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

Theorem	dochkrshp2 35001	Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LSHyp LFnl LKer $/\$ $/\$

Theorem	dochkrshp3 35002	Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $/\$

Theorem	dochkrshp4 35003	Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $\/$

Theorem	dochdmj1 35004	De Morgan-like law for subspace orthocomplement. (Contributed by NM, 5-Aug-2014.)
$/\$ $/\$ $/\$

Theorem	dochnoncon 35005	Law of noncontradiction. The intersection of a subspace and its orthocomplement is the zero subspace. (Contributed by NM, 16-Apr-2014.)
$/\$ $/\$

Theorem	dochnel2 35006	A nonzero member of a subspace doesn't belong to the orthocomplement of the subspace. (Contributed by NM, 28-Feb-2015.)
$/\$ $\$

Theorem	dochnel 35007	A nonzero vector doesn't belong to the orthocomplement of its singleton. (Contributed by NM, 27-Oct-2014.)
$/\$ $\$

Syntax	cdjh 35008	Extend class notation with subspace join for vector space.
joinH

Definition	df-djh 35009*	Define (closed) subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theorem	djhffval 35010*	Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theorem	djhfval 35011*	Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhval 35012	Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhval2 35013	Value of subspace join for vector space. (Contributed by NM, 6-Aug-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhcl 35014	Closure of subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhlj 35015	Transfer lattice join to vector space closed subspace join. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhljjN 35016	Lattice join in terms of vector space closed subspace join. (Contributed by NM, 17-Aug-2014.) (New usage is discouraged.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhjlj 35017	vector space closed subspace join in terms of lattice join. (Contributed by NM, 9-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhj 35018	vector space closed subspace join in terms of lattice join. (Contributed by NM, 17-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhcom 35019	Subspace join commutes. (Contributed by NM, 8-Aug-2014.)
$.\/$ joinH $/\$ $.\/$ $.\/$

Theorem	djhspss 35020	Subspace span of union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhsumss 35021	Subspace sum is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	dihsumssj 35022	The subspace sum of two isomorphisms of lattice elements is less than the isomorphism of their lattice join. (Contributed by NM, 23-Sep-2014.)
$.\/$ $/\$ $.\/$

Theorem	djhunssN 35023	Subspace union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.) (New usage is discouraged.)
$.\/$ joinH $/\$ $.\/$

Theorem	dochdmm1 35024	De Morgan-like law for closed subspace orthocomplement. (Contributed by NM, 13-Jan-2015.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhexmid 35025	Excluded middle property of vector space closed subspace join. (Contributed by NM, 22-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $.\/$

Theorem	djh01 35026	Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djh02 35027	Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhlsmcl 35028	A closed subspace sum equals subspace join. (shjshseli 27202 analog.) (Contributed by NM, 13-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhcvat42 35029*	A covering property. (cvrat42 33055 analog.) (Contributed by NM, 17-Aug-2014.)
$.\/$ joinH $/\$ $\$ $\$ $/\$ $.\/$ $\$ $/\$ $.\/$

Theorem	dihjatb 35030	Isomorphism H of lattice join of two atoms under the fiducial hyperplane. (Contributed by NM, 23-Sep-2014.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjatc 35031	Isomorphism H of lattice join of an element under the fiducial hyperplane with atom not under it. (Contributed by NM, 26-Aug-2014.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjatcclem1 35032	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjatcclem2 35033	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjatcclem3 35034*	Lemma for dihjatcc 35036. (Contributed by NM, 28-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$

Theorem	dihjatcclem4 35035*	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$

Theorem	dihjatcc 35036	Isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjat 35037	Isomorphism H of lattice join of two atoms. (Contributed by NM, 29-Sep-2014.)
$.\/$ $/\$ $.\/$

Theorem	dihprrnlem1N 35038	Lemma for dihprrn 35040, showing one of 4 cases. (Contributed by NM, 30-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dihprrnlem2 35039	Lemma for dihprrn 35040. (Contributed by NM, 29-Sep-2014.)
$/\$

Theorem	dihprrn 35040	The span of a vector pair belongs to the range of isomorphism H i.e. is a closed subspace. (Contributed by NM, 29-Sep-2014.)
$/\$

Theorem	djhlsmat 35041	The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 35040; should we directly use dihjat 35037? (Contributed by NM, 13-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	dihjat1lem 35042	Subspace sum of a closed subspace and an atom. (pmapjat1 33464 analog.) TODO: merge into dihjat1 35043? (Contributed by NM, 18-Aug-2014.)
$.\/$ joinH $/\$ $\$ $.\/$

Theorem	dihjat1 35043	Subspace sum of a closed subspace and an atom. (pmapjat1 33464 analog.) (Contributed by NM, 1-Oct-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	dihsmsprn 35044	Subspace sum of a closed subspace and the span of a singleton. (Contributed by NM, 17-Jan-2015.)
$/\$

