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Theorem cvlatl 36460
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 2821 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2821 . . 3 (le‘𝐾) = (le‘𝐾)
3 eqid 2821 . . 3 (join‘𝐾) = (join‘𝐾)
4 eqid 2821 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
51, 2, 3, 4iscvlat 36458 . 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 500 1 (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wcel 2110  wral 3138   class class class wbr 5065  cfv 6354  (class class class)co 7155  Basecbs 16482  lecple 16571  joincjn 17553  Atomscatm 36398  AtLatcal 36399  CvLatclc 36400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-iota 6313  df-fv 6362  df-ov 7158  df-cvlat 36457
This theorem is referenced by:  cvllat  36461  cvlexch3  36467  cvlexch4N  36468  cvlatexchb1  36469  cvlcvr1  36474  cvlcvrp  36475  cvlatcvr1  36476  cvlsupr2  36478  hlatl  36495
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