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Theorem List for Metamath Proof Explorer - 29401-29500   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremcmtvalN 29401 Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 22163 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)

Theoremisolat 29402 The predicate "is an ortholattice." (Contributed by NM, 18-Sep-2011.)

Theoremollat 29403 An ortholattice is a lattice. (Contributed by NM, 18-Sep-2011.)

Theoremolop 29404 An ortholattice is an orthoposet. (Contributed by NM, 18-Sep-2011.)

TheoremolposN 29405 An ortholattice is a poset. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)

TheoremisolatiN 29406 Properties that determine an ortholattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)

Theoremoldmm1 29407 DeMorgan's law for meet in an ortholattice. (chdmm1 22104 analog.) (Contributed by NM, 6-Nov-2011.)

Theoremoldmm2 29408 DeMorgan's law for meet in an ortholattice. (chdmm2 22105 analog.) (Contributed by NM, 6-Nov-2011.)

Theoremoldmm3N 29409 DeMorgan's law for meet in an ortholattice. (chdmm3 22106 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremoldmm4 29410 DeMorgan's law for meet in an ortholattice. (chdmm4 22107 analog.) (Contributed by NM, 6-Nov-2011.)

Theoremoldmj1 29411 DeMorgan's law for join in an ortholattice. (chdmj1 22108 analog.) (Contributed by NM, 6-Nov-2011.)

Theoremoldmj2 29412 DeMorgan's law for join in an ortholattice. (chdmj2 22109 analog.) (Contributed by NM, 7-Nov-2011.)

Theoremoldmj3 29413 DeMorgan's law for join in an ortholattice. (chdmj3 22110 analog.) (Contributed by NM, 7-Nov-2011.)

Theoremoldmj4 29414 DeMorgan's law for join in an ortholattice. (chdmj4 22111 analog.) (Contributed by NM, 7-Nov-2011.)

Theoremolj01 29415 An ortholattice element joined with zero equals itself. (chj0 22076 analog.) (Contributed by NM, 19-Oct-2011.)

Theoremolj02 29416 An ortholattice element joined with zero equals itself. (Contributed by NM, 28-Jan-2012.)

Theoremolm11 29417 The meet of an ortholattice element with one equals itself. (chm1i 22035 analog.) (Contributed by NM, 22-May-2012.)

Theoremolm12 29418 The meet of an ortholattice element with one equals itself. (Contributed by NM, 22-May-2012.)

TheoremlatmassOLD 29419 Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 3379 analog.) (Contributed by NM, 7-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremlatm12 29420 A rearrangement of lattice meet. (in12 3380 analog.) (Contributed by NM, 8-Nov-2011.)

Theoremlatm32 29421 A rearrangement of lattice meet. (in12 3380 analog.) (Contributed by NM, 13-Nov-2012.)

Theoremlatmrot 29422 Rotate lattice meet of 3 classes. (Contributed by NM, 9-Oct-2012.)

Theoremlatm4 29423 Rearrangement of lattice meet of 4 classes. (in4 3385 analog.) (Contributed by NM, 8-Nov-2011.)

TheoremlatmmdiN 29424 Lattice meet distributes over itself. (inindi 3386 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremlatmmdir 29425 Lattice meet distributes over itself. (inindir 3387 analog.) (Contributed by NM, 6-Jun-2012.)

Theoremolm01 29426 Meet with lattice zero is zero. (chm0 22070 analog.) (Contributed by NM, 8-Nov-2011.)

Theoremolm02 29427 Meet with lattice zero is zero. (Contributed by NM, 9-Oct-2012.)

Theoremisoml 29428* The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)

TheoremisomliN 29429* Properties that determine an orthomodular lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)

Theoremomlol 29430 An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)

Theoremomlop 29431 An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.)

Theoremomllat 29432 An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.)

Theoremomllaw 29433 The orthomodular law. (Contributed by NM, 18-Sep-2011.)

