P. 375 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 37401-37500 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	dihord4 37401	The isomorphism H for a lattice is order-preserving in the region not under co-atom . TODO: reformat q e. A /\ -. q .<_ W to eliminate adant*. (Contributed by NM, 6-Mar-2014.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	dihord5b 37402	Part of proof that isomorphism H is order-preserving. TODO: eliminate 3ad2ant1; combine w/ other way to have one lhpmcvr2 (Contributed by NM, 7-Mar-2014.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	dihord6b 37403	Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	dihord6a 37404	Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	dihord5apre 37405	Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	dihord5a 37406	Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	dihord 37407	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 37408	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 37409	Functionality of the isomorphism H. (Contributed by NM, 6-Mar-2014.)
$/\$

Theorem	dihf11 37410	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 37411	Functionality and domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)
$/\$

Theorem	dihdm 37412	Domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)
$/\$

Theorem	dihcl 37413	Closure of isomorphism H. (Contributed by NM, 8-Mar-2014.)
$/\$ $/\$

Theorem	dihcnvcl 37414	Closure of isomorphism H converse. (Contributed by NM, 8-Mar-2014.)
$/\$ $/\$

Theorem	dihcnvid1 37415	The converse isomorphism of an isomorphism. (Contributed by NM, 5-Aug-2014.)
$/\$ $/\$

Theorem	dihcnvid2 37416	The isomorphism of a converse isomorphism. (Contributed by NM, 5-Aug-2014.)
$/\$ $/\$

Theorem	dihcnvord 37417	Ordering property for converse of isomorphism H. (Contributed by NM, 17-Aug-2014.)
$/\$

Theorem	dihcnv11 37418	The converse of isomorphism H is one-to-one. (Contributed by NM, 17-Aug-2014.)
$/\$

Theorem	dihsslss 37419	The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)
$/\$

Theorem	dihrnlss 37420	The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)
$/\$ $/\$

Theorem	dihrnss 37421	The isomorphism H maps to a set of vectors. (Contributed by NM, 14-Mar-2014.)
$/\$ $/\$

Theorem	dihvalrel 37422	The value of isomorphism H is a relation. (Contributed by NM, 9-Mar-2014.)
$/\$

Theorem	dih0 37423	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 37424	A lattice element is zero iff its isomorphism is the zero subspace. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dih0vbN 37425	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 37426	The isomorphism H converse value of the zero subspace is the lattice zero. (Contributed by NM, 19-Jun-2014.)
$/\$

Theorem	dih0rn 37427	The zero subspace belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)
$/\$

Theorem	dih0sb 37428	A subspace is zero iff the converse of its isomorphism is lattice zero. (Contributed by NM, 17-Aug-2014.)
$/\$

Theorem	dih1 37429	The value of isomorphism H at the lattice unit is the set of all vectors. (Contributed by NM, 13-Mar-2014.)
$/\$

Theorem	dih1rn 37430	The full vector space belongs to the range of isomorphism H. (Contributed by NM, 19-Jun-2014.)
$/\$

Theorem	dih1cnv 37431	The isomorphism H converse value of the full vector space is the lattice one. (Contributed by NM, 19-Jun-2014.)
$/\$

Theorem	dihwN 37432*	Value of isomorphism H at the fiducial hyperplane . (Contributed by NM, 25-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dihmeetlem1N 37433*	Isomorphism H of a conjunction. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
$./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihglblem5apreN 37434*	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 37435	A conjunction property of isomorphism H. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $./\$

Theorem	dihglblem2aN 37436*	Lemma for isomorphism H of a GLB. (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)
$./\$ $./\$ $/\$ $/\$ $/\$

Theorem	dihglblem2N 37437*	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 37438*	Isomorphism H of a lattice glb. (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.)
$./\$ $./\$ $/\$ $/\$ $/\$ $/\$

Theorem	dihglblem3aN 37439*	Isomorphism H of a lattice glb. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$./\$ $./\$ $/\$ $/\$ $/\$ $/\$

Theorem	dihglblem4 37440*	Isomorphism H of a lattice glb. (Contributed by NM, 21-Mar-2014.)
$./\$ $./\$ $/\$ $/\$ $/\$

