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**Theorem List for Metamath Proof Explorer -** 32201-32300 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	lkreqN 32201	Proportional functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
Scalar LFnl LKer LDual $\$

Theorem	lkrlspeqN 32202	Condition for colinear functionals to have equal kernels. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
LFnl LKer LDual $\$

21.21.9 Ortholattices and orthomodular lattices

Syntax	cops 32203	Extend class notation with orthoposets.


Syntax	ccmtN 32204	Extend class notation with the commutes relation.


Syntax	col 32205	Extend class notation with orthlattices.


Syntax	coml 32206	Extend class notation with orthomodular lattices.


Definition	df-oposet 32207*	Define the class of orthoposets, which are bounded posets with an orthocomplementation operation. Note that p ) e. dom ( lub means there is an upper bound , and similarly for the element. (Contributed by NM, 20-Oct-2011.) (Revised by NM, 13-Sep-2018.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Definition	df-cmtN 32208*	Define the commutes relation for orthoposets. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Nov-2011.)
$/\$ $/\$

Definition	df-ol 32209	Define the class of ortholattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)


Definition	df-oml 32210*	Define the class of orthomodular lattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)


Theorem	isopos 32211*	The predicate "is an orthoposet." (Contributed by NM, 20-Oct-2011.) (Revised by NM, 14-Sep-2018.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $./\$

Theorem	opposet 32212	Every orthoposet is a poset. (Contributed by NM, 12-Oct-2011.)


Theorem	oposlem 32213	Lemma for orthoposet properties. (Contributed by NM, 20-Oct-2011.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $./\$

Theorem	op01dm 32214	Conditions necessary for zero and unit elements to exist. (Contributed by NM, 14-Sep-2018.)
$/\$

Theorem	op0cl 32215	An orthoposet has a zero element. (h0elch 26600 analog.) (Contributed by NM, 12-Oct-2011.)


Theorem	op1cl 32216	An orthoposet has a unit element. (helch 26588 analog.) (Contributed by NM, 22-Oct-2011.)


Theorem	op0le 32217	Orthoposet zero is less than or equal to any element. (ch0le 26786 analog.) (Contributed by NM, 12-Oct-2011.)
$/\$

Theorem	ople0 32218	An element less than or equal to zero equals zero. (chle0 26788 analog.) (Contributed by NM, 21-Oct-2011.)
$/\$

Theorem	opnlen0 32219	An element not less than another is nonzero. TODO: Look for uses of necon3bd 2617 and op0le 32217 to see if this is useful elsewhere. (Contributed by NM, 5-May-2013.)
$/\$ $/\$ $/\$

Theorem	lub0N 32220	The least upper bound of the empty set is the zero element. (Contributed by NM, 15-Sep-2013.) (New usage is discouraged.)


Theorem	opltn0 32221	A lattice element greater than zero is nonzero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
$/\$

Theorem	ople1 32222	Any element is less than the orthoposet unit. (chss 26574 analog.) (Contributed by NM, 23-Oct-2011.)
$/\$

Theorem	op1le 32223	If the orthoposet unit is less than or equal to an element, the element equals the unit. (chle0 26788 analog.) (Contributed by NM, 5-Dec-2011.)
$/\$

Theorem	glb0N 32224	The greatest lower bound of the empty set is the unit element. (Contributed by NM, 5-Dec-2011.) (New usage is discouraged.)


Theorem	opoccl 32225	Closure of orthocomplement operation. (choccl 26651 analog.) (Contributed by NM, 20-Oct-2011.)
$/\$

Theorem	opococ 32226	Double negative law for orthoposets. (ococ 26751 analog.) (Contributed by NM, 13-Sep-2011.)
$/\$

Theorem	opcon3b 32227	Contraposition law for orthoposets. (chcon3i 26811 analog.) (Contributed by NM, 8-Nov-2011.)
$/\$ $/\$

Theorem	opcon2b 32228	Orthocomplement contraposition law. (negcon2 9910 analog.) (Contributed by NM, 16-Jan-2012.)
$/\$ $/\$

Theorem	opcon1b 32229	Orthocomplement contraposition law. (negcon1 9909 analog.) (Contributed by NM, 24-Jan-2012.)
$/\$ $/\$

Theorem	oplecon3 32230	Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011.)
$/\$ $/\$

Theorem	oplecon3b 32231	Contraposition law for orthoposets. (chsscon3 26845 analog.) (Contributed by NM, 4-Nov-2011.)
$/\$ $/\$

Theorem	oplecon1b 32232	Contraposition law for strict ordering in orthoposets. (chsscon1 26846 analog.) (Contributed by NM, 6-Nov-2011.)
$/\$ $/\$

Theorem	opoc1 32233	Orthocomplement of orthoposet unit. (Contributed by NM, 24-Jan-2012.)


