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**Theorem List for Metamath Proof Explorer -** 33001-33100 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	atleneN 33001	Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	atltcvr 33002	An equivalence of less-than ordering and covers relation. (Contributed by NM, 7-Feb-2012.)
$.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	atle 33003*	Any non-zero element has an atom under it. (Contributed by NM, 28-Jun-2012.)
$/\$ $/\$

Theorem	atlt 33004	Two atoms are unequal iff their join is greater than one of them. (Contributed by NM, 6-May-2012.)
$.\/$ $/\$ $/\$ $.\/$

Theorem	atlelt 33005	Transfer less-than relation from one atom to another. (Contributed by NM, 7-May-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	2atlt 33006*	Given an atom less than an element, there is another atom less than the element. (Contributed by NM, 6-May-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	atexchcvrN 33007	Atom exchange property. Version of hlatexch2 32963 with covers relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	atexchltN 33008	Atom exchange property. Version of hlatexch2 32963 with less-than ordering. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cvrat3 33009	A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 28061 analog.) (Contributed by NM, 30-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cvrat4 33010*	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 28062 analog.) (Contributed by NM, 30-Nov-2011.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$

Theorem	cvrat42 33011*	Commuted version of cvrat4 33010. (Contributed by NM, 28-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$

Theorem	2atjm 33012	The meet of a line (expressed with 2 atoms) and a lattice element. (Contributed by NM, 30-Jul-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	atbtwn 33013	Property of a 3rd atom on a line $.\/$ intersecting element at . (Contributed by NM, 30-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	atbtwnexOLDN 33014*	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.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	atbtwnex 33015*	Given atoms in and not in , there exists an atom not in such that the line $.\/$ intersects at . (Contributed by NM, 1-Aug-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	3noncolr2 33016	Two ways to express 3 non-colinear atoms (rotated right 2 places). (Contributed by NM, 12-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$

Theorem	3noncolr1N 33017	Two ways to express 3 non-colinear atoms (rotated right 1 place). (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$

Theorem	hlatcon3 33018	Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	hlatcon2 33019	Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	4noncolr3 33020	A way to express 4 non-colinear atoms (rotated right 3 places). (Contributed by NM, 11-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4noncolr2 33021	A way to express 4 non-colinear atoms (rotated right 2 places). (Contributed by NM, 11-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4noncolr1 33022	A way to express 4 non-colinear atoms (rotated right 1 places). (Contributed by NM, 11-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	athgt 33023*	A Hilbert lattice, whose height is at least 4, has a chain of 4 successively covering atom joins. (Contributed by NM, 3-May-2012.)
$.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	3dim0 33024*	There exists a 3-dimensional (height-4) element i.e. a volume. (Contributed by NM, 25-Jul-2012.)
$.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	3dimlem1 33025	Lemma for 3dim1 33034. (Contributed by NM, 25-Jul-2012.)
$.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	3dimlem2 33026	Lemma for 3dim1 33034. (Contributed by NM, 25-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	3dimlem3a 33027	Lemma for 3dim3 33036. (Contributed by NM, 27-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	3dimlem3 33028	Lemma for 3dim1 33034. (Contributed by NM, 25-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	3dimlem3OLDN 33029	Lemma for 3dim1 33034. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	3dimlem4a 33030	Lemma for 3dim3 33036. (Contributed by NM, 27-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	3dimlem4 33031	Lemma for 3dim1 33034. (Contributed by NM, 25-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	3dimlem4OLDN 33032	Lemma for 3dim1 33034. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	3dim1lem5 33033*	Lemma for 3dim1 33034. (Contributed by NM, 26-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	3dim1 33034*	Construct a 3-dimensional volume (height-4 element) on top of a given atom . (Contributed by NM, 25-Jul-2012.)
$.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	3dim2 33035*	Construct 2 new layers on top of 2 given atoms. (Contributed by NM, 27-Jul-2012.)
$.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	3dim3 33036*	Construct a new layer on top of 3 given atoms. (Contributed by NM, 27-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	2dim 33037*	Generate a height-3 element (2-dimensional plane) from an atom. (Contributed by NM, 3-May-2012.)
$.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	1dimN 33038*	An atom is covered by a height-2 element (1-dimensional line). (Contributed by NM, 3-May-2012.) (New usage is discouraged.)
$.\/$ $/\$ $.\/$

Theorem	1cvrco 33039	The orthocomplement of an element covered by 1 is an atom. (Contributed by NM, 7-May-2012.)
$/\$

Theorem	1cvratex 33040*	There exists an atom less than an element covered by 1. (Contributed by NM, 7-May-2012.) (Revised by Mario Carneiro, 13-Jun-2014.)
$/\$ $/\$

Theorem	1cvratlt 33041	An atom less than or equal to an element covered by 1 is less than the element. (Contributed by NM, 7-May-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	1cvrjat 33042	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.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	1cvrat 33043	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.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	ps-1 33044	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.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	ps-2 33045*	Lattice analogue 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.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$

Theorem	2atjlej 33046	Two atoms are different if their join majorizes the join of two different atoms. (Contributed by NM, 4-Jun-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	hlatexch3N 33047	Rearrange join of atoms in an equality. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	hlatexch4 33048	Exchange 2 atoms. (Contributed by NM, 13-May-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	ps-2b 33049	Variation of projective geometry axiom ps-2 33045. (Contributed by NM, 3-Jul-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	3atlem1 33050	Lemma for 3at 33057. (Contributed by NM, 22-Jun-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	3atlem2 33051	Lemma for 3at 33057. (Contributed by NM, 22-Jun-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	3atlem3 33052	Lemma for 3at 33057. (Contributed by NM, 23-Jun-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	3atlem4 33053	Lemma for 3at 33057. (Contributed by NM, 23-Jun-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	3atlem5 33054	Lemma for 3at 33057. (Contributed by NM, 23-Jun-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	3atlem6 33055	Lemma for 3at 33057. (Contributed by NM, 23-Jun-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	3atlem7 33056	Lemma for 3at 33057. (Contributed by NM, 23-Jun-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	3at 33057	Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analogue of ps-1 33044 for lines and 4at 33180 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.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

21.21.12 Projective geometries based on Hilbert lattices

Syntax	clln 33058	Extend class notation with set of all "lattice lines" (lattice elements which cover an atom) in a Hilbert lattice.


