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Theorem atl0dm 35920
Description: Condition necessary for zero element to exist. (Contributed by NM, 14-Sep-2018.)
Hypotheses
Ref Expression
atl01dm.b 𝐵 = (Base‘𝐾)
atl01dm.u 𝑈 = (lub‘𝐾)
atl01dm.g 𝐺 = (glb‘𝐾)
Assertion
Ref Expression
atl0dm (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺)

Proof of Theorem atl0dm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 atl01dm.b . . 3 𝐵 = (Base‘𝐾)
2 atl01dm.g . . 3 𝐺 = (glb‘𝐾)
3 eqid 2771 . . 3 (le‘𝐾) = (le‘𝐾)
4 eqid 2771 . . 3 (0.‘𝐾) = (0.‘𝐾)
5 eqid 2771 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
61, 2, 3, 4, 5isatl 35917 . 2 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥 ≠ (0.‘𝐾) → ∃𝑦 ∈ (Atoms‘𝐾)𝑦(le‘𝐾)𝑥)))
76simp2bi 1127 1 (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1508  wcel 2051  wne 2960  wral 3081  wrex 3082   class class class wbr 4925  dom cdm 5403  cfv 6185  Basecbs 16337  lecple 16426  lubclub 17422  glbcglb 17423  0.cp0 17517  Latclat 17525  Atomscatm 35881  AtLatcal 35882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-dm 5413  df-iota 6149  df-fv 6193  df-atl 35916
This theorem is referenced by:  atl0cl  35921  atl0le  35922
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