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Theorem List for Metamath Proof Explorer - 28301-28400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltltncvr 28301 A chained strong ordering is not a covers relation. (Contributed by NM, 18-Jun-2012.)

Theoremltcvrntr 28302 Non-transitive condition for the covers relation. (Contributed by NM, 18-Jun-2012.)

Theoremcvrntr 28303 The covers relation is not transitive. (cvntr 22702 analog.) (Contributed by NM, 18-Jun-2012.)

Theorematcvr0eq 28304 The covers relation is not transitive. (atcv0eq 22789 analog.) (Contributed by NM, 29-Nov-2011.)

Theoremlnnat 28305 A line (the join of two distinct atoms) is not an atom. (Contributed by NM, 14-Jun-2012.)

Theorematcvrj0 28306 Two atoms covering the zero subspace are equal. (atcv1 22790 analog.) (Contributed by NM, 29-Nov-2011.)

Theoremcvrat2 28307 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 22797 analog.) (Contributed by NM, 30-Nov-2011.)

TheorematcvrneN 28308 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

Theorematcvrj1 28309 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)

Theorematcvrj2b 28310 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)

Theorematcvrj2 28311 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)

TheorematleneN 28312 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

Theorematltcvr 28313 An equivalence of less-than ordering and covers relation. (Contributed by NM, 7-Feb-2012.)

Theorematle 28314* Any non-zero element has an atom under it. (Contributed by NM, 28-Jun-2012.)

Theorematlt 28315 Two atoms are unequal iff their join is greater than one of them. (Contributed by NM, 6-May-2012.)

Theorematlelt 28316 Transfer less-than relation from one atom to another. (Contributed by NM, 7-May-2012.)

Theorem2atlt 28317* Given an atom less than an element, there is another atom less than the element. (Contributed by NM, 6-May-2012.)

TheorematexchcvrN 28318 Atom exchange property. Version of hlatexch2 28274 with covers relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

TheorematexchltN 28319 Atom exchange property. Version of hlatexch2 28274 with less-than ordering. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

Theoremcvrat3 28320 A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 22806 analog.) (Contributed by NM, 30-Nov-2011.)

Theoremcvrat4 28321* A condition implying existence of an atom with the properties shown. Lemma 3.2.20 in [PtakPulmannova] p. 68. Also Lemma 9.2(delta) in [MaedaMaeda] p. 41. (atcvat4i 22807 analog.) (Contributed by NM, 30-Nov-2011.)

Theoremcvrat42 28322* Commuted version of cvrat4 28321. (Contributed by NM, 28-Jan-2012.)

Theorem2atjm 28323 The meet of a line (expressed with 2 atoms) and a lattice element. (Contributed by NM, 30-Jul-2012.)

Theorematbtwn 28324 Property of a 3rd atom on a line intersecting element at . (Contributed by NM, 30-Jul-2012.)

TheorematbtwnexOLDN 28325* There exists a 3rd atom on a line intersecting element at , such that is different from and not in . (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)

Theorematbtwnex 28326* Given atoms in and not in , there exists an atom not in such that the line intersects at . (Contributed by NM, 1-Aug-2012.)

Theorem3noncolr2 28327 Two ways to express 3 non-colinear atoms (rotated right 2 places). (Contributed by NM, 12-Jul-2012.)

Theorem3noncolr1N 28328 Two ways to express 3 non-colinear atoms (rotated right 1 place). (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)

Theoremhlatcon3 28329 Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.)

Theoremhlatcon2 28330 Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.)

Theorem4noncolr3 28331 A way to express 4 non-colinear atoms (rotated right 3 places). (Contributed by NM, 11-Jul-2012.)

Theorem4noncolr2 28332 A way to express 4 non-colinear atoms (rotated right 2 places). (Contributed by NM, 11-Jul-2012.)

Theorem4noncolr1 28333 A way to express 4 non-colinear atoms (rotated right 1 places). (Contributed by NM, 11-Jul-2012.)

Theoremathgt 28334* A Hilbert lattice, whose height is at least 4, has a chain of 4 successively covering atom joins. (Contributed by NM, 3-May-2012.)

Theorem3dim0 28335* There exists a 3-dimensional (height-4) element i.e. a volume. (Contributed by NM, 25-Jul-2012.)

Theorem3dimlem1 28336 Lemma for 3dim1 28345. (Contributed by NM, 25-Jul-2012.)

Theorem3dimlem2 28337 Lemma for 3dim1 28345. (Contributed by NM, 25-Jul-2012.)

