P. 350 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 34901-35000 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	dihmeetlem15N 34901	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihmeetlem16N 34902	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem17N 34903	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$

Theorem	dihmeetlem18N 34904	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihmeetlem19N 34905	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem20N 34906	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetALTN 34907	Isomorphism H of a lattice meet. This version does not depend on the atomisticity of the constructed vector space. TODO: Delete? (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $./\$

Theorem	dih1dimatlem0 34908*	Lemma for dih1dimat 34910. (Contributed by NM, 11-Apr-2014.)
LSAtoms Scalar $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	dih1dimatlem 34909*	Lemma for dih1dimat 34910. (Contributed by NM, 10-Apr-2014.)
LSAtoms Scalar $/\$ $/\$

Theorem	dih1dimat 34910	Any 1-dimensional subspace is a value of isomorphism H. (Contributed by NM, 11-Apr-2014.)
LSAtoms $/\$ $/\$

Theorem	dihlsprn 34911	The span of a vector belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)
$/\$ $/\$

Theorem	dihlspsnssN 34912	A subspace included in a 1-dim subspace belongs to the range of isomorphism H. (Contributed by NM, 26-Apr-2014.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	dihlspsnat 34913	The inverse isomorphism H of the span of a singleton is a Hilbert lattice atom. (Contributed by NM, 27-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dihatlat 34914	The isomorphism H of an atom is a 1-dim subspace. (Contributed by NM, 28-Apr-2014.)
LSAtoms $/\$ $/\$

Theorem	dihat 34915	There exists at least one atom in the subspaces of vector space H. (Contributed by NM, 12-Aug-2014.)
LSAtoms $/\$

Theorem	dihpN 34916*	The value of isomorphism H at the fiducial atom is determined by the vector (the zero translation ltrnid 33712 and a nonzero member of the endomorphism ring). In particular, can be replaced with the ring unit . (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.)
$/\$ $/\$

Theorem	dihlatat 34917	The reverse isomorphism H of a 1-dim subspace is an atom. (Contributed by NM, 28-Apr-2014.)
LSAtoms $/\$ $/\$

Theorem	dihatexv 34918*	There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 16-Aug-2014.)
$/\$ $\$

Theorem	dihatexv2 34919*	There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 17-Aug-2014.)
$/\$ $\$

Theorem	dihglblem6 34920*	Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)
$./\$ LSAtoms $/\$ $/\$ $/\$

Theorem	dihglb 34921*	Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dihglb2 34922*	Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)
$/\$ $/\$

Theorem	dihmeet 34923	Isomorphism H of a lattice meet. (Contributed by NM, 13-Apr-2014.)
$./\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihintcl 34924	The intersection of closed subspaces (the range of isomorphism H) is a closed subspace. (Contributed by NM, 14-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dihmeetcl 34925	Closure of closed subspace meet for vector space. (Contributed by NM, 5-Aug-2014.)
$/\$ $/\$ $/\$

Theorem	dihmeet2 34926	Reverse isomorphism H of a closed subspace intersection. (Contributed by NM, 15-Jan-2015.)
$./\$ $/\$ $./\$

Syntax	coch 34927	Extend class notation with subspace orthocomplement for vector space.


Definition	df-doch 34928*	Define subspace orthocomplement for vector space. Temporarily, we are using the range of the isomorphism instead of the set of closed subspaces. Later, when inner product is introduced, we will show that these are the same. (Contributed by NM, 14-Mar-2014.)


Theorem	dochffval 34929*	Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)


Theorem	dochfval 34930*	Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)
$/\$

Theorem	dochval 34931*	Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)
$/\$ $/\$

Theorem	dochval2 34932*	Subspace orthocomplement for vector space. (Contributed by NM, 14-Apr-2014.)
$/\$ $/\$

Theorem	dochcl 34933	Closure of subspace orthocomplement for vector space. (Contributed by NM, 9-Mar-2014.)
$/\$ $/\$

Theorem	dochlss 34934	A subspace orthocomplement is a subspace of the vector space. (Contributed by NM, 22-Jul-2014.)
$/\$ $/\$

Theorem	dochssv 34935	A subspace orthocomplement belongs to the vector space. (Contributed by NM, 22-Jul-2014.)
$/\$ $/\$

Theorem	dochfN 34936	Domain and codomain of the subspace orthocomplement for the vector space. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)
$/\$

Theorem	dochvalr 34937	Orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)
$/\$ $/\$

Theorem	doch0 34938	Orthocomplement of the zero subspace. (Contributed by NM, 19-Jun-2014.)
$/\$

Theorem	doch1 34939	Orthocomplement of the unit subspace (all vectors). (Contributed by NM, 19-Jun-2014.)
$/\$

Theorem	dochoc0 34940	The zero subspace is closed. (Contributed by NM, 16-Feb-2015.)
$/\$

Theorem	dochoc1 34941	The unit subspace (all vectors) is closed. (Contributed by NM, 16-Feb-2015.)
$/\$

Theorem	dochvalr2 34942	Orthocomplement of a closed subspace. (Contributed by NM, 21-Jul-2014.)
$/\$ $/\$

Theorem	dochvalr3 34943	Orthocomplement of a closed subspace. (Contributed by NM, 15-Jan-2015.)
$/\$

Theorem	doch2val2 34944*	Double orthocomplement for vector space. (Contributed by NM, 26-Jul-2014.)
$/\$

Theorem	dochss 34945	Subset law for orthocomplement. (Contributed by NM, 16-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dochocss 34946	Double negative law for orthocomplement of an arbitrary set of vectors. (Contributed by NM, 16-Apr-2014.)
$/\$ $/\$

Theorem	dochoc 34947	Double negative law for orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)
$/\$ $/\$

