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Theorem List for Metamath Proof Explorer - 36401-36500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcvrle 36401 The covers relation implies the "less than or equal to" relation. (Contributed by NM, 12-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       (((𝐾𝐴𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 𝑌)

Theoremcvrnbtwn4 36402 The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 30058 analog.) (Contributed by NM, 18-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 𝑌) ↔ (𝑋 = 𝑍𝑍 = 𝑌)))

Theoremcvrnle 36403 The covers relation implies the negation of the converse "less than or equal to" relation. (Contributed by NM, 18-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → ¬ 𝑌 𝑋)

Theoremcvrne 36404 The covers relation implies inequality. (Contributed by NM, 13-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       (((𝐾𝐴𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋𝑌)

TheoremcvrnrefN 36405 The covers relation is not reflexive. (cvnref 30060 analog.) (Contributed by NM, 1-Nov-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾𝐴𝑋𝐵) → ¬ 𝑋𝐶𝑋)

Theoremcvrcmp 36406 If two lattice elements that cover a third are comparable, then they are equal. (Contributed by NM, 6-Feb-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑋 𝑌𝑋 = 𝑌))

Theoremcvrcmp2 36407 If two lattice elements covered by a third are comparable, then they are equal. (Contributed by NM, 20-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (𝑋 𝑌𝑋 = 𝑌))

Theorempats 36408* The set of atoms in a poset. (Contributed by NM, 18-Sep-2011.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾𝐷𝐴 = {𝑥𝐵0 𝐶𝑥})

Theoremisat 36409 The predicate "is an atom". (ela 30108 analog.) (Contributed by NM, 18-Sep-2011.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾𝐷 → (𝑃𝐴 ↔ (𝑃𝐵0 𝐶𝑃)))

Theoremisat2 36410 The predicate "is an atom". (elatcv0 30110 analog.) (Contributed by NM, 18-Jun-2012.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾𝐷𝑃𝐵) → (𝑃𝐴0 𝐶𝑃))

Theorematcvr0 36411 An atom covers zero. (atcv0 30111 analog.) (Contributed by NM, 4-Nov-2011.)
0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾𝐷𝑃𝐴) → 0 𝐶𝑃)

Theorematbase 36412 An atom is a member of the lattice base set (i.e. a lattice element). (atelch 30113 analog.) (Contributed by NM, 10-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝑃𝐴𝑃𝐵)

Theorematssbase 36413 The set of atoms is a subset of the base set. (atssch 30112 analog.) (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝐴 = (Atoms‘𝐾)       𝐴𝐵

Theorem0ltat 36414 An atom is greater than zero. (Contributed by NM, 4-Jul-2012.)
0 = (0.‘𝐾)    &    < = (lt‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ OP ∧ 𝑃𝐴) → 0 < 𝑃)

Theoremleatb 36415 A poset element less than or equal to an atom equals either zero or the atom. (atss 30115 analog.) (Contributed by NM, 17-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴) → (𝑋 𝑃 ↔ (𝑋 = 𝑃𝑋 = 0 )))

Theoremleat 36416 A poset element less than or equal to an atom equals either zero or the atom. (Contributed by NM, 15-Oct-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴) ∧ 𝑋 𝑃) → (𝑋 = 𝑃𝑋 = 0 ))

Theoremleat2 36417 A nonzero poset element less than or equal to an atom equals the atom. (Contributed by NM, 6-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴) ∧ (𝑋0𝑋 𝑃)) → 𝑋 = 𝑃)

Theoremleat3 36418 A poset element less than or equal to an atom is either an atom or zero. (Contributed by NM, 2-Dec-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴) ∧ 𝑋 𝑃) → (𝑋𝐴𝑋 = 0 ))

Theoremmeetat 36419 The meet of any element with an atom is either the atom or zero. (Contributed by NM, 28-Aug-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑃𝐴) → ((𝑋 𝑃) = 𝑃 ∨ (𝑋 𝑃) = 0 ))

Theoremmeetat2 36420 The meet of any element with an atom is either the atom or zero. (Contributed by NM, 30-Aug-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑃𝐴) → ((𝑋 𝑃) ∈ 𝐴 ∨ (𝑋 𝑃) = 0 ))

