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	ldual0vcl 32801	The dual zero vector is a functional. (Contributed by NM, 5-Mar-2015.)
LFnl LDual

Theorem	lduallmodlem 32802	Lemma for lduallmod 32803. (Contributed by NM, 22-Oct-2014.)
LDual LFnl Scalar opp_r

Theorem	lduallmod 32803	The dual of a left module is also a left module. (Contributed by NM, 22-Oct-2014.)
LDual

Theorem	lduallvec 32804	The dual of a left vector space is also a left vector space. Note that scalar multiplication is reversed by df-oppr 16720; otherwise, the dual would be a right vector space as is sometimes the case in the literature. (Contributed by NM, 22-Oct-2014.)
LDual

Theorem	ldualvsub 32805	The value of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.)
Scalar LFnl LDual

Theorem	ldualvsubcl 32806	Closure of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.)
LFnl LDual

Theorem	ldualvsubval 32807	The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 32805? (Requires to opp_r conversion.) (Contributed by NM, 26-Feb-2015.)
Scalar LFnl LDual

Theorem	ldualssvscl 32808	Closure of scalar product in a dual subspace.) (Contributed by NM, 5-Feb-2015.)
Scalar LDual

Theorem	ldualssvsubcl 32809	Closure of vector subtraction in a dual subspace.) (Contributed by NM, 9-Mar-2015.)
LDual

Theorem	ldual0vs 32810	Scalar zero times a functional is the zero functional. (Contributed by NM, 17-Feb-2015.)
LFnl Scalar LDual

Theorem	lkr0f2 32811	The kernel of the zero functional is the set of all vectors. (Contributed by NM, 4-Feb-2015.)
LFnl LKer LDual

Theorem	lduallkr3 32812	The kernels of nonzero functionals are hyperplanes. (Contributed by NM, 22-Feb-2015.)
LSHyp LFnl LKer LDual

Theorem	lkrpssN 32813	Proper subset relation between kernels. (Contributed by NM, 16-Feb-2015.) (New usage is discouraged.)
LFnl LKer LDual $/\$

Theorem	lkrin 32814	Intersection of the kernels of 2 functionals is included in the kernel of their sum. (Contributed by NM, 7-Jan-2015.)
LFnl LKer LDual

Theorem	eqlkr4 32815*	Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 4-Feb-2015.)
Scalar LFnl LKer LDual

Theorem	ldual1dim 32816*	Equivalent expressions for a 1-dim subspace (ray) of functionals. (Contributed by NM, 24-Oct-2014.)
LFnl LKer LDual

Theorem	ldualkrsc 32817	The kernel of a non-zero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 28-Dec-2014.)
Scalar LFnl LKer LDual

Theorem	lkrss 32818	The kernel of a scalar product of a functional includes the kernel of the functional. (Contributed by NM, 27-Jan-2015.)
Scalar LFnl LKer LDual

Theorem	lkrss2N 32819*	Two functionals with kernels in a subset relationship. (Contributed by NM, 17-Feb-2015.) (New usage is discouraged.)
Scalar LFnl LKer LDual

Theorem	lkreqN 32820	Proportional functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
Scalar LFnl LKer LDual $\$

Theorem	lkrlspeqN 32821	Condition for colinear functionals to have equal kernels. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
LFnl LKer LDual $\$

21.30.6 Ortholattices and orthomodular lattices

Syntax	cops 32822	Extend class notation with orthoposets.


Syntax	ccmtN 32823	Extend class notation with the commutes relation.


Syntax	col 32824	Extend class notation with orthlattices.


Syntax	coml 32825	Extend class notation with orthomodular lattices.


Definition	df-oposet 32826*	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 32827*	Define the commutes relation for orthoposets. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Nov-2011.)
$/\$ $/\$

Definition	df-ol 32828	Define the class of ortholattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)


Definition	df-oml 32829*	Define the class of orthomodular lattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)


Theorem	isopos 32830*	The predicate "is an orthoposet." (Contributed by NM, 20-Oct-2011.) (Revised by NM, 14-Sep-2018.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $./\$

Theorem	opposet 32831	Every orthoposet is a poset. (Contributed by NM, 12-Oct-2011.)


Theorem	oposlem 32832	Lemma for orthoposet properties. (Contributed by NM, 20-Oct-2011.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $./\$

Theorem	op01dm 32833	Conditions necessary for zero and unit elements to exist. (Contributed by NM, 14-Sep-2018.)
$/\$

Theorem	op0cl 32834	An orthoposet has a zero element. (h0elch 24663 analog.) (Contributed by NM, 12-Oct-2011.)


