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Theorem iscvlat 33628
 Description: The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
iscvlat.b 𝐵 = (Base‘𝐾)
iscvlat.l = (le‘𝐾)
iscvlat.j = (join‘𝐾)
iscvlat.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
iscvlat (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝𝐴𝑞𝐴𝑥𝐵 ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝))))
Distinct variable groups:   𝑞,𝑝,𝐴   𝑥,𝐵   𝑥,𝑝,𝐾,𝑞
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑞,𝑝)   (𝑥,𝑞,𝑝)   (𝑥,𝑞,𝑝)

Proof of Theorem iscvlat
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6103 . . . 4 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
2 iscvlat.a . . . 4 𝐴 = (Atoms‘𝐾)
31, 2syl6eqr 2662 . . 3 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
4 fveq2 6103 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
5 iscvlat.b . . . . . 6 𝐵 = (Base‘𝐾)
64, 5syl6eqr 2662 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
7 fveq2 6103 . . . . . . . . . 10 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
8 iscvlat.l . . . . . . . . . 10 = (le‘𝐾)
97, 8syl6eqr 2662 . . . . . . . . 9 (𝑘 = 𝐾 → (le‘𝑘) = )
109breqd 4594 . . . . . . . 8 (𝑘 = 𝐾 → (𝑝(le‘𝑘)𝑥𝑝 𝑥))
1110notbid 307 . . . . . . 7 (𝑘 = 𝐾 → (¬ 𝑝(le‘𝑘)𝑥 ↔ ¬ 𝑝 𝑥))
12 eqidd 2611 . . . . . . . 8 (𝑘 = 𝐾𝑝 = 𝑝)
13 fveq2 6103 . . . . . . . . . 10 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
14 iscvlat.j . . . . . . . . . 10 = (join‘𝐾)
1513, 14syl6eqr 2662 . . . . . . . . 9 (𝑘 = 𝐾 → (join‘𝑘) = )
1615oveqd 6566 . . . . . . . 8 (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑞) = (𝑥 𝑞))
1712, 9, 16breq123d 4597 . . . . . . 7 (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞) ↔ 𝑝 (𝑥 𝑞)))
1811, 17anbi12d 743 . . . . . 6 (𝑘 = 𝐾 → ((¬ 𝑝(le‘𝑘)𝑥𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞)) ↔ (¬ 𝑝 𝑥𝑝 (𝑥 𝑞))))
19 eqidd 2611 . . . . . . 7 (𝑘 = 𝐾𝑞 = 𝑞)
2015oveqd 6566 . . . . . . 7 (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑝) = (𝑥 𝑝))
2119, 9, 20breq123d 4597 . . . . . 6 (𝑘 = 𝐾 → (𝑞(le‘𝑘)(𝑥(join‘𝑘)𝑝) ↔ 𝑞 (𝑥 𝑝)))
2218, 21imbi12d 333 . . . . 5 (𝑘 = 𝐾 → (((¬ 𝑝(le‘𝑘)𝑥𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞)) → 𝑞(le‘𝑘)(𝑥(join‘𝑘)𝑝)) ↔ ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝))))
236, 22raleqbidv 3129 . . . 4 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)((¬ 𝑝(le‘𝑘)𝑥𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞)) → 𝑞(le‘𝑘)(𝑥(join‘𝑘)𝑝)) ↔ ∀𝑥𝐵 ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝))))
243, 23raleqbidv 3129 . . 3 (𝑘 = 𝐾 → (∀𝑞 ∈ (Atoms‘𝑘)∀𝑥 ∈ (Base‘𝑘)((¬ 𝑝(le‘𝑘)𝑥𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞)) → 𝑞(le‘𝑘)(𝑥(join‘𝑘)𝑝)) ↔ ∀𝑞𝐴𝑥𝐵 ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝))))
253, 24raleqbidv 3129 . 2 (𝑘 = 𝐾 → (∀𝑝 ∈ (Atoms‘𝑘)∀𝑞 ∈ (Atoms‘𝑘)∀𝑥 ∈ (Base‘𝑘)((¬ 𝑝(le‘𝑘)𝑥𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞)) → 𝑞(le‘𝑘)(𝑥(join‘𝑘)𝑝)) ↔ ∀𝑝𝐴𝑞𝐴𝑥𝐵 ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝))))
26 df-cvlat 33627 . 2 CvLat = {𝑘 ∈ AtLat ∣ ∀𝑝 ∈ (Atoms‘𝑘)∀𝑞 ∈ (Atoms‘𝑘)∀𝑥 ∈ (Base‘𝑘)((¬ 𝑝(le‘𝑘)𝑥𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞)) → 𝑞(le‘𝑘)(𝑥(join‘𝑘)𝑝))}
2725, 26elrab2 3333 1 (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝𝐴𝑞𝐴𝑥𝐵 ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  Atomscatm 33568  AtLatcal 33569  CvLatclc 33570 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-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-iota 5768  df-fv 5812  df-ov 6552  df-cvlat 33627 This theorem is referenced by:  iscvlat2N  33629  cvlatl  33630  cvlexch1  33633  ishlat2  33658
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