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Theorem List for Metamath Proof Explorer - 29901-30000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcvrexchlem 29901 Lemma for cvrexch 29902. (cvexchlem 23824 analog.) (Contributed by NM, 18-Nov-2011.)

Theoremcvrexch 29902 A Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of [Kalmbach] p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (cvexchi 23825 analog.) (Contributed by NM, 18-Nov-2011.)

Theoremcvratlem 29903 Lemma for cvrat 29904. (atcvatlem 23841 analog.) (Contributed by NM, 22-Nov-2011.)

Theoremcvrat 29904 A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 23842 analog.) (Contributed by NM, 22-Nov-2011.)

Theoremltltncvr 29905 A chained strong ordering is not a covers relation. (Contributed by NM, 18-Jun-2012.)

Theoremltcvrntr 29906 Non-transitive condition for the covers relation. (Contributed by NM, 18-Jun-2012.)

Theoremcvrntr 29907 The covers relation is not transitive. (cvntr 23748 analog.) (Contributed by NM, 18-Jun-2012.)

Theorematcvr0eq 29908 The covers relation is not transitive. (atcv0eq 23835 analog.) (Contributed by NM, 29-Nov-2011.)

Theoremlnnat 29909 A line (the join of two distinct atoms) is not an atom. (Contributed by NM, 14-Jun-2012.)

Theorematcvrj0 29910 Two atoms covering the zero subspace are equal. (atcv1 23836 analog.) (Contributed by NM, 29-Nov-2011.)

Theoremcvrat2 29911 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 23843 analog.) (Contributed by NM, 30-Nov-2011.)

TheorematcvrneN 29912 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

Theorematcvrj1 29913 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)

Theorematcvrj2b 29914 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)

Theorematcvrj2 29915 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)

TheorematleneN 29916 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

Theorematltcvr 29917 An equivalence of less-than ordering and covers relation. (Contributed by NM, 7-Feb-2012.)

Theorematle 29918* Any non-zero element has an atom under it. (Contributed by NM, 28-Jun-2012.)

Theorematlt 29919 Two atoms are unequal iff their join is greater than one of them. (Contributed by NM, 6-May-2012.)

Theorematlelt 29920 Transfer less-than relation from one atom to another. (Contributed by NM, 7-May-2012.)

Theorem2atlt 29921* Given an atom less than an element, there is another atom less than the element. (Contributed by NM, 6-May-2012.)

TheorematexchcvrN 29922 Atom exchange property. Version of hlatexch2 29878 with covers relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

TheorematexchltN 29923 Atom exchange property. Version of hlatexch2 29878 with less-than ordering. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

Theoremcvrat3 29924 A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 23852 analog.) (Contributed by NM, 30-Nov-2011.)

Theoremcvrat4 29925* 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 23853 analog.) (Contributed by NM, 30-Nov-2011.)

Theoremcvrat42 29926* Commuted version of cvrat4 29925. (Contributed by NM, 28-Jan-2012.)

Theorem2atjm 29927 The meet of a line (expressed with 2 atoms) and a lattice element. (Contributed by NM, 30-Jul-2012.)

Theorematbtwn 29928 Property of a 3rd atom on a line intersecting element at . (Contributed by NM, 30-Jul-2012.)

TheorematbtwnexOLDN 29929* 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 29930* 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 29931 Two ways to express 3 non-colinear atoms (rotated right 2 places). (Contributed by NM, 12-Jul-2012.)

Theorem3noncolr1N 29932 Two ways to express 3 non-colinear atoms (rotated right 1 place). (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)

Theoremhlatcon3 29933 Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.)

Theoremhlatcon2 29934 Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.)

Theorem4noncolr3 29935 A way to express 4 non-colinear atoms (rotated right 3 places). (Contributed by NM, 11-Jul-2012.)

Theorem4noncolr2 29936 A way to express 4 non-colinear atoms (rotated right 2 places). (Contributed by NM, 11-Jul-2012.)

Theorem4noncolr1 29937 A way to express 4 non-colinear atoms (rotated right 1 places). (Contributed by NM, 11-Jul-2012.)

Theoremathgt 29938* 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 29939* There exists a 3-dimensional (height-4) element i.e. a volume. (Contributed by NM, 25-Jul-2012.)

Theorem3dimlem1 29940 Lemma for 3dim1 29949. (Contributed by NM, 25-Jul-2012.)

Theorem3dimlem2 29941 Lemma for 3dim1 29949. (Contributed by NM, 25-Jul-2012.)

Theorem3dimlem3a 29942 Lemma for 3dim3 29951. (Contributed by NM, 27-Jul-2012.)

Theorem3dimlem3 29943 Lemma for 3dim1 29949. (Contributed by NM, 25-Jul-2012.)

Theorem3dimlem3OLDN 29944 Lemma for 3dim1 29949. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem3dimlem4a 29945 Lemma for 3dim3 29951. (Contributed by NM, 27-Jul-2012.)

Theorem3dimlem4 29946 Lemma for 3dim1 29949. (Contributed by NM, 25-Jul-2012.)