Theorem	opoc0 32234	Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012.)


Theorem	opltcon3b 32235	Contraposition law for strict ordering in orthoposets. (chpsscon3 26848 analog.) (Contributed by NM, 4-Nov-2011.)
$/\$ $/\$

Theorem	opltcon1b 32236	Contraposition law for strict ordering in orthoposets. (chpsscon1 26849 analog.) (Contributed by NM, 5-Nov-2011.)
$/\$ $/\$

Theorem	opltcon2b 32237	Contraposition law for strict ordering in orthoposets. (chsscon2 26847 analog.) (Contributed by NM, 5-Nov-2011.)
$/\$ $/\$

Theorem	opexmid 32238	Law of excluded middle for orthoposets. (chjo 26860 analog.) (Contributed by NM, 13-Sep-2011.)
$.\/$ $/\$ $.\/$

Theorem	opnoncon 32239	Law of contradiction for orthoposets. (chocin 26840 analog.) (Contributed by NM, 13-Sep-2011.)
$./\$ $/\$ $./\$

Theorem	riotaocN 32240*	The orthocomplement of the unique poset element such that . (riotaneg 10560 analog.) (Contributed by NM, 16-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	cmtfvalN 32241*	Value of commutes relation. (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$ $./\$

Theorem	cmtvalN 32242	Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 26929 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$ $./\$

Theorem	isolat 32243	The predicate "is an ortholattice." (Contributed by NM, 18-Sep-2011.)
$/\$

Theorem	ollat 32244	An ortholattice is a lattice. (Contributed by NM, 18-Sep-2011.)


Theorem	olop 32245	An ortholattice is an orthoposet. (Contributed by NM, 18-Sep-2011.)


Theorem	olposN 32246	An ortholattice is a poset. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)


Theorem	isolatiN 32247	Properties that determine an ortholattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)


Theorem	oldmm1 32248	De Morgan's law for meet in an ortholattice. (chdmm1 26870 analog.) (Contributed by NM, 6-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	oldmm2 32249	De Morgan's law for meet in an ortholattice. (chdmm2 26871 analog.) (Contributed by NM, 6-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	oldmm3N 32250	De Morgan's law for meet in an ortholattice. (chdmm3 26872 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	oldmm4 32251	De Morgan's law for meet in an ortholattice. (chdmm4 26873 analog.) (Contributed by NM, 6-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	oldmj1 32252	De Morgan's law for join in an ortholattice. (chdmj1 26874 analog.) (Contributed by NM, 6-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	oldmj2 32253	De Morgan's law for join in an ortholattice. (chdmj2 26875 analog.) (Contributed by NM, 7-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	oldmj3 32254	De Morgan's law for join in an ortholattice. (chdmj3 26876 analog.) (Contributed by NM, 7-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	oldmj4 32255	De Morgan's law for join in an ortholattice. (chdmj4 26877 analog.) (Contributed by NM, 7-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	olj01 32256	An ortholattice element joined with zero equals itself. (chj0 26842 analog.) (Contributed by NM, 19-Oct-2011.)
$.\/$ $/\$ $.\/$

Theorem	olj02 32257	An ortholattice element joined with zero equals itself. (Contributed by NM, 28-Jan-2012.)
$.\/$ $/\$ $.\/$

Theorem	olm11 32258	The meet of an ortholattice element with one equals itself. (chm1i 26801 analog.) (Contributed by NM, 22-May-2012.)
$./\$ $/\$ $./\$

Theorem	olm12 32259	The meet of an ortholattice element with one equals itself. (Contributed by NM, 22-May-2012.)
$./\$ $/\$ $./\$

