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Theorem List for Metamath Proof Explorer - 29801-29900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorematnem0 29801 The meet of distinct atoms is zero. (atnemeq0 23833 analog.) (Contributed by NM, 5-Nov-2012.)

Theorematlatmstc 29802* 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 23818 analog.) (Contributed by NM, 5-Nov-2012.)

Theorematlatle 29803* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 23827 analog.) (Contributed by NM, 5-Nov-2012.)

Theorematlrelat1 29804* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 23819, with swapped, analog.) (Contributed by NM, 4-Dec-2011.)

Definitiondf-cvlat 29805* 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.)

Theoremiscvlat 29806* The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.)

Theoremiscvlat2N 29807* The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)

Theoremcvlatl 29808 An atomic lattice with the covering property is an atomic lattice. (Contributed by NM, 5-Nov-2012.)

Theoremcvllat 29809 An atomic lattice with the covering property is a lattice. (Contributed by NM, 5-Nov-2012.)

TheoremcvlposN 29810 An atomic lattice with the covering property is a poset. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)

Theoremcvlexch1 29811 An atomic covering lattice has the exchange property. (Contributed by NM, 6-Nov-2011.)

Theoremcvlexch2 29812 An atomic covering lattice has the exchange property. (Contributed by NM, 6-May-2012.)

Theoremcvlexchb1 29813 An atomic covering lattice has the exchange property. (Contributed by NM, 16-Nov-2011.)

Theoremcvlexchb2 29814 An atomic covering lattice has the exchange property. (Contributed by NM, 22-Jun-2012.)

Theoremcvlexch3 29815 An atomic covering lattice has the exchange property. (atexch 23837 analog.) (Contributed by NM, 5-Nov-2012.)

Theoremcvlexch4N 29816 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.)

Theoremcvlatexchb1 29817 A version of cvlexchb1 29813 for atoms. (Contributed by NM, 5-Nov-2012.)

Theoremcvlatexchb2 29818 A version of cvlexchb2 29814 for atoms. (Contributed by NM, 5-Nov-2012.)

Theoremcvlatexch1 29819 Atom exchange property. (Contributed by NM, 5-Nov-2012.)

Theoremcvlatexch2 29820 Atom exchange property. (Contributed by NM, 5-Nov-2012.)

Theoremcvlatexch3 29821 Atom exchange property. (Contributed by NM, 29-Nov-2012.)

Theoremcvlcvr1 29822 The covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 23811 analog.) (Contributed by NM, 5-Nov-2012.)

Theoremcvlcvrp 29823 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 23831 analog.) (Contributed by NM, 5-Nov-2012.)

Theoremcvlatcvr1 29824 An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.)

Theoremcvlatcvr2 29825 An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.)

Theoremcvlsupr2 29826 Two equivalent ways of expressing that is a superposition of and . (Contributed by NM, 5-Nov-2012.)

Theoremcvlsupr3 29827 Two equivalent ways of expressing that is a superposition of and , which can replace the superposition part of ishlat1 29835, , with the simpler as shown in ishlat3N 29837. (Contributed by NM, 5-Nov-2012.)

Theoremcvlsupr4 29828 Consequence of superposition condition . (Contributed by NM, 9-Nov-2012.)

Theoremcvlsupr5 29829 Consequence of superposition condition . (Contributed by NM, 9-Nov-2012.)

Theoremcvlsupr6 29830 Consequence of superposition condition . (Contributed by NM, 9-Nov-2012.)

Theoremcvlsupr7 29831 Consequence of superposition condition . (Contributed by NM, 24-Nov-2012.)

Theoremcvlsupr8 29832 Consequence of superposition condition . (Contributed by NM, 24-Nov-2012.)

19.26.8  Hilbert lattices

Syntaxchlt 29833 Extend class notation with Hilbert lattices.

Definitiondf-hlat 29834* Define the class of Hilbert lattices, which are complete, atomic lattices satisfying the superposition principle and minimum height. (Contributed by NM, 5-Nov-2012.)

Theoremishlat1 29835* The predicate "is a Hilbert lattice," which is orthomodular ( ), complete ( ), atomic and satisfying the exchange (or covering) property ( ), satisfies the superposition principle, and has a minimum height of 4. (Contributed by NM, 5-Nov-2012.)

Theoremishlat2 29836* The predicate "is a Hilbert lattice". Here we replace with the weaker and show the exchange property explicitly. (Contributed by NM, 5-Nov-2012.)

