P. 366 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 36501-36600 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	dihmeetbN 36501	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 36502	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem3N 36503	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	dihmeetlem4preN 36504*	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihmeetlem4N 36505	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihmeetlem5 36506	Part of proof that isomorphism H is order-preserving . (Contributed by NM, 6-Apr-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$

Theorem	dihmeetlem6 36507	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$

Theorem	dihmeetlem7N 36508	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$

Theorem	dihjatc1 36509	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 36510	Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$

Theorem	dihjatc3 36511	Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$ $.\/$ $./\$

Theorem	dihmeetlem8N 36512	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 36513	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem10N 36514	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $.\/$

Theorem	dihmeetlem11N 36515	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$

Theorem	dihmeetlem12N 36516	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem13N 36517*	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	dihmeetlem14N 36518	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem15N 36519	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihmeetlem16N 36520	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem17N 36521	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$

Theorem	dihmeetlem18N 36522	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihmeetlem19N 36523	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetlem20N 36524	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	dihmeetALTN 36525	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 36526*	Lemma for dih1dimat 36528. (Contributed by NM, 11-Apr-2014.)
LSAtoms Scalar $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	dih1dimatlem 36527*	Lemma for dih1dimat 36528. (Contributed by NM, 10-Apr-2014.)
LSAtoms Scalar $/\$ $/\$

Theorem	dih1dimat 36528	Any 1-dimensional subspace is a value of isomorphism H. (Contributed by NM, 11-Apr-2014.)
LSAtoms $/\$ $/\$

Theorem	dihlsprn 36529	The span of a vector belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)
$/\$ $/\$

Theorem	dihlspsnssN 36530	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 36531	The inverse isomorphism H of the span of a singleton is a Hilbert lattice atom. (Contributed by NM, 27-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dihatlat 36532	The isomorphism H of an atom is a 1-dim subspace. (Contributed by NM, 28-Apr-2014.)
LSAtoms $/\$ $/\$

Theorem	dihat 36533	There exists at least one atom in the subspaces of vector space H. (Contributed by NM, 12-Aug-2014.)
LSAtoms $/\$

Theorem	dihpN 36534*	The value of isomorphism H at the fiducial atom is determined by the vector (the zero translation ltrnid 35332 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 36535	The reverse isomorphism H of a 1-dim subspace is an atom. (Contributed by NM, 28-Apr-2014.)
LSAtoms $/\$ $/\$

Theorem	dihatexv 36536*	There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 16-Aug-2014.)
$/\$ $\$

Theorem	dihatexv2 36537*	There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 17-Aug-2014.)
$/\$ $\$

Theorem	dihglblem6 36538*	Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)
$./\$ LSAtoms $/\$ $/\$ $/\$

Theorem	dihglb 36539*	Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dihglb2 36540*	Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)
$/\$ $/\$

Theorem	dihmeet 36541	Isomorphism H of a lattice meet. (Contributed by NM, 13-Apr-2014.)
$./\$ $/\$ $/\$ $/\$ $./\$

Theorem	dihintcl 36542	The intersection of closed subspaces (the range of isomorphism H) is a closed subspace. (Contributed by NM, 14-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dihmeetcl 36543	Closure of closed subspace meet for vector space. (Contributed by NM, 5-Aug-2014.)
$/\$ $/\$ $/\$

Theorem	dihmeet2 36544	Reverse isomorphism H of a closed subspace intersection. (Contributed by NM, 15-Jan-2015.)
$./\$ $/\$ $./\$

Syntax	coch 36545	Extend class notation with subspace orthocomplement for vector space.


Definition	df-doch 36546*	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 36547*	Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)


Theorem	dochfval 36548*	Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)
$/\$

Theorem	dochval 36549*	Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)
$/\$ $/\$

Theorem	dochval2 36550*	Subspace orthocomplement for vector space. (Contributed by NM, 14-Apr-2014.)
$/\$ $/\$

Theorem	dochcl 36551	Closure of subspace orthocomplement for vector space. (Contributed by NM, 9-Mar-2014.)
$/\$ $/\$

Theorem	dochlss 36552	A subspace orthocomplement is a subspace of the vector space. (Contributed by NM, 22-Jul-2014.)
$/\$ $/\$

Theorem	dochssv 36553	A subspace orthocomplement belongs to the vector space. (Contributed by NM, 22-Jul-2014.)
$/\$ $/\$

Theorem	dochfN 36554	Domain and codomain of the subspace orthocomplement for the vector space. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)
$/\$

Theorem	dochvalr 36555	Orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)
$/\$ $/\$

Theorem	doch0 36556	Orthocomplement of the zero subspace. (Contributed by NM, 19-Jun-2014.)
$/\$

Theorem	doch1 36557	Orthocomplement of the unit subspace (all vectors). (Contributed by NM, 19-Jun-2014.)
$/\$

Theorem	dochoc0 36558	The zero subspace is closed. (Contributed by NM, 16-Feb-2015.)
$/\$