Theoremomllaw2N 29434 Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 22164 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)

Theoremomllaw3 29435 Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 22015 analog.) (Contributed by NM, 19-Oct-2011.)

Theoremomllaw4 29436 Orthomodular law equivalent. Remark in [Holland95] p. 223. (Contributed by NM, 19-Oct-2011.)

Theoremomllaw5N 29437 The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 22192 analog.) (Contributed by NM, 14-Nov-2011.) (New usage is discouraged.)

TheoremcmtcomlemN 29438 Lemma for cmtcomN 29439. (cmcmlem 22170 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)

TheoremcmtcomN 29439 Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 22171 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)

Theoremcmt2N 29440 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 22172 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremcmt3N 29441 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 22174 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremcmt4N 29442 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 22174 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremcmtbr2N 29443 Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 22175 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremcmtbr3N 29444 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 22187 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremcmtbr4N 29445 Alternate definition for the commutes relation. (cmbr4i 22180 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)

TheoremlecmtN 29446 Ordered elements commute. (lecmi 22181 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)

TheoremcmtidN 29447 Any element commutes with itself. (cmidi 22189 analog.) (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Theoremomlfh1N 29448 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 22197 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremomlfh3N 29449 Foulis-Holland Theorem, part 3. Dual of omlfh1N 29448. (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremomlmod1i2N 29450 Analog of modular law atmod1i2 30048 that holds in any OML. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremomlspjN 29451 Contraction of a Sasaki projection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

18.27.7  Atomic lattices with covering property

Syntaxccvr 29452 Extend class notation with covers relation.

Syntaxcatm 29453 Extend class notation with atoms.

Syntaxcal 29454 Extend class notation with atomic lattices.

Syntaxclc 29455 Extend class notation with lattices with the covering property.

Definitiondf-covers 29456* 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 29459 for the relation form. (Contributed by NM, 18-Sep-2011.)

Definitiondf-ats 29457* Define the class of poset atoms. (Contributed by NM, 18-Sep-2011.)

Theoremcvrfval 29458* Value of covers relation "is covered by". (Contributed by NM, 18-Sep-2011.)

Theoremcvrval 29459* 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 22862 analog.) (Contributed by NM, 18-Sep-2011.)

Theoremcvrlt 29460 The covers relation implies the less-than relation. (cvpss 22865 analog.) (Contributed by NM, 8-Oct-2011.)

Theoremcvrnbtwn 29461 There is no element between the two arguments of the covers relation. (cvnbtwn 22866 analog.) (Contributed by NM, 18-Oct-2011.)

Theoremncvr1 29462 No element covers the lattice unit. (Contributed by NM, 8-Jul-2013.)

TheoremcvrletrN 29463 Property of an element above a covering. (Contributed by NM, 7-Dec-2012.) (New usage is discouraged.)

Theoremcvrval2 29464* Binary relation expressing covers . Definition of covers in [Kalmbach] p. 15. (cvbr2 22863 analog.) (Contributed by NM, 16-Nov-2011.)

Theoremcvrnbtwn2 29465 The covers relation implies no in-betweenness. (cvnbtwn2 22867 analog.) (Contributed by NM, 17-Nov-2011.)

Theoremcvrnbtwn3 29466 The covers relation implies no in-betweenness. (cvnbtwn3 22868 analog.) (Contributed by NM, 4-Nov-2011.)

Theoremcvrcon3b 29467 Contraposition law for the covers relation. (cvcon3 22864 analog.) (Contributed by NM, 4-Nov-2011.)

Theoremcvrle 29468 The covers relation implies the less-than-or-equal relation. (Contributed by NM, 12-Oct-2011.)

Theoremcvrnbtwn4 29469 The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 22869 analog.) (Contributed by NM, 18-Oct-2011.)

Theoremcvrnle 29470 The covers relation implies the negation of the reverse less-than-or-equal relation. (Contributed by NM, 18-Oct-2011.)

