P. 367 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 36601-36700 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	djhjlj 36601	vector space closed subspace join in terms of lattice join. (Contributed by NM, 9-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhj 36602	vector space closed subspace join in terms of lattice join. (Contributed by NM, 17-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhcom 36603	Subspace join commutes. (Contributed by NM, 8-Aug-2014.)
$.\/$ joinH $/\$ $.\/$ $.\/$

Theorem	djhspss 36604	Subspace span of union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhsumss 36605	Subspace sum is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	dihsumssj 36606	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 36607	Subspace union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.) (New usage is discouraged.)
$.\/$ joinH $/\$ $.\/$

Theorem	dochdmm1 36608	De Morgan-like law for closed subspace orthocomplement. (Contributed by NM, 13-Jan-2015.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhexmid 36609	Excluded middle property of vector space closed subspace join. (Contributed by NM, 22-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $.\/$

Theorem	djh01 36610	Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djh02 36611	Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhlsmcl 36612	A closed subspace sum equals subspace join. (shjshseli 26234 analog.) (Contributed by NM, 13-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhcvat42 36613*	A covering property. (cvrat42 34641 analog.) (Contributed by NM, 17-Aug-2014.)
$.\/$ joinH $/\$ $\$ $\$ $/\$ $.\/$ $\$ $/\$ $.\/$

Theorem	dihjatb 36614	Isomorphism H of lattice join of two atoms under the fiducial hyperplane. (Contributed by NM, 23-Sep-2014.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjatc 36615	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 36616	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjatcclem2 36617	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjatcclem3 36618*	Lemma for dihjatcc 36620. (Contributed by NM, 28-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$

Theorem	dihjatcclem4 36619*	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$

Theorem	dihjatcc 36620	Isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjat 36621	Isomorphism H of lattice join of two atoms. (Contributed by NM, 29-Sep-2014.)
$.\/$ $/\$ $.\/$

Theorem	dihprrnlem1N 36622	Lemma for dihprrn 36624, showing one of 4 cases. (Contributed by NM, 30-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dihprrnlem2 36623	Lemma for dihprrn 36624. (Contributed by NM, 29-Sep-2014.)
$/\$

Theorem	dihprrn 36624	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 36625	The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 36624; should we directly use dihjat 36621? (Contributed by NM, 13-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	dihjat1lem 36626	Subspace sum of a closed subspace and an atom. (pmapjat1 35050 analog.) TODO: merge into dihjat1 36627? (Contributed by NM, 18-Aug-2014.)
$.\/$ joinH $/\$ $\$ $.\/$

Theorem	dihjat1 36627	Subspace sum of a closed subspace and an atom. (pmapjat1 35050 analog.) (Contributed by NM, 1-Oct-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	dihsmsprn 36628	Subspace sum of a closed subspace and the span of a singleton. (Contributed by NM, 17-Jan-2015.)
$/\$

Theorem	dihjat2 36629	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 36630	Isomorphism H of lattice join with an atom. (Contributed by NM, 25-Apr-2015.)
$.\/$ $/\$ $.\/$

Theorem	dihjat4 36631	Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
$.\/$ LSAtoms $/\$ $.\/$

Theorem	dihjat6 36632	Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
$.\/$ LSAtoms $/\$ $.\/$

Theorem	dihsmsnrn 36633	The subspace sum of two singleton spans is closed. (Contributed by NM, 27-Feb-2015.)
$/\$

Theorem	dihsmatrn 36634	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 36629. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dihjat5N 36635	Transfer lattice join with atom to subspace sum. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
$.\/$ $/\$ $.\/$

Theorem	dvh4dimat 36636*	There is an atom that is outside the subspace sum of 3 others. (Contributed by NM, 25-Apr-2015.)
LSAtoms $/\$

Theorem	dvh3dimatN 36637*	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 36638*	Given an atom, there exists another. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
LSAtoms $/\$

Theorem	dvh1dimat 36639*	There exists an atom. (Contributed by NM, 25-Apr-2015.)
LSAtoms $/\$

Theorem	dvh1dim 36640*	There exists a nonzero vector. (Contributed by NM, 26-Apr-2015.)
$/\$

Theorem	dvh4dimlem 36641*	Lemma for dvh4dimN 36645. (Contributed by NM, 22-May-2015.)
$/\$

Theorem	dvhdimlem 36642*	Lemma for dvh2dim 36643 and dvh3dim 36644. TODO: make this obsolete and use dvh4dimlem 36641 directly? (Contributed by NM, 24-Apr-2015.)
$/\$

Theorem	dvh2dim 36643*	There is a vector that is outside the span of another. (Contributed by NM, 25-Apr-2015.)
$/\$

Theorem	dvh3dim 36644*	There is a vector that is outside the span of 2 others. (Contributed by NM, 24-Apr-2015.)
$/\$

Theorem	dvh4dimN 36645*	There is a vector that is outside the span of 3 others. (Contributed by NM, 22-May-2015.) (New usage is discouraged.)
$/\$

Theorem	dvh3dim2 36646*	There is a vector that is outside of 2 spans with a common vector. (Contributed by NM, 13-May-2015.)
$/\$ $/\$