Theoremishlat3N 29837* The predicate "is a Hilbert lattice". Note that the superposition principle is expressed in the compact form . The exchange property and atomicity are provided by , and "minimum height 4" is shown explicitly. (Contributed by NM, 8-Nov-2012.) (New usage is discouraged.)

TheoremishlatiN 29838* Properties that determine a Hilbert lattice. (Contributed by NM, 13-Nov-2011.) (New usage is discouraged.)

Theoremhlomcmcv 29839 A Hilbert lattice is orthomodular, complete, and has the covering (exchange) property. (Contributed by NM, 5-Nov-2012.)

Theoremhloml 29840 A Hilbert lattice is orthomodular. (Contributed by NM, 20-Oct-2011.)

Theoremhlclat 29841 A Hilbert lattice is complete. (Contributed by NM, 20-Oct-2011.)

Theoremhlcvl 29842 A Hilbert lattice is an atomic lattice with the covering property. (Contributed by NM, 5-Nov-2012.)

Theoremhlatl 29843 A Hilbert lattice is atomic. (Contributed by NM, 20-Oct-2011.)

Theoremhlol 29844 A Hilbert lattice is an ortholattice. (Contributed by NM, 20-Oct-2011.)

Theoremhlop 29845 A Hilbert lattice is an orthoposet. (Contributed by NM, 20-Oct-2011.)

Theoremhllat 29846 A Hilbert lattice is a lattice. (Contributed by NM, 20-Oct-2011.)

Theoremhlomcmat 29847 A Hilbert lattice is orthomodular, complete, and atomic. (Contributed by NM, 5-Nov-2012.)

Theoremhlpos 29848 A Hilbert lattice is a poset. (Contributed by NM, 20-Oct-2011.)

Theoremhlatjcl 29849 Closure of join operation. Frequently-used special case of latjcl 14434 for atoms. (Contributed by NM, 15-Jun-2012.)

Theoremhlatjcom 29850 Commutatitivity of join operation. Frequently-used special case of latjcom 14443 for atoms. (Contributed by NM, 15-Jun-2012.)

Theoremhlatjidm 29851 Idempotence of join operation. Frequently-used special case of latjcom 14443 for atoms. (Contributed by NM, 15-Jul-2012.)

Theoremhlatjass 29852 Lattice join is associative. Frequently-used special case of latjass 14479 for atoms. (Contributed by NM, 27-Jul-2012.)

Theoremhlatj12 29853 Swap 1st and 2nd members of lattice join. Frequently-used special case of latj32 14481 for atoms. (Contributed by NM, 4-Jun-2012.)

Theoremhlatj32 29854 Swap 2nd and 3rd members of lattice join. Frequently-used special case of latj32 14481 for atoms. (Contributed by NM, 21-Jul-2012.)

Theoremhlatjrot 29855 Rotate lattice join of 3 classes. Frequently-used special case of latjrot 14484 for atoms. (Contributed by NM, 2-Aug-2012.)

Theoremhlatj4 29856 Rearrangement of lattice join of 4 classes. Frequently-used special case of latj4 14485 for atoms. (Contributed by NM, 9-Aug-2012.)

Theoremhlatlej1 29857 A join's first argument is less than or equal to the join. Special case of latlej1 14444 to show an atom is on a line. (Contributed by NM, 15-May-2013.)

Theoremhlatlej2 29858 A join's second argument is less than or equal to the join. Special case of latlej2 14445 to show an atom is on a line. (Contributed by NM, 15-May-2013.)

TheoremglbconN 29859* De Morgan's law for GLB and LUB. This holds in any complete ortholattice, although we assume for convenience. (Contributed by NM, 17-Jan-2012.) (New usage is discouraged.)

TheoremglbconxN 29860* De Morgan's law for GLB and LUB. Index-set version of glbconN 29859, where we read as . (Contributed by NM, 17-Jan-2012.) (New usage is discouraged.)

Theorematnlej1 29861 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.)

Theorematnlej2 29862 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.)

Theoremhlsuprexch 29863* A Hilbert lattice has the superposition and exchange properties. (Contributed by NM, 13-Nov-2011.)

Theoremhlexch1 29864 A Hilbert lattice has the exchange property. (Contributed by NM, 13-Nov-2011.)

Theoremhlexch2 29865 A Hilbert lattice has the exchange property. (Contributed by NM, 6-May-2012.)

