P. 349 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 34801-34900 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	dihord 34801	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 34802	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 34803	Functionality of the isomorphism H. (Contributed by NM, 6-Mar-2014.)
$/\$

Theorem	dihf11 34804	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 34805	Functionality and domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)
$/\$

Theorem	dihdm 34806	Domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)
$/\$

Theorem	dihcl 34807	Closure of isomorphism H. (Contributed by NM, 8-Mar-2014.)
$/\$ $/\$

Theorem	dihcnvcl 34808	Closure of isomorphism H converse. (Contributed by NM, 8-Mar-2014.)
$/\$ $/\$

Theorem	dihcnvid1 34809	The converse isomorphism of an isomorphism. (Contributed by NM, 5-Aug-2014.)
$/\$ $/\$

Theorem	dihcnvid2 34810	The isomorphism of a converse isomorphism. (Contributed by NM, 5-Aug-2014.)
$/\$ $/\$

Theorem	dihcnvord 34811	Ordering property for converse of isomorphism H. (Contributed by NM, 17-Aug-2014.)
$/\$

Theorem	dihcnv11 34812	The converse of isomorphism H is one-to-one. (Contributed by NM, 17-Aug-2014.)
$/\$

Theorem	dihsslss 34813	The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)
$/\$

Theorem	dihrnlss 34814	The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)
$/\$ $/\$

Theorem	dihrnss 34815	The isomorphism H maps to a set of vectors. (Contributed by NM, 14-Mar-2014.)
$/\$ $/\$

Theorem	dihvalrel 34816	The value of isomorphism H is a relation. (Contributed by NM, 9-Mar-2014.)
$/\$

Theorem	dih0 34817	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 34818	A lattice element is zero iff its isomorphism is the zero subspace. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dih0vbN 34819	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 34820	The isomorphism H converse value of the zero subspace is the lattice zero. (Contributed by NM, 19-Jun-2014.)
$/\$

Theorem	dih0rn 34821	The zero subspace belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)
$/\$

Theorem	dih0sb 34822	A subspace is zero iff the converse of its isomorphism is lattice zero. (Contributed by NM, 17-Aug-2014.)
$/\$

Theorem	dih1 34823	The value of isomorphism H at the lattice unit is the set of all vectors. (Contributed by NM, 13-Mar-2014.)
$/\$

Theorem	dih1rn 34824	The full vector space belongs to the range of isomorphism H. (Contributed by NM, 19-Jun-2014.)
$/\$

Theorem	dih1cnv 34825	The isomorphism H converse value of the full vector space is the lattice one. (Contributed by NM, 19-Jun-2014.)
$/\$

Theorem	dihwN 34826*	Value of isomorphism H at the fiducial hyperplane . (Contributed by NM, 25-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dihmeetlem1N 34827*	Isomorphism H of a conjunction. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
$./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihglblem5apreN 34828*	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 34829	A conjunction property of isomorphism H. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $./\$

Theorem	dihglblem2aN 34830*	Lemma for isomorphism H of a GLB. (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)
$./\$ $./\$ $/\$ $/\$ $/\$

Theorem	dihglblem2N 34831*	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 34832*	Isomorphism H of a lattice glb. (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.)
$./\$ $./\$ $/\$ $/\$ $/\$ $/\$

Theorem	dihglblem3aN 34833*	Isomorphism H of a lattice glb. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$./\$ $./\$ $/\$ $/\$ $/\$ $/\$

Theorem	dihglblem4 34834*	Isomorphism H of a lattice glb. (Contributed by NM, 21-Mar-2014.)
$./\$ $./\$ $/\$ $/\$ $/\$

Theorem	dihglblem5 34835*	Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dihmeetlem2N 34836	Isomorphism H of a conjunction. (Contributed by NM, 22-Mar-2014.) (New usage is discouraged.)
$./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihglbcpreN 34837*	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 34838*	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 34839	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 34840	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 34841	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem3N 34842	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	dihmeetlem4preN 34843*	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihmeetlem4N 34844	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihmeetlem5 34845	Part of proof that isomorphism H is order-preserving . (Contributed by NM, 6-Apr-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$

Theorem	dihmeetlem6 34846	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$

Theorem	dihmeetlem7N 34847	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$

Theorem	dihjatc1 34848	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 34849	Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$

Theorem	dihjatc3 34850	Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$ $.\/$ $./\$

Theorem	dihmeetlem8N 34851	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 34852	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem10N 34853	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $.\/$

Theorem	dihmeetlem11N 34854	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$

