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

Theorem atl0dm 38676
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 2724 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
4 eqid 2724 . . 3 (0.β€˜πΎ) = (0.β€˜πΎ)
5 eqid 2724 . . 3 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
61, 2, 3, 4, 5isatl 38673 . 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 1143 1 (𝐾 ∈ AtLat β†’ 𝐡 ∈ dom 𝐺)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098   β‰  wne 2932  βˆ€wral 3053  βˆƒwrex 3062   class class class wbr 5139  dom cdm 5667  β€˜cfv 6534  Basecbs 17149  lecple 17209  lubclub 18270  glbcglb 18271  0.cp0 18384  Latclat 18392  Atomscatm 38637  AtLatcal 38638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-dm 5677  df-iota 6486  df-fv 6542  df-atl 38672
This theorem is referenced by:  atl0cl  38677  atl0le  38678
  Copyright terms: Public domain W3C validator