Theoremhlexchb1 29866 A Hilbert lattice has the exchange property. (Contributed by NM, 16-Nov-2011.)

Theoremhlexchb2 29867 A Hilbert lattice has the exchange property. (Contributed by NM, 22-Jun-2012.)

Theoremhlsupr 29868* A Hilbert lattice has the superposition property. Theorem 13.2 in [Crawley] p. 107. (Contributed by NM, 30-Jan-2012.)

Theoremhlsupr2 29869* A Hilbert lattice has the superposition property. (Contributed by NM, 25-Nov-2012.)

Theoremhlhgt4 29870* A Hilbert lattice has a height of at least 4. (Contributed by NM, 4-Dec-2011.)

Theoremhlhgt2 29871* A Hilbert lattice has a height of at least 2. (Contributed by NM, 4-Dec-2011.)

Theoremhl0lt1N 29872 Lattice 0 is less than lattice 1 in a Hilbert lattice. (Contributed by NM, 4-Dec-2011.) (New usage is discouraged.)

Theoremhlexch3 29873 A Hilbert lattice has the exchange property. (atexch 23837 analog.) (Contributed by NM, 15-Nov-2011.)

Theoremhlexch4N 29874 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.)

Theoremhlatexchb1 29875 A version of hlexchb1 29866 for atoms. (Contributed by NM, 15-Nov-2011.)

Theoremhlatexchb2 29876 A version of hlexchb2 29867 for atoms. (Contributed by NM, 7-Feb-2012.)

Theoremhlatexch1 29877 Atom exchange property. (Contributed by NM, 7-Jan-2012.)

Theoremhlatexch2 29878 Atom exchange property. (Contributed by NM, 8-Jan-2012.)

TheoremhlatmstcOLDN 29879* 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 23818 analog.) (Contributed by NM, 21-Oct-2011.) (New usage is discouraged.)

Theoremhlatle 29880* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 23827 analog.) (Contributed by NM, 4-Nov-2011.)

Theoremhlateq 29881* The equality of two Hilbert lattice elements is determined by the atoms under them. (chrelat4i 23829 analog.) (Contributed by NM, 24-May-2012.)

Theoremhlrelat1 29882* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 23819, with swapped, analog.) (Contributed by NM, 4-Dec-2011.)

Theoremhlrelat5N 29883* An atomistic lattice with 0 is relatively atomic, using the definition in Remark 2 of [Kalmbach] p. 149. (Contributed by NM, 21-Oct-2011.) (New usage is discouraged.)

Theoremhlrelat 29884* A Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 23820 analog.) (Contributed by NM, 4-Feb-2012.)

Theoremhlrelat2 29885* A consequence of relative atomicity. (chrelat2i 23821 analog.) (Contributed by NM, 5-Feb-2012.)

TheoremexatleN 29886 A condition for an atom to be less than or equal to a lattice element. Part of proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)

Theoremhl2at 29887* A Hilbert lattice has at least 2 atoms. (Contributed by NM, 5-Dec-2011.)

Theorematex 29888 At least one atom exists. (Contributed by NM, 15-Jul-2012.)

TheoremintnatN 29889 If the intersection with a non-majorizing element is an atom, the intersecting element is not an atom. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)

Theorem2llnne2N 29890 Condition implying that two intersecting lines are different. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)

Theorem2llnneN 29891 Condition implying that two intersecting lines are different. (Contributed by NM, 29-May-2012.) (New usage is discouraged.)

Theoremcvr1 29892 A Hilbert lattice has the covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 23811 analog.) (Contributed by NM, 17-Nov-2011.)

Theoremcvr2N 29893 Less-than and covers equivalence in a Hilbert lattice. (chcv2 23812 analog.) (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

Theoremhlrelat3 29894* The Hilbert lattice is relatively atomic. Stronger version of hlrelat 29884. (Contributed by NM, 2-May-2012.)

Theoremcvrval3 29895* Binary relation expressing covers . (Contributed by NM, 16-Jun-2012.)

Theoremcvrval4N 29896* Binary relation expressing covers . (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)

Theoremcvrval5 29897* Binary relation expressing covers . (Contributed by NM, 7-Dec-2012.)

Theoremcvrp 29898 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 23831 analog.) (Contributed by NM, 18-Nov-2011.)

Theorematcvr1 29899 An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012.)

Theorematcvr2 29900 An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012.)

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