P. 329 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 32801-32900 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	2llnmat 32801	Two intersecting lines intersect at an atom. (Contributed by NM, 30-Apr-2012.)
$./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	2at0mat0 32802	Special case of 2atmat0 32803 where one atom could be zero. (Contributed by NM, 30-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $\/$ $.\/$ $./\$ $.\/$

Theorem	2atmat0 32803	The meet of two unequal lines (expressed as joins of atoms) is an atom or zero. (Contributed by NM, 2-Dec-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $\/$ $.\/$ $./\$ $.\/$

Theorem	2atm 32804	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 32805	Variation of projective geometry axiom ps-2 32755. (Contributed by NM, 3-Jul-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	lplnset 32806*	The set of lattice planes in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)


Theorem	islpln 32807*	The predicate "is a lattice plane". (Contributed by NM, 16-Jun-2012.)
$/\$

Theorem	islpln4 32808*	The predicate "is a lattice plane". (Contributed by NM, 17-Jun-2012.)
$/\$

Theorem	lplni 32809	Condition implying a lattice plane. (Contributed by NM, 20-Jun-2012.)
$/\$ $/\$ $/\$

Theorem	islpln3 32810*	The predicate "is a lattice plane". (Contributed by NM, 17-Jun-2012.)
$.\/$ $/\$ $/\$ $.\/$

Theorem	lplnbase 32811	A lattice plane is a lattice element. (Contributed by NM, 17-Jun-2012.)


Theorem	islpln5 32812*	The predicate "is a lattice plane" in terms of atoms. (Contributed by NM, 24-Jun-2012.)
$.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	islpln2 32813*	The predicate "is a lattice plane" in terms of atoms. (Contributed by NM, 25-Jun-2012.)
$.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	lplni2 32814	The join of 3 different atoms is a lattice plane. (Contributed by NM, 4-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	lvolex3N 32815*	There is an atom outside of a lattice plane i.e. a 3-dimensional lattice volume exists. (Contributed by NM, 28-Jul-2012.) (New usage is discouraged.)
$/\$

Theorem	llnmlplnN 32816	The intersection of a line with a plane not containing it is an atom. (Contributed by NM, 29-Jun-2012.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	lplnle 32817*	Any element greater than 0 and not an atom and not a lattice line majorizes a lattice plane. (Contributed by NM, 28-Jun-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	lplnnle2at 32818	A lattice line (or atom) cannot majorize a lattice plane. (Contributed by NM, 8-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	lplnnleat 32819	A lattice plane cannot majorize an atom. (Contributed by NM, 14-Jul-2012.)
$/\$ $/\$

Theorem	lplnnlelln 32820	A lattice plane is not less than or equal to a lattice line. (Contributed by NM, 14-Jul-2012.)
$/\$ $/\$

Theorem	2atnelpln 32821	The join of two atoms is not a lattice plane. (Contributed by NM, 16-Jul-2012.)
$.\/$ $/\$ $/\$ $.\/$

Theorem	lplnneat 32822	No lattice plane is an atom. (Contributed by NM, 15-Jul-2012.)
$/\$

Theorem	lplnnelln 32823	No lattice plane is a lattice line. (Contributed by NM, 19-Jun-2012.)
$/\$

Theorem	lplnn0N 32824	A lattice plane is non-zero. (Contributed by NM, 15-Jul-2012.) (New usage is discouraged.)
$/\$

Theorem	islpln2a 32825	The predicate "is a lattice plane" for join of atoms. (Contributed by NM, 16-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$

Theorem	islpln2ah 32826	The predicate "is a lattice plane" for join of atoms. Version of islpln2a 32825 expressed with an abbreviation hypothesis. (Contributed by NM, 30-Jul-2012.)
$.\/$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	lplnriaN 32827	Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
$.\/$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	lplnribN 32828	Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
$.\/$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	lplnric 32829	Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.)
$.\/$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	lplnri1 32830	Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.)
$.\/$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$

Theorem	lplnri2N 32831	Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
$.\/$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$

Theorem	lplnri3N 32832	Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
$.\/$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$

