P. 376 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 37501-37600 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	doch0 37501	Orthocomplement of the zero subspace. (Contributed by NM, 19-Jun-2014.)
$/\$

Theorem	doch1 37502	Orthocomplement of the unit subspace (all vectors). (Contributed by NM, 19-Jun-2014.)
$/\$

Theorem	dochoc0 37503	The zero subspace is closed. (Contributed by NM, 16-Feb-2015.)
$/\$

Theorem	dochoc1 37504	The unit subspace (all vectors) is closed. (Contributed by NM, 16-Feb-2015.)
$/\$

Theorem	dochvalr2 37505	Orthocomplement of a closed subspace. (Contributed by NM, 21-Jul-2014.)
$/\$ $/\$

Theorem	dochvalr3 37506	Orthocomplement of a closed subspace. (Contributed by NM, 15-Jan-2015.)
$/\$

Theorem	doch2val2 37507*	Double orthocomplement for vector space. (Contributed by NM, 26-Jul-2014.)
$/\$

Theorem	dochss 37508	Subset law for orthocomplement. (Contributed by NM, 16-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dochocss 37509	Double negative law for orthocomplement of an arbitrary set of vectors. (Contributed by NM, 16-Apr-2014.)
$/\$ $/\$

Theorem	dochoc 37510	Double negative law for orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)
$/\$ $/\$

Theorem	dochsscl 37511	If a set of vectors is included in a closed set, so is its closure. (Contributed by NM, 17-Jun-2015.)
$/\$

Theorem	dochoccl 37512	A set of vectors is closed iff it equals its double orthocomplent. (Contributed by NM, 1-Jan-2015.)
$/\$

Theorem	dochord 37513	Ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.)
$/\$

Theorem	dochord2N 37514	Ordering law for orthocomplement. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.)
$/\$

Theorem	dochord3 37515	Ordering law for orthocomplement. (Contributed by NM, 9-Mar-2015.)
$/\$

Theorem	doch11 37516	Orthocomplement is one-to-one. (Contributed by NM, 12-Aug-2014.)
$/\$

Theorem	dochsordN 37517	Strict ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dochn0nv 37518	An orthocomplement is nonzero iff the double orthocomplement is not the whole vector space. (Contributed by NM, 1-Jan-2015.)
$/\$

Theorem	dihoml4c 37519	Version of dihoml4 37520 with closed subspaces. (Contributed by NM, 15-Jan-2015.)
$/\$

Theorem	dihoml4 37520	Orthomodular law for constructed vector space H. Lemma 3.3(1) in [Holland95] p. 215. (poml4N 36093 analog.) (Contributed by NM, 15-Jan-2015.)
$/\$

Theorem	dochspss 37521	The span of a set of vectors is included in their double orthocomplement. (Contributed by NM, 26-Jul-2014.)
$/\$

Theorem	dochocsp 37522	The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.)
$/\$

Theorem	dochspocN 37523	The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dochocsn 37524	The double orthocomplement of a singleton is its span. (Contributed by NM, 13-Jan-2015.)
$/\$

Theorem	dochsncom 37525	Swap vectors in an orthocomplement of a singleton. (Contributed by NM, 17-Jun-2015.)
$/\$

Theorem	dochsat 37526	The double orthocomplement of an atom is an atom. (Contributed by NM, 29-Oct-2014.)
LSAtoms $/\$

Theorem	dochshpncl 37527	If a hyperplane is not closed, its closure equals the vector space. (Contributed by NM, 29-Oct-2014.)
LSHyp $/\$

Theorem	dochlkr 37528	Equivalent conditions for the closure of a kernel to be a hyperplane. (Contributed by NM, 29-Oct-2014.)
LFnl LSHyp LKer $/\$ $/\$

Theorem	dochkrshp 37529	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 37530	Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LSHyp LFnl LKer $/\$ $/\$

Theorem	dochkrshp3 37531	Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $/\$

Theorem	dochkrshp4 37532	Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $\/$

Theorem	dochdmj1 37533	De Morgan-like law for subspace orthocomplement. (Contributed by NM, 5-Aug-2014.)
$/\$ $/\$ $/\$

Theorem	dochnoncon 37534	Law of noncontradiction. The intersection of a subspace and its orthocomplement is the zero subspace. (Contributed by NM, 16-Apr-2014.)
$/\$ $/\$