Theorem	dvh3dim3N 36647*	There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 36646 everywhere. If this one is needed, make dvh3dim2 36646 into a lemma. (Contributed by NM, 21-May-2015.) (New usage is discouraged.)
$/\$ $/\$

Theorem	dochsnnz 36648	The orthocomplement of a singleton is nonzero. (Contributed by NM, 13-Jun-2015.)
$/\$

Theorem	dochsatshp 36649	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 36650	The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 29-Oct-2014.)
LSAtoms LSHyp $/\$

Theorem	dochsnshp 36651	The orthocomplement of a nonzero singleton is a hyperplane. (Contributed by NM, 3-Jan-2015.)
LSHyp $/\$ $\$

Theorem	dochshpsat 36652	A hyperplane is closed iff its orthocomplement is an atom. (Contributed by NM, 29-Oct-2014.)
LSAtoms LSHyp $/\$

Theorem	dochkrsat 36653	The orthocomplement of a kernel is an atom iff it is nonzero. (Contributed by NM, 1-Nov-2014.)
LSAtoms LFnl LKer $/\$

Theorem	dochkrsat2 36654	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 36655	The orthocomplement of a kernel is either an atom or zero. (Contributed by NM, 29-Jan-2015.)
LSAtoms LFnl LKer $/\$ $\/$

Theorem	dochkrsm 36656	The subspace sum of a closed subspace and a kernel orthocomplement is closed. (djhlsmcl 36612 can be used to convert sum to join.) (Contributed by NM, 29-Jan-2015.)
LFnl LKer $/\$

Theorem	dochexmidat 36657	Special case of excluded middle for the singleton of a vector. (Contributed by NM, 27-Oct-2014.)
$/\$ $\$

Theorem	dochexmidlem1 36658	Lemma for dochexmid 36666. Holland's proof implicitly requires , which we prove here. (Contributed by NM, 14-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem2 36659	Lemma for dochexmid 36666. (Contributed by NM, 14-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem3 36660	Lemma for dochexmid 36666. Use atom exchange lsatexch1 34244 to swap and . (Contributed by NM, 14-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem4 36661	Lemma for dochexmid 36666. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem5 36662	Lemma for dochexmid 36666. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem6 36663	Lemma for dochexmid 36666. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem7 36664	Lemma for dochexmid 36666. Contradict dochexmidlem6 36663. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem8 36665	Lemma for dochexmid 36666. The contradiction of dochexmidlem6 36663 and dochexmidlem7 36664 shows that there can be no atom that is not in , which is therefore the whole atom space. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmid 36666	Excluded middle law for closed subspaces, which is equivalent to (and derived from) the orthomodular law dihoml4 36575. Lemma 3.3(2) in [Holland95] p. 215. In our proof, we use the variables , , , , in place of Hollands' l, m, P, Q, L respectively. (pexmidALTN 35175 analog.) (Contributed by NM, 15-Jan-2015.)
$/\$

Theorem	dochsnkrlem1 36667	Lemma for dochsnkr 36670. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $\$

Theorem	dochsnkrlem2 36668	Lemma for dochsnkr 36670. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $\$ LSAtoms

Theorem	dochsnkrlem3 36669	Lemma for dochsnkr 36670. (Contributed by NM, 2-Jan-2015.)
LFnl LKer $/\$ $\$

Theorem	dochsnkr 36670	A (closed) kernel expressed in terms of a nonzero vector in its orthocomplement. TODO: consolidate lemmas unless they're needed for something else (in which case break out as theorems) (Contributed by NM, 2-Jan-2015.)
LFnl LKer $/\$ $\$

Theorem	dochsnkr2 36671*	Kernel of the explicit functional determined by a nonzero vector . Compare the more general lshpkr 34315. (Contributed by NM, 27-Oct-2014.)
LKer Scalar $/\$ $\$

Theorem	dochsnkr2cl 36672*	The determining functional belongs to the atom formed by the orthocomplement of the kernel. (Contributed by NM, 4-Jan-2015.)
LKer Scalar $/\$ $\$ $\$

Theorem	dochflcl 36673*	Closure of the explicit functional determined by a nonzero vector . Compare the more general lshpkrcl 34314. (Contributed by NM, 27-Oct-2014.)
LFnl Scalar $/\$ $\$

Theorem	dochfl1 36674*	The value of the explicit functional is 1 at the that determines it. (Contributed by NM, 27-Oct-2014.)
Scalar $/\$ $\$

Theorem	dochfln0 36675	The value of a functional is nonzero at a nonzero vector in the orthocomplement of its kernel. (Contributed by NM, 2-Jan-2015.)
Scalar LFnl LKer $/\$ $\$

Theorem	dochkr1 36676*	A non-zero functional has a value of 1 at some argument belonging to the orthocomplement of its kernel (when its kernel is a closed hyperplane). Tighter version of lfl1 34268. (Contributed by NM, 2-Jan-2015.)
Scalar LFnl LKer $/\$ $\$

Theorem	dochkr1OLDN 36677*	A non-zero functional has a value of 1 at some argument belonging to the orthocomplement of its kernel (when its kernel is a closed hyperplane). Tighter version of lfl1 34268. (Contributed by NM, 2-Jan-2015.) (New usage is discouraