P. 351 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 35001-35100 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	ldualsbase 35001	Base set of scalar ring for the dual of a vector space. (Contributed by NM, 24-Oct-2014.)
Scalar LDual Scalar

Theorem	ldualsaddN 35002	Scalar addition for the dual of a vector space. (Contributed by NM, 24-Oct-2014.) (New usage is discouraged.)
Scalar LDual Scalar

Theorem	ldualsmul 35003	Scalar multiplication for the dual of a vector space. (Contributed by NM, 19-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Scalar LDual Scalar

Theorem	ldualfvs 35004*	Scalar product operation for the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
LFnl Scalar LDual

Theorem	ldualvs 35005	Scalar product operation value (which is a functional) for the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
LFnl Scalar LDual

Theorem	ldualvsval 35006	Value of scalar product operation value for the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
LFnl Scalar LDual

Theorem	ldualvscl 35007	The scalar product operation value is a functional. (Contributed by NM, 18-Oct-2014.)
LFnl Scalar LDual

Theorem	ldualvaddcom 35008	Commutative law for vector (functional) addition. (Contributed by NM, 17-Jan-2015.)
LFnl LDual

Theorem	ldualvsass 35009	Associative law for scalar product operation. (Contributed by NM, 20-Oct-2014.)
LFnl Scalar LDual

Theorem	ldualvsass2 35010	Associative law for scalar product operation, using operations from the dual space. (Contributed by NM, 20-Oct-2014.)
LFnl Scalar LDual Scalar

Theorem	ldualvsdi1 35011	Distributive law for scalar product operation, using operations from the dual space. (Contributed by NM, 21-Oct-2014.)
LFnl Scalar LDual

Theorem	ldualvsdi2 35012	Reverse distributive law for scalar product operation, using operations from the dual space. (Contributed by NM, 21-Oct-2014.)
LFnl Scalar LDual

Theorem	ldualgrplem 35013	Lemma for ldualgrp 35014. (Contributed by NM, 22-Oct-2014.)
LDual LFnl Scalar opp_r

Theorem	ldualgrp 35014	The dual of a vector space is a group. (Contributed by NM, 21-Oct-2014.)
LDual

Theorem	ldual0 35015	The zero scalar of the dual of a vector space. (Contributed by NM, 28-Dec-2014.)
Scalar LDual Scalar

Theorem	ldual1 35016	The unit scalar of the dual of a vector space. (Contributed by NM, 26-Feb-2015.)
Scalar LDual Scalar

Theorem	ldualneg 35017	The negative of a scalar of the dual of a vector space. (Contributed by NM, 26-Feb-2015.)
Scalar LDual Scalar

Theorem	ldual0v 35018	The zero vector of the dual of a vector space. (Contributed by NM, 24-Oct-2014.)
Scalar LDual

Theorem	ldual0vcl 35019	The dual zero vector is a functional. (Contributed by NM, 5-Mar-2015.)
LFnl LDual

Theorem	lduallmodlem 35020	Lemma for lduallmod 35021. (Contributed by NM, 22-Oct-2014.)
LDual LFnl Scalar opp_r

Theorem	lduallmod 35021	The dual of a left module is also a left module. (Contributed by NM, 22-Oct-2014.)
LDual

Theorem	lduallvec 35022	The dual of a left vector space is also a left vector space. Note that scalar multiplication is reversed by df-oppr 17399; 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 35023	The value of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.)
Scalar LFnl LDual

Theorem	ldualvsubcl 35024	Closure of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.)
LFnl LDual

Theorem	ldualvsubval 35025	The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 35023? (Requires to opp_r conversion.) (Contributed by NM, 26-Feb-2015.)
Scalar LFnl LDual

Theorem	ldualssvscl 35026	Closure of scalar product in a dual subspace.) (Contributed by NM, 5-Feb-2015.)
Scalar LDual

Theorem	ldualssvsubcl 35027	Closure of vector subtraction in a dual subspace.) (Contributed by NM, 9-Mar-2015.)
LDual

Theorem	ldual0vs 35028	Scalar zero times a functional is the zero functional. (Contributed by NM, 17-Feb-2015.)
LFnl Scalar LDual

Theorem	lkr0f2 35029	The kernel of the zero functional is the set of all vectors. (Contributed by NM, 4-Feb-2015.)
LFnl LKer LDual

Theorem	lduallkr3 35030	The kernels of nonzero functionals are hyperplanes. (Contributed by NM, 22-Feb-2015.)
LSHyp LFnl LKer LDual

