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Theorem List for Metamath Proof Explorer - 30201-30300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhl0lt1N 30201 Lattice 0 is less than lattice 1 in a Hilbert lattice. (Contributed by NM, 4-Dec-2011.) (New usage is discouraged.)

Theoremhlexch3 30202 A Hilbert lattice has the exchange property. (atexch 22977 analog.) (Contributed by NM, 15-Nov-2011.)

Theoremhlexch4N 30203 A Hilbert lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (Contributed by NM, 15-Nov-2011.) (New usage is discouraged.)

Theoremhlatexchb1 30204 A version of hlexchb1 30195 for atoms. (Contributed by NM, 15-Nov-2011.)

Theoremhlatexchb2 30205 A version of hlexchb2 30196 for atoms. (Contributed by NM, 7-Feb-2012.)

Theoremhlatexch1 30206 Atom exchange property. (Contributed by NM, 7-Jan-2012.)

Theoremhlatexch2 30207 Atom exchange property. (Contributed by NM, 8-Jan-2012.)

TheoremhlatmstcOLDN 30208* 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 22958 analog.) (Contributed by NM, 21-Oct-2011.) (New usage is discouraged.)

Theoremhlatle 30209* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 22967 analog.) (Contributed by NM, 4-Nov-2011.)

Theoremhlateq 30210* The equality of two Hilbert lattice elements is determined by the atoms under them. (chrelat4i 22969 analog.) (Contributed by NM, 24-May-2012.)

Theoremhlrelat1 30211* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 22959, with swapped, analog.) (Contributed by NM, 4-Dec-2011.)

Theoremhlrelat5N 30212* An atomistic lattice with 0 is relatively atomic, using the definition in Remark 2 of [Kalmbach] p. 149. (Contributed by NM, 21-Oct-2011.) (New usage is discouraged.)

Theoremhlrelat 30213* A Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 22960 analog.) (Contributed by NM, 4-Feb-2012.)

Theoremhlrelat2 30214* A consequence of relative atomicity. (chrelat2i 22961 analog.) (Contributed by NM, 5-Feb-2012.)

TheoremexatleN 30215 A condition for an atom to be less than or equal to a lattice element. Part of proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)

Theoremhl2at 30216* A Hilbert lattice has at least 2 atoms. (Contributed by NM, 5-Dec-2011.)

Theorematex 30217 At least one atom exists. (Contributed by NM, 15-Jul-2012.)

TheoremintnatN 30218 If the intersection with a non-majorizing element is an atom, the intersecting element is not an atom. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)

Theorem2llnne2N 30219 Condition implying that two intersecting lines are different. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)

Theorem2llnneN 30220 Condition implying that two intersecting lines are different. (Contributed by NM, 29-May-2012.) (New usage is discouraged.)

Theoremcvr1 30221 A Hilbert lattice has the covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 22951 analog.) (Contributed by NM, 17-Nov-2011.)

Theoremcvr2N 30222 Less-than and covers equivalence in a Hilbert lattice. (chcv2 22952 analog.) (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

Theoremhlrelat3 30223* The Hilbert lattice is relatively atomic. Stronger version of hlrelat 30213. (Contributed by NM, 2-May-2012.)

Theoremcvrval3 30224* Binary relation expressing covers . (Contributed by NM, 16-Jun-2012.)

Theoremcvrval4N 30225* Binary relation expressing covers . (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)

Theoremcvrval5 30226* Binary relation expressing covers . (Contributed by NM, 7-Dec-2012.)

Theoremcvrp 30227 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 22971 analog.) (Contributed by NM, 18-Nov-2011.)

Theorematcvr1 30228 An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012.)

Theorematcvr2 30229 An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012.)

Theoremcvrexchlem 30230 Lemma for cvrexch 30231. (cvexchlem 22964 analog.) (Contributed by NM, 18-Nov-2011.)

Theoremcvrexch 30231 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 22965 analog.) (Contributed by NM, 18-Nov-2011.)

Theoremcvratlem 30232 Lemma for cvrat 30233. (atcvatlem 22981 analog.) (Contributed by NM, 22-Nov-2011.)

Theoremcvrat 30233 A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 22982 analog.) (Contributed by NM, 22-Nov-2011.)

Theoremltltncvr 30234 A chained strong ordering is not a covers relation. (Contributed by NM, 18-Jun-2012.)

Theoremltcvrntr 30235 Non-transitive condition for the covers relation. (Contributed by NM, 18-Jun-2012.)

Theoremcvrntr 30236 The covers relation is not transitive. (cvntr 22888 analog.) (Contributed by NM, 18-Jun-2012.)

Theorematcvr0eq 30237 The covers relation is not transitive. (atcv0eq 22975 analog.) (Contributed by NM, 29-Nov-2011.)

Theoremlnnat 30238 A line (the join of two distinct atoms) is not an atom. (Contributed by NM, 14-Jun-2012.)

Theorematcvrj0 30239 Two atoms covering the zero subspace are equal. (atcv1 22976 analog.) (Contributed by NM, 29-Nov-2011.)

Theoremcvrat2 30240 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 22983 analog.) (Contributed by NM, 30-Nov-2011.)

TheorematcvrneN 30241 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

Theorematcvrj1 30242 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)

Theorematcvrj2b 30243 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)

Theorematcvrj2 30244 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)

TheorematleneN 30245 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

Theorematltcvr 30246 An equivalence of less-than ordering and covers relation. (Contributed by NM, 7-Feb-2012.)

Theorematle 30247* Any non-zero element has an atom under it. (Contributed by NM, 28-Jun-2012.)

