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Theorem atllat 36554
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 2822 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2822 . . 3 (glb‘𝐾) = (glb‘𝐾)
3 eqid 2822 . . 3 (le‘𝐾) = (le‘𝐾)
4 eqid 2822 . . 3 (0.‘𝐾) = (0.‘𝐾)
5 eqid 2822 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
61, 2, 3, 4, 5isatl 36553 . 2 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ (Base‘𝐾) ∈ dom (glb‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)(𝑥 ≠ (0.‘𝐾) → ∃𝑝 ∈ (Atoms‘𝐾)𝑝(le‘𝐾)𝑥)))
76simp1bi 1142 1 (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  wne 3011  wral 3130  wrex 3131   class class class wbr 5042  dom cdm 5532  cfv 6334  Basecbs 16474  lecple 16563  glbcglb 17544  0.cp0 17638  Latclat 17646  Atomscatm 36517  AtLatcal 36518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-un 3913  df-in 3915  df-ss 3925  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-dm 5542  df-iota 6293  df-fv 6342  df-atl 36552
This theorem is referenced by:  atlpos  36555  atnle  36571  atlatmstc  36573  cvllat  36580  hllat  36617  snatpsubN  37004
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