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	dihord5apre 34901	Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	dihord5a 34902	Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	dihord 34903	The isomorphism H is order-preserving. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)
$/\$ $/\$ $/\$

Theorem	dih11 34904	The isomorphism H is one-to-one. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)
$/\$ $/\$ $/\$

Theorem	dihf11lem 34905	Functionality of the isomorphism H. (Contributed by NM, 6-Mar-2014.)
$/\$

Theorem	dihf11 34906	The isomorphism H for a lattice is a one-to-one function. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)
$/\$

Theorem	dihfn 34907	Functionality and domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)
$/\$

Theorem	dihdm 34908	Domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)
$/\$

Theorem	dihcl 34909	Closure of isomorphism H. (Contributed by NM, 8-Mar-2014.)
$/\$ $/\$

Theorem	dihcnvcl 34910	Closure of isomorphism H converse. (Contributed by NM, 8-Mar-2014.)
$/\$ $/\$

Theorem	dihcnvid1 34911	The converse isomorphism of an isomorphism. (Contributed by NM, 5-Aug-2014.)
$/\$ $/\$

Theorem	dihcnvid2 34912	The isomorphism of a converse isomorphism. (Contributed by NM, 5-Aug-2014.)
$/\$ $/\$

Theorem	dihcnvord 34913	Ordering property for converse of isomorphism H. (Contributed by NM, 17-Aug-2014.)
$/\$

Theorem	dihcnv11 34914	The converse of isomorphism H is one-to-one. (Contributed by NM, 17-Aug-2014.)
$/\$

Theorem	dihsslss 34915	The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)
$/\$

Theorem	dihrnlss 34916	The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)
$/\$ $/\$

Theorem	dihrnss 34917	The isomorphism H maps to a set of vectors. (Contributed by NM, 14-Mar-2014.)
$/\$ $/\$

Theorem	dihvalrel 34918	The value of isomorphism H is a relation. (Contributed by NM, 9-Mar-2014.)
$/\$

Theorem	dih0 34919	The value of isomorphism H at the lattice zero is the singleton of the zero vector i.e. the zero subspace. (Contributed by NM, 9-Mar-2014.)
$/\$

Theorem	dih0bN 34920	A lattice element is zero iff its isomorphism is the zero subspace. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dih0vbN 34921	A vector is zero iff its span is the isomorphism of lattice zero. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dih0cnv 34922	The isomorphism H converse value of the zero subspace is the lattice zero. (Contributed by NM, 19-Jun-2014.)
$/\$

Theorem	dih0rn 34923	The zero subspace belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)
$/\$

Theorem	dih0sb 34924	A subspace is zero iff the converse of its isomorphism is lattice zero. (Contributed by NM, 17-Aug-2014.)
$/\$

Theorem	dih1 34925	The value of isomorphism H at the lattice unit is the set of all vectors. (Contributed by NM, 13-Mar-2014.)
$/\$

Theorem	dih1rn 34926	The full vector space belongs to the range of isomorphism H. (Contributed by NM, 19-Jun-2014.)
$/\$

Theorem	dih1cnv 34927	The isomorphism H converse value of the full vector space is the lattice one. (Contributed by NM, 19-Jun-2014.)
$/\$

Theorem	dihwN 34928*	Value of isomorphism H at the fiducial hyperplane . (Contributed by NM, 25-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dihmeetlem1N 34929*	Isomorphism H of a conjunction. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
$./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihglblem5apreN 34930*	A conjunction property of isomorphism H. TODO: reduce antecedent size; general review for shorter proof. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
$./\$ $.\/$ $/\$ $/\$ $/\$ $./\$

Theorem	dihglblem5aN 34931	A conjunction property of isomorphism H. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $./\$

Theorem	dihglblem2aN 34932*	Lemma for isomorphism H of a GLB. (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)
$./\$ $./\$ $/\$ $/\$ $/\$

Theorem	dihglblem2N 34933*	The GLB of a set of lattice elements is the same as that of the set with elements of cut down to be under . (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)
$./\$ $./\$ $/\$ $/\$ $/\$

Theorem	dihglblem3N 34934*	Isomorphism H of a lattice glb. (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.)
$./\$ $./\$ $/\$ $/\$ $/\$ $/\$

Theorem	dihglblem3aN 34935*	Isomorphism H of a lattice glb. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$./\$ $./\$ $/\$ $/\$ $/\$ $/\$

Theorem	dihglblem4 34936*	Isomorphism H of a lattice glb. (Contributed by NM, 21-Mar-2014.)
$./\$ $./\$ $/\$ $/\$ $/\$

Theorem	dihglblem5 34937*	Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dihmeetlem2N 34938	Isomorphism H of a conjunction. (Contributed by NM, 22-Mar-2014.) (New usage is discouraged.)
$./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihglbcpreN 34939*	Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane . (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$

Theorem	dihglbcN 34940*	Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane . (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$

Theorem	dihmeetcN 34941	Isomorphism H of a lattice meet when the meet is not under the fiducial hyperplane . (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetbN 34942	Isomorphism H of a lattice meet when one element is under the fiducial hyperplane . (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihmeetbclemN 34943	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem3N 34944	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	dihmeetlem4preN 34945*	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihmeetlem4N 34946	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihmeetlem5 34947	Part of proof that isomorphism H is order-preserving . (Contributed by NM, 6-Apr-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$

