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Theorem isatl 33604
 Description: The predicate "is an atomic lattice." Every nonzero element is less than or equal to an atom. (Contributed by NM, 18-Sep-2011.) (Revised by NM, 14-Sep-2018.)
Hypotheses
Ref Expression
isatlat.b 𝐵 = (Base‘𝐾)
isatlat.g 𝐺 = (glb‘𝐾)
isatlat.l = (le‘𝐾)
isatlat.z 0 = (0.‘𝐾)
isatlat.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
isatl (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦,𝐾
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐺(𝑥,𝑦)   (𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem isatl
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6103 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2 isatlat.b . . . . . 6 𝐵 = (Base‘𝐾)
31, 2syl6eqr 2662 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4 fveq2 6103 . . . . . . 7 (𝑘 = 𝐾 → (glb‘𝑘) = (glb‘𝐾))
5 isatlat.g . . . . . . 7 𝐺 = (glb‘𝐾)
64, 5syl6eqr 2662 . . . . . 6 (𝑘 = 𝐾 → (glb‘𝑘) = 𝐺)
76dmeqd 5248 . . . . 5 (𝑘 = 𝐾 → dom (glb‘𝑘) = dom 𝐺)
83, 7eleq12d 2682 . . . 4 (𝑘 = 𝐾 → ((Base‘𝑘) ∈ dom (glb‘𝑘) ↔ 𝐵 ∈ dom 𝐺))
9 fveq2 6103 . . . . . . . 8 (𝑘 = 𝐾 → (0.‘𝑘) = (0.‘𝐾))
10 isatlat.z . . . . . . . 8 0 = (0.‘𝐾)
119, 10syl6eqr 2662 . . . . . . 7 (𝑘 = 𝐾 → (0.‘𝑘) = 0 )
1211neeq2d 2842 . . . . . 6 (𝑘 = 𝐾 → (𝑥 ≠ (0.‘𝑘) ↔ 𝑥0 ))
13 fveq2 6103 . . . . . . . 8 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
14 isatlat.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
1513, 14syl6eqr 2662 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
16 fveq2 6103 . . . . . . . . 9 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
17 isatlat.l . . . . . . . . 9 = (le‘𝐾)
1816, 17syl6eqr 2662 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = )
1918breqd 4594 . . . . . . 7 (𝑘 = 𝐾 → (𝑦(le‘𝑘)𝑥𝑦 𝑥))
2015, 19rexeqbidv 3130 . . . . . 6 (𝑘 = 𝐾 → (∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥 ↔ ∃𝑦𝐴 𝑦 𝑥))
2112, 20imbi12d 333 . . . . 5 (𝑘 = 𝐾 → ((𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥) ↔ (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
223, 21raleqbidv 3129 . . . 4 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥) ↔ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
238, 22anbi12d 743 . . 3 (𝑘 = 𝐾 → (((Base‘𝑘) ∈ dom (glb‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥)) ↔ (𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥))))
24 df-atl 33603 . . 3 AtLat = {𝑘 ∈ Lat ∣ ((Base‘𝑘) ∈ dom (glb‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥))}
2523, 24elrab2 3333 . 2 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ (𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥))))
26 3anass 1035 . 2 ((𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)) ↔ (𝐾 ∈ Lat ∧ (𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥))))
2725, 26bitr4i 266 1 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ 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:  atllat  33605  atl0dm  33607  atlex  33621
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