Theorem	lplnllnneN 32833	Two lattice lines defined by atoms defining a lattice plane are not equal. (Contributed by NM, 9-Oct-2012.) (New usage is discouraged.)
$.\/$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	llncvrlpln2 32834	A lattice line under a lattice plane is covered by it. (Contributed by NM, 24-Jun-2012.)
$/\$ $/\$ $/\$

Theorem	llncvrlpln 32835	An element covering a lattice line is a lattice plane and vice-versa. (Contributed by NM, 26-Jun-2012.)
$/\$ $/\$ $/\$

Theorem	2lplnmN 32836	If the join of two lattice planes covers one of them, their meet is a lattice line. (Contributed by NM, 30-Jun-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	2llnmj 32837	The meet of two lattice lines is an atom iff their join is a lattice plane. (Contributed by NM, 27-Jun-2012.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	2atmat 32838	The meet of two intersecting lines (expressed as joins of atoms) is an atom. (Contributed by NM, 21-Nov-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	lplncmp 32839	If two lattice planes are comparable, they are equal. (Contributed by NM, 24-Jun-2012.)
$/\$ $/\$

Theorem	lplnexatN 32840*	Given a lattice line on a lattice plane, there is an atom whose join with the line equals the plane. (Contributed by NM, 29-Jun-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	lplnexllnN 32841*	Given an atom on a lattice plane, there is a lattice line whose join with the atom equals the plane. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	lplnnlt 32842	Two lattice planes cannot satisfy the less than relation. (Contributed by NM, 7-Jul-2012.)
$/\$ $/\$

Theorem	2llnjaN 32843	The join of two different lattice lines in a lattice plane equals the plane (version of 2llnjN 32844 in terms of atoms). (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	2llnjN 32844	The join of two different lattice lines in a lattice plane equals the plane. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	2llnm2N 32845	The meet of two different lattice lines in a lattice plane is an atom. (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	2llnm3N 32846	Two lattice lines in a lattice plane always meet. (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	2llnm4 32847	Two lattice lines that majorize the same atom always meet. (Contributed by NM, 20-Jul-2012.)
$./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	2llnmeqat 32848	An atom equals the intersection of two majorizing lines. (Contributed by NM, 3-Apr-2013.)
$./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	lvolset 32849*	The set of 3-dim lattice volumes in a Hilbert lattice. (Contributed by NM, 1-Jul-2012.)


Theorem	islvol 32850*	The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)
$/\$

Theorem	islvol4 32851*	The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)
$/\$

Theorem	lvoli 32852	Condition implying a 3-dim lattice volume. (Contributed by NM, 1-Jul-2012.)
$/\$ $/\$ $/\$

Theorem	islvol3 32853*	The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)
$.\/$ $/\$ $/\$ $.\/$

Theorem	lvoli3 32854	Condition implying a 3-dim lattice volume. (Contributed by NM, 2-Aug-2012.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	lvolbase 32855	A 3-dim lattice volume is a lattice element. (Contributed by NM, 1-Jul-2012.)


Theorem	islvol5 32856*	The predicate "is a 3-dim lattice volume" in terms of atoms. (Contributed by NM, 1-Jul-2012.)
$.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	islvol2 32857*	The predicate "is a 3-dim lattice volume" in terms of atoms. (Contributed by NM, 1-Jul-2012.)
$.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	lvoli2 32858	The join of 4 different atoms is a lattice volume. (Contributed by NM, 8-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	lvolnle3at 32859	A lattice plane (or lattice line or atom) cannot majorize a lattice volume. (Contributed by NM, 8-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	lvolnleat 32860	An atom cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)
$/\$ $/\$

Theorem	lvolnlelln 32861	A lattice line cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)
$/\$ $/\$

Theorem	lvolnlelpln 32862	A lattice plane cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)
$/\$ $/\$

Theorem	3atnelvolN 32863	The join of 3 atoms is not a lattice volume. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	2atnelvolN 32864	The join of two atoms is not a lattice volume. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $.\/$

Theorem	lvolneatN 32865	No lattice volume is an atom. (Contributed by NM, 15-Jul-2012.) (New usage is discouraged.)
$/\$

Theorem	lvolnelln 32866	No lattice volume is a lattice line. (Contributed by NM, 15-Jul-2012.)
$/\$