Definitiondf-atl 36421* Define the class of atomic lattices, in which every nonzero element is greater than or equal to an atom. We also ensure the existence of a lattice zero, since a lattice by itself may not have a zero. (Contributed by NM, 18-Sep-2011.) (Revised by NM, 14-Sep-2018.)
AtLat = {𝑘 ∈ Lat ∣ ((Base‘𝑘) ∈ dom (glb‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑝 ∈ (Atoms‘𝑘)𝑝(le‘𝑘)𝑥))}

Theoremisatl 36422* 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.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))

Theorematllat 36423 An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.)
(𝐾 ∈ AtLat → 𝐾 ∈ Lat)

Theorematlpos 36424 An atomic lattice is a poset. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ AtLat → 𝐾 ∈ Poset)

Theorematl0dm 36425 Condition necessary for zero element to exist. (Contributed by NM, 14-Sep-2018.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐺 = (glb‘𝐾)       (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺)

Theorematl0cl 36426 An atomic lattice has a zero element. We can use this in place of op0cl 36307 for lattices without orthocomplements. (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)       (𝐾 ∈ AtLat → 0𝐵)

Theorematl0le 36427 Orthoposet zero is less than or equal to any element. (ch0le 29210 analog.) (Contributed by NM, 12-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑋𝐵) → 0 𝑋)

Theorematlle0 36428 An element less than or equal to zero equals zero. (chle0 29212 analog.) (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑋𝐵) → (𝑋 0𝑋 = 0 ))

Theorematlltn0 36429 A lattice element greater than zero is nonzero. (Contributed by NM, 1-Jun-2012.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑋𝐵) → ( 0 < 𝑋𝑋0 ))

Theoremisat3 36430* The predicate "is an atom". (elat2 30109 analog.) (Contributed by NM, 27-Apr-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾 ∈ AtLat → (𝑃𝐴 ↔ (𝑃𝐵𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )))))

Theorematn0 36431 An atom is not zero. (atne0 30114 analog.) (Contributed by NM, 5-Nov-2012.)
0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 𝑃0 )

Theorematnle0 36432 An atom is not less than or equal to zero. (Contributed by NM, 17-Oct-2011.)
= (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → ¬ 𝑃 0 )

Theorematlen0 36433 A lattice element is nonzero if an atom is under it. (Contributed by NM, 26-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ AtLat ∧ 𝑋𝐵𝑃𝐴) ∧ 𝑃 𝑋) → 𝑋0 )

Theorematcmp 36434 If two atoms are comparable, they are equal. (atsseq 30116 analog.) (Contributed by NM, 13-Oct-2011.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄𝑃 = 𝑄))

Theorematncmp 36435 Frequently-used variation of atcmp 36434. (Contributed by NM, 29-Jun-2012.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴) → (¬ 𝑃 𝑄𝑃𝑄))

Theorematnlt 36436 Two atoms cannot satisfy the less than relation. (Contributed by NM, 7-Feb-2012.)
< = (lt‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴) → ¬ 𝑃 < 𝑄)

Theorematcvreq0 36437 An element covered by an atom must be zero. (atcveq0 30117 analog.) (Contributed by NM, 4-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑋𝐵𝑃𝐴) → (𝑋𝐶𝑃𝑋 = 0 ))

TheorematncvrN 36438 Two atoms cannot satisfy the covering relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴) → ¬ 𝑃𝐶𝑄)

Theorematlex 36439* Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 30129 analog.) (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑋𝐵𝑋0 ) → ∃𝑦𝐴 𝑦 𝑋)

Theorematnle 36440 Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 30145 analog.) (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (¬ 𝑃 𝑋 ↔ (𝑃 𝑋) = 0 ))

Theorematnem0 36441 The meet of distinct atoms is zero. (atnemeq0 30146 analog.) (Contributed by NM, 5-Nov-2012.)
= (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴) → (𝑃𝑄 ↔ (𝑃 𝑄) = 0 ))

Theorematlatmstc 36442* An atomic, complete, orthomodular lattice is atomistic i.e. every element is the join of the atoms under it. See remark before Proposition 1 in [Kalmbach] p. 140; also remark in [BeltramettiCassinelli] p. 98. (hatomistici 30131 analog.) (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    1 = (lub‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋𝐵) → ( 1 ‘{𝑦𝐴𝑦 𝑋}) = 𝑋)