Theorem	lkrpssN 35031	Proper subset relation between kernels. (Contributed by NM, 16-Feb-2015.) (New usage is discouraged.)
LFnl LKer LDual $/\$

Theorem	lkrin 35032	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 35033*	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 35034*	Equivalent expressions for a 1-dim subspace (ray) of functionals. (Contributed by NM, 24-Oct-2014.)
LFnl LKer LDual

Theorem	ldualkrsc 35035	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 35036	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 35037*	Two functionals with kernels in a subset relationship. (Contributed by NM, 17-Feb-2015.) (New usage is discouraged.)
Scalar LFnl LKer LDual

Theorem	lkreqN 35038	Proportional functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
Scalar LFnl LKer LDual $\$

Theorem	lkrlspeqN 35039	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 35040	Extend class notation with orthoposets.


Syntax	ccmtN 35041	Extend class notation with the commutes relation.


Syntax	col 35042	Extend class notation with orthlattices.


Syntax	coml 35043	Extend class notation with orthomodular lattices.


Definition	df-oposet 35044*	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 35045*	Define the commutes relation for orthoposets. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Nov-2011.)
$/\$ $/\$

Definition	df-ol 35046	Define the class of ortholattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)


Definition	df-oml 35047*	Define the class of orthomodular lattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)


Theorem	isopos 35048*	The predicate "is an orthoposet." (Contributed by NM, 20-Oct-2011.) (Revised by NM, 14-Sep-2018.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $./\$

Theorem	opposet 35049	Every orthoposet is a poset. (Contributed by NM, 12-Oct-2011.)


Theorem	oposlem 35050	Lemma for orthoposet properties. (Contributed by NM, 20-Oct-2011.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $./\$

Theorem	op01dm 35051	Conditions necessary for zero and unit elements to exist. (Contributed by NM, 14-Sep-2018.)
$/\$

Theorem	op0cl 35052	An orthoposet has a zero element. (h0elch 26300 analog.) (Contributed by NM, 12-Oct-2011.)


Theorem	op1cl 35053	An orthoposet has a unit element. (helch 26288 analog.) (Contributed by NM, 22-Oct-2011.)


Theorem	op0le 35054	Orthoposet zero is less than or equal to any element. (ch0le 26486 analog.) (Contributed by NM, 12-Oct-2011.)
$/\$

Theorem	ople0 35055	An element less than or equal to zero equals zero. (chle0 26488 analog.) (Contributed by NM, 21-Oct-2011.)
$/\$

Theorem	opnlen0 35056	An element not less than another is nonzero. TODO: Look for uses of necon3bd 2669 and op0le 35054 to see if this is useful elsewhere. (Contributed by NM, 5-May-2013.)
$/\$ $/\$ $/\$

Theorem	lub0N 35057	The least upper bound of the empty set is the zero element. (Contributed by NM, 15-Sep-2013.) (New usage is discouraged.)


Theorem	opltn0 35058	A lattice element greater than zero is nonzero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
$/\$

Theorem	ople1 35059	Any element is less than the orthoposet unit. (chss 26274 analog.) (Contributed by NM, 23-Oct-2011.)
$/\$

Theorem	op1le 35060	If the orthoposet unit is less than or equal to an element, the element equals the unit. (chle0 26488 analog.) (Contributed by NM, 5-Dec-2011.)
$/\$

Theorem	glb0N 35061	The greatest lower bound of the empty set is the unit element. (Contributed by NM, 5-Dec-2011.) (New usage is discouraged.)


Theorem	opoccl 35062	Closure of orthocomplement operation. (choccl 26351 analog.) (Contributed by NM, 20-Oct-2011.)
$/\$

Theorem	opococ 35063	Double negative law for orthoposets. (ococ 26451 analog.) (Contributed by NM, 13-Sep-2011.)
$/\$

Theorem	opcon3b 35064	Contraposition law for orthoposets. (chcon3i 26511 analog.) (Contributed by NM, 8-Nov-2011.)
$/\$ $/\$

Theorem	opcon2b 35065	Orthocomplement contraposition law. (negcon2 9891 analog.) (Contributed by NM, 16-Jan-2012.)
$/\$ $/\$

Theorem	opcon1b 35066	Orthocomplement contraposition law. (negcon1 9890 analog.) (Contributed by NM, 24-Jan-2012.)
$/\$ $/\$

Theorem	oplecon3 35067	Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011.)
$/\$ $/\$

Theorem	oplecon3b 35068	Contraposition law for orthoposets. (chsscon3 26545 analog.) (Contributed by NM, 4-Nov-2011.)
$/\$ $/\$

Theorem	oplecon1b 35069	Contraposition law for strict ordering in orthoposets. (chsscon1 26546 analog.) (Contributed by NM, 6-Nov-2011.)
$/\$ $/\$

Theorem	opoc1 35070	Orthocomplement of orthoposet unit. (Contributed by NM, 24-Jan-2012.)