Theorem3dimlem4OLDN 29947 Lemma for 3dim1 29949. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem3dim1lem5 29948* Lemma for 3dim1 29949. (Contributed by NM, 26-Jul-2012.)

Theorem3dim1 29949* Construct a 3-dimensional volume (height-4 element) on top of a given atom . (Contributed by NM, 25-Jul-2012.)

Theorem3dim2 29950* Construct 2 new layers on top of 2 given atoms. (Contributed by NM, 27-Jul-2012.)

Theorem3dim3 29951* Construct a new layer on top of 3 given atoms. (Contributed by NM, 27-Jul-2012.)

Theorem2dim 29952* Generate a height-3 element (2-dimensional plane) from an atom. (Contributed by NM, 3-May-2012.)

Theorem1dimN 29953* An atom is covered by a height-2 element (1-dimensional line). (Contributed by NM, 3-May-2012.) (New usage is discouraged.)

Theorem1cvrco 29954 The orthocomplement of an element covered by 1 is an atom. (Contributed by NM, 7-May-2012.)

Theorem1cvratex 29955* 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 29956 An atom less than or equal to an element covered by 1 is less than the element. (Contributed by NM, 7-May-2012.)

Theorem1cvrjat 29957 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 29958 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 29959 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 29960* 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 29961 Two atoms are different if their join majorizes the join of two different atoms. (Contributed by NM, 4-Jun-2013.)

Theoremhlatexch3N 29962 Rearrange join of atoms in an equality. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)

Theoremhlatexch4 29963 Exchange 2 atoms. (Contributed by NM, 13-May-2013.)

Theoremps-2b 29964 Variation of projective geometry axiom ps-2 29960. (Contributed by NM, 3-Jul-2012.)

Theorem3atlem1 29965 Lemma for 3at 29972. (Contributed by NM, 22-Jun-2012.)

Theorem3atlem2 29966 Lemma for 3at 29972. (Contributed by NM, 22-Jun-2012.)

Theorem3atlem3 29967 Lemma for 3at 29972. (Contributed by NM, 23-Jun-2012.)

Theorem3atlem4 29968 Lemma for 3at 29972. (Contributed by NM, 23-Jun-2012.)

Theorem3atlem5 29969 Lemma for 3at 29972. (Contributed by NM, 23-Jun-2012.)

Theorem3atlem6 29970 Lemma for 3at 29972. (Contributed by NM, 23-Jun-2012.)

Theorem3atlem7 29971 Lemma for 3at 29972. (Contributed by NM, 23-Jun-2012.)

Theorem3at 29972 Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analog of ps-1 29959 for lines and 4at 30095 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 29973 Extend class notation with set of all "lattice lines" (lattice elements which cover an atom) in a Hilbert lattice.

Syntaxclpl 29974 Extend class notation with set of all "lattice planes" (lattice elements which cover a line) in a Hilbert lattice.

Syntaxclvol 29975 Extend class notation with set of all 3-dimensional "lattice volumes" (lattice elements which cover a plane) in a Hilbert lattice.

Syntaxclines 29976 Extend class notation with set of all projective lines for a Hilbert lattice.

SyntaxcpointsN 29977 Extend class notation with set of all projective points.

Syntaxcpsubsp 29978 Extend class notation with set of all projective subspaces.

Syntaxcpmap 29979 Extend class notation with projective map.

Definitiondf-llines 29980* 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 29981* 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 29982* 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 29983* 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 29984* 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 29985* Define set of all projective subspaces. Based on definition of subspace in [Holland95] p. 212. (Contributed by NM, 2-Oct-2011.)

Definitiondf-pmap 29986* Define projective map for at . Definition in Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 2-Oct-2011.)

Theoremllnset 29987* The set of lattice lines in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)

Theoremislln 29988* The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.)

Theoremislln4 29989* The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.)

Theoremllni 29990 Condition implying a lattice line. (Contributed by NM, 17-Jun-2012.)

Theoremllnbase 29991 A lattice line is a lattice element. (Contributed by NM, 16-Jun-2012.)

Theoremislln3 29992* The predicate "is a lattice line". (Contributed by NM, 17-Jun-2012.)

Theoremislln2 29993* The predicate "is a lattice line". (Contributed by NM, 23-Jun-2012.)

Theoremllni2 29994 The join of two different atoms is a lattice line. (Contributed by NM, 26-Jun-2012.)

Theoremllnnleat 29995 An atom cannot majorize a lattice line. (Contributed by NM, 8-Jul-2012.)

Theoremllnneat 29996 A lattice line is not an atom. (Contributed by NM, 19-Jun-2012.)

Theorem2atneat 29997 The join of two distinct atoms is not an atom. (Contributed by NM, 12-Oct-2012.)

Theoremllnn0 29998 A lattice line is non-zero. (Contributed by NM, 15-Jul-2012.)

Theoremislln2a 29999 The predicate "is a lattice line" in terms of atoms. (Contributed by NM, 15-Jul-2012.)

Theoremllnle 30000* Any element greater than 0 and not an atom majorizes a lattice line. (Contributed by NM, 28-Jun-2012.)

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