Theorem	dochsscl 34948	If a set of vectors is included in a closed set, so is its closure. (Contributed by NM, 17-Jun-2015.)
$/\$

Theorem	dochoccl 34949	A set of vectors is closed iff it equals its double orthocomplent. (Contributed by NM, 1-Jan-2015.)
$/\$

Theorem	dochord 34950	Ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.)
$/\$

Theorem	dochord2N 34951	Ordering law for orthocomplement. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.)
$/\$

Theorem	dochord3 34952	Ordering law for orthocomplement. (Contributed by NM, 9-Mar-2015.)
$/\$

Theorem	doch11 34953	Orthocomplement is one-to-one. (Contributed by NM, 12-Aug-2014.)
$/\$

Theorem	dochsordN 34954	Strict ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dochn0nv 34955	An orthocomplement is nonzero iff the double orthocomplement is not the whole vector space. (Contributed by NM, 1-Jan-2015.)
$/\$

Theorem	dihoml4c 34956	Version of dihoml4 34957 with closed subspaces. (Contributed by NM, 15-Jan-2015.)
$/\$

Theorem	dihoml4 34957	Orthomodular law for constructed vector space H. Lemma 3.3(1) in [Holland95] p. 215. (poml4N 33530 analog.) (Contributed by NM, 15-Jan-2015.)
$/\$

Theorem	dochspss 34958	The span of a set of vectors is included in their double orthocomplement. (Contributed by NM, 26-Jul-2014.)
$/\$

Theorem	dochocsp 34959	The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.)
$/\$

Theorem	dochspocN 34960	The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dochocsn 34961	The double orthocomplement of a singleton is its span. (Contributed by NM, 13-Jan-2015.)
$/\$

Theorem	dochsncom 34962	Swap vectors in an orthocomplement of a singleton. (Contributed by NM, 17-Jun-2015.)
$/\$

Theorem	dochsat 34963	The double orthocomplement of an atom is an atom. (Contributed by NM, 29-Oct-2014.)
LSAtoms $/\$

Theorem	dochshpncl 34964	If a hyperplane is not closed, its closure equals the vector space. (Contributed by NM, 29-Oct-2014.)
LSHyp $/\$

Theorem	dochlkr 34965	Equivalent conditions for the closure of a kernel to be a hyperplane. (Contributed by NM, 29-Oct-2014.)
LFnl LSHyp LKer $/\$ $/\$

Theorem	dochkrshp 34966	The closure of a kernel is a hyperplane iff it doesn't contain all vectors. (Contributed by NM, 1-Nov-2014.)
LSHyp LFnl LKer $/\$

Theorem	dochkrshp2 34967	Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LSHyp LFnl LKer $/\$ $/\$

Theorem	dochkrshp3 34968	Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $/\$

Theorem	dochkrshp4 34969	Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $\/$

Theorem	dochdmj1 34970	De Morgan-like law for subspace orthocomplement. (Contributed by NM, 5-Aug-2014.)
$/\$ $/\$ $/\$

Theorem	dochnoncon 34971	Law of noncontradiction. The intersection of a subspace and its orthocomplement is the zero subspace. (Contributed by NM, 16-Apr-2014.)
$/\$ $/\$

Theorem	dochnel2 34972	A nonzero member of a subspace doesn't belong to the orthocomplement of the subspace. (Contributed by NM, 28-Feb-2015.)
$/\$ $\$

Theorem	dochnel 34973	A nonzero vector doesn't belong to the orthocomplement of its singleton. (Contributed by NM, 27-Oct-2014.)
$/\$ $\$

Syntax	cdjh 34974	Extend class notation with subspace join for vector space.
joinH

Definition	df-djh 34975*	Define (closed) subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theorem	djhffval 34976*	Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theorem	djhfval 34977*	Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhval 34978	Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhval2 34979	Value of subspace join for vector space. (Contributed by NM, 6-Aug-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhcl 34980	Closure of subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhlj 34981	Transfer lattice join to vector space closed subspace join. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhljjN 34982	Lattice join in terms of vector space closed subspace join. (Contributed by NM, 17-Aug-2014.) (New usage is discouraged.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhjlj 34983	vector space closed subspace join in terms of lattice join. (Contributed by NM, 9-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhj 34984	vector space closed subspace join in terms of lattice join. (Contributed by NM, 17-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhcom 34985	Subspace join commutes. (Contributed by NM, 8-Aug-2014.)
$.\/$ joinH $/\$ $.\/$ $.\/$

Theorem	djhspss 34986	Subspace span of union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhsumss 34987	Subspace sum is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	dihsumssj 34988	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 34989	Subspace union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.) (New usage is discouraged.)
$.\/$ joinH $/\$ $.\/$

Theorem	dochdmm1 34990	De Morgan-like law for closed subspace orthocomplement. (Contributed by NM, 13-Jan-2015.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhexmid 34991	Excluded middle property of vector space closed subspace join. (Contributed by NM, 22-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $.\/$

Theorem	djh01 34992	Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djh02 34993	Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhlsmcl 34994	A closed subspace sum equals subspace join. (shjshseli 27158 analog.) (Contributed by NM, 13-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhcvat42 34995*	A covering property. (cvrat42 33021 analog.) (Contributed by NM, 17-Aug-2014.)
$.\/$ joinH $/\$ $\$ $\$ $/\$ $.\/$ $\$ $/\$ $.\/$

Theorem	dihjatb 34996	Isomorphism H of lattice join of two atoms under the fiducial hyperplane. (Contributed by NM, 23-Sep-2014.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjatc 34997	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 34998	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjatcclem2 34999	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjatcclem3 35000*	Lemma for dihjatcc 35002. (Contributed by NM, 28-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$

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