P. 352 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 35101-35200 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	dochoc0 35101	The zero subspace is closed. (Contributed by NM, 16-Feb-2015.)
$/\$

Theorem	dochoc1 35102	The unit subspace (all vectors) is closed. (Contributed by NM, 16-Feb-2015.)
$/\$

Theorem	dochvalr2 35103	Orthocomplement of a closed subspace. (Contributed by NM, 21-Jul-2014.)
$/\$ $/\$

Theorem	dochvalr3 35104	Orthocomplement of a closed subspace. (Contributed by NM, 15-Jan-2015.)
$/\$

Theorem	doch2val2 35105*	Double orthocomplement for vector space. (Contributed by NM, 26-Jul-2014.)
$/\$

Theorem	dochss 35106	Subset law for orthocomplement. (Contributed by NM, 16-Apr-2014.)
$/\$ $/\$ $/\$

Theorem	dochocss 35107	Double negative law for orthocomplement of an arbitrary set of vectors. (Contributed by NM, 16-Apr-2014.)
$/\$ $/\$

Theorem	dochoc 35108	Double negative law for orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)
$/\$ $/\$

Theorem	dochsscl 35109	If a set of vectors is included in a closed set, so is its closure. (Contributed by NM, 17-Jun-2015.)
$/\$

Theorem	dochoccl 35110	A set of vectors is closed iff it equals its double orthocomplent. (Contributed by NM, 1-Jan-2015.)
$/\$

Theorem	dochord 35111	Ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.)
$/\$

Theorem	dochord2N 35112	Ordering law for orthocomplement. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.)
$/\$

Theorem	dochord3 35113	Ordering law for orthocomplement. (Contributed by NM, 9-Mar-2015.)
$/\$

Theorem	doch11 35114	Orthocomplement is one-to-one. (Contributed by NM, 12-Aug-2014.)
$/\$

Theorem	dochsordN 35115	Strict ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dochn0nv 35116	An orthocomplement is nonzero iff the double orthocomplement is not the whole vector space. (Contributed by NM, 1-Jan-2015.)
$/\$

Theorem	dihoml4c 35117	Version of dihoml4 35118 with closed subspaces. (Contributed by NM, 15-Jan-2015.)
$/\$

Theorem	dihoml4 35118	Orthomodular law for constructed vector space H. Lemma 3.3(1) in [Holland95] p. 215. (poml4N 33693 analog.) (Contributed by NM, 15-Jan-2015.)
$/\$

Theorem	dochspss 35119	The span of a set of vectors is included in their double orthocomplement. (Contributed by NM, 26-Jul-2014.)
$/\$

Theorem	dochocsp 35120	The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.)
$/\$

Theorem	dochspocN 35121	The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dochocsn 35122	The double orthocomplement of a singleton is its span. (Contributed by NM, 13-Jan-2015.)
$/\$

Theorem	dochsncom 35123	Swap vectors in an orthocomplement of a singleton. (Contributed by NM, 17-Jun-2015.)
$/\$

Theorem	dochsat 35124	The double orthocomplement of an atom is an atom. (Contributed by NM, 29-Oct-2014.)
LSAtoms $/\$

Theorem	dochshpncl 35125	If a hyperplane is not closed, its closure equals the vector space. (Contributed by NM, 29-Oct-2014.)
LSHyp $/\$

Theorem	dochlkr 35126	Equivalent conditions for the closure of a kernel to be a hyperplane. (Contributed by NM, 29-Oct-2014.)
LFnl LSHyp LKer $/\$ $/\$

Theorem	dochkrshp 35127	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 35128	Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LSHyp LFnl LKer $/\$ $/\$

Theorem	dochkrshp3 35129	Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $/\$

Theorem	dochkrshp4 35130	Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $\/$

Theorem	dochdmj1 35131	De Morgan-like law for subspace orthocomplement. (Contributed by NM, 5-Aug-2014.)
$/\$ $/\$ $/\$

