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Theorem List for Metamath Proof Explorer - 33601-33700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmeetat 33601 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 33602 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 33603* 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 33604* 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 33605 An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.)
(𝐾 ∈ AtLat → 𝐾 ∈ Lat)

Theorematlpos 33606 An atomic lattice is a poset. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ AtLat → 𝐾 ∈ Poset)

Theorematl0dm 33607 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 33608 An atomic lattice has a zero element. We can use this in place of op0cl 33489 for lattices without orthocomplements. (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)       (𝐾 ∈ AtLat → 0𝐵)

Theorematl0le 33609 Orthoposet zero is less than or equal to any element. (ch0le 27684 analog.) (Contributed by NM, 12-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑋𝐵) → 0 𝑋)

Theorematlle0 33610 An element less than or equal to zero equals zero. (chle0 27686 analog.) (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑋𝐵) → (𝑋 0𝑋 = 0 ))

Theorematlltn0 33611 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 33612* The predicate "is an atom". (elat2 28583 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 33613 An atom is not zero. (atne0 28588 analog.) (Contributed by NM, 5-Nov-2012.)
0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 𝑃0 )

Theorematnle0 33614 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 33615 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 33616 If two atoms are comparable, they are equal. (atsseq 28590 analog.) (Contributed by NM, 13-Oct-2011.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄𝑃 = 𝑄))

Theorematncmp 33617 Frequently-used variation of atcmp 33616. (Contributed by NM, 29-Jun-2012.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴) → (¬ 𝑃 𝑄𝑃𝑄))

Theorematnlt 33618 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 33619 An element covered by an atom must be zero. (atcveq0 28591 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 33620 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 33621* Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 28603 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 33622 Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 28619 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 33623 The meet of distinct atoms is zero. (atnemeq0 28620 analog.) (Contributed by NM, 5-Nov-2012.)
= (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴) → (𝑃𝑄 ↔ (𝑃 𝑄) = 0 ))

Theorematlatmstc 33624* 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 28605 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 33625* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 28614 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 33626* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 28606, 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 33627* 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 33628* 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 33629* 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 33630 An atomic lattice with the covering property is an atomic lattice. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ CvLat → 𝐾 ∈ AtLat)

Theoremcvllat 33631 An atomic lattice with the covering property is a lattice. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ CvLat → 𝐾 ∈ Lat)

TheoremcvlposN 33632 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 33633 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 33634 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 33635 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 33636 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 33637 An atomic covering lattice has the exchange property. (atexch 28624 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 33638 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 33639 A version of cvlexchb1 33635 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 33640 A version of cvlexchb2 33636 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 33641 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 33642 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 33643 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 33644 The covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 28598 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 33645 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 28618 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 33646 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 33647 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 33648 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 33649 Two equivalent ways of expressing that 𝑅 is a superposition of 𝑃 and 𝑄, which can replace the superposition part of ishlat1 33657, (𝑥𝑦 → ∃𝑧𝐴(𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦)) ), with the simpler 𝑧𝐴(𝑥 𝑧) = (𝑦 𝑧) as shown in ishlat3N 33659. (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 33650 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 33651 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 33652 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 33653 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 33654 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))

21.22.11  Hilbert lattices

Syntaxchlt 33655 Extend class notation with Hilbert lattices.
class HL

Definitiondf-hlat 33656* 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 33657* The predicate "is a Hilbert lattice," which is orthomodular (𝐾 ∈ OML), complete (𝐾 ∈ CLat), atomic and satisfying the exchange (or covering) property (𝐾 ∈ CvLat), satisfies the superposition principle, and has a minimum height of 4. (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 33658* 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 33659* 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 33660* 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 33661 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 33662 A Hilbert lattice is orthomodular. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ OML)

Theoremhlclat 33663 A Hilbert lattice is complete. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ CLat)

Theoremhlcvl 33664 A Hilbert lattice is an atomic lattice with the covering property. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ HL → 𝐾 ∈ CvLat)

Theoremhlatl 33665 A Hilbert lattice is atomic. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ AtLat)

Theoremhlol 33666 A Hilbert lattice is an ortholattice. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ OL)

Theoremhlop 33667 A Hilbert lattice is an orthoposet. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ OP)

Theoremhllat 33668 A Hilbert lattice is a lattice. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ Lat)

Theoremhlomcmat 33669 A Hilbert lattice is orthomodular, complete, and atomic. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))

Theoremhlpos 33670 A Hilbert lattice is a poset. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ Poset)

Theoremhlatjcl 33671 Closure of join operation. Frequently-used special case of latjcl 16874 for atoms. (Contributed by NM, 15-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) → (𝑋 𝑌) ∈ 𝐵)

Theoremhlatjcom 33672 Commutatitivity of join operation. Frequently-used special case of latjcom 16882 for atoms. (Contributed by NM, 15-Jun-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) → (𝑋 𝑌) = (𝑌 𝑋))

Theoremhlatjidm 33673 Idempotence of join operation. Frequently-used special case of latjcom 16882 for atoms. (Contributed by NM, 15-Jul-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑋 𝑋) = 𝑋)

