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

Theorem	lub0N 32801	The least upper bound of the empty set is the zero element. (Contributed by NM, 15-Sep-2013.) (New usage is discouraged.)


Theorem	opltn0 32802	A lattice element greater than zero is nonzero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
$/\$

Theorem	ople1 32803	Any element is less than the orthoposet unit. (chss 26938 analog.) (Contributed by NM, 23-Oct-2011.)
$/\$

Theorem	op1le 32804	If the orthoposet unit is less than or equal to an element, the element equals the unit. (chle0 27152 analog.) (Contributed by NM, 5-Dec-2011.)
$/\$

Theorem	glb0N 32805	The greatest lower bound of the empty set is the unit element. (Contributed by NM, 5-Dec-2011.) (New usage is discouraged.)


Theorem	opoccl 32806	Closure of orthocomplement operation. (choccl 27015 analog.) (Contributed by NM, 20-Oct-2011.)
$/\$

Theorem	opococ 32807	Double negative law for orthoposets. (ococ 27115 analog.) (Contributed by NM, 13-Sep-2011.)
$/\$

Theorem	opcon3b 32808	Contraposition law for orthoposets. (chcon3i 27175 analog.) (Contributed by NM, 8-Nov-2011.)
$/\$ $/\$

Theorem	opcon2b 32809	Orthocomplement contraposition law. (negcon2 9958 analog.) (Contributed by NM, 16-Jan-2012.)
$/\$ $/\$

Theorem	opcon1b 32810	Orthocomplement contraposition law. (negcon1 9957 analog.) (Contributed by NM, 24-Jan-2012.)
$/\$ $/\$

Theorem	oplecon3 32811	Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011.)
$/\$ $/\$

Theorem	oplecon3b 32812	Contraposition law for orthoposets. (chsscon3 27209 analog.) (Contributed by NM, 4-Nov-2011.)
$/\$ $/\$

Theorem	oplecon1b 32813	Contraposition law for strict ordering in orthoposets. (chsscon1 27210 analog.) (Contributed by NM, 6-Nov-2011.)
$/\$ $/\$

Theorem	opoc1 32814	Orthocomplement of orthoposet unit. (Contributed by NM, 24-Jan-2012.)


Theorem	opoc0 32815	Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012.)


Theorem	opltcon3b 32816	Contraposition law for strict ordering in orthoposets. (chpsscon3 27212 analog.) (Contributed by NM, 4-Nov-2011.)
$/\$ $/\$

Theorem	opltcon1b 32817	Contraposition law for strict ordering in orthoposets. (chpsscon1 27213 analog.) (Contributed by NM, 5-Nov-2011.)
$/\$ $/\$

Theorem	opltcon2b 32818	Contraposition law for strict ordering in orthoposets. (chsscon2 27211 analog.) (Contributed by NM, 5-Nov-2011.)
$/\$ $/\$

Theorem	opexmid 32819	Law of excluded middle for orthoposets. (chjo 27224 analog.) (Contributed by NM, 13-Sep-2011.)
$.\/$ $/\$ $.\/$

Theorem	opnoncon 32820	Law of contradiction for orthoposets. (chocin 27204 analog.) (Contributed by NM, 13-Sep-2011.)
$./\$ $/\$ $./\$

Theorem	riotaocN 32821*	The orthocomplement of the unique poset element such that . (riotaneg 10619 analog.) (Contributed by NM, 16-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	cmtfvalN 32822*	Value of commutes relation. (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$ $./\$

Theorem	cmtvalN 32823	Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 27293 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$ $./\$

Theorem	isolat 32824	The predicate "is an ortholattice." (Contributed by NM, 18-Sep-2011.)
$/\$

Theorem	ollat 32825	An ortholattice is a lattice. (Contributed by NM, 18-Sep-2011.)


Theorem	olop 32826	An ortholattice is an orthoposet. (Contributed by NM, 18-Sep-2011.)


