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Theorem atl0dm 37810
Description: Condition necessary for zero element to exist. (Contributed by NM, 14-Sep-2018.)
Hypotheses
Ref Expression
atl01dm.b 𝐡 = (Baseβ€˜πΎ)
atl01dm.u π‘ˆ = (lubβ€˜πΎ)
atl01dm.g 𝐺 = (glbβ€˜πΎ)
Assertion
Ref Expression
atl0dm (𝐾 ∈ AtLat β†’ 𝐡 ∈ dom 𝐺)

Proof of Theorem atl0dm
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 atl01dm.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 atl01dm.g . . 3 𝐺 = (glbβ€˜πΎ)
3 eqid 2733 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
4 eqid 2733 . . 3 (0.β€˜πΎ) = (0.β€˜πΎ)
5 eqid 2733 . . 3 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
61, 2, 3, 4, 5isatl 37807 . 2 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐡 ∈ dom 𝐺 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ β‰  (0.β€˜πΎ) β†’ βˆƒπ‘¦ ∈ (Atomsβ€˜πΎ)𝑦(leβ€˜πΎ)π‘₯)))
76simp2bi 1147 1 (𝐾 ∈ AtLat β†’ 𝐡 ∈ dom 𝐺)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070   class class class wbr 5106  dom cdm 5634  β€˜cfv 6497  Basecbs 17088  lecple 17145  lubclub 18203  glbcglb 18204  0.cp0 18317  Latclat 18325  Atomscatm 37771  AtLatcal 37772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-dm 5644  df-iota 6449  df-fv 6505  df-atl 37806
This theorem is referenced by:  atl0cl  37811  atl0le  37812
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