Theorem	dochnoncon 35132	Law of noncontradiction. The intersection of a subspace and its orthocomplement is the zero subspace. (Contributed by NM, 16-Apr-2014.)
$/\$ $/\$

Theorem	dochnel2 35133	A nonzero member of a subspace doesn't belong to the orthocomplement of the subspace. (Contributed by NM, 28-Feb-2015.)
$/\$ $\$

Theorem	dochnel 35134	A nonzero vector doesn't belong to the orthocomplement of its singleton. (Contributed by NM, 27-Oct-2014.)
$/\$ $\$

Syntax	cdjh 35135	Extend class notation with subspace join for vector space.
joinH

Definition	df-djh 35136*	Define (closed) subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theorem	djhffval 35137*	Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theorem	djhfval 35138*	Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhval 35139	Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhval2 35140	Value of subspace join for vector space. (Contributed by NM, 6-Aug-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhcl 35141	Closure of subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhlj 35142	Transfer lattice join to vector space closed subspace join. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhljjN 35143	Lattice join in terms of vector space closed subspace join. (Contributed by NM, 17-Aug-2014.) (New usage is discouraged.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhjlj 35144	vector space closed subspace join in terms of lattice join. (Contributed by NM, 9-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhj 35145	vector space closed subspace join in terms of lattice join. (Contributed by NM, 17-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhcom 35146	Subspace join commutes. (Contributed by NM, 8-Aug-2014.)
$.\/$ joinH $/\$ $.\/$ $.\/$

Theorem	djhspss 35147	Subspace span of union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhsumss 35148	Subspace sum is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	dihsumssj 35149	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 35150	Subspace union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.) (New usage is discouraged.)
$.\/$ joinH $/\$ $.\/$

Theorem	dochdmm1 35151	De Morgan-like law for closed subspace orthocomplement. (Contributed by NM, 13-Jan-2015.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhexmid 35152	Excluded middle property of vector space closed subspace join. (Contributed by NM, 22-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $.\/$

Theorem	djh01 35153	Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djh02 35154	Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhlsmcl 35155	A closed subspace sum equals subspace join. (shjshseli 24918 analog.) (Contributed by NM, 13-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhcvat42 35156*	A covering property. (cvrat42 33184 analog.) (Contributed by NM, 17-Aug-2014.)
$.\/$ joinH $/\$ $\$ $\$ $/\$ $.\/$ $\$ $/\$ $.\/$

Theorem	dihjatb 35157	Isomorphism H of lattice join of two atoms under the fiducial hyperplane. (Contributed by NM, 23-Sep-2014.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjatc 35158	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 35159	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjatcclem2 35160	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjatcclem3 35161*	Lemma for dihjatcc 35163. (Contributed by NM, 28-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$

Theorem	dihjatcclem4 35162*	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$

Theorem	dihjatcc 35163	Isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjat 35164	Isomorphism H of lattice join of two atoms. (Contributed by NM, 29-Sep-2014.)
$.\/$ $/\$ $.\/$

Theorem	dihprrnlem1N 35165	Lemma for dihprrn 35167, showing one of 4 cases. (Contributed by NM, 30-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dihprrnlem2 35166	Lemma for dihprrn 35167. (Contributed by NM, 29-Sep-2014.)
$/\$

Theorem	dihprrn 35167	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 35168	The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 35167; should we directly use dihjat 35164? (Contributed by NM, 13-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	dihjat1lem 35169	Subspace sum of a closed subspace and an atom. (pmapjat1 33593 analog.) TODO: merge into dihjat1 35170? (Contributed by NM, 18-Aug-2014.)
$.\/$ joinH $/\$ $\$ $.\/$

Theorem	dihjat1 35170	Subspace sum of a closed subspace and an atom. (pmapjat1 33593 analog.) (Contributed by NM, 1-Oct-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	dihsmsprn 35171	Subspace sum of a closed subspace and the span of a singleton. (Contributed by NM, 17-Jan-2015.)
$/\$

