P. 333 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 33201-33300 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	oldmm1 33201	De Morgan's law for meet in an ortholattice. (chdmm1 25081 analog.) (Contributed by NM, 6-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	oldmm2 33202	De Morgan's law for meet in an ortholattice. (chdmm2 25082 analog.) (Contributed by NM, 6-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	oldmm3N 33203	De Morgan's law for meet in an ortholattice. (chdmm3 25083 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	oldmm4 33204	De Morgan's law for meet in an ortholattice. (chdmm4 25084 analog.) (Contributed by NM, 6-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	oldmj1 33205	De Morgan's law for join in an ortholattice. (chdmj1 25085 analog.) (Contributed by NM, 6-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	oldmj2 33206	De Morgan's law for join in an ortholattice. (chdmj2 25086 analog.) (Contributed by NM, 7-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	oldmj3 33207	De Morgan's law for join in an ortholattice. (chdmj3 25087 analog.) (Contributed by NM, 7-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	oldmj4 33208	De Morgan's law for join in an ortholattice. (chdmj4 25088 analog.) (Contributed by NM, 7-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	olj01 33209	An ortholattice element joined with zero equals itself. (chj0 25053 analog.) (Contributed by NM, 19-Oct-2011.)
$.\/$ $/\$ $.\/$

Theorem	olj02 33210	An ortholattice element joined with zero equals itself. (Contributed by NM, 28-Jan-2012.)
$.\/$ $/\$ $.\/$

Theorem	olm11 33211	The meet of an ortholattice element with one equals itself. (chm1i 25012 analog.) (Contributed by NM, 22-May-2012.)
$./\$ $/\$ $./\$

Theorem	olm12 33212	The meet of an ortholattice element with one equals itself. (Contributed by NM, 22-May-2012.)
$./\$ $/\$ $./\$

Theorem	latmassOLD 33213	Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 3669 analog.) (Contributed by NM, 7-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$

Theorem	latm12 33214	A rearrangement of lattice meet. (in12 3670 analog.) (Contributed by NM, 8-Nov-2011.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$

Theorem	latm32 33215	A rearrangement of lattice meet. (in12 3670 analog.) (Contributed by NM, 13-Nov-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$

Theorem	latmrot 33216	Rotate lattice meet of 3 classes. (Contributed by NM, 9-Oct-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$

Theorem	latm4 33217	Rearrangement of lattice meet of 4 classes. (in4 3675 analog.) (Contributed by NM, 8-Nov-2011.)
$./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$ $./\$ $./\$

Theorem	latmmdiN 33218	Lattice meet distributes over itself. (inindi 3676 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$ $./\$

Theorem	latmmdir 33219	Lattice meet distributes over itself. (inindir 3677 analog.) (Contributed by NM, 6-Jun-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$ $./\$

Theorem	olm01 33220	Meet with lattice zero is zero. (chm0 25047 analog.) (Contributed by NM, 8-Nov-2011.)
$./\$ $/\$ $./\$

Theorem	olm02 33221	Meet with lattice zero is zero. (Contributed by NM, 9-Oct-2012.)
$./\$ $/\$ $./\$

Theorem	isoml 33222*	The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)
$.\/$ $./\$ $/\$ $.\/$ $./\$

Theorem	isomliN 33223*	Properties that determine an orthomodular lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $.\/$ $./\$

Theorem	omlol 33224	An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)


Theorem	omlop 33225	An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.)


Theorem	omllat 33226	An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.)


Theorem	omllaw 33227	The orthomodular law. (Contributed by NM, 18-Sep-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	omllaw2N 33228	Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 25141 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	omllaw3 33229	Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 24992 analog.) (Contributed by NM, 19-Oct-2011.)
$./\$ $/\$ $/\$ $/\$ $./\$

Theorem	omllaw4 33230	Orthomodular law equivalent. Remark in [Holland95] p. 223. (Contributed by NM, 19-Oct-2011.)
$./\$ $/\$ $/\$ $./\$ $./\$

Theorem	omllaw5N 33231	The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 25169 analog.) (Contributed by NM, 14-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$

Theorem	cmtcomlemN 33232	Lemma for cmtcomN 33233. (cmcmlem 25147 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

Theorem	cmtcomN 33233	Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 25148 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

Theorem	cmt2N 33234	Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 25149 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

Theorem	cmt3N 33235	Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 25151 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

Theorem	cmt4N 33236	Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 25151 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

Theorem	cmtbr2N 33237	Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 25152 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cmtbr3N 33238	Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 25164 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$ $./\$