Theorem	olposN 32827	An ortholattice is a poset. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)


Theorem	isolatiN 32828	Properties that determine an ortholattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)


Theorem	oldmm1 32829	De Morgan's law for meet in an ortholattice. (chdmm1 27234 analog.) (Contributed by NM, 6-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	oldmm2 32830	De Morgan's law for meet in an ortholattice. (chdmm2 27235 analog.) (Contributed by NM, 6-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	oldmm3N 32831	De Morgan's law for meet in an ortholattice. (chdmm3 27236 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	oldmm4 32832	De Morgan's law for meet in an ortholattice. (chdmm4 27237 analog.) (Contributed by NM, 6-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	oldmj1 32833	De Morgan's law for join in an ortholattice. (chdmj1 27238 analog.) (Contributed by NM, 6-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	oldmj2 32834	De Morgan's law for join in an ortholattice. (chdmj2 27239 analog.) (Contributed by NM, 7-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	oldmj3 32835	De Morgan's law for join in an ortholattice. (chdmj3 27240 analog.) (Contributed by NM, 7-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	oldmj4 32836	De Morgan's law for join in an ortholattice. (chdmj4 27241 analog.) (Contributed by NM, 7-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	olj01 32837	An ortholattice element joined with zero equals itself. (chj0 27206 analog.) (Contributed by NM, 19-Oct-2011.)
$.\/$ $/\$ $.\/$

Theorem	olj02 32838	An ortholattice element joined with zero equals itself. (Contributed by NM, 28-Jan-2012.)
$.\/$ $/\$ $.\/$

Theorem	olm11 32839	The meet of an ortholattice element with one equals itself. (chm1i 27165 analog.) (Contributed by NM, 22-May-2012.)
$./\$ $/\$ $./\$

Theorem	olm12 32840	The meet of an ortholattice element with one equals itself. (Contributed by NM, 22-May-2012.)
$./\$ $/\$ $./\$

Theorem	latmassOLD 32841	Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 3654 analog.) (Contributed by NM, 7-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$

Theorem	latm12 32842	A rearrangement of lattice meet. (in12 3655 analog.) (Contributed by NM, 8-Nov-2011.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$

Theorem	latm32 32843	A rearrangement of lattice meet. (in12 3655 analog.) (Contributed by NM, 13-Nov-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$

Theorem	latmrot 32844	Rotate lattice meet of 3 classes. (Contributed by NM, 9-Oct-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$

Theorem	latm4 32845	Rearrangement of lattice meet of 4 classes. (in4 3660 analog.) (Contributed by NM, 8-Nov-2011.)
$./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$ $./\$ $./\$

Theorem	latmmdiN 32846	Lattice meet distributes over itself. (inindi 3661 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$ $./\$

Theorem	latmmdir 32847	Lattice meet distributes over itself. (inindir 3662 analog.) (Contributed by NM, 6-Jun-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$ $./\$

Theorem	olm01 32848	Meet with lattice zero is zero. (chm0 27200 analog.) (Contributed by NM, 8-Nov-2011.)
$./\$ $/\$ $./\$

Theorem	olm02 32849	Meet with lattice zero is zero. (Contributed by NM, 9-Oct-2012.)
$./\$ $/\$ $./\$

Theorem	isoml 32850*	The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)
$.\/$ $./\$ $/\$ $.\/$ $./\$

Theorem	isomliN 32851*	Properties that determine an orthomodular lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $.\/$ $./\$

Theorem	omlol 32852	An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)


Theorem	omlop 32853	An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.)


Theorem	omllat 32854	An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.)


Theorem	omllaw 32855	The orthomodular law. (Contributed by NM, 18-Sep-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	omllaw2N 32856	Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 27294 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	omllaw3 32857	Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 27145 analog.) (Contributed by NM, 19-Oct-2011.)
$./\$ $/\$ $/\$ $/\$ $./\$

Theorem	omllaw4 32858	Orthomodular law equivalent. Remark in [Holland95] p. 223. (Contributed by NM, 19-Oct-2011.)
$./\$ $/\$ $/\$ $./\$ $./\$

Theorem	omllaw5N 32859	The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 27322 analog.) (Contributed by NM, 14-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$

Theorem	cmtcomlemN 32860	Lemma for cmtcomN 32861. (cmcmlem 27300 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

Theorem	cmtcomN 32861	Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 27301 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

Theorem	cmt2N 32862	Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 27302 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

Theorem	cmt3N 32863	Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 27304 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

Theorem	cmt4N 32864	Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 27304 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

Theorem	cmtbr2N 32865	Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 27305 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cmtbr3N 32866	Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 27317 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$ $./\$

Theorem	cmtbr4N 32867	Alternate definition for the commutes relation. (cmbr4i 27310 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	lecmtN 32868	Ordered elements commute. (lecmi 27311 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

Theorem	cmtidN 32869	Any element commutes with itself. (cmidi 27319 analog.) (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
$/\$

Theorem	omlfh1N 32870	Foulis-Holland Theorem, part 1. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Part of Theorem 5 in [Kalmbach] p. 25. (fh1 27327 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	omlfh3N 32871	Foulis-Holland Theorem, part 3. Dual of omlfh1N 32870. (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$

Theorem	omlmod1i2N 32872	Analogue of modular law atmod1i2 33470 that holds in any OML. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	omlspjN 32873	Contraction of a Sasaki projection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$ $./\$

21.21.10 Atomic lattices with covering property

Syntax	ccvr 32874	Extend class notation with covers relation.


