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Theorem List for Metamath Proof Explorer - 30101-30200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremomlop 30101 An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.)

Theoremomllat 30102 An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.)

Theoremomllaw 30103 The orthomodular law. (Contributed by NM, 18-Sep-2011.)

Theoremomllaw2N 30104 Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 23089 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)

Theoremomllaw3 30105 Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 22940 analog.) (Contributed by NM, 19-Oct-2011.)

Theoremomllaw4 30106 Orthomodular law equivalent. Remark in [Holland95] p. 223. (Contributed by NM, 19-Oct-2011.)

Theoremomllaw5N 30107 The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 23117 analog.) (Contributed by NM, 14-Nov-2011.) (New usage is discouraged.)

TheoremcmtcomlemN 30108 Lemma for cmtcomN 30109. (cmcmlem 23095 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)

TheoremcmtcomN 30109 Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 23096 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)

Theoremcmt2N 30110 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 23097 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremcmt3N 30111 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 23099 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremcmt4N 30112 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 23099 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremcmtbr2N 30113 Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 23100 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremcmtbr3N 30114 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 23112 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremcmtbr4N 30115 Alternate definition for the commutes relation. (cmbr4i 23105 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)

TheoremlecmtN 30116 Ordered elements commute. (lecmi 23106 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)

TheoremcmtidN 30117 Any element commutes with itself. (cmidi 23114 analog.) (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Theoremomlfh1N 30118 Foulis-Holland Theorem, part 1. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Part of Theorem 5 in [Kalmbach] p. 25. (fh1 23122 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremomlfh3N 30119 Foulis-Holland Theorem, part 3. Dual of omlfh1N 30118. (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremomlmod1i2N 30120 Analog of modular law atmod1i2 30718 that holds in any OML. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremomlspjN 30121 Contraction of a Sasaki projection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

19.26.7  Atomic lattices with covering property

Syntaxccvr 30122 Extend class notation with covers relation.

Syntaxcatm 30123 Extend class notation with atoms.

Syntaxcal 30124 Extend class notation with atomic lattices.

Syntaxclc 30125 Extend class notation with lattices with the covering property.

Definitiondf-covers 30126* Define the covers relation ("is covered by") for posets. " is covered by " means that is strictly less than and there is nothing in between. See cvrval 30129 for the relation form. (Contributed by NM, 18-Sep-2011.)

Definitiondf-ats 30127* Define the class of poset atoms. (Contributed by NM, 18-Sep-2011.)

Theoremcvrfval 30128* Value of covers relation "is covered by". (Contributed by NM, 18-Sep-2011.)

Theoremcvrval 30129* Binary relation expressing covers , which means that is larger than and there is nothing in between. Definition 3.2.18 of [PtakPulmannova] p. 68. (cvbr 23787 analog.) (Contributed by NM, 18-Sep-2011.)

Theoremcvrlt 30130 The covers relation implies the less-than relation. (cvpss 23790 analog.) (Contributed by NM, 8-Oct-2011.)

Theoremcvrnbtwn 30131 There is no element between the two arguments of the covers relation. (cvnbtwn 23791 analog.) (Contributed by NM, 18-Oct-2011.)

Theoremncvr1 30132 No element covers the lattice unit. (Contributed by NM, 8-Jul-2013.)

TheoremcvrletrN 30133 Property of an element above a covering. (Contributed by NM, 7-Dec-2012.) (New usage is discouraged.)

Theoremcvrval2 30134* Binary relation expressing covers . Definition of covers in [Kalmbach] p. 15. (cvbr2 23788 analog.) (Contributed by NM, 16-Nov-2011.)

Theoremcvrnbtwn2 30135 The covers relation implies no in-betweenness. (cvnbtwn2 23792 analog.) (Contributed by NM, 17-Nov-2011.)

Theoremcvrnbtwn3 30136 The covers relation implies no in-betweenness. (cvnbtwn3 23793 analog.) (Contributed by NM, 4-Nov-2011.)