Theorem	cmtbr4N 33239	Alternate definition for the commutes relation. (cmbr4i 25157 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	lecmtN 33240	Ordered elements commute. (lecmi 25158 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

Theorem	cmtidN 33241	Any element commutes with itself. (cmidi 25166 analog.) (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
$/\$

Theorem	omlfh1N 33242	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 25174 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	omlfh3N 33243	Foulis-Holland Theorem, part 3. Dual of omlfh1N 33242. (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$

Theorem	omlmod1i2N 33244	Analog of modular law atmod1i2 33842 that holds in any OML. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	omlspjN 33245	Contraction of a Sasaki projection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$ $./\$

21.30.7 Atomic lattices with covering property

Syntax	ccvr 33246	Extend class notation with covers relation.


Syntax	catm 33247	Extend class notation with atoms.


Syntax	cal 33248	Extend class notation with atomic lattices.


Syntax	clc 33249	Extend class notation with lattices with the covering property.


Definition	df-covers 33250*	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 33253 for the relation form. (Contributed by NM, 18-Sep-2011.)
$/\$ $/\$ $/\$ $/\$

Definition	df-ats 33251*	Define the class of poset atoms. (Contributed by NM, 18-Sep-2011.)


Theorem	cvrfval 33252*	Value of covers relation "is covered by". (Contributed by NM, 18-Sep-2011.)
$/\$ $/\$ $/\$ $/\$

Theorem	cvrval 33253*	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 25839 analog.) (Contributed by NM, 18-Sep-2011.)
$/\$ $/\$ $/\$ $/\$

Theorem	cvrlt 33254	The covers relation implies the less-than relation. (cvpss 25842 analog.) (Contributed by NM, 8-Oct-2011.)
$/\$ $/\$ $/\$

Theorem	cvrnbtwn 33255	There is no element between the two arguments of the covers relation. (cvnbtwn 25843 analog.) (Contributed by NM, 18-Oct-2011.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	ncvr1 33256	No element covers the lattice unit. (Contributed by NM, 8-Jul-2013.)
$/\$

Theorem	cvrletrN 33257	Property of an element above a covering. (Contributed by NM, 7-Dec-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$

Theorem	cvrval2 33258*	Binary relation expressing covers . Definition of covers in [Kalmbach] p. 15. (cvbr2 25840 analog.) (Contributed by NM, 16-Nov-2011.)
$/\$ $/\$ $/\$ $/\$

Theorem	cvrnbtwn2 33259	The covers relation implies no in-betweenness. (cvnbtwn2 25844 analog.) (Contributed by NM, 17-Nov-2011.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cvrnbtwn3 33260	The covers relation implies no in-betweenness. (cvnbtwn3 25845 analog.) (Contributed by NM, 4-Nov-2011.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cvrcon3b 33261	Contraposition law for the covers relation. (cvcon3 25841 analog.) (Contributed by NM, 4-Nov-2011.)
$/\$ $/\$

Theorem	cvrle 33262	The covers relation implies the less-than-or-equal relation. (Contributed by NM, 12-Oct-2011.)
$/\$ $/\$ $/\$

Theorem	cvrnbtwn4 33263	The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 25846 analog.) (Contributed by NM, 18-Oct-2011.)
$/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Theorem	cvrnle 33264	The covers relation implies the negation of the reverse less-than-or-equal relation. (Contributed by NM, 18-Oct-2011.)
$/\$ $/\$ $/\$

Theorem	cvrne 33265	The covers relation implies inequality. (Contributed by NM, 13-Oct-2011.)
$/\$ $/\$ $/\$

Theorem	cvrnrefN 33266	The covers relation is not reflexive. (cvnref 25848 analog.) (Contributed by NM, 1-Nov-2012.) (New usage is discouraged.)
$/\$

Theorem	cvrcmp 33267	If two lattice elements that cover a third are comparable, then they are equal. (Contributed by NM, 6-Feb-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cvrcmp2 33268	If two lattice elements covered by a third are comparable, then they are equal. (Contributed by NM, 20-Jun-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	pats 33269*	The set of atoms in a poset. (Contributed by NM, 18-Sep-2011.)


Theorem	isat 33270	The predicate "is an atom". (ela 25896 analog.) (Contributed by NM, 18-Sep-2011.)
$/\$

Theorem	isat2 33271	The predicate "is an atom". (elatcv0 25898 analog.) (Contributed by NM, 18-Jun-2012.)
$/\$

Theorem	atcvr0 33272	An atom covers zero. (atcv0 25899 analog.) (Contributed by NM, 4-Nov-2011.)
$/\$

Theorem	atbase 33273	An atom is a member of the lattice base set (i.e. a lattice element). (atelch 25901 analog.) (Contributed by NM, 10-Oct-2011.)