Theorem	opoc0 35071	Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012.)


Theorem	opltcon3b 35072	Contraposition law for strict ordering in orthoposets. (chpsscon3 26548 analog.) (Contributed by NM, 4-Nov-2011.)
$/\$ $/\$

Theorem	opltcon1b 35073	Contraposition law for strict ordering in orthoposets. (chpsscon1 26549 analog.) (Contributed by NM, 5-Nov-2011.)
$/\$ $/\$

Theorem	opltcon2b 35074	Contraposition law for strict ordering in orthoposets. (chsscon2 26547 analog.) (Contributed by NM, 5-Nov-2011.)
$/\$ $/\$

Theorem	opexmid 35075	Law of excluded middle for orthoposets. (chjo 26560 analog.) (Contributed by NM, 13-Sep-2011.)
$.\/$ $/\$ $.\/$

Theorem	opnoncon 35076	Law of contradiction for orthoposets. (chocin 26540 analog.) (Contributed by NM, 13-Sep-2011.)
$./\$ $/\$ $./\$

Theorem	riotaocN 35077*	The orthocomplement of the unique poset element such that . (riotaneg 10538 analog.) (Contributed by NM, 16-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	cmtfvalN 35078*	Value of commutes relation. (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$ $./\$

Theorem	cmtvalN 35079	Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 26629 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$ $./\$

Theorem	isolat 35080	The predicate "is an ortholattice." (Contributed by NM, 18-Sep-2011.)
$/\$

Theorem	ollat 35081	An ortholattice is a lattice. (Contributed by NM, 18-Sep-2011.)


Theorem	olop 35082	An ortholattice is an orthoposet. (Contributed by NM, 18-Sep-2011.)


Theorem	olposN 35083	An ortholattice is a poset. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)


Theorem	isolatiN 35084	Properties that determine an ortholattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)


Theorem	oldmm1 35085	De Morgan's law for meet in an ortholattice. (chdmm1 26570 analog.) (Contributed by NM, 6-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	oldmm2 35086	De Morgan's law for meet in an ortholattice. (chdmm2 26571 analog.) (Contributed by NM, 6-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	oldmm3N 35087	De Morgan's law for meet in an ortholattice. (chdmm3 26572 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	oldmm4 35088	De Morgan's law for meet in an ortholattice. (chdmm4 26573 analog.) (Contributed by NM, 6-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$

Theorem	oldmj1 35089	De Morgan's law for join in an ortholattice. (chdmj1 26574 analog.) (Contributed by NM, 6-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	oldmj2 35090	De Morgan's law for join in an ortholattice. (chdmj2 26575 analog.) (Contributed by NM, 7-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	oldmj3 35091	De Morgan's law for join in an ortholattice. (chdmj3 26576 analog.) (Contributed by NM, 7-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	oldmj4 35092	De Morgan's law for join in an ortholattice. (chdmj4 26577 analog.) (Contributed by NM, 7-Nov-2011.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	olj01 35093	An ortholattice element joined with zero equals itself. (chj0 26542 analog.) (Contributed by NM, 19-Oct-2011.)
$.\/$ $/\$ $.\/$

Theorem	olj02 35094	An ortholattice element joined with zero equals itself. (Contributed by NM, 28-Jan-2012.)
$.\/$ $/\$ $.\/$

Theorem	olm11 35095	The meet of an ortholattice element with one equals itself. (chm1i 26501 analog.) (Contributed by NM, 22-May-2012.)
$./\$ $/\$ $./\$

Theorem	olm12 35096	The meet of an ortholattice element with one equals itself. (Contributed by NM, 22-May-2012.)
$./\$ $/\$ $./\$

Theorem	latmassOLD 35097	Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 3704 analog.) (Contributed by NM, 7-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$

Theorem	latm12 35098	A rearrangement of lattice meet. (in12 3705 analog.) (Contributed by NM, 8-Nov-2011.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$

Theorem	latm32 35099	A rearrangement of lattice meet. (in12 3705 analog.) (Contributed by NM, 13-Nov-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$

Theorem	latmrot 35100	Rotate lattice meet of 3 classes. (Contributed by NM, 9-Oct-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$

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