Theoremcvrcon3b 30137 Contraposition law for the covers relation. (cvcon3 23789 analog.) (Contributed by NM, 4-Nov-2011.)

Theoremcvrle 30138 The covers relation implies the less-than-or-equal relation. (Contributed by NM, 12-Oct-2011.)

Theoremcvrnbtwn4 30139 The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 23794 analog.) (Contributed by NM, 18-Oct-2011.)

Theoremcvrnle 30140 The covers relation implies the negation of the reverse less-than-or-equal relation. (Contributed by NM, 18-Oct-2011.)

Theoremcvrne 30141 The covers relation implies inequality. (Contributed by NM, 13-Oct-2011.)

TheoremcvrnrefN 30142 The covers relation is not reflexive. (cvnref 23796 analog.) (Contributed by NM, 1-Nov-2012.) (New usage is discouraged.)

Theoremcvrcmp 30143 If two lattice elements that cover a third are comparable, then they are equal. (Contributed by NM, 6-Feb-2012.)

Theoremcvrcmp2 30144 If two lattice elements covered by a third are comparable, then they are equal. (Contributed by NM, 20-Jun-2012.)

Theorempats 30145* The set of atoms in a poset. (Contributed by NM, 18-Sep-2011.)

Theoremisat 30146 The predicate "is an atom". (ela 23844 analog.) (Contributed by NM, 18-Sep-2011.)

Theoremisat2 30147 The predicate "is an atom". (elatcv0 23846 analog.) (Contributed by NM, 18-Jun-2012.)

Theorematcvr0 30148 An atom covers zero. (atcv0 23847 analog.) (Contributed by NM, 4-Nov-2011.)

Theorematbase 30149 An atom is a member of the lattice base set (i.e. a lattice element). (atelch 23849 analog.) (Contributed by NM, 10-Oct-2011.)

Theorematssbase 30150 The set of atoms is a subset of the base set. (atssch 23848 analog.) (Contributed by NM, 21-Oct-2011.)

Theorem0ltat 30151 An atom is greater than zero. (Contributed by NM, 4-Jul-2012.)

Theoremleatb 30152 A poset element less than or equal to an atom equals either zero or the atom. (atss 23851 analog.) (Contributed by NM, 17-Nov-2011.)

Theoremleat 30153 A poset element less than or equal to an atom equals either zero or the atom. (Contributed by NM, 15-Oct-2013.)

Theoremleat2 30154 A nonzero poset element less than or equal to an atom equals the atom. (Contributed by NM, 6-Mar-2013.)

Theoremleat3 30155 A poset element less than or equal to an atom is either an atom or zero. (Contributed by NM, 2-Dec-2012.)

Theoremmeetat 30156 The meet of any element with an atom is either the atom or zero. (Contributed by NM, 28-Aug-2012.)

Theoremmeetat2 30157 The meet of any element with an atom is either the atom or zero. (Contributed by NM, 30-Aug-2012.)

Definitiondf-atl 30158* 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.)

Theoremisatl 30159* The predicate "is an atomic lattice." Every nonzero element is less than or equal to an atom. (Contributed by NM, 18-Sep-2011.)

Theorematllat 30160 An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.)

Theorematlpos 30161 An atomic lattice is a poset. (Contributed by NM, 5-Nov-2012.)

TheoremisatliN 30162* Properties that determine an atomic lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)

Theorematl0cl 30163 An atomic lattice has a zero element. We can use this in place of op0cl 30044 for lattices without orthocomplements. (Contributed by NM, 5-Nov-2012.)

Theorematl0le 30164 Orthoposet zero is less than or equal to any element. (ch0le 22945 analog.) (Contributed by NM, 12-Oct-2011.)

Theorematlle0 30165 An element less than or equal to zero equals zero. (chle0 22947 analog.) (Contributed by NM, 21-Oct-2011.)