Theorem3dimlem3a 28338 Lemma for 3dim3 28347. (Contributed by NM, 27-Jul-2012.)

Theorem3dimlem3 28339 Lemma for 3dim1 28345. (Contributed by NM, 25-Jul-2012.)

Theorem3dimlem3OLDN 28340 Lemma for 3dim1 28345. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem3dimlem4a 28341 Lemma for 3dim3 28347. (Contributed by NM, 27-Jul-2012.)

Theorem3dimlem4 28342 Lemma for 3dim1 28345. (Contributed by NM, 25-Jul-2012.)

Theorem3dimlem4OLDN 28343 Lemma for 3dim1 28345. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem3dim1lem5 28344* Lemma for 3dim1 28345. (Contributed by NM, 26-Jul-2012.)

Theorem3dim1 28345* Construct a 3-dimensional volume (height-4 element) on top of a given atom . (Contributed by NM, 25-Jul-2012.)

Theorem3dim2 28346* Construct 2 new layers on top of 2 given atoms. (Contributed by NM, 27-Jul-2012.)

Theorem3dim3 28347* Construct a new layer on top of 3 given atoms. (Contributed by NM, 27-Jul-2012.)

Theorem2dim 28348* Generate a height-3 element (2-dimensional plane) from an atom. (Contributed by NM, 3-May-2012.)

Theorem1dimN 28349* An atom is covered by a height-2 element (1-dimensional line). (Contributed by NM, 3-May-2012.) (New usage is discouraged.)

Theorem1cvrco 28350 The orthocomplement of an element covered by 1 is an atom. (Contributed by NM, 7-May-2012.)

Theorem1cvratex 28351* There exists an atom less than an element covered by 1. (Contributed by NM, 7-May-2012.) (Revised by Mario Carneiro, 13-Jun-2014.)

Theorem1cvratlt 28352 An atom less than or equal to an element covered by 1 is less than the element. (Contributed by NM, 7-May-2012.)

Theorem1cvrjat 28353 An element covered by the lattice unit, when joined with an atom not under it, equals the lattice unit. (Contributed by NM, 30-Apr-2012.)

Theorem1cvrat 28354 Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 30-Apr-2012.)

Theoremps-1 28355 The join of two atoms (specifying a projective geometry line) is determined uniquely by any two atoms (specifying two points) less than or equal to that join. Part of Lemma 16.4 of [MaedaMaeda] p. 69, showing projective space postulate PS1 in [MaedaMaeda] p. 67. (Contributed by NM, 15-Nov-2011.)

Theoremps-2 28356* Lattice analog for the projective geometry axiom, "if a line intersects two sides of a triangle at different points then it also intersects the third side." Projective space condition PS2 in [MaedaMaeda] p. 68 and part of Theorem 16.4 in [MaedaMaeda] p. 69. (Contributed by NM, 1-Dec-2011.)

Theorem2atjlej 28357 Two atoms are be different if their join majorizes the join of two different atoms. (Contributed by NM, 4-Jun-2013.)

Theoremhlatexch3N 28358 Rearrange join of atoms in an equality. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)

Theoremhlatexch4 28359 Exchange 2 atoms. (Contributed by NM, 13-May-2013.)

Theoremps-2b 28360 Variation of projective geometry axiom ps-2 28356. (Contributed by NM, 3-Jul-2012.)

Theorem3atlem1 28361 Lemma for 3at 28368. (Contributed by NM, 22-Jun-2012.)

Theorem3atlem2 28362 Lemma for 3at 28368. (Contributed by NM, 22-Jun-2012.)

Theorem3atlem3 28363 Lemma for 3at 28368. (Contributed by NM, 23-Jun-2012.)

Theorem3atlem4 28364 Lemma for 3at 28368. (Contributed by NM, 23-Jun-2012.)

Theorem3atlem5 28365 Lemma for 3at 28368. (Contributed by NM, 23-Jun-2012.)

Theorem3atlem6 28366 Lemma for 3at 28368. (Contributed by NM, 23-Jun-2012.)

Theorem3atlem7 28367 Lemma for 3at 28368. (Contributed by NM, 23-Jun-2012.)

Theorem3at 28368 Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analog of ps-1 28355 for lines and 4at 28491 for volumes. I could not find this proof in the literature on projective geometry (where it is either given as an axiom or stated as an unproved fact), but it is similar to Theorem 15 of Veblen, "The Foundations of Geometry" (1911), p. 18, which uses different axioms. This proof was written before I became aware of Veblen's, and it is possible that a shorter proof could be obtained by using Veblen's proof for hints. (Contributed by NM, 23-Jun-2012.)