Theoremhlatjass 33674 Lattice join is associative. Frequently-used special case of latjass 16918 for atoms. (Contributed by NM, 27-Jul-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) = (𝑃 (𝑄 𝑅)))

Theoremhlatj12 33675 Swap 1st and 2nd members of lattice join. Frequently-used special case of latj32 16920 for atoms. (Contributed by NM, 4-Jun-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃 (𝑄 𝑅)) = (𝑄 (𝑃 𝑅)))

Theoremhlatj32 33676 Swap 2nd and 3rd members of lattice join. Frequently-used special case of latj32 16920 for atoms. (Contributed by NM, 21-Jul-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) = ((𝑃 𝑅) 𝑄))

Theoremhlatjrot 33677 Rotate lattice join of 3 classes. Frequently-used special case of latjrot 16923 for atoms. (Contributed by NM, 2-Aug-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) = ((𝑅 𝑃) 𝑄))

Theoremhlatj4 33678 Rearrangement of lattice join of 4 classes. Frequently-used special case of latj4 16924 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 33679 A join's first argument is less than or equal to the join. Special case of latlej1 16883 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 33680 A join's second argument is less than or equal to the join. Special case of latlej2 16884 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 33681* 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})))

TheoremglbconxN 33682* De Morgan's law for GLB and LUB. Index-set version of glbconN 33681, where we read 𝑆 as 𝑆(𝑖). (Contributed by NM, 17-Jan-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐺 = (glb‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐺‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}) = ( ‘(𝑈‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)})))

Theorematnlej1 33683 If an atom is not less than or equal to the join of two others, it is not equal to either. (This also holds for non-atoms, but in this form it is convenient.) (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑃𝑄)

Theorematnlej2 33684 If an atom is not less than or equal to the join of two others, it is not equal to either. (This also holds for non-atoms, but in this form it is convenient.) (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑃𝑅)

Theoremhlsuprexch 33685* A Hilbert lattice has the superposition and exchange properties. (Contributed by NM, 13-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → ((𝑃𝑄 → ∃𝑧𝐴 (𝑧𝑃𝑧𝑄𝑧 (𝑃 𝑄))) ∧ ∀𝑧𝐵 ((¬ 𝑃 𝑧𝑃 (𝑧 𝑄)) → 𝑄 (𝑧 𝑃))))

Theoremhlexch1 33686 A Hilbert lattice has the exchange property. (Contributed by NM, 13-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))

Theoremhlexch2 33687 A Hilbert lattice has the exchange property. (Contributed by NM, 6-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑄 𝑋) → 𝑄 (𝑃 𝑋)))

Theoremhlexchb1 33688 A Hilbert lattice has the exchange property. (Contributed by NM, 16-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) ↔ (𝑋 𝑃) = (𝑋 𝑄)))

Theoremhlexchb2 33689 A Hilbert lattice has the exchange property. (Contributed by NM, 22-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑄 𝑋) ↔ (𝑃 𝑋) = (𝑄 𝑋)))

Theoremhlsupr 33690* A Hilbert lattice has the superposition property. Theorem 13.2 in [Crawley] p. 107. (Contributed by NM, 30-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → ∃𝑟𝐴 (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄)))

Theoremhlsupr2 33691* A Hilbert lattice has the superposition property. (Contributed by NM, 25-Nov-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → ∃𝑟𝐴 (𝑃 𝑟) = (𝑄 𝑟))

Theoremhlhgt4 33692* A Hilbert lattice has a height of at least 4. (Contributed by NM, 4-Dec-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)       (𝐾 ∈ HL → ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))

Theoremhlhgt2 33693* A Hilbert lattice has a height of at least 2. (Contributed by NM, 4-Dec-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)       (𝐾 ∈ HL → ∃𝑥𝐵 ( 0 < 𝑥𝑥 < 1 ))

Theoremhl0lt1N 33694 Lattice 0 is less than lattice 1 in a Hilbert lattice. (Contributed by NM, 4-Dec-2011.) (New usage is discouraged.)
< = (lt‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)       (𝐾 ∈ HL → 0 < 1 )

Theoremhlexch3 33695 A Hilbert lattice has the exchange property. (atexch 28624 analog.) (Contributed by NM, 15-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))

Theoremhlexch4N 33696 A Hilbert lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (Contributed by NM, 15-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 (𝑋 𝑄) ↔ (𝑋 𝑃) = (𝑋 𝑄)))

Theoremhlatexchb1 33697 A version of hlexchb1 33688 for atoms. (Contributed by NM, 15-Nov-2011.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑅 𝑄) ↔ (𝑅 𝑃) = (𝑅 𝑄)))

Theoremhlatexchb2 33698 A version of hlexchb2 33689 for atoms. (Contributed by NM, 7-Feb-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑄 𝑅) ↔ (𝑃 𝑅) = (𝑄 𝑅)))

Theoremhlatexch1 33699 Atom exchange property. (Contributed by NM, 7-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑅 𝑄) → 𝑄 (𝑅 𝑃)))

Theoremhlatexch2 33700 Atom exchange property. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑄 𝑅) → 𝑄 (𝑃 𝑅)))

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