Theorematlltn0 30166 A lattice element greater than zero is nonzero. (Contributed by NM, 1-Jun-2012.)

Theoremisat3 30167* The predicate "is an atom". (elat2 23845 analog.) (Contributed by NM, 27-Apr-2014.)

Theorematn0 30168 An atom is not zero. (atne0 23850 analog.) (Contributed by NM, 5-Nov-2012.)

Theorematnle0 30169 An atom is not less than or equal to zero. (Contributed by NM, 17-Oct-2011.)

Theorematlen0 30170 A lattice element is nonzero if an atom is under it. (Contributed by NM, 26-May-2012.)

Theorematcmp 30171 If two atoms are comparable, they are equal. (atsseq 23852 analog.) (Contributed by NM, 13-Oct-2011.)

Theorematncmp 30172 Frequently-used variation of atcmp 30171. (Contributed by NM, 29-Jun-2012.)

Theorematnlt 30173 Two atoms cannot satisfy the less than relation. (Contributed by NM, 7-Feb-2012.)

Theorematcvreq0 30174 An element covered by an atom must be zero. (atcveq0 23853 analog.) (Contributed by NM, 4-Nov-2011.)

TheorematncvrN 30175 Two atoms cannot satisfy the covering relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

Theorematlex 30176* Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 23865 analog.) (Contributed by NM, 21-Oct-2011.)

Theorematnle 30177 Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 23881 analog.) (Contributed by NM, 5-Nov-2012.)

Theorematnem0 30178 The meet of distinct atoms is zero. (atnemeq0 23882 analog.) (Contributed by NM, 5-Nov-2012.)

Theorematlatmstc 30179* 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 23867 analog.) (Contributed by NM, 5-Nov-2012.)

Theorematlatle 30180* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 23876 analog.) (Contributed by NM, 5-Nov-2012.)

Theorematlrelat1 30181* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 23868, with swapped, analog.) (Contributed by NM, 4-Dec-2011.)

Definitiondf-cvlat 30182* 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 30183* The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.)

Theoremiscvlat2N 30184* The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)

Theoremcvlatl 30185 An atomic lattice with the covering property is an atomic lattice. (Contributed by NM, 5-Nov-2012.)

Theoremcvllat 30186 An atomic lattice with the covering property is a lattice. (Contributed by NM, 5-Nov-2012.)

TheoremcvlposN 30187 An atomic lattice with the covering property is a poset. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)

Theoremcvlexch1 30188 An atomic covering lattice has the exchange property. (Contributed by NM, 6-Nov-2011.)

Theoremcvlexch2 30189 An atomic covering lattice has the exchange property. (Contributed by NM, 6-May-2012.)

Theoremcvlexchb1 30190 An atomic covering lattice has the exchange property. (Contributed by NM, 16-Nov-2011.)

Theoremcvlexchb2 30191 An atomic covering lattice has the exchange property. (Contributed by NM, 22-Jun-2012.)

Theoremcvlexch3 30192 An atomic covering lattice has the exchange property. (atexch 23886 analog.) (Contributed by NM, 5-Nov-2012.)

Theoremcvlexch4N 30193 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 30194 A version of cvlexchb1 30190 for atoms. (Contributed by NM, 5-Nov-2012.)

Theoremcvlatexchb2 30195 A version of cvlexchb2 30191 for atoms. (Contributed by NM, 5-Nov-2012.)

Theoremcvlatexch1 30196 Atom exchange property. (Contributed by NM, 5-Nov-2012.)

Theoremcvlatexch2 30197 Atom exchange property. (Contributed by NM, 5-Nov-2012.)

Theoremcvlatexch3 30198 Atom exchange property. (Contributed by NM, 29-Nov-2012.)

Theoremcvlcvr1 30199 The covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 23860 analog.) (Contributed by NM, 5-Nov-2012.)

Theoremcvlcvrp 30200 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 23880 analog.) (Contributed by NM, 5-Nov-2012.)

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