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

Theorem atlex 37097
Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 30468 analog.) (Contributed by NM, 21-Oct-2011.)
Hypotheses
Ref Expression
atlex.b 𝐵 = (Base‘𝐾)
atlex.l = (le‘𝐾)
atlex.z 0 = (0.‘𝐾)
atlex.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
atlex ((𝐾 ∈ AtLat ∧ 𝑋𝐵𝑋0 ) → ∃𝑦𝐴 𝑦 𝑋)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐾   𝑦,𝑋
Allowed substitution hints:   𝐵(𝑦)   (𝑦)   0 (𝑦)

Proof of Theorem atlex
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 atlex.b . . . . 5 𝐵 = (Base‘𝐾)
2 eqid 2738 . . . . 5 (glb‘𝐾) = (glb‘𝐾)
3 atlex.l . . . . 5 = (le‘𝐾)
4 atlex.z . . . . 5 0 = (0.‘𝐾)
5 atlex.a . . . . 5 𝐴 = (Atoms‘𝐾)
61, 2, 3, 4, 5isatl 37080 . . . 4 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom (glb‘𝐾) ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
76simp3bi 1149 . . 3 (𝐾 ∈ AtLat → ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥))
8 neeq1 3004 . . . . 5 (𝑥 = 𝑋 → (𝑥0𝑋0 ))
9 breq2 5072 . . . . . 6 (𝑥 = 𝑋 → (𝑦 𝑥𝑦 𝑋))
109rexbidv 3224 . . . . 5 (𝑥 = 𝑋 → (∃𝑦𝐴 𝑦 𝑥 ↔ ∃𝑦𝐴 𝑦 𝑋))
118, 10imbi12d 348 . . . 4 (𝑥 = 𝑋 → ((𝑥0 → ∃𝑦𝐴 𝑦 𝑥) ↔ (𝑋0 → ∃𝑦𝐴 𝑦 𝑋)))
1211rspccv 3547 . . 3 (∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥) → (𝑋𝐵 → (𝑋0 → ∃𝑦𝐴 𝑦 𝑋)))
137, 12syl 17 . 2 (𝐾 ∈ AtLat → (𝑋𝐵 → (𝑋0 → ∃𝑦𝐴 𝑦 𝑋)))
14133imp 1113 1 ((𝐾 ∈ AtLat ∧ 𝑋𝐵𝑋0 ) → ∃𝑦𝐴 𝑦 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1089   = wceq 1543  wcel 2111  wne 2941  wral 3062  wrex 3063   class class class wbr 5068  dom cdm 5566  cfv 6398  Basecbs 16788  lecple 16837  glbcglb 17845  0.cp0 17957  Latclat 17965  Atomscatm 37044  AtLatcal 37045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2942  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4835  df-br 5069  df-dm 5576  df-iota 6356  df-fv 6406  df-atl 37079
This theorem is referenced by:  atnle  37098  atlatmstc  37100  cvratlem  37202  cvrat4  37224  2llnmat  37305  2lnat  37565
  Copyright terms: Public domain W3C validator