Theorem	dochoc1 36559	The unit subspace (all vectors) is closed. (Contributed by NM, 16-Feb-2015.)
$/\$

Theorem	dochvalr2 36560	Orthocomplement of a closed subspace. (Contributed by NM, 21-Jul-2014.)
$/\$ $/\$

Theorem	dochvalr3 36561	Orthocomplement of a closed subspace. (Contributed by NM, 15-Jan-2015.)
$/\$

Theorem	doch2val2 36562*	Double orthocomplement for vector space. (Contributed by NM, 26-Jul-2014.)
$/\$

Theorem	dochss 36563	Subset law for orthocomplement. (Contributed by NM, 16-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dochocss 36564	Double negative law for orthocomplement of an arbitrary set of vectors. (Contributed by NM, 16-Apr-2014.)
$/\$ $/\$

Theorem	dochoc 36565	Double negative law for orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)
$/\$ $/\$

Theorem	dochsscl 36566	If a set of vectors is included in a closed set, so is its closure. (Contributed by NM, 17-Jun-2015.)
$/\$

Theorem	dochoccl 36567	A set of vectors is closed iff it equals its double orthocomplent. (Contributed by NM, 1-Jan-2015.)
$/\$

Theorem	dochord 36568	Ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.)
$/\$

Theorem	dochord2N 36569	Ordering law for orthocomplement. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.)
$/\$

Theorem	dochord3 36570	Ordering law for orthocomplement. (Contributed by NM, 9-Mar-2015.)
$/\$

Theorem	doch11 36571	Orthocomplement is one-to-one. (Contributed by NM, 12-Aug-2014.)
$/\$

Theorem	dochsordN 36572	Strict ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dochn0nv 36573	An orthocomplement is nonzero iff the double orthocomplement is not the whole vector space. (Contributed by NM, 1-Jan-2015.)
$/\$

Theorem	dihoml4c 36574	Version of dihoml4 36575 with closed subspaces. (Contributed by NM, 15-Jan-2015.)
$/\$

Theorem	dihoml4 36575	Orthomodular law for constructed vector space H. Lemma 3.3(1) in [Holland95] p. 215. (poml4N 35150 analog.) (Contributed by NM, 15-Jan-2015.)
$/\$

Theorem	dochspss 36576	The span of a set of vectors is included in their double orthocomplement. (Contributed by NM, 26-Jul-2014.)
$/\$

Theorem	dochocsp 36577	The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.)
$/\$

Theorem	dochspocN 36578	The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dochocsn 36579	The double orthocomplement of a singleton is its span. (Contributed by NM, 13-Jan-2015.)
$/\$

Theorem	dochsncom 36580	Swap vectors in an orthocomplement of a singleton. (Contributed by NM, 17-Jun-2015.)
$/\$

Theorem	dochsat 36581	The double orthocomplement of an atom is an atom. (Contributed by NM, 29-Oct-2014.)
LSAtoms $/\$

Theorem	dochshpncl 36582	If a hyperplane is not closed, its closure equals the vector space. (Contributed by NM, 29-Oct-2014.)
LSHyp $/\$

Theorem	dochlkr 36583	Equivalent conditions for the closure of a kernel to be a hyperplane. (Contributed by NM, 29-Oct-2014.)
LFnl LSHyp LKer $/\$ $/\$

Theorem	dochkrshp 36584	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 36585	Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LSHyp LFnl LKer $/\$ $/\$

Theorem	dochkrshp3 36586	Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $/\$

Theorem	dochkrshp4 36587	Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $\/$

Theorem	dochdmj1 36588	De Morgan-like law for subspace orthocomplement. (Contributed by NM, 5-Aug-2014.)
$/\$ $/\$ $/\$

Theorem	dochnoncon 36589	Law of noncontradiction. The intersection of a subspace and its orthocomplement is the zero subspace. (Contributed by NM, 16-Apr-2014.)
$/\$ $/\$

Theorem	dochnel2 36590	A nonzero member of a subspace doesn't belong to the orthocomplement of the subspace. (Contributed by NM, 28-Feb-2015.)
$/\$ $\$

Theorem	dochnel 36591	A nonzero vector doesn't belong to the orthocomplement of its singleton. (Contributed by NM, 27-Oct-2014.)
$/\$ $\$

Syntax	cdjh 36592	Extend class notation with subspace join for vector space.
joinH

Definition	df-djh 36593*	Define (closed) subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theorem	djhffval 36594*	Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theorem	djhfval 36595*	Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhval 36596	Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhval2 36597	Value of subspace join for vector space. (Contributed by NM, 6-Aug-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhcl 36598	Closure of subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhlj 36599	Transfer lattice join to vector space closed subspace join. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhljjN 36600	Lattice join in terms of vector space closed subspace join. (Contributed by NM, 17-Aug-2014.) (New usage is discouraged.)
$.\/$ joinH $/\$ $.\/$

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