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Theorem atllat 39282
Description: An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.)
Assertion
Ref Expression
atllat (𝐾 ∈ AtLat → 𝐾 ∈ Lat)

Proof of Theorem atllat
Dummy variables 𝑥 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2735 . . 3 (glb‘𝐾) = (glb‘𝐾)
3 eqid 2735 . . 3 (le‘𝐾) = (le‘𝐾)
4 eqid 2735 . . 3 (0.‘𝐾) = (0.‘𝐾)
5 eqid 2735 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
61, 2, 3, 4, 5isatl 39281 . 2 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ (Base‘𝐾) ∈ dom (glb‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)(𝑥 ≠ (0.‘𝐾) → ∃𝑝 ∈ (Atoms‘𝐾)𝑝(le‘𝐾)𝑥)))
76simp1bi 1144 1 (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wne 2938  wral 3059  wrex 3068   class class class wbr 5148  dom cdm 5689  cfv 6563  Basecbs 17245  lecple 17305  glbcglb 18368  0.cp0 18481  Latclat 18489  Atomscatm 39245  AtLatcal 39246
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-dm 5699  df-iota 6516  df-fv 6571  df-atl 39280
This theorem is referenced by:  atlpos  39283  atnle  39299  atlatmstc  39301  cvllat  39308  hllat  39345  snatpsubN  39733
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