Syntax	clpl 33059	Extend class notation with set of all "lattice planes" (lattice elements which cover a line) in a Hilbert lattice.


Syntax	clvol 33060	Extend class notation with set of all 3-dimensional "lattice volumes" (lattice elements which cover a plane) in a Hilbert lattice.


Syntax	clines 33061	Extend class notation with set of all projective lines for a Hilbert lattice.


Syntax	cpointsN 33062	Extend class notation with set of all projective points.


Syntax	cpsubsp 33063	Extend class notation with set of all projective subspaces.


Syntax	cpmap 33064	Extend class notation with projective map.


Definition	df-llines 33065*	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.)


Definition	df-lplanes 33066*	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.)


Definition	df-lvols 33067*	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.)


Definition	df-lines 33068*	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.)
$/\$

Definition	df-pointsN 33069*	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.)


Definition	df-psubsp 33070*	Define set of all projective subspaces. Based on definition of subspace in [Holland95] p. 212. (Contributed by NM, 2-Oct-2011.)
$/\$

Definition	df-pmap 33071*	Define projective map for at . Definition in Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 2-Oct-2011.)


Theorem	llnset 33072*	The set of lattice lines in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)


Theorem	islln 33073*	The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.)
$/\$

Theorem	islln4 33074*	The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.)
$/\$

Theorem	llni 33075	Condition implying a lattice line. (Contributed by NM, 17-Jun-2012.)
$/\$ $/\$ $/\$

Theorem	llnbase 33076	A lattice line is a lattice element. (Contributed by NM, 16-Jun-2012.)


Theorem	islln3 33077*	The predicate "is a lattice line". (Contributed by NM, 17-Jun-2012.)
$.\/$ $/\$ $/\$ $.\/$

Theorem	islln2 33078*	The predicate "is a lattice line". (Contributed by NM, 23-Jun-2012.)
$.\/$ $/\$ $/\$ $.\/$

Theorem	llni2 33079	The join of two different atoms is a lattice line. (Contributed by NM, 26-Jun-2012.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	llnnleat 33080	An atom cannot majorize a lattice line. (Contributed by NM, 8-Jul-2012.)
$/\$ $/\$

Theorem	llnneat 33081	A lattice line is not an atom. (Contributed by NM, 19-Jun-2012.)
$/\$

Theorem	2atneat 33082	The join of two distinct atoms is not an atom. (Contributed by NM, 12-Oct-2012.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	llnn0 33083	A lattice line is non-zero. (Contributed by NM, 15-Jul-2012.)
$/\$

Theorem	islln2a 33084	The predicate "is a lattice line" in terms of atoms. (Contributed by NM, 15-Jul-2012.)
$.\/$ $/\$ $/\$ $.\/$

Theorem	llnle 33085*	Any element greater than 0 and not an atom majorizes a lattice line. (Contributed by NM, 28-Jun-2012.)
$/\$ $/\$ $/\$

Theorem	atcvrlln2 33086	An atom under a line is covered by it. (Contributed by NM, 2-Jul-2012.)
$/\$ $/\$ $/\$

Theorem	atcvrlln 33087	An element covering an atom is a lattice line and vice-versa. (Contributed by NM, 18-Jun-2012.)
$/\$ $/\$ $/\$

Theorem	llnexatN 33088*	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.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	llncmp 33089	If two lattice lines are comparable, they are equal. (Contributed by NM, 19-Jun-2012.)
$/\$ $/\$

Theorem	llnnlt 33090	Two lattice lines cannot satisfy the less than relation. (Contributed by NM, 26-Jun-2012.)
$/\$ $/\$

Theorem	2llnmat 33091	Two intersecting lines intersect at an atom. (Contributed by NM, 30-Apr-2012.)
$./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	2at0mat0 33092	Special case of 2atmat0 33093 where one atom could be zero. (Contributed by NM, 30-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $\/$ $.\/$ $./\$ $.\/$

Theorem	2atmat0 33093	The meet of two unequal lines (expressed as joins of atoms) is an atom or zero. (Contributed by NM, 2-Dec-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $\/$ $.\/$ $./\$ $.\/$

Theorem	2atm 33094	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.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	ps-2c 33095	Variation of projective geometry axiom ps-2 33045. (Contributed by NM, 3-Jul-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	lplnset 33096*	The set of lattice planes in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)


Theorem	islpln 33097*	The predicate "is a lattice plane". (Contributed by NM, 16-Jun-2012.)
$/\$

Theorem	islpln4 33098*	The predicate "is a lattice plane". (Contributed by NM, 17-Jun-2012.)
$/\$

Theorem	lplni 33099	Condition implying a lattice plane. (Contributed by NM, 20-Jun-2012.)
$/\$ $/\$ $/\$

Theorem	islpln3 33100*	The predicate "is a lattice plane". (Contributed by NM, 17-Jun-2012.)
$.\/$ $/\$ $/\$ $.\/$

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7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40865

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