Theorem	op1cl 32835	An orthoposet has a unit element. (helch 24651 analog.) (Contributed by NM, 22-Oct-2011.)


Theorem	op0le 32836	Orthoposet zero is less than or equal to any element. (ch0le 24849 analog.) (Contributed by NM, 12-Oct-2011.)
$/\$

Theorem	ople0 32837	An element less than or equal to zero equals zero. (chle0 24851 analog.) (Contributed by NM, 21-Oct-2011.)
$/\$

Theorem	opnlen0 32838	An element not less than another is nonzero. TODO: Look for uses of necon3bd 2650 and op0le 32836 to see if this is useful elsewhere. (Contributed by NM, 5-May-2013.)
$/\$ $/\$ $/\$

Theorem	lub0N 32839	The least upper bound of the empty set is the zero element. (Contributed by NM, 15-Sep-2013.) (New usage is discouraged.)


Theorem	opltn0 32840	A lattice element greater than zero is nonzero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
$/\$

Theorem	ople1 32841	Any element is less than the orthoposet unit. (chss 24637 analog.) (Contributed by NM, 23-Oct-2011.)
$/\$

Theorem	op1le 32842	If the orthoposet unit is less than or equal to an element, the element equals the unit. (chle0 24851 analog.) (Contributed by NM, 5-Dec-2011.)
$/\$

Theorem	glb0N 32843	The greatest lower bound of the empty set is the unit element. (Contributed by NM, 5-Dec-2011.) (New usage is discouraged.)


Theorem	opoccl 32844	Closure of orthocomplement operation. (choccl 24714 analog.) (Contributed by NM, 20-Oct-2011.)
$/\$

Theorem	opococ 32845	Double negative law for orthoposets. (ococ 24814 analog.) (Contributed by NM, 13-Sep-2011.)
$/\$

Theorem	opcon3b 32846	Contraposition law for orthoposets. (chcon3i 24874 analog.) (Contributed by NM, 8-Nov-2011.)
$/\$ $/\$

Theorem	opcon2b 32847	Orthocomplement contraposition law. (negcon2 9667 analog.) (Contributed by NM, 16-Jan-2012.)
$/\$ $/\$

Theorem	opcon1b 32848	Orthocomplement contraposition law. (negcon1 9666 analog.) (Contributed by NM, 24-Jan-2012.)
$/\$ $/\$

Theorem	oplecon3 32849	Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011.)
$/\$ $/\$

Theorem	oplecon3b 32850	Contraposition law for orthoposets. (chsscon3 24908 analog.) (Contributed by NM, 4-Nov-2011.)
$/\$ $/\$

Theorem	oplecon1b 32851	Contraposition law for strict ordering in orthoposets. (chsscon1 24909 analog.) (Contributed by NM, 6-Nov-2011.)
$/\$ $/\$

Theorem	opoc1 32852	Orthocomplement of orthoposet unit. (Contributed by NM, 24-Jan-2012.)


Theorem	opoc0 32853	Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012.)


Theorem	opltcon3b 32854	Contraposition law for strict ordering in orthoposets. (chpsscon3 24911 analog.) (Contributed by NM, 4-Nov-2011.)
$/\$ $/\$

Theorem	opltcon1b 32855	Contraposition law for strict ordering in orthoposets. (chpsscon1 24912 analog.) (Contributed by NM, 5-Nov-2011.)
$/\$ $/\$

Theorem	opltcon2b 32856	Contraposition law for strict ordering in orthoposets. (chsscon2 24910 analog.) (Contributed by NM, 5-Nov-2011.)
$/\$ $/\$

Theorem	opexmid 32857	Law of excluded middle for orthoposets. (chjo 24923 analog.) (Contributed by NM, 13-Sep-2011.)
$.\/$ $/\$ $.\/$

Theorem	opnoncon 32858	Law of contradiction for orthoposets. (chocin 24903 analog.) (Contributed by NM, 13-Sep-2011.)
$./\$ $/\$ $./\$

Theorem	riotaocN 32859*	The orthocomplement of the unique poset element such that . (riotaneg 10310 analog.) (Contributed by NM, 16-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	cmtfvalN 32860*	Value of commutes relation. (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$ $./\$

Theorem	cmtvalN 32861	Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 24992 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$ $./\$

Theorem	isolat 32862	The predicate "is an ortholattice." (Contributed by NM, 18-Sep-2011.)
$/\$

Theorem	ollat 32863	An ortholattice is a lattice. (Contributed by NM, 18-Sep-2011.)