Theorematlatle 36443* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 30140 analog.) (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ∀𝑝𝐴 (𝑝 𝑋𝑝 𝑌)))

Theorematlrelat1 36444* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 30132, with swapped, analog.) (Contributed by NM, 4-Dec-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 → ∃𝑝𝐴𝑝 𝑋𝑝 𝑌)))

Definitiondf-cvlat 36445* Define the class of atomic lattices with the covering property. (This is actually the exchange property, but they are equivalent. The literature usually uses the covering property terminology.) (Contributed by NM, 5-Nov-2012.)
CvLat = {𝑘 ∈ AtLat ∣ ∀𝑎 ∈ (Atoms‘𝑘)∀𝑏 ∈ (Atoms‘𝑘)∀𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))}

Theoremiscvlat 36446* The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝𝐴𝑞𝐴𝑥𝐵 ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝))))

Theoremiscvlat2N 36447* The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝𝐴𝑞𝐴𝑥𝐵 (((𝑝 𝑥) = 0𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝))))

Theoremcvlatl 36448 An atomic lattice with the covering property is an atomic lattice. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ CvLat → 𝐾 ∈ AtLat)

Theoremcvllat 36449 An atomic lattice with the covering property is a lattice. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ CvLat → 𝐾 ∈ Lat)

TheoremcvlposN 36450 An atomic lattice with the covering property is a poset. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
(𝐾 ∈ CvLat → 𝐾 ∈ Poset)

Theoremcvlexch1 36451 An atomic covering lattice has the exchange property. (Contributed by NM, 6-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))

Theoremcvlexch2 36452 An atomic covering lattice has the exchange property. (Contributed by NM, 6-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑄 𝑋) → 𝑄 (𝑃 𝑋)))

Theoremcvlexchb1 36453 An atomic covering lattice has the exchange property. (Contributed by NM, 16-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) ↔ (𝑋 𝑃) = (𝑋 𝑄)))

Theoremcvlexchb2 36454 An atomic covering lattice has the exchange property. (Contributed by NM, 22-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑄 𝑋) ↔ (𝑃 𝑋) = (𝑄 𝑋)))

Theoremcvlexch3 36455 An atomic covering lattice has the exchange property. (atexch 30150 analog.) (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))

Theoremcvlexch4N 36456 An atomic covering lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 (𝑋 𝑄) ↔ (𝑋 𝑃) = (𝑋 𝑄)))

Theoremcvlatexchb1 36457 A version of cvlexchb1 36453 for atoms. (Contributed by NM, 5-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑅 𝑄) ↔ (𝑅 𝑃) = (𝑅 𝑄)))

Theoremcvlatexchb2 36458 A version of cvlexchb2 36454 for atoms. (Contributed by NM, 5-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑄 𝑅) ↔ (𝑃 𝑅) = (𝑄 𝑅)))

Theoremcvlatexch1 36459 Atom exchange property. (Contributed by NM, 5-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑅 𝑄) → 𝑄 (𝑅 𝑃)))

Theoremcvlatexch2 36460 Atom exchange property. (Contributed by NM, 5-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑄 𝑅) → 𝑄 (𝑃 𝑅)))

Theoremcvlatexch3 36461 Atom exchange property. (Contributed by NM, 29-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄𝑃𝑅)) → (𝑃 (𝑄 𝑅) → (𝑃 𝑄) = (𝑃 𝑅)))

Theoremcvlcvr1 36462 The covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 30124 analog.) (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋𝐵𝑃𝐴) → (¬ 𝑃 𝑋𝑋𝐶(𝑋 𝑃)))

Theoremcvlcvrp 36463 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 30144 analog.) (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋𝐵𝑃𝐴) → ((𝑋 𝑃) = 0𝑋𝐶(𝑋 𝑃)))

Theoremcvlatcvr1 36464 An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.)
= (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑃𝐴𝑄𝐴) → (𝑃𝑄𝑃𝐶(𝑃 𝑄)))

Theoremcvlatcvr2 36465 An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.)
= (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑃𝐴𝑄𝐴) → (𝑃𝑄𝑃𝐶(𝑄 𝑃)))