Theorem	dihmeetlem6 34948	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$

Theorem	dihmeetlem7N 34949	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$

Theorem	dihjatc1 34950	Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change $.\/$ order of $./\$ $.\/$ here and down? (Contributed by NM, 6-Apr-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$ $.\/$ $./\$

Theorem	dihjatc2N 34951	Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$

Theorem	dihjatc3 34952	Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$ $.\/$ $./\$

Theorem	dihmeetlem8N 34953	Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change $.\/$ order of $./\$ $.\/$ here and down? (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$ $.\/$ $./\$

Theorem	dihmeetlem9N 34954	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem10N 34955	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $.\/$

Theorem	dihmeetlem11N 34956	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$

Theorem	dihmeetlem12N 34957	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem13N 34958*	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	dihmeetlem14N 34959	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem15N 34960	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihmeetlem16N 34961	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem17N 34962	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$

Theorem	dihmeetlem18N 34963	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihmeetlem19N 34964	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem20N 34965	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetALTN 34966	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 34967*	Lemma for dih1dimat 34969. (Contributed by NM, 11-Apr-2014.)
LSAtoms Scalar $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	dih1dimatlem 34968*	Lemma for dih1dimat 34969. (Contributed by NM, 10-Apr-2014.)
LSAtoms Scalar $/\$ $/\$

Theorem	dih1dimat 34969	Any 1-dimensional subspace is a value of isomorphism H. (Contributed by NM, 11-Apr-2014.)
LSAtoms $/\$ $/\$

Theorem	dihlsprn 34970	The span of a vector belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)
$/\$ $/\$

Theorem	dihlspsnssN 34971	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 34972	The inverse isomorphism H of the span of a singleton is a Hilbert lattice atom. (Contributed by NM, 27-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dihatlat 34973	The isomorphism H of an atom is a 1-dim subspace. (Contributed by NM, 28-Apr-2014.)
LSAtoms $/\$ $/\$

Theorem	dihat 34974	There exists at least one atom in the subspaces of vector space H. (Contributed by NM, 12-Aug-2014.)
LSAtoms $/\$

Theorem	dihpN 34975*	The value of isomorphism H at the fiducial atom is determined by the vector (the zero translation ltrnid 33771 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 34976	The reverse isomorphism H of a 1-dim subspace is an atom. (Contributed by NM, 28-Apr-2014.)
LSAtoms $/\$ $/\$

Theorem	dihatexv 34977*	There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 16-Aug-2014.)
$/\$ $\$

Theorem	dihatexv2 34978*	There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 17-Aug-2014.)
$/\$ $\$

Theorem	dihglblem6 34979*	Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)
$./\$ LSAtoms $/\$ $/\$ $/\$

Theorem	dihglb 34980*	Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dihglb2 34981*	Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)
$/\$ $/\$

Theorem	dihmeet 34982	Isomorphism H of a lattice meet. (Contributed by NM, 13-Apr-2014.)
$./\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihintcl 34983	The intersection of closed subspaces (the range of isomorphism H) is a closed subspace. (Contributed by NM, 14-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dihmeetcl 34984	Closure of closed subspace meet for vector space. (Contributed by NM, 5-Aug-2014.)
$/\$ $/\$ $/\$

Theorem	dihmeet2 34985	Reverse isomorphism H of a closed subspace intersection. (Contributed by NM, 15-Jan-2015.)
$./\$ $/\$ $./\$

Syntax	coch 34986	Extend class notation with subspace orthocomplement for vector space.


Definition	df-doch 34987*	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 34988*	Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)


Theorem	dochfval 34989*	Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)
$/\$

Theorem	dochval 34990*	Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)
$/\$ $/\$

Theorem	dochval2 34991*	Subspace orthocomplement for vector space. (Contributed by NM, 14-Apr-2014.)
$/\$ $/\$

Theorem	dochcl 34992	Closure of subspace orthocomplement for vector space. (Contributed by NM, 9-Mar-2014.)
$/\$ $/\$

Theorem	dochlss 34993	A subspace orthocomplement is a subspace of the vector space. (Contributed by NM, 22-Jul-2014.)
$/\$ $/\$

Theorem	dochssv 34994	A subspace orthocomplement belongs to the vector space. (Contributed by NM, 22-Jul-2014.)
$/\$ $/\$

Theorem	dochfN 34995	Domain and codomain of the subspace orthocomplement for the vector space. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)
$/\$

Theorem	dochvalr 34996	Orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)
$/\$ $/\$

Theorem	doch0 34997	Orthocomplement of the zero subspace. (Contributed by NM, 19-Jun-2014.)
$/\$

Theorem	doch1 34998	Orthocomplement of the unit subspace (all vectors). (Contributed by NM, 19-Jun-2014.)
$/\$

Theorem	dochoc0 34999	The zero subspace is closed. (Contributed by NM, 16-Feb-2015.)
$/\$

Theorem	dochoc1 35000	The unit subspace (all vectors) is closed. (Contributed by NM, 16-Feb-2015.)
$/\$

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