Theorem	dihglblem5 37441*	Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dihmeetlem2N 37442	Isomorphism H of a conjunction. (Contributed by NM, 22-Mar-2014.) (New usage is discouraged.)
$./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihglbcpreN 37443*	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 37444*	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 37445	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 37446	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 37447	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem3N 37448	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	dihmeetlem4preN 37449*	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihmeetlem4N 37450	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihmeetlem5 37451	Part of proof that isomorphism H is order-preserving . (Contributed by NM, 6-Apr-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$

Theorem	dihmeetlem6 37452	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$

Theorem	dihmeetlem7N 37453	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$

Theorem	dihjatc1 37454	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 37455	Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$

Theorem	dihjatc3 37456	Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$ $.\/$ $./\$

Theorem	dihmeetlem8N 37457	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 37458	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem10N 37459	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $.\/$

Theorem	dihmeetlem11N 37460	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$

Theorem	dihmeetlem12N 37461	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem13N 37462*	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	dihmeetlem14N 37463	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem15N 37464	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihmeetlem16N 37465	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem17N 37466	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$

Theorem	dihmeetlem18N 37467	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihmeetlem19N 37468	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem20N 37469	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetALTN 37470	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 37471*	Lemma for dih1dimat 37473. (Contributed by NM, 11-Apr-2014.)
LSAtoms Scalar $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	dih1dimatlem 37472*	Lemma for dih1dimat 37473. (Contributed by NM, 10-Apr-2014.)
LSAtoms Scalar $/\$ $/\$

Theorem	dih1dimat 37473	Any 1-dimensional subspace is a value of isomorphism H. (Contributed by NM, 11-Apr-2014.)
LSAtoms $/\$ $/\$

Theorem	dihlsprn 37474	The span of a vector belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)
$/\$ $/\$

Theorem	dihlspsnssN 37475	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 37476	The inverse isomorphism H of the span of a singleton is a Hilbert lattice atom. (Contributed by NM, 27-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dihatlat 37477	The isomorphism H of an atom is a 1-dim subspace. (Contributed by NM, 28-Apr-2014.)
LSAtoms $/\$ $/\$

Theorem	dihat 37478	There exists at least one atom in the subspaces of vector space H. (Contributed by NM, 12-Aug-2014.)
LSAtoms $/\$

Theorem	dihpN 37479*	The value of isomorphism H at the fiducial atom is determined by the vector (the zero translation ltrnid 36275 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 37480	The reverse isomorphism H of a 1-dim subspace is an atom. (Contributed by NM, 28-Apr-2014.)
LSAtoms $/\$ $/\$

Theorem	dihatexv 37481*	There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 16-Aug-2014.)
$/\$ $\$

Theorem	dihatexv2 37482*	There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 17-Aug-2014.)
$/\$ $\$

Theorem	dihglblem6 37483*	Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)
$./\$ LSAtoms $/\$ $/\$ $/\$

Theorem	dihglb 37484*	Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dihglb2 37485*	Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)
$/\$ $/\$

Theorem	dihmeet 37486	Isomorphism H of a lattice meet. (Contributed by NM, 13-Apr-2014.)
$./\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihintcl 37487	The intersection of closed subspaces (the range of isomorphism H) is a closed subspace. (Contributed by NM, 14-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dihmeetcl 37488	Closure of closed subspace meet for vector space. (Contributed by NM, 5-Aug-2014.)
$/\$ $/\$ $/\$

Theorem	dihmeet2 37489	Reverse isomorphism H of a closed subspace intersection. (Contributed by NM, 15-Jan-2015.)
$./\$ $/\$ $./\$

Syntax	coch 37490	Extend class notation with subspace orthocomplement for vector space.


Definition	df-doch 37491*	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 37492*	Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)


Theorem	dochfval 37493*	Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)
$/\$

Theorem	dochval 37494*	Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)
$/\$ $/\$

Theorem	dochval2 37495*	Subspace orthocomplement for vector space. (Contributed by NM, 14-Apr-2014.)
$/\$ $/\$

Theorem	dochcl 37496	Closure of subspace orthocomplement for vector space. (Contributed by NM, 9-Mar-2014.)
$/\$ $/\$

Theorem	dochlss 37497	A subspace orthocomplement is a subspace of the vector space. (Contributed by NM, 22-Jul-2014.)
$/\$ $/\$

Theorem	dochssv 37498	A subspace orthocomplement belongs to the vector space. (Contributed by NM, 22-Jul-2014.)
$/\$ $/\$

Theorem	dochfN 37499	Domain and codomain of the subspace orthocomplement for the vector space. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)
$/\$

Theorem	dochvalr 37500	Orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)
$/\$ $/\$

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