Theorem	olop 32864	An ortholattice is an orthoposet. (Contributed by NM, 18-Sep-2011.)


Theorem	olposN 32865	An ortholattice is a poset. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)


Theorem	isolatiN 32866	Properties that determine an ortholattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)


Theorem	oldmm1 32867	De Morgan's law for meet in an ortholattice. (chdmm1 24933 analog.) (Contributed by NM, 6-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	oldmm2 32868	De Morgan's law for meet in an ortholattice. (chdmm2 24934 analog.) (Contributed by NM, 6-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	oldmm3N 32869	De Morgan's law for meet in an ortholattice. (chdmm3 24935 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	oldmm4 32870	De Morgan's law for meet in an ortholattice. (chdmm4 24936 analog.) (Contributed by NM, 6-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	oldmj1 32871	De Morgan's law for join in an ortholattice. (chdmj1 24937 analog.) (Contributed by NM, 6-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	oldmj2 32872	De Morgan's law for join in an ortholattice. (chdmj2 24938 analog.) (Contributed by NM, 7-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	oldmj3 32873	De Morgan's law for join in an ortholattice. (chdmj3 24939 analog.) (Contributed by NM, 7-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	oldmj4 32874	De Morgan's law for join in an ortholattice. (chdmj4 24940 analog.) (Contributed by NM, 7-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	olj01 32875	An ortholattice element joined with zero equals itself. (chj0 24905 analog.) (Contributed by NM, 19-Oct-2011.)
$.\/$ $/\$ $.\/$

Theorem	olj02 32876	An ortholattice element joined with zero equals itself. (Contributed by NM, 28-Jan-2012.)
$.\/$ $/\$ $.\/$

Theorem	olm11 32877	The meet of an ortholattice element with one equals itself. (chm1i 24864 analog.) (Contributed by NM, 22-May-2012.)
$./\$ $/\$ $./\$

Theorem	olm12 32878	The meet of an ortholattice element with one equals itself. (Contributed by NM, 22-May-2012.)
$./\$ $/\$ $./\$

Theorem	latmassOLD 32879	Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 3565 analog.) (Contributed by NM, 7-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$

Theorem	latm12 32880	A rearrangement of lattice meet. (in12 3566 analog.) (Contributed by NM, 8-Nov-2011.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$

Theorem	latm32 32881	A rearrangement of lattice meet. (in12 3566 analog.) (Contributed by NM, 13-Nov-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$

Theorem	latmrot 32882	Rotate lattice meet of 3 classes. (Contributed by NM, 9-Oct-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$

Theorem	latm4 32883	Rearrangement of lattice meet of 4 classes. (in4 3571 analog.) (Contributed by NM, 8-Nov-2011.)
$./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$ $./\$ $./\$

Theorem	latmmdiN 32884	Lattice meet distributes over itself. (inindi 3572 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$ $./\$

Theorem	latmmdir 32885	Lattice meet distributes over itself. (inindir 3573 analog.) (Contributed by NM, 6-Jun-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$ $./\$

Theorem	olm01 32886	Meet with lattice zero is zero. (chm0 24899 analog.) (Contributed by NM, 8-Nov-2011.)
$./\$ $/\$ $./\$

Theorem	olm02 32887	Meet with lattice zero is zero. (Contributed by NM, 9-Oct-2012.)
$./\$ $/\$ $./\$

Theorem	isoml 32888*	The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)
$.\/$ $./\$ $/\$ $.\/$ $./\$

Theorem	isomliN 32889*	Properties that determine an orthomodular lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $.\/$ $./\$

Theorem	omlol 32890	An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)


Theorem	omlop 32891	An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.)


Theorem	omllat 32892	An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.)


Theorem	omllaw 32893	The orthomodular law. (Contributed by NM, 18-Sep-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	omllaw2N 32894	Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 24993 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	omllaw3 32895	Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 24844 analog.) (Contributed by NM, 19-Oct-2011.)
$./\$ $/\$ $/\$ $/\$ $./\$

Theorem	omllaw4 32896	Orthomodular law equivalent. Remark in [Holland95] p. 223. (Contributed by NM, 19-Oct-2011.)
$./\$ $/\$ $/\$ $./\$ $./\$

Theorem	omllaw5N 32897	The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 25021 analog.) (Contributed by NM, 14-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$

Theorem	cmtcomlemN 32898	Lemma for cmtcomN 32899. (cmcmlem 24999 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

Theorem	cmtcomN 32899	Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 25000 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

Theorem	cmt2N 32900	Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 25001 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$/\$ $/\$

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