Theorem	atssbase 33274	The set of atoms is a subset of the base set. (atssch 25900 analog.) (Contributed by NM, 21-Oct-2011.)


Theorem	0ltat 33275	An atom is greater than zero. (Contributed by NM, 4-Jul-2012.)
$/\$

Theorem	leatb 33276	A poset element less than or equal to an atom equals either zero or the atom. (atss 25903 analog.) (Contributed by NM, 17-Nov-2011.)
$/\$ $/\$ $\/$

Theorem	leat 33277	A poset element less than or equal to an atom equals either zero or the atom. (Contributed by NM, 15-Oct-2013.)
$/\$ $/\$ $/\$ $\/$

Theorem	leat2 33278	A nonzero poset element less than or equal to an atom equals the atom. (Contributed by NM, 6-Mar-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	leat3 33279	A poset element less than or equal to an atom is either an atom or zero. (Contributed by NM, 2-Dec-2012.)
$/\$ $/\$ $/\$ $\/$

Theorem	meetat 33280	The meet of any element with an atom is either the atom or zero. (Contributed by NM, 28-Aug-2012.)
$./\$ $/\$ $/\$ $./\$ $\/$ $./\$

Theorem	meetat2 33281	The meet of any element with an atom is either the atom or zero. (Contributed by NM, 30-Aug-2012.)
$./\$ $/\$ $/\$ $./\$ $\/$ $./\$

Definition	df-atl 33282*	Define the class of atomic lattices, in which every nonzero element is greater than or equal to an atom. We also ensure the existence of a lattice zero, since a lattice by itself may not have a zero. (Contributed by NM, 18-Sep-2011.) (Revised by NM, 14-Sep-2018.)
$/\$

Theorem	isatl 33283*	The predicate "is an atomic lattice." Every nonzero element is less than or equal to an atom. (Contributed by NM, 18-Sep-2011.) (Revised by NM, 14-Sep-2018.)
$/\$ $/\$

Theorem	atllat 33284	An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.)


Theorem	atlpos 33285	An atomic lattice is a poset. (Contributed by NM, 5-Nov-2012.)


Theorem	atl0dm 33286	Condition necessary for zero element to exist. (Contributed by NM, 14-Sep-2018.)


Theorem	atl0cl 33287	An atomic lattice has a zero element. We can use this in place of op0cl 33168 for lattices without orthocomplements. (Contributed by NM, 5-Nov-2012.)


Theorem	atl0le 33288	Orthoposet zero is less than or equal to any element. (ch0le 24997 analog.) (Contributed by NM, 12-Oct-2011.)
$/\$

Theorem	atlle0 33289	An element less than or equal to zero equals zero. (chle0 24999 analog.) (Contributed by NM, 21-Oct-2011.)
$/\$

Theorem	atlltn0 33290	A lattice element greater than zero is nonzero. (Contributed by NM, 1-Jun-2012.)
$/\$

Theorem	isat3 33291*	The predicate "is an atom". (elat2 25897 analog.) (Contributed by NM, 27-Apr-2014.)
$/\$ $/\$ $\/$

Theorem	atn0 33292	An atom is not zero. (atne0 25902 analog.) (Contributed by NM, 5-Nov-2012.)
$/\$

Theorem	atnle0 33293	An atom is not less than or equal to zero. (Contributed by NM, 17-Oct-2011.)
$/\$

Theorem	atlen0 33294	A lattice element is nonzero if an atom is under it. (Contributed by NM, 26-May-2012.)
$/\$ $/\$ $/\$

Theorem	atcmp 33295	If two atoms are comparable, they are equal. (atsseq 25904 analog.) (Contributed by NM, 13-Oct-2011.)
$/\$ $/\$

Theorem	atncmp 33296	Frequently-used variation of atcmp 33295. (Contributed by NM, 29-Jun-2012.)
$/\$ $/\$

Theorem	atnlt 33297	Two atoms cannot satisfy the less than relation. (Contributed by NM, 7-Feb-2012.)
$/\$ $/\$

Theorem	atcvreq0 33298	An element covered by an atom must be zero. (atcveq0 25905 analog.) (Contributed by NM, 4-Nov-2011.)
$/\$ $/\$

Theorem	atncvrN 33299	Two atoms cannot satisfy the covering relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	atlex 33300*	Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 25917 analog.) (Contributed by NM, 21-Oct-2011.)
$/\$ $/\$

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