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Theorem List for Metamath Proof Explorer - 30301-30400   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheorematcvrneN 30301 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

Theorematcvrj1 30302 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)

Theorematcvrj2b 30303 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)

Theorematcvrj2 30304 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)

TheorematleneN 30305 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

Theorematltcvr 30306 An equivalence of less-than ordering and covers relation. (Contributed by NM, 7-Feb-2012.)

Theorematle 30307* Any non-zero element has an atom under it. (Contributed by NM, 28-Jun-2012.)

Theorematlt 30308 Two atoms are unequal iff their join is greater than one of them. (Contributed by NM, 6-May-2012.)

Theorematlelt 30309 Transfer less-than relation from one atom to another. (Contributed by NM, 7-May-2012.)

Theorem2atlt 30310* Given an atom less than an element, there is another atom less than the element. (Contributed by NM, 6-May-2012.)

TheorematexchcvrN 30311 Atom exchange property. Version of hlatexch2 30267 with covers relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

TheorematexchltN 30312 Atom exchange property. Version of hlatexch2 30267 with less-than ordering. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

Theoremcvrat3 30313 A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 23904 analog.) (Contributed by NM, 30-Nov-2011.)

Theoremcvrat4 30314* 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 23905 analog.) (Contributed by NM, 30-Nov-2011.)

Theoremcvrat42 30315* Commuted version of cvrat4 30314. (Contributed by NM, 28-Jan-2012.)

Theorem2atjm 30316 The meet of a line (expressed with 2 atoms) and a lattice element. (Contributed by NM, 30-Jul-2012.)

Theorematbtwn 30317 Property of a 3rd atom on a line intersecting element at . (Contributed by NM, 30-Jul-2012.)

TheorematbtwnexOLDN 30318* 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 30319* 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 30320 Two ways to express 3 non-colinear atoms (rotated right 2 places). (Contributed by NM, 12-Jul-2012.)

Theorem3noncolr1N 30321 Two ways to express 3 non-colinear atoms (rotated right 1 place). (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)

Theoremhlatcon3 30322 Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.)

Theoremhlatcon2 30323 Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.)

Theorem4noncolr3 30324 A way to express 4 non-colinear atoms (rotated right 3 places). (Contributed by NM, 11-Jul-2012.)

Theorem4noncolr2 30325 A way to express 4 non-colinear atoms (rotated right 2 places). (Contributed by NM, 11-Jul-2012.)

Theorem4noncolr1 30326 A way to express 4 non-colinear atoms (rotated right 1 places). (Contributed by NM, 11-Jul-2012.)

Theoremathgt 30327* 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 30328* There exists a 3-dimensional (height-4) element i.e. a volume. (Contributed by NM, 25-Jul-2012.)

Theorem3dimlem1 30329 Lemma for 3dim1 30338. (Contributed by NM, 25-Jul-2012.)

Theorem3dimlem2 30330 Lemma for 3dim1 30338. (Contributed by NM, 25-Jul-2012.)

Theorem3dimlem3a 30331 Lemma for 3dim3 30340. (Contributed by NM, 27-Jul-2012.)

Theorem3dimlem3 30332 Lemma for 3dim1 30338. (Contributed by NM, 25-Jul-2012.)

Theorem3dimlem3OLDN 30333 Lemma for 3dim1 30338. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem3dimlem4a 30334 Lemma for 3dim3 30340. (Contributed by NM, 27-Jul-2012.)

Theorem3dimlem4 30335 Lemma for 3dim1 30338. (Contributed by NM, 25-Jul-2012.)

Theorem3dimlem4OLDN 30336 Lemma for 3dim1 30338. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem3dim1lem5 30337* Lemma for 3dim1 30338. (Contributed by NM, 26-Jul-2012.)

Theorem3dim1 30338* Construct a 3-dimensional volume (height-4 element) on top of a given atom . (Contributed by NM, 25-Jul-2012.)

Theorem3dim2 30339* Construct 2 new layers on top of 2 given atoms. (Contributed by NM, 27-Jul-2012.)

Theorem3dim3 30340* Construct a new layer on top of 3 given atoms. (Contributed by NM, 27-Jul-2012.)

Theorem2dim 30341* Generate a height-3 element (2-dimensional plane) from an atom. (Contributed by NM, 3-May-2012.)

Theorem1dimN 30342* An atom is covered by a height-2 element (1-dimensional line). (Contributed by NM, 3-May-2012.) (New usage is discouraged.)

Theorem1cvrco 30343 The orthocomplement of an element covered by 1 is an atom. (Contributed by NM, 7-May-2012.)

Theorem1cvratex 30344* 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 30345 An atom less than or equal to an element covered by 1 is less than the element. (Contributed by NM, 7-May-2012.)

Theorem1cvrjat 30346 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 30347 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 30348 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 30349* 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 30350 Two atoms are different if their join majorizes the join of two different atoms. (Contributed by NM, 4-Jun-2013.)

