Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atl0dm Structured version   Visualization version   GIF version

Theorem atl0dm 33607
 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 2610 . . 3 (le‘𝐾) = (le‘𝐾)
4 eqid 2610 . . 3 (0.‘𝐾) = (0.‘𝐾)
5 eqid 2610 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
61, 2, 3, 4, 5isatl 33604 . 2 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥 ≠ (0.‘𝐾) → ∃𝑦 ∈ (Atoms‘𝐾)𝑦(le‘𝐾)𝑥)))
76simp2bi 1070 1 (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   class class class wbr 4583  dom cdm 5038  ‘cfv 5804  Basecbs 15695  lecple 15775  lubclub 16765  glbcglb 16766  0.cp0 16860  Latclat 16868  Atomscatm 33568  AtLatcal 33569 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-dm 5048  df-iota 5768  df-fv 5812  df-atl 33603 This theorem is referenced by:  atl0cl  33608  atl0le  33609
 Copyright terms: Public domain W3C validator