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Theorem cvlatl 35400
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 2825 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2825 . . 3 (le‘𝐾) = (le‘𝐾)
3 eqid 2825 . . 3 (join‘𝐾) = (join‘𝐾)
4 eqid 2825 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
51, 2, 3, 4iscvlat 35398 . 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 493 1 (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  wcel 2166  wral 3117   class class class wbr 4873  cfv 6123  (class class class)co 6905  Basecbs 16222  lecple 16312  joincjn 17297  Atomscatm 35338  AtLatcal 35339  CvLatclc 35340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131  df-ov 6908  df-cvlat 35397
This theorem is referenced by:  cvllat  35401  cvlexch3  35407  cvlexch4N  35408  cvlatexchb1  35409  cvlcvr1  35414  cvlcvrp  35415  cvlatcvr1  35416  cvlsupr2  35418  hlatl  35435
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