Theorem	lvolnelpln 32867	No lattice volume is a lattice plane. (Contributed by NM, 19-Jun-2012.)
$/\$

Theorem	lvoln0N 32868	A lattice volume is non-zero. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)
$/\$

Theorem	islvol2aN 32869	The predicate "is a lattice volume". (Contributed by NM, 16-Jul-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atlem0a 32870	Lemma for 4at 32890. (Contributed by NM, 10-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	4atlem0ae 32871	Lemma for 4at 32890. (Contributed by NM, 10-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	4atlem0be 32872	Lemma for 4at 32890. (Contributed by NM, 10-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	4atlem3 32873	Lemma for 4at 32890. Break inequality into 4 cases. (Contributed by NM, 8-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $\/$ $.\/$ $.\/$ $\/$ $.\/$ $.\/$ $\/$ $.\/$ $.\/$

Theorem	4atlem3a 32874	Lemma for 4at 32890. Break inequality into 3 cases. (Contributed by NM, 9-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $\/$ $.\/$ $.\/$ $\/$ $.\/$ $.\/$

Theorem	4atlem3b 32875	Lemma for 4at 32890. Break inequality into 2 cases. (Contributed by NM, 9-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $\/$ $.\/$ $.\/$

Theorem	4atlem4a 32876	Lemma for 4at 32890. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	4atlem4b 32877	Lemma for 4at 32890. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	4atlem4c 32878	Lemma for 4at 32890. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	4atlem4d 32879	Lemma for 4at 32890. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	4atlem9 32880	Lemma for 4at 32890. Substitute for . (Contributed by NM, 9-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	4atlem10a 32881	Lemma for 4at 32890. Substitute for . (Contributed by NM, 9-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	4atlem10b 32882	Lemma for 4at 32890. Substitute for (cont.). (Contributed by NM, 10-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	4atlem10 32883	Lemma for 4at 32890. Combine both possible cases. (Contributed by NM, 9-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	4atlem11a 32884	Lemma for 4at 32890. Substitute for . (Contributed by NM, 9-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	4atlem11b 32885	Lemma for 4at 32890. Substitute for (cont.). (Contributed by NM, 10-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	4atlem11 32886	Lemma for 4at 32890. Combine all three possible cases. (Contributed by NM, 10-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	4atlem12a 32887	Lemma for 4at 32890. Substitute for . (Contributed by NM, 9-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	4atlem12b 32888	Lemma for 4at 32890. Substitute for (cont.). (Contributed by NM, 11-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	4atlem12 32889	Lemma for 4at 32890. Combine all four possible cases. (Contributed by NM, 11-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	4at 32890	Four atoms determine a lattice volume uniquely. Three-dimensional analog of ps-1 32754 and 3at 32767. (Contributed by NM, 11-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	4at2 32891	Four atoms determine a lattice volume uniquely. (Contributed by NM, 11-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	lplncvrlvol2 32892	A lattice line under a lattice plane is covered by it. (Contributed by NM, 12-Jul-2012.)
$/\$ $/\$ $/\$

Theorem	lplncvrlvol 32893	An element covering a lattice plane is a lattice volume and vice-versa. (Contributed by NM, 15-Jul-2012.)
$/\$ $/\$ $/\$

Theorem	lvolcmp 32894	If two lattice planes are comparable, they are equal. (Contributed by NM, 12-Jul-2012.)
$/\$ $/\$

Theorem	lvolnltN 32895	Two lattice volumes cannot satisfy the less than relation. (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	2lplnja 32896	The join of two different lattice planes in a lattice volume equals the volume (version of 2lplnj 32897 in terms of atoms). (Contributed by NM, 12-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$

Theorem	2lplnj 32897	The join of two different lattice planes in a (3-dimensional) lattice volume equals the volume. (Contributed by NM, 12-Jul-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	2lplnm2N 32898	The meet of two different lattice planes in a lattice volume is a lattice line. (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	2lplnmj 32899	The meet of two lattice planes is a lattice line iff their join is a lattice volume. (Contributed by NM, 13-Jul-2012.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	dalemkehl 32900	Lemma for dath 33013. Frequently-used utility lemma. (Contributed by NM, 13-Aug-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 39301-39313

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