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Theorem atl0dm 36591
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 2801 . . 3 (le‘𝐾) = (le‘𝐾)
4 eqid 2801 . . 3 (0.‘𝐾) = (0.‘𝐾)
5 eqid 2801 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
61, 2, 3, 4, 5isatl 36588 . 2 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥 ≠ (0.‘𝐾) → ∃𝑦 ∈ (Atoms‘𝐾)𝑦(le‘𝐾)𝑥)))
76simp2bi 1143 1 (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  wne 2990  wral 3109  wrex 3110   class class class wbr 5033  dom cdm 5523  cfv 6328  Basecbs 16478  lecple 16567  lubclub 17547  glbcglb 17548  0.cp0 17642  Latclat 17650  Atomscatm 36552  AtLatcal 36553
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
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 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-dm 5533  df-iota 6287  df-fv 6336  df-atl 36587
This theorem is referenced by:  atl0cl  36592  atl0le  36593
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