16.22.9  Projective geometries based on Hilbert lattices

Syntaxclln 28369 Extend class notation with set of all "lattice lines" (lattice elements which cover an atom) in a Hilbert lattice.

Syntaxclpl 28370 Extend class notation with set of all "lattice planes" (lattice elements which cover a line) in a Hilbert lattice.

Syntaxclvol 28371 Extend class notation with set of all 3-dimensional "lattice volumes" (lattice elements which cover a plane) in a Hilbert lattice.

Syntaxclines 28372 Extend class notation with set of all projective lines for a Hilbert lattice.

SyntaxcpointsN 28373 Extend class notation with set of all projective points.

Syntaxcpsubsp 28374 Extend class notation with set of all projective subspaces.

Syntaxcpmap 28375 Extend class notation with projective map.

Definitiondf-llines 28376* Define the set of all "lattice lines" (lattice elements which cover an atom) in a Hilbert lattice , in other words all elements of height 2 (or lattice dimension 2 or projective dimension 1). (Contributed by NM, 16-Jun-2012.)

Definitiondf-lplanes 28377* Define the set of all "lattice planes" (lattice elements which cover a line) in a Hilbert lattice , in other words all elements of height 3 (or lattice dimension 3 or projective dimension 2). (Contributed by NM, 16-Jun-2012.)

Definitiondf-lvols 28378* Define the set of all 3-dimensional "lattice volumes" (lattice elements which cover a plane) in a Hilbert lattice , in other words all elements of height 4 (or lattice dimension 4 or projective dimension 3). (Contributed by NM, 1-Jul-2012.)

Definitiondf-lines 28379* Define set of all projective lines for a Hilbert lattice (actually in any set at all, for simplicity). The join of two distinct atoms equals a line. Definition of lines in item 1 of [Holland95] p. 222. (Contributed by NM, 19-Sep-2011.)

Definitiondf-pointsN 28380* Define set of all projective points in a Hilbert lattice (actually in any set at all, for simplicity). A projective point is the singleton of a lattice atom. Definition 15.1 of [MaedaMaeda] p. 61. Note that item 1 in [Holland95] p. 222 defines a point as the atom itself, but this leads to a complicated subspace ordering that may be either membership or inclusion based on its arguments. (Contributed by NM, 2-Oct-2011.)

Definitiondf-psubsp 28381* Define set of all projective subspaces. Based on definition of subspace in [Holland95] p. 212. (Contributed by NM, 2-Oct-2011.)

Definitiondf-pmap 28382* Define projective map for at . Definition in Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 2-Oct-2011.)

Theoremllnset 28383* The set of lattice lines in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)

Theoremislln 28384* The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.)

Theoremislln4 28385* The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.)

Theoremllni 28386 Condition implying a lattice line. (Contributed by NM, 17-Jun-2012.)

Theoremllnbase 28387 A lattice line is a lattice element. (Contributed by NM, 16-Jun-2012.)

Theoremislln3 28388* The predicate "is a lattice line". (Contributed by NM, 17-Jun-2012.)

Theoremislln2 28389* The predicate "is a lattice line". (Contributed by NM, 23-Jun-2012.)

Theoremllni2 28390 The join of two different atoms is a lattice line. (Contributed by NM, 26-Jun-2012.)

Theoremllnnleat 28391 An atom cannot majorize a lattice line. (Contributed by NM, 8-Jul-2012.)

Theoremllnneat 28392 A lattice line is not an atom. (Contributed by NM, 19-Jun-2012.)

Theorem2atneat 28393 The join of two distinct atoms is not an atom. (Contributed by NM, 12-Oct-2012.)

Theoremllnn0 28394 A lattice line is non-zero. (Contributed by NM, 15-Jul-2012.)

Theoremislln2a 28395 The predicate "is a lattice line" in terms of atoms. (Contributed by NM, 15-Jul-2012.)

Theoremllnle 28396* Any element greater than 0 and not an atom majorizes a lattice line. (Contributed by NM, 28-Jun-2012.)

Theorematcvrlln2 28397 An atom under a line is covered by it. (Contributed by NM, 2-Jul-2012.)

Theorematcvrlln 28398 An element covering an atom is a lattice line and vice-versa. (Contributed by NM, 18-Jun-2012.)

TheoremllnexatN 28399* Given an atom on a line, there is another atom whose join equals the line. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)

Theoremllncmp 28400 If two lattice lines are comparable, they are equal. (Contributed by NM, 19-Jun-2012.)

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