Theorem	dochnel2 37535	A nonzero member of a subspace doesn't belong to the orthocomplement of the subspace. (Contributed by NM, 28-Feb-2015.)
$/\$ $\$

Theorem	dochnel 37536	A nonzero vector doesn't belong to the orthocomplement of its singleton. (Contributed by NM, 27-Oct-2014.)
$/\$ $\$

Syntax	cdjh 37537	Extend class notation with subspace join for vector space.
joinH

Definition	df-djh 37538*	Define (closed) subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theorem	djhffval 37539*	Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theorem	djhfval 37540*	Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhval 37541	Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhval2 37542	Value of subspace join for vector space. (Contributed by NM, 6-Aug-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhcl 37543	Closure of subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhlj 37544	Transfer lattice join to vector space closed subspace join. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhljjN 37545	Lattice join in terms of vector space closed subspace join. (Contributed by NM, 17-Aug-2014.) (New usage is discouraged.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhjlj 37546	vector space closed subspace join in terms of lattice join. (Contributed by NM, 9-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhj 37547	vector space closed subspace join in terms of lattice join. (Contributed by NM, 17-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhcom 37548	Subspace join commutes. (Contributed by NM, 8-Aug-2014.)
$.\/$ joinH $/\$ $.\/$ $.\/$

Theorem	djhspss 37549	Subspace span of union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhsumss 37550	Subspace sum is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	dihsumssj 37551	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 37552	Subspace union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.) (New usage is discouraged.)
$.\/$ joinH $/\$ $.\/$

Theorem	dochdmm1 37553	De Morgan-like law for closed subspace orthocomplement. (Contributed by NM, 13-Jan-2015.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhexmid 37554	Excluded middle property of vector space closed subspace join. (Contributed by NM, 22-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $.\/$

Theorem	djh01 37555	Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djh02 37556	Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhlsmcl 37557	A closed subspace sum equals subspace join. (shjshseli 26612 analog.) (Contributed by NM, 13-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhcvat42 37558*	A covering property. (cvrat42 35584 analog.) (Contributed by NM, 17-Aug-2014.)
$.\/$ joinH $/\$ $\$ $\$ $/\$ $.\/$ $\$ $/\$ $.\/$

Theorem	dihjatb 37559	Isomorphism H of lattice join of two atoms under the fiducial hyperplane. (Contributed by NM, 23-Sep-2014.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjatc 37560	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 37561	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjatcclem2 37562	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjatcclem3 37563*	Lemma for dihjatcc 37565. (Contributed by NM, 28-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$

Theorem	dihjatcclem4 37564*	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$

Theorem	dihjatcc 37565	Isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjat 37566	Isomorphism H of lattice join of two atoms. (Contributed by NM, 29-Sep-2014.)
$.\/$ $/\$ $.\/$

Theorem	dihprrnlem1N 37567	Lemma for dihprrn 37569, showing one of 4 cases. (Contributed by NM, 30-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dihprrnlem2 37568	Lemma for dihprrn 37569. (Contributed by NM, 29-Sep-2014.)
$/\$

Theorem	dihprrn 37569	The span of a vector pair belongs to the range of isomorphism H i.e. is a closed subspace. (Contributed by NM, 29-Sep-2014.)
$/\$

Theorem	djhlsmat 37570	The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 37569; should we directly use dihjat 37566? (Contributed by NM, 13-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	dihjat1lem 37571	Subspace sum of a closed subspace and an atom. (pmapjat1 35993 analog.) TODO: merge into dihjat1 37572? (Contributed by NM, 18-Aug-2014.)
$.\/$ joinH $/\$ $\$ $.\/$

Theorem	dihjat1 37572	Subspace sum of a closed subspace and an atom. (pmapjat1 35993 analog.) (Contributed by NM, 1-Oct-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	dihsmsprn 37573	Subspace sum of a closed subspace and the span of a singleton. (Contributed by NM, 17-Jan-2015.)
$/\$

Theorem	dihjat2 37574	The subspace sum of a closed subspace and an atom is the same as their subspace join. (Contributed by NM, 1-Oct-2014.)
$.\/$ joinH LSAtoms $/\$ $.\/$