Theorem	dihmeetlem12N 34855	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem13N 34856*	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	dihmeetlem14N 34857	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem15N 34858	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihmeetlem16N 34859	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem17N 34860	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$

Theorem	dihmeetlem18N 34861	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihmeetlem19N 34862	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem20N 34863	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetALTN 34864	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 34865*	Lemma for dih1dimat 34867. (Contributed by NM, 11-Apr-2014.)
LSAtoms Scalar $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	dih1dimatlem 34866*	Lemma for dih1dimat 34867. (Contributed by NM, 10-Apr-2014.)
LSAtoms Scalar $/\$ $/\$

Theorem	dih1dimat 34867	Any 1-dimensional subspace is a value of isomorphism H. (Contributed by NM, 11-Apr-2014.)
LSAtoms $/\$ $/\$

Theorem	dihlsprn 34868	The span of a vector belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)
$/\$ $/\$

Theorem	dihlspsnssN 34869	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 34870	The inverse isomorphism H of the span of a singleton is a Hilbert lattice atom. (Contributed by NM, 27-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dihatlat 34871	The isomorphism H of an atom is a 1-dim subspace. (Contributed by NM, 28-Apr-2014.)
LSAtoms $/\$ $/\$

Theorem	dihat 34872	There exists at least one atom in the subspaces of vector space H. (Contributed by NM, 12-Aug-2014.)
LSAtoms $/\$

Theorem	dihpN 34873*	The value of isomorphism H at the fiducial atom is determined by the vector (the zero translation ltrnid 33669 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 34874	The reverse isomorphism H of a 1-dim subspace is an atom. (Contributed by NM, 28-Apr-2014.)
LSAtoms $/\$ $/\$

Theorem	dihatexv 34875*	There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 16-Aug-2014.)
$/\$ $\$

Theorem	dihatexv2 34876*	There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 17-Aug-2014.)
$/\$ $\$

Theorem	dihglblem6 34877*	Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)
$./\$ LSAtoms $/\$ $/\$ $/\$

Theorem	dihglb 34878*	Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dihglb2 34879*	Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)
$/\$ $/\$

Theorem	dihmeet 34880	Isomorphism H of a lattice meet. (Contributed by NM, 13-Apr-2014.)
$./\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihintcl 34881	The intersection of closed subspaces (the range of isomorphism H) is a closed subspace. (Contributed by NM, 14-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dihmeetcl 34882	Closure of closed subspace meet for vector space. (Contributed by NM, 5-Aug-2014.)
$/\$ $/\$ $/\$

Theorem	dihmeet2 34883	Reverse isomorphism H of a closed subspace intersection. (Contributed by NM, 15-Jan-2015.)
$./\$ $/\$ $./\$

Syntax	coch 34884	Extend class notation with subspace orthocomplement for vector space.


Definition	df-doch 34885*	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 34886*	Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)


Theorem	dochfval 34887*	Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)
$/\$

Theorem	dochval 34888*	Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)
$/\$ $/\$

Theorem	dochval2 34889*	Subspace orthocomplement for vector space. (Contributed by NM, 14-Apr-2014.)
$/\$ $/\$

Theorem	dochcl 34890	Closure of subspace orthocomplement for vector space. (Contributed by NM, 9-Mar-2014.)
$/\$ $/\$

Theorem	dochlss 34891	A subspace orthocomplement is a subspace of the vector space. (Contributed by NM, 22-Jul-2014.)
$/\$ $/\$

Theorem	dochssv 34892	A subspace orthocomplement belongs to the vector space. (Contributed by NM, 22-Jul-2014.)
$/\$ $/\$

Theorem	dochfN 34893	Domain and codomain of the subspace orthocomplement for the vector space. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)
$/\$

Theorem	dochvalr 34894	Orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)
$/\$ $/\$

Theorem	doch0 34895	Orthocomplement of the zero subspace. (Contributed by NM, 19-Jun-2014.)
$/\$

Theorem	doch1 34896	Orthocomplement of the unit subspace (all vectors). (Contributed by NM, 19-Jun-2014.)
$/\$

Theorem	dochoc0 34897	The zero subspace is closed. (Contributed by NM, 16-Feb-2015.)
$/\$

Theorem	dochoc1 34898	The unit subspace (all vectors) is closed. (Contributed by NM, 16-Feb-2015.)
$/\$

Theorem	dochvalr2 34899	Orthocomplement of a closed subspace. (Contributed by NM, 21-Jul-2014.)
$/\$ $/\$

Theorem	dochvalr3 34900	Orthocomplement of a closed subspace. (Contributed by NM, 15-Jan-2015.)
$/\$

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