Theoremcvrne 29471 The covers relation implies inequality. (Contributed by NM, 13-Oct-2011.)

TheoremcvrnrefN 29472 The covers relation is not reflexive. (cvnref 22871 analog.) (Contributed by NM, 1-Nov-2012.) (New usage is discouraged.)

Theoremcvrcmp 29473 If two lattice elements that cover a third are comparable, then they are equal. (Contributed by NM, 6-Feb-2012.)

Theoremcvrcmp2 29474 If two lattice elements covered by a third are comparable, then they are equal. (Contributed by NM, 20-Jun-2012.)

Theorempats 29475* The set of atoms in a poset. (Contributed by NM, 18-Sep-2011.)

Theoremisat 29476 The predicate "is an atom". (ela 22919 analog.) (Contributed by NM, 18-Sep-2011.)

Theoremisat2 29477 The predicate "is an atom". (elatcv0 22921 analog.) (Contributed by NM, 18-Jun-2012.)

Theorematcvr0 29478 An atom covers zero. (atcv0 22922 analog.) (Contributed by NM, 4-Nov-2011.)

Theorematbase 29479 An atom is a member of the lattice base set (i.e. a lattice element). (atelch 22924 analog.) (Contributed by NM, 10-Oct-2011.)

Theorematssbase 29480 The set of atoms is a subset of the base set. (atssch 22923 analog.) (Contributed by NM, 21-Oct-2011.)

Theorem0ltat 29481 An atom is greater than zero. (Contributed by NM, 4-Jul-2012.)

Theoremleatb 29482 A poset element less than or equal to an atom equals either zero or the atom. (atss 22926 analog.) (Contributed by NM, 17-Nov-2011.)

Theoremleat 29483 A poset element less than or equal to an atom equals either zero or the atom. (Contributed by NM, 15-Oct-2013.)

Theoremleat2 29484 A nonzero poset element less than or equal to an atom equals the atom. (Contributed by NM, 6-Mar-2013.)

Theoremleat3 29485 A poset element less than or equal to an atom is either an atom or zero. (Contributed by NM, 2-Dec-2012.)

Theoremmeetat 29486 The meet of any element with an atom is either the atom or zero. (Contributed by NM, 28-Aug-2012.)

Theoremmeetat2 29487 The meet of any element with an atom is either the atom or zero. (Contributed by NM, 30-Aug-2012.)

Definitiondf-atl 29488* 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 29489* The predicate "is an atomic lattice." Every nonzero element is less than or equal to an atom. (Contributed by NM, 18-Sep-2011.)

Theorematllat 29490 An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.)

Theorematlpos 29491 An atomic lattice is a poset. (Contributed by NM, 5-Nov-2012.)

TheoremisatliN 29492* Properties that determine an atomic lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)

Theorematl0cl 29493 An atomic lattice has a zero element. We can use this in place of op0cl 29374 for lattices without orthocomplements. (Contributed by NM, 5-Nov-2012.)

Theorematl0le 29494 Orthoposet zero is less than or equal to any element. (ch0le 22020 analog.) (Contributed by NM, 12-Oct-2011.)

Theorematlle0 29495 An element less than or equal to zero equals zero. (chle0 22022 analog.) (Contributed by NM, 21-Oct-2011.)

Theorematlltn0 29496 A lattice element greater than zero is nonzero. (Contributed by NM, 1-Jun-2012.)

Theoremisat3 29497* The predicate "is an atom". (elat2 22920 analog.) (Contributed by NM, 27-Apr-2014.)

Theorematn0 29498 An atom is not zero. (atne0 22925 analog.) (Contributed by NM, 5-Nov-2012.)

Theorematnle0 29499 An atom is not less than or equal to zero. (Contributed by NM, 17-Oct-2011.)

Theorematlen0 29500 A lattice element is nonzero if an atom is under it. (Contributed by NM, 26-May-2012.)

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