Theorem	dihjat3 37575	Isomorphism H of lattice join with an atom. (Contributed by NM, 25-Apr-2015.)
$.\/$ $/\$ $.\/$

Theorem	dihjat4 37576	Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
$.\/$ LSAtoms $/\$ $.\/$

Theorem	dihjat6 37577	Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
$.\/$ LSAtoms $/\$ $.\/$

Theorem	dihsmsnrn 37578	The subspace sum of two singleton spans is closed. (Contributed by NM, 27-Feb-2015.)
$/\$

Theorem	dihsmatrn 37579	The subspace sum of a closed subspace and an atom is closed. TODO: see if proof at http://math.stackexchange.com/a/1233211/50776 and Mon, 13 Apr 2015 20:44:07 -0400 email could be used instead of this and dihjat2 37574. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dihjat5N 37580	Transfer lattice join with atom to subspace sum. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
$.\/$ $/\$ $.\/$

Theorem	dvh4dimat 37581*	There is an atom that is outside the subspace sum of 3 others. (Contributed by NM, 25-Apr-2015.)
LSAtoms $/\$

Theorem	dvh3dimatN 37582*	There is an atom that is outside the subspace sum of 2 others. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
LSAtoms $/\$

Theorem	dvh2dimatN 37583*	Given an atom, there exists another. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
LSAtoms $/\$

Theorem	dvh1dimat 37584*	There exists an atom. (Contributed by NM, 25-Apr-2015.)
LSAtoms $/\$

Theorem	dvh1dim 37585*	There exists a nonzero vector. (Contributed by NM, 26-Apr-2015.)
$/\$

Theorem	dvh4dimlem 37586*	Lemma for dvh4dimN 37590. (Contributed by NM, 22-May-2015.)
$/\$

Theorem	dvhdimlem 37587*	Lemma for dvh2dim 37588 and dvh3dim 37589. TODO: make this obsolete and use dvh4dimlem 37586 directly? (Contributed by NM, 24-Apr-2015.)
$/\$

Theorem	dvh2dim 37588*	There is a vector that is outside the span of another. (Contributed by NM, 25-Apr-2015.)
$/\$

Theorem	dvh3dim 37589*	There is a vector that is outside the span of 2 others. (Contributed by NM, 24-Apr-2015.)
$/\$

Theorem	dvh4dimN 37590*	There is a vector that is outside the span of 3 others. (Contributed by NM, 22-May-2015.) (New usage is discouraged.)
$/\$

Theorem	dvh3dim2 37591*	There is a vector that is outside of 2 spans with a common vector. (Contributed by NM, 13-May-2015.)
$/\$ $/\$

Theorem	dvh3dim3N 37592*	There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 37591 everywhere. If this one is needed, make dvh3dim2 37591 into a lemma. (Contributed by NM, 21-May-2015.) (New usage is discouraged.)
$/\$ $/\$

Theorem	dochsnnz 37593	The orthocomplement of a singleton is nonzero. (Contributed by NM, 13-Jun-2015.)
$/\$

Theorem	dochsatshp 37594	The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 27-Jul-2014.) (Revised by Mario Carneiro, 1-Oct-2014.)
LSAtoms LSHyp $/\$

Theorem	dochsatshpb 37595	The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 29-Oct-2014.)
LSAtoms LSHyp $/\$

Theorem	dochsnshp 37596	The orthocomplement of a nonzero singleton is a hyperplane. (Contributed by NM, 3-Jan-2015.)
LSHyp $/\$ $\$

Theorem	dochshpsat 37597	A hyperplane is closed iff its orthocomplement is an atom. (Contributed by NM, 29-Oct-2014.)
LSAtoms LSHyp $/\$

Theorem	dochkrsat 37598	The orthocomplement of a kernel is an atom iff it is nonzero. (Contributed by NM, 1-Nov-2014.)
LSAtoms LFnl LKer $/\$

Theorem	dochkrsat2 37599	The orthocomplement of a kernel is an atom iff the double orthocomplement is not the vector space. (Contributed by NM, 1-Jan-2015.)
LSAtoms LFnl LKer $/\$

Theorem	dochsat0 37600	The orthocomplement of a kernel is either an atom or zero. (Contributed by NM, 29-Jan-2015.)
LSAtoms LFnl LKer $/\$ $\/$

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