Theoremcvlsupr2 36466 Two equivalent ways of expressing that 𝑅 is a superposition of 𝑃 and 𝑄. (Contributed by NM, 5-Nov-2012.)
𝐴 = (Atoms‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))

Theoremcvlsupr3 36467 Two equivalent ways of expressing that 𝑅 is a superposition of 𝑃 and 𝑄, which can replace the superposition part of ishlat1 36475, (𝑥𝑦 → ∃𝑧𝐴(𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦)) ), with the simpler 𝑧𝐴(𝑥 𝑧) = (𝑦 𝑧) as shown in ishlat3N 36477. (Contributed by NM, 5-Nov-2012.)
𝐴 = (Atoms‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑃𝑄 → (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄)))))

Theoremcvlsupr4 36468 Consequence of superposition condition (𝑃 𝑅) = (𝑄 𝑅). (Contributed by NM, 9-Nov-2012.)
𝐴 = (Atoms‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑅 (𝑃 𝑄))

Theoremcvlsupr5 36469 Consequence of superposition condition (𝑃 𝑅) = (𝑄 𝑅). (Contributed by NM, 9-Nov-2012.)
𝐴 = (Atoms‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑅𝑃)

Theoremcvlsupr6 36470 Consequence of superposition condition (𝑃 𝑅) = (𝑄 𝑅). (Contributed by NM, 9-Nov-2012.)
𝐴 = (Atoms‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑅𝑄)

Theoremcvlsupr7 36471 Consequence of superposition condition (𝑃 𝑅) = (𝑄 𝑅). (Contributed by NM, 24-Nov-2012.)
𝐴 = (Atoms‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → (𝑃 𝑄) = (𝑅 𝑄))

Theoremcvlsupr8 36472 Consequence of superposition condition (𝑃 𝑅) = (𝑄 𝑅). (Contributed by NM, 24-Nov-2012.)
𝐴 = (Atoms‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → (𝑃 𝑄) = (𝑃 𝑅))

20.24.11  Hilbert lattices

Syntaxchlt 36473 Extend class notation with Hilbert lattices.
class HL

Definitiondf-hlat 36474* Define the class of Hilbert lattices, which are complete, atomic lattices satisfying the superposition principle and minimum height. (Contributed by NM, 5-Nov-2012.)
HL = {𝑙 ∈ ((OML ∩ CLat) ∩ CvLat) ∣ (∀𝑎 ∈ (Atoms‘𝑙)∀𝑏 ∈ (Atoms‘𝑙)(𝑎𝑏 → ∃𝑐 ∈ (Atoms‘𝑙)(𝑐𝑎𝑐𝑏𝑐(le‘𝑙)(𝑎(join‘𝑙)𝑏))) ∧ ∃𝑎 ∈ (Base‘𝑙)∃𝑏 ∈ (Base‘𝑙)∃𝑐 ∈ (Base‘𝑙)(((0.‘𝑙)(lt‘𝑙)𝑎𝑎(lt‘𝑙)𝑏) ∧ (𝑏(lt‘𝑙)𝑐𝑐(lt‘𝑙)(1.‘𝑙))))}

Theoremishlat1 36475* The predicate "is a Hilbert lattice", which is: is orthomodular (𝐾 ∈ OML), complete (𝐾 ∈ CLat), atomic and satisfies the exchange (or covering) property (𝐾 ∈ CvLat), satisfies the superposition principle, and has a minimum height of 4 (witnessed here by 0, x, y, z, 1). (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))))

Theoremishlat2 36476* The predicate "is a Hilbert lattice". Here we replace 𝐾 ∈ CvLat with the weaker 𝐾 ∈ AtLat and show the exchange property explicitly. (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∀𝑧𝐵 ((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) → 𝑦 (𝑧 𝑥))) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))))

Theoremishlat3N 36477* The predicate "is a Hilbert lattice". Note that the superposition principle is expressed in the compact form 𝑧𝐴(𝑥 𝑧) = (𝑦 𝑧). The exchange property and atomicity are provided by 𝐾 ∈ CvLat, and "minimum height 4" is shown explicitly. (Contributed by NM, 8-Nov-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥 𝑧) = (𝑦 𝑧) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))))

