is referenced by:  cvllat  39327  cvlexch3  39333  cvlexch4N  39334  cvlatexchb1  39335  cvlcvr1  39340  cvlcvrp  39341  cvlatcvr1  39342  cvlsupr2  39344  hlatl  39361
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