Theorem	dihjat2 35045	The subspace sum of a closed subspace and an atom is the same as their subspace join. (Contributed by NM, 1-Oct-2014.)
$.\/$ joinH LSAtoms $/\$ $.\/$

Theorem	dihjat3 35046	Isomorphism H of lattice join with an atom. (Contributed by NM, 25-Apr-2015.)
$.\/$ $/\$ $.\/$

Theorem	dihjat4 35047	Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
$.\/$ LSAtoms $/\$ $.\/$

Theorem	dihjat6 35048	Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
$.\/$ LSAtoms $/\$ $.\/$

Theorem	dihsmsnrn 35049	The subspace sum of two singleton spans is closed. (Contributed by NM, 27-Feb-2015.)
$/\$

Theorem	dihsmatrn 35050	The subspace sum of a closed subspace and an atom is closed. TODO: see if proof at http://math.stackexchange.com/a/1233211/50776 and Mon, 13 Apr 2015 20:44:07 -0400 email could be used instead of this and dihjat2 35045. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dihjat5N 35051	Transfer lattice join with atom to subspace sum. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
$.\/$ $/\$ $.\/$

Theorem	dvh4dimat 35052*	There is an atom that is outside the subspace sum of 3 others. (Contributed by NM, 25-Apr-2015.)
LSAtoms $/\$

Theorem	dvh3dimatN 35053*	There is an atom that is outside the subspace sum of 2 others. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
LSAtoms $/\$

Theorem	dvh2dimatN 35054*	Given an atom, there exists another. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
LSAtoms $/\$

Theorem	dvh1dimat 35055*	There exists an atom. (Contributed by NM, 25-Apr-2015.)
LSAtoms $/\$

Theorem	dvh1dim 35056*	There exists a nonzero vector. (Contributed by NM, 26-Apr-2015.)
$/\$

Theorem	dvh4dimlem 35057*	Lemma for dvh4dimN 35061. (Contributed by NM, 22-May-2015.)
$/\$

Theorem	dvhdimlem 35058*	Lemma for dvh2dim 35059 and dvh3dim 35060. TODO: make this obsolete and use dvh4dimlem 35057 directly? (Contributed by NM, 24-Apr-2015.)
$/\$

Theorem	dvh2dim 35059*	There is a vector that is outside the span of another. (Contributed by NM, 25-Apr-2015.)
$/\$

Theorem	dvh3dim 35060*	There is a vector that is outside the span of 2 others. (Contributed by NM, 24-Apr-2015.)
$/\$

Theorem	dvh4dimN 35061*	There is a vector that is outside the span of 3 others. (Contributed by NM, 22-May-2015.) (New usage is discouraged.)
$/\$

Theorem	dvh3dim2 35062*	There is a vector that is outside of 2 spans with a common vector. (Contributed by NM, 13-May-2015.)
$/\$ $/\$

Theorem	dvh3dim3N 35063*	There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 35062 everywhere. If this one is needed, make dvh3dim2 35062 into a lemma. (Contributed by NM, 21-May-2015.) (New usage is discouraged.)
$/\$ $/\$

Theorem	dochsnnz 35064	The orthocomplement of a singleton is nonzero. (Contributed by NM, 13-Jun-2015.)
$/\$

Theorem	dochsatshp 35065	The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 27-Jul-2014.) (Revised by Mario Carneiro, 1-Oct-2014.)
LSAtoms LSHyp $/\$

Theorem	dochsatshpb 35066	The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 29-Oct-2014.)
LSAtoms LSHyp $/\$

Theorem	dochsnshp 35067	The orthocomplement of a nonzero singleton is a hyperplane. (Contributed by NM, 3-Jan-2015.)
LSHyp $/\$ $\$

Theorem	dochshpsat 35068	A hyperplane is closed iff its orthocomplement is an atom. (Contributed by NM, 29-Oct-2014.)
LSAtoms LSHyp $/\$

Theorem	dochkrsat 35069	The orthocomplement of a kernel is an atom iff it is nonzero. (Contributed by NM, 1-Nov-2014.)
LSAtoms LFnl LKer $/\$

Theorem	dochkrsat2 35070	The orthocomplement of a kernel is an atom iff the double orthocomplement is not the vector space. (Contributed by NM, 1-Jan-2015.)
LSAtoms LFnl LKer $/\$

Theorem	dochsat0 35071	The orthocomplement of a kernel is either an atom or zero. (Contributed by NM, 29-Jan-2015.)
LSAtoms LFnl LKer $/\$ $\/$

Theorem	dochkrsm 35072	The subspace sum of a closed subspace and a kernel orthocomplement is closed. (djhlsmcl 35028 can be used to convert sum to join.) (Contributed by NM, 29-Jan-2015.)
LFnl LKer $/\$