Theorem	dihjat2 35172	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 35173	Isomorphism H of lattice join with an atom. (Contributed by NM, 25-Apr-2015.)
$.\/$ $/\$ $.\/$

Theorem	dihjat4 35174	Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
$.\/$ LSAtoms $/\$ $.\/$

Theorem	dihjat6 35175	Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
$.\/$ LSAtoms $/\$ $.\/$

Theorem	dihsmsnrn 35176	The subspace sum of two singleton spans is closed. (Contributed by NM, 27-Feb-2015.)
$/\$

Theorem	dihsmatrn 35177	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 35172. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dihjat5N 35178	Transfer lattice join with atom to subspace sum. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
$.\/$ $/\$ $.\/$

Theorem	dvh4dimat 35179*	There is an atom that is outside the subspace sum of 3 others. (Contributed by NM, 25-Apr-2015.)
LSAtoms $/\$

Theorem	dvh3dimatN 35180*	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 35181*	Given an atom, there exists another. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
LSAtoms $/\$

Theorem	dvh1dimat 35182*	There exists an atom. (Contributed by NM, 25-Apr-2015.)
LSAtoms $/\$

Theorem	dvh1dim 35183*	There exists a nonzero vector. (Contributed by NM, 26-Apr-2015.)
$/\$

Theorem	dvh4dimlem 35184*	Lemma for dvh4dimN 35188. (Contributed by NM, 22-May-2015.)
$/\$

Theorem	dvhdimlem 35185*	Lemma for dvh2dim 35186 and dvh3dim 35187. TODO: make this obsolete and use dvh4dimlem 35184 directly? (Contributed by NM, 24-Apr-2015.)
$/\$

Theorem	dvh2dim 35186*	There is a vector that is outside the span of another. (Contributed by NM, 25-Apr-2015.)
$/\$

Theorem	dvh3dim 35187*	There is a vector that is outside the span of 2 others. (Contributed by NM, 24-Apr-2015.)
$/\$

Theorem	dvh4dimN 35188*	There is a vector that is outside the span of 3 others. (Contributed by NM, 22-May-2015.) (New usage is discouraged.)
$/\$

Theorem	dvh3dim2 35189*	There is a vector that is outside of 2 spans with a common vector. (Contributed by NM, 13-May-2015.)
$/\$ $/\$

Theorem	dvh3dim3N 35190*	There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 35189 everywhere. If this one is needed, make dvh3dim2 35189 into a lemma. (Contributed by NM, 21-May-2015.) (New usage is discouraged.)
$/\$ $/\$

Theorem	dochsnnz 35191	The orthocomplement of a singleton is nonzero. (Contributed by NM, 13-Jun-2015.)
$/\$

Theorem	dochsatshp 35192	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 35193	The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 29-Oct-2014.)
LSAtoms LSHyp $/\$

Theorem	dochsnshp 35194	The orthocomplement of a nonzero singleton is a hyperplane. (Contributed by NM, 3-Jan-2015.)
LSHyp $/\$ $\$

Theorem	dochshpsat 35195	A hyperplane is closed iff its orthocomplement is an atom. (Contributed by NM, 29-Oct-2014.)
LSAtoms LSHyp $/\$

Theorem	dochkrsat 35196	The orthocomplement of a kernel is an atom iff it is nonzero. (Contributed by NM, 1-Nov-2014.)
LSAtoms LFnl LKer $/\$

Theorem	dochkrsat2 35197	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 35198	The orthocomplement of a kernel is either an atom or zero. (Contributed by NM, 29-Jan-2015.)
LSAtoms LFnl LKer $/\$ $\/$

Theorem	dochkrsm 35199	The subspace sum of a closed subspace and a kernel orthocomplement is closed. (djhlsmcl 35155 can be used to convert sum to join.) (Contributed by NM, 29-Jan-2015.)
LFnl LKer $/\$

Theorem	dochexmidat 35200	Special case of excluded middle for the singleton of a vector. (Contributed by NM, 27-Oct-2014.)
$/\$ $\$

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