Theoremhlatexch3N 30351 Rearrange join of atoms in an equality. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)

Theoremhlatexch4 30352 Exchange 2 atoms. (Contributed by NM, 13-May-2013.)

Theoremps-2b 30353 Variation of projective geometry axiom ps-2 30349. (Contributed by NM, 3-Jul-2012.)

Theorem3atlem1 30354 Lemma for 3at 30361. (Contributed by NM, 22-Jun-2012.)

Theorem3atlem2 30355 Lemma for 3at 30361. (Contributed by NM, 22-Jun-2012.)

Theorem3atlem3 30356 Lemma for 3at 30361. (Contributed by NM, 23-Jun-2012.)

Theorem3atlem4 30357 Lemma for 3at 30361. (Contributed by NM, 23-Jun-2012.)

Theorem3atlem5 30358 Lemma for 3at 30361. (Contributed by NM, 23-Jun-2012.)

Theorem3atlem6 30359 Lemma for 3at 30361. (Contributed by NM, 23-Jun-2012.)

Theorem3atlem7 30360 Lemma for 3at 30361. (Contributed by NM, 23-Jun-2012.)

Theorem3at 30361 Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analog of ps-1 30348 for lines and 4at 30484 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.)

19.26.9  Projective geometries based on Hilbert lattices

Syntaxclln 30362 Extend class notation with set of all "lattice lines" (lattice elements which cover an atom) in a Hilbert lattice.

Syntaxclpl 30363 Extend class notation with set of all "lattice planes" (lattice elements which cover a line) in a Hilbert lattice.

Syntaxclvol 30364 Extend class notation with set of all 3-dimensional "lattice volumes" (lattice elements which cover a plane) in a Hilbert lattice.

Syntaxclines 30365 Extend class notation with set of all projective lines for a Hilbert lattice.

SyntaxcpointsN 30366 Extend class notation with set of all projective points.

Syntaxcpsubsp 30367 Extend class notation with set of all projective subspaces.

Syntaxcpmap 30368 Extend class notation with projective map.

Definitiondf-llines 30369* 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 30370* 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 30371* 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 30372* 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 30373* 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 30374* Define set of all projective subspaces. Based on definition of subspace in [Holland95] p. 212. (Contributed by NM, 2-Oct-2011.)

Definitiondf-pmap 30375* Define projective map for at . Definition in Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 2-Oct-2011.)

Theoremllnset 30376* The set of lattice lines in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)

Theoremislln 30377* The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.)

Theoremislln4 30378* The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.)

Theoremllni 30379 Condition implying a lattice line. (Contributed by NM, 17-Jun-2012.)

Theoremllnbase 30380 A lattice line is a lattice element. (Contributed by NM, 16-Jun-2012.)

Theoremislln3 30381* The predicate "is a lattice line". (Contributed by NM, 17-Jun-2012.)

Theoremislln2 30382* The predicate "is a lattice line". (Contributed by NM, 23-Jun-2012.)

Theoremllni2 30383 The join of two different atoms is a lattice line. (Contributed by NM, 26-Jun-2012.)

Theoremllnnleat 30384 An atom cannot majorize a lattice line. (Contributed by NM, 8-Jul-2012.)

Theoremllnneat 30385 A lattice line is not an atom. (Contributed by NM, 19-Jun-2012.)

Theorem2atneat 30386 The join of two distinct atoms is not an atom. (Contributed by NM, 12-Oct-2012.)

Theoremllnn0 30387 A lattice line is non-zero. (Contributed by NM, 15-Jul-2012.)

Theoremislln2a 30388 The predicate "is a lattice line" in terms of atoms. (Contributed by NM, 15-Jul-2012.)

Theoremllnle 30389* Any element greater than 0 and not an atom majorizes a lattice line. (Contributed by NM, 28-Jun-2012.)

Theorematcvrlln2 30390 An atom under a line is covered by it. (Contributed by NM, 2-Jul-2012.)

Theorematcvrlln 30391 An element covering an atom is a lattice line and vice-versa. (Contributed by NM, 18-Jun-2012.)

TheoremllnexatN 30392* 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 30393 If two lattice lines are comparable, they are equal. (Contributed by NM, 19-Jun-2012.)

Theoremllnnlt 30394 Two lattice lines cannot satisfy the less than relation. (Contributed by NM, 26-Jun-2012.)

Theorem2llnmat 30395 Two intersecting lines intersect at an atom. (Contributed by NM, 30-Apr-2012.)

Theorem2at0mat0 30396 Special case of 2atmat0 30397 where one atom could be zero. (Contributed by NM, 30-May-2013.)

Theorem2atmat0 30397 The meet of two unequal lines (expressed as joins of atoms) is an atom or zero. (Contributed by NM, 2-Dec-2012.)

Theorem2atm 30398 An atom majorized by two different atom joins (which could be atoms or lines) is equal to their intersection. (Contributed by NM, 30-Jun-2013.)

Theoremps-2c 30399 Variation of projective geometry axiom ps-2 30349. (Contributed by NM, 3-Jul-2012.)

Theoremlplnset 30400* The set of lattice planes in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)

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