TheoremishlatiN 36478* Properties that determine a Hilbert lattice. (Contributed by NM, 13-Nov-2011.) (New usage is discouraged.)
𝐾 ∈ OML    &   𝐾 ∈ CLat    &   𝐾 ∈ AtLat    &   𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∀𝑧𝐵 ((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) → 𝑦 (𝑧 𝑥)))    &   𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 ))       𝐾 ∈ HL

Theoremhlomcmcv 36479 A Hilbert lattice is orthomodular, complete, and has the covering (exchange) property. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))

Theoremhloml 36480 A Hilbert lattice is orthomodular. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ OML)

Theoremhlclat 36481 A Hilbert lattice is complete. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ CLat)

Theoremhlcvl 36482 A Hilbert lattice is an atomic lattice with the covering property. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ HL → 𝐾 ∈ CvLat)

Theoremhlatl 36483 A Hilbert lattice is atomic. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ AtLat)

Theoremhlol 36484 A Hilbert lattice is an ortholattice. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ OL)

Theoremhlop 36485 A Hilbert lattice is an orthoposet. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ OP)

Theoremhllat 36486 A Hilbert lattice is a lattice. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ Lat)

Theoremhllatd 36487 Deduction form of hllat 36486. A Hilbert lattice is a lattice. (Contributed by BJ, 14-Aug-2022.)
(𝜑𝐾 ∈ HL)       (𝜑𝐾 ∈ Lat)

Theoremhlomcmat 36488 A Hilbert lattice is orthomodular, complete, and atomic. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))

Theoremhlpos 36489 A Hilbert lattice is a poset. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ Poset)

Theoremhlatjcl 36490 Closure of join operation. Frequently-used special case of latjcl 17653 for atoms. (Contributed by NM, 15-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) → (𝑋 𝑌) ∈ 𝐵)

Theoremhlatjcom 36491 Commutatitivity of join operation. Frequently-used special case of latjcom 17661 for atoms. (Contributed by NM, 15-Jun-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) → (𝑋 𝑌) = (𝑌 𝑋))

Theoremhlatjidm 36492 Idempotence of join operation. Frequently-used special case of latjcom 17661 for atoms. (Contributed by NM, 15-Jul-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑋 𝑋) = 𝑋)

Theoremhlatjass 36493 Lattice join is associative. Frequently-used special case of latjass 17697 for atoms. (Contributed by NM, 27-Jul-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) = (𝑃 (𝑄 𝑅)))

Theoremhlatj12 36494 Swap 1st and 2nd members of lattice join. Frequently-used special case of latj32 17699 for atoms. (Contributed by NM, 4-Jun-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃 (𝑄 𝑅)) = (𝑄 (𝑃 𝑅)))

Theoremhlatj32 36495 Swap 2nd and 3rd members of lattice join. Frequently-used special case of latj32 17699 for atoms. (Contributed by NM, 21-Jul-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) = ((𝑃 𝑅) 𝑄))

Theoremhlatjrot 36496 Rotate lattice join of 3 classes. Frequently-used special case of latjrot 17702 for atoms. (Contributed by NM, 2-Aug-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) = ((𝑅 𝑃) 𝑄))

Theoremhlatj4 36497 Rearrangement of lattice join of 4 classes. Frequently-used special case of latj4 17703 for atoms. (Contributed by NM, 9-Aug-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑃 𝑅) (𝑄 𝑆)))

Theoremhlatlej1 36498 A join's first argument is less than or equal to the join. Special case of latlej1 17662 to show an atom is on a line. (Contributed by NM, 15-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑃 (𝑃 𝑄))

Theoremhlatlej2 36499 A join's second argument is less than or equal to the join. Special case of latlej2 17663 to show an atom is on a line. (Contributed by NM, 15-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄 (𝑃 𝑄))

TheoremglbconN 36500* De Morgan's law for GLB and LUB. This holds in any complete ortholattice, although we assume HL for convenience. (Contributed by NM, 17-Jan-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐺 = (glb‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝐺𝑆) = ( ‘(𝑈‘{𝑥𝐵 ∣ ( 𝑥) ∈ 𝑆})))

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44891
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