Syntax	catm 32875	Extend class notation with atoms.


Syntax	cal 32876	Extend class notation with atomic lattices.


Syntax	clc 32877	Extend class notation with lattices with the covering property.


Definition	df-covers 32878*	Define the covers relation ("is covered by") for posets. " is covered by " means that is strictly less than and there is nothing in between. See cvrval 32881 for the relation form. (Contributed by NM, 18-Sep-2011.)
$/\$ $/\$ $/\$ $/\$

Definition	df-ats 32879*	Define the class of poset atoms. (Contributed by NM, 18-Sep-2011.)


Theorem	cvrfval 32880*	Value of covers relation "is covered by". (Contributed by NM, 18-Sep-2011.)
$/\$ $/\$ $/\$ $/\$

Theorem	cvrval 32881*	Binary relation expressing covers , which means that is larger than and there is nothing in between. Definition 3.2.18 of [PtakPulmannova] p. 68. (cvbr 27991 analog.) (Contributed by NM, 18-Sep-2011.)
$/\$ $/\$ $/\$ $/\$

Theorem	cvrlt 32882	The covers relation implies the less-than relation. (cvpss 27994 analog.) (Contributed by NM, 8-Oct-2011.)
$/\$ $/\$ $/\$

Theorem	cvrnbtwn 32883	There is no element between the two arguments of the covers relation. (cvnbtwn 27995 analog.) (Contributed by NM, 18-Oct-2011.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	ncvr1 32884	No element covers the lattice unit. (Contributed by NM, 8-Jul-2013.)
$/\$

Theorem	cvrletrN 32885	Property of an element above a covering. (Contributed by NM, 7-Dec-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$

Theorem	cvrval2 32886*	Binary relation expressing covers . Definition of covers in [Kalmbach] p. 15. (cvbr2 27992 analog.) (Contributed by NM, 16-Nov-2011.)
$/\$ $/\$ $/\$ $/\$

Theorem	cvrnbtwn2 32887	The covers relation implies no in-betweenness. (cvnbtwn2 27996 analog.) (Contributed by NM, 17-Nov-2011.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cvrnbtwn3 32888	The covers relation implies no in-betweenness. (cvnbtwn3 27997 analog.) (Contributed by NM, 4-Nov-2011.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cvrcon3b 32889	Contraposition law for the covers relation. (cvcon3 27993 analog.) (Contributed by NM, 4-Nov-2011.)
$/\$ $/\$

Theorem	cvrle 32890	The covers relation implies the less-than-or-equal relation. (Contributed by NM, 12-Oct-2011.)
$/\$ $/\$ $/\$

Theorem	cvrnbtwn4 32891	The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 27998 analog.) (Contributed by NM, 18-Oct-2011.)
$/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Theorem	cvrnle 32892	The covers relation implies the negation of the reverse less-than-or-equal relation. (Contributed by NM, 18-Oct-2011.)
$/\$ $/\$ $/\$

Theorem	cvrne 32893	The covers relation implies inequality. (Contributed by NM, 13-Oct-2011.)
$/\$ $/\$ $/\$

Theorem	cvrnrefN 32894	The covers relation is not reflexive. (cvnref 28000 analog.) (Contributed by NM, 1-Nov-2012.) (New usage is discouraged.)
$/\$

Theorem	cvrcmp 32895	If two lattice elements that cover a third are comparable, then they are equal. (Contributed by NM, 6-Feb-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cvrcmp2 32896	If two lattice elements covered by a third are comparable, then they are equal. (Contributed by NM, 20-Jun-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	pats 32897*	The set of atoms in a poset. (Contributed by NM, 18-Sep-2011.)


Theorem	isat 32898	The predicate "is an atom". (ela 28048 analog.) (Contributed by NM, 18-Sep-2011.)
$/\$

Theorem	isat2 32899	The predicate "is an atom". (elatcv0 28050 analog.) (Contributed by NM, 18-Jun-2012.)
$/\$

Theorem	atcvr0 32900	An atom covers zero. (atcv0 28051 analog.) (Contributed by NM, 4-Nov-2011.)
$/\$

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