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Theorem atlex 33621
 Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 28603 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 2610 . . . . 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 33604 . . . 4 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom (glb‘𝐾) ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
76simp3bi 1071 . . 3 (𝐾 ∈ AtLat → ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥))
8 neeq1 2844 . . . . 5 (𝑥 = 𝑋 → (𝑥0𝑋0 ))
9 breq2 4587 . . . . . 6 (𝑥 = 𝑋 → (𝑦 𝑥𝑦 𝑋))
109rexbidv 3034 . . . . 5 (𝑥 = 𝑋 → (∃𝑦𝐴 𝑦 𝑥 ↔ ∃𝑦𝐴 𝑦 𝑋))
118, 10imbi12d 333 . . . 4 (𝑥 = 𝑋 → ((𝑥0 → ∃𝑦𝐴 𝑦 𝑥) ↔ (𝑋0 → ∃𝑦𝐴 𝑦 𝑋)))
1211rspccv 3279 . . 3 (∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥) → (𝑋𝐵 → (𝑋0 → ∃𝑦𝐴 𝑦 𝑋)))
137, 12syl 17 . 2 (𝐾 ∈ AtLat → (𝑋𝐵 → (𝑋0 → ∃𝑦𝐴 𝑦 𝑋)))
14133imp 1249 1 ((𝐾 ∈ AtLat ∧ 𝑋𝐵𝑋0 ) → ∃𝑦𝐴 𝑦 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   class class class wbr 4583  dom cdm 5038  ‘cfv 5804  Basecbs 15695  lecple 15775  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:  atnle  33622  atlatmstc  33624  cvratlem  33725  cvrat4  33747  2llnmat  33828  2lnat  34088
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