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Theorem atl0dm 36318
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 2818 . . 3 (le‘𝐾) = (le‘𝐾)
4 eqid 2818 . . 3 (0.‘𝐾) = (0.‘𝐾)
5 eqid 2818 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
61, 2, 3, 4, 5isatl 36315 . 2 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥 ≠ (0.‘𝐾) → ∃𝑦 ∈ (Atoms‘𝐾)𝑦(le‘𝐾)𝑥)))
76simp2bi 1138 1 (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136   class class class wbr 5057  dom cdm 5548  cfv 6348  Basecbs 16471  lecple 16560  lubclub 17540  glbcglb 17541  0.cp0 17635  Latclat 17643  Atomscatm 36279  AtLatcal 36280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-dm 5558  df-iota 6307  df-fv 6356  df-atl 36314
This theorem is referenced by:  atl0cl  36319  atl0le  36320
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