Theorem	dochexmidat 35073	Special case of excluded middle for the singleton of a vector. (Contributed by NM, 27-Oct-2014.)
$/\$ $\$

Theorem	dochexmidlem1 35074	Lemma for dochexmid 35082. Holland's proof implicitly requires , which we prove here. (Contributed by NM, 14-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem2 35075	Lemma for dochexmid 35082. (Contributed by NM, 14-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem3 35076	Lemma for dochexmid 35082. Use atom exchange lsatexch1 32658 to swap and . (Contributed by NM, 14-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem4 35077	Lemma for dochexmid 35082. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem5 35078	Lemma for dochexmid 35082. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem6 35079	Lemma for dochexmid 35082. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem7 35080	Lemma for dochexmid 35082. Contradict dochexmidlem6 35079. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem8 35081	Lemma for dochexmid 35082. The contradiction of dochexmidlem6 35079 and dochexmidlem7 35080 shows that there can be no atom that is not in , which is therefore the whole atom space. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmid 35082	Excluded middle law for closed subspaces, which is equivalent to (and derived from) the orthomodular law dihoml4 34991. Lemma 3.3(2) in [Holland95] p. 215. In our proof, we use the variables , , , , in place of Hollands' l, m, P, Q, L respectively. (pexmidALTN 33589 analog.) (Contributed by NM, 15-Jan-2015.)
$/\$

Theorem	dochsnkrlem1 35083	Lemma for dochsnkr 35086. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $\$

Theorem	dochsnkrlem2 35084	Lemma for dochsnkr 35086. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $\$ LSAtoms

Theorem	dochsnkrlem3 35085	Lemma for dochsnkr 35086. (Contributed by NM, 2-Jan-2015.)
LFnl LKer $/\$ $\$

Theorem	dochsnkr 35086	A (closed) kernel expressed in terms of a nonzero vector in its orthocomplement. TODO: consolidate lemmas unless they're needed for something else (in which case break out as theorems). (Contributed by NM, 2-Jan-2015.)
LFnl LKer $/\$ $\$

Theorem	dochsnkr2 35087*	Kernel of the explicit functional determined by a nonzero vector . Compare the more general lshpkr 32729. (Contributed by NM, 27-Oct-2014.)
LKer Scalar $/\$ $\$

Theorem	dochsnkr2cl 35088*	The determining functional belongs to the atom formed by the orthocomplement of the kernel. (Contributed by NM, 4-Jan-2015.)
LKer Scalar $/\$ $\$ $\$

Theorem	dochflcl 35089*	Closure of the explicit functional determined by a nonzero vector . Compare the more general lshpkrcl 32728. (Contributed by NM, 27-Oct-2014.)
LFnl Scalar $/\$ $\$

Theorem	dochfl1 35090*	The value of the explicit functional is 1 at the that determines it. (Contributed by NM, 27-Oct-2014.)
Scalar $/\$ $\$

Theorem	dochfln0 35091	The value of a functional is nonzero at a nonzero vector in the orthocomplement of its kernel. (Contributed by NM, 2-Jan-2015.)
Scalar LFnl LKer $/\$ $\$

Theorem	dochkr1 35092*	A non-zero functional has a value of 1 at some argument belonging to the orthocomplement of its kernel (when its kernel is a closed hyperplane). Tighter version of lfl1 32682. (Contributed by NM, 2-Jan-2015.)
Scalar LFnl LKer $/\$ $\$

Theorem	dochkr1OLDN 35093*	A non-zero functional has a value of 1 at some argument belonging to the orthocomplement of its kernel (when its kernel is a closed hyperplane). Tighter version of lfl1 32682. (Contributed by NM, 2-Jan-2015.) (New usage is discouraged.)
Scalar LFnl LKer $/\$

21.21.14 Construction of involution and inner product from a Hilbert lattice

Syntax	clpoN 35094	Extend class notation with all polarities of a left module or left vector space.
LPol

Definition	df-lpolN 35095*	Define the set of all polarities of a left module or left vector space. A polarity is a kind of complementation operation on a subspace. The double polarity of a subspace is a closure operation. Based on Definition 3.2 of [Holland95] p. 214 for projective geometry polarities. For convenience, we open up the domain to include all vector subsets and not just subspaces, but any more restricted polarity can be converted to this one by taking the span of its argument. (Contributed by NM, 24-Nov-2014.)
LPol $/\$ $/\$ $/\$ $/\$ LSAtoms LSHyp $/\$

Theorem	lpolsetN 35096*	The set of polarities of a left module or left vector space. (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)
LSAtoms LSHyp LPol $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	islpolN 35097*	The predicate "is a polarity". (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)
LSAtoms LSHyp LPol $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	islpoldN 35098*	Properties that determine a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LSAtoms LSHyp LPol $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	lpolfN 35099	Functionality of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LPol

Theorem	lpolvN 35100	The polarity of the whole space is the zero subspace. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LPol

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