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Theorem atl0dm 38774
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 2728 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
4 eqid 2728 . . 3 (0.β€˜πΎ) = (0.β€˜πΎ)
5 eqid 2728 . . 3 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
61, 2, 3, 4, 5isatl 38771 . 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 1144 1 (𝐾 ∈ AtLat β†’ 𝐡 ∈ dom 𝐺)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534   ∈ wcel 2099   β‰  wne 2937  βˆ€wral 3058  βˆƒwrex 3067   class class class wbr 5148  dom cdm 5678  β€˜cfv 6548  Basecbs 17180  lecple 17240  lubclub 18301  glbcglb 18302  0.cp0 18415  Latclat 18423  Atomscatm 38735  AtLatcal 38736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-dm 5688  df-iota 6500  df-fv 6556  df-atl 38770
This theorem is referenced by:  atl0cl  38775  atl0le  38776
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