Theorem	latmassOLD 32260	Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 3651 analog.) (Contributed by NM, 7-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$

Theorem	latm12 32261	A rearrangement of lattice meet. (in12 3652 analog.) (Contributed by NM, 8-Nov-2011.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$

Theorem	latm32 32262	A rearrangement of lattice meet. (in12 3652 analog.) (Contributed by NM, 13-Nov-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$

Theorem	latmrot 32263	Rotate lattice meet of 3 classes. (Contributed by NM, 9-Oct-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$

Theorem	latm4 32264	Rearrangement of lattice meet of 4 classes. (in4 3657 analog.) (Contributed by NM, 8-Nov-2011.)
$./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$ $./\$ $./\$

Theorem	latmmdiN 32265	Lattice meet distributes over itself. (inindi 3658 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$ $./\$

Theorem	latmmdir 32266	Lattice meet distributes over itself. (inindir 3659 analog.) (Contributed by NM, 6-Jun-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$ $./\$

Theorem	olm01 32267	Meet with lattice zero is zero. (chm0 26836 analog.) (Contributed by NM, 8-Nov-2011.)
$./\$ $/\$ $./\$

Theorem	olm02 32268	Meet with lattice zero is zero. (Contributed by NM, 9-Oct-2012.)
$./\$ $/\$ $./\$

Theorem	isoml 32269*	The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)
$.\/$ $./\$ $/\$ $.\/$ $./\$

Theorem	isomliN 32270*	Properties that determine an orthomodular lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $.\/$ $./\$

Theorem	omlol 32271	An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)


Theorem	omlop 32272	An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.)


Theorem	omllat 32273	An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.)


Theorem	omllaw 32274	The orthomodular law. (Contributed by NM, 18-Sep-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	omllaw2N 32275	Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 26930 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	omllaw3 32276	Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 26781 analog.) (Contributed by NM, 19-Oct-2011.)
$./\$ $/\$ $/\$ $/\$ $./\$

Theorem	omllaw4 32277	Orthomodular law equivalent. Remark in [Holland95] p. 223. (Contributed by NM, 19-Oct-2011.)
$./\$ $/\$ $/\$ $./\$ $./\$

Theorem	omllaw5N 32278	The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 26958 analog.) (Contributed by NM, 14-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$

Theorem	cmtcomlemN 32279	Lemma for cmtcomN 32280. (cmcmlem 26936 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

Theorem	cmtcomN 32280	Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 26937 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

Theorem	cmt2N 32281	Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 26938 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

Theorem	cmt3N 32282	Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 26940 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

Theorem	cmt4N 32283	Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 26940 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

Theorem	cmtbr2N 32284	Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 26941 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cmtbr3N 32285	Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 26953 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$ $./\$

Theorem	cmtbr4N 32286	Alternate definition for the commutes relation. (cmbr4i 26946 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	lecmtN 32287	Ordered elements commute. (lecmi 26947 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

Theorem	cmtidN 32288	Any element commutes with itself. (cmidi 26955 analog.) (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
$/\$

Theorem	omlfh1N 32289	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 26963 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	omlfh3N 32290	Foulis-Holland Theorem, part 3. Dual of omlfh1N 32289. (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$

Theorem	omlmod1i2N 32291	Analog of modular law atmod1i2 32889 that holds in any OML. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	omlspjN 32292	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 32293	Extend class notation with covers relation.


Syntax	catm 32294	Extend class notation with atoms.


Syntax	cal 32295	Extend class notation with atomic lattices.


Syntax	clc 32296	Extend class notation with lattices with the covering property.


Definition	df-covers 32297*	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 32300 for the relation form. (Contributed by NM, 18-Sep-2011.)
$/\$ $/\$ $/\$ $/\$

Definition	df-ats 32298*	Define the class of poset atoms. (Contributed by NM, 18-Sep-2011.)


Theorem	cvrfval 32299*	Value of covers relation "is covered by". (Contributed by NM, 18-Sep-2011.)
$/\$ $/\$ $/\$ $/\$

Theorem	cvrval 32300*	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 27627 analog.) (Contributed by NM, 18-Sep-2011.)
$/\$ $/\$ $/\$ $/\$

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