Theorematlt 30248 Two atoms are unequal iff their join is greater than one of them. (Contributed by NM, 6-May-2012.)

Theorematlelt 30249 Transfer less-than relation from one atom to another. (Contributed by NM, 7-May-2012.)

Theorem2atlt 30250* Given an atom less than an element, there is another atom less than the element. (Contributed by NM, 6-May-2012.)

TheorematexchcvrN 30251 Atom exchange property. Version of hlatexch2 30207 with covers relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

TheorematexchltN 30252 Atom exchange property. Version of hlatexch2 30207 with less-than ordering. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

Theoremcvrat3 30253 A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 22992 analog.) (Contributed by NM, 30-Nov-2011.)

Theoremcvrat4 30254* 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 22993 analog.) (Contributed by NM, 30-Nov-2011.)

Theoremcvrat42 30255* Commuted version of cvrat4 30254. (Contributed by NM, 28-Jan-2012.)

Theorem2atjm 30256 The meet of a line (expressed with 2 atoms) and a lattice element. (Contributed by NM, 30-Jul-2012.)

Theorematbtwn 30257 Property of a 3rd atom on a line intersecting element at . (Contributed by NM, 30-Jul-2012.)

TheorematbtwnexOLDN 30258* 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 30259* 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 30260 Two ways to express 3 non-colinear atoms (rotated right 2 places). (Contributed by NM, 12-Jul-2012.)

Theorem3noncolr1N 30261 Two ways to express 3 non-colinear atoms (rotated right 1 place). (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)

Theoremhlatcon3 30262 Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.)

Theoremhlatcon2 30263 Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.)

Theorem4noncolr3 30264 A way to express 4 non-colinear atoms (rotated right 3 places). (Contributed by NM, 11-Jul-2012.)

Theorem4noncolr2 30265 A way to express 4 non-colinear atoms (rotated right 2 places). (Contributed by NM, 11-Jul-2012.)

Theorem4noncolr1 30266 A way to express 4 non-colinear atoms (rotated right 1 places). (Contributed by NM, 11-Jul-2012.)

Theoremathgt 30267* 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 30268* There exists a 3-dimensional (height-4) element i.e. a volume. (Contributed by NM, 25-Jul-2012.)

Theorem3dimlem1 30269 Lemma for 3dim1 30278. (Contributed by NM, 25-Jul-2012.)

Theorem3dimlem2 30270 Lemma for 3dim1 30278. (Contributed by NM, 25-Jul-2012.)

Theorem3dimlem3a 30271 Lemma for 3dim3 30280. (Contributed by NM, 27-Jul-2012.)

Theorem3dimlem3 30272 Lemma for 3dim1 30278. (Contributed by NM, 25-Jul-2012.)

Theorem3dimlem3OLDN 30273 Lemma for 3dim1 30278. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem3dimlem4a 30274 Lemma for 3dim3 30280. (Contributed by NM, 27-Jul-2012.)

Theorem3dimlem4 30275 Lemma for 3dim1 30278. (Contributed by NM, 25-Jul-2012.)

Theorem3dimlem4OLDN 30276 Lemma for 3dim1 30278. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem3dim1lem5 30277* Lemma for 3dim1 30278. (Contributed by NM, 26-Jul-2012.)

Theorem3dim1 30278* Construct a 3-dimensional volume (height-4 element) on top of a given atom . (Contributed by NM, 25-Jul-2012.)

Theorem3dim2 30279* Construct 2 new layers on top of 2 given atoms. (Contributed by NM, 27-Jul-2012.)

Theorem3dim3 30280* Construct a new layer on top of 3 given atoms. (Contributed by NM, 27-Jul-2012.)

Theorem2dim 30281* Generate a height-3 element (2-dimensional plane) from an atom. (Contributed by NM, 3-May-2012.)

Theorem1dimN 30282* An atom is covered by a height-2 element (1-dimensional line). (Contributed by NM, 3-May-2012.) (New usage is discouraged.)

Theorem1cvrco 30283 The orthocomplement of an element covered by 1 is an atom. (Contributed by NM, 7-May-2012.)

Theorem1cvratex 30284* 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 30285 An atom less than or equal to an element covered by 1 is less than the element. (Contributed by NM, 7-May-2012.)

Theorem1cvrjat 30286 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 30287 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 30288 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 30289* 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 30290 Two atoms are different if their join majorizes the join of two different atoms. (Contributed by NM, 4-Jun-2013.)

Theoremhlatexch3N 30291 Rearrange join of atoms in an equality. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)

Theoremhlatexch4 30292 Exchange 2 atoms. (Contributed by NM, 13-May-2013.)

Theoremps-2b 30293 Variation of projective geometry axiom ps-2 30289. (Contributed by NM, 3-Jul-2012.)

Theorem3atlem1 30294 Lemma for 3at 30301. (Contributed by NM, 22-Jun-2012.)

Theorem3atlem2 30295 Lemma for 3at 30301. (Contributed by NM, 22-Jun-2012.)

Theorem3atlem3 30296 Lemma for 3at 30301. (Contributed by NM, 23-Jun-2012.)

Theorem3atlem4 30297 Lemma for 3at 30301. (Contributed by NM, 23-Jun-2012.)

Theorem3atlem5 30298 Lemma for 3at 30301. (Contributed by NM, 23-Jun-2012.)

Theorem3atlem6 30299 Lemma for 3at 30301. (Contributed by NM, 23-Jun-2012.)

Theorem3atlem7 30300 Lemma for 3at 30301. (Contributed by NM, 23-Jun-2012.)

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