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Theorem atl0dm 39303
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 2737 . . 3 (le‘𝐾) = (le‘𝐾)
4 eqid 2737 . . 3 (0.‘𝐾) = (0.‘𝐾)
5 eqid 2737 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
61, 2, 3, 4, 5isatl 39300 . 2 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥 ≠ (0.‘𝐾) → ∃𝑦 ∈ (Atoms‘𝐾)𝑦(le‘𝐾)𝑥)))
76simp2bi 1147 1 (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070   class class class wbr 5143  dom cdm 5685  cfv 6561  Basecbs 17247  lecple 17304  lubclub 18355  glbcglb 18356  0.cp0 18468  Latclat 18476  Atomscatm 39264  AtLatcal 39265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-dm 5695  df-iota 6514  df-fv 6569  df-atl 39299
This theorem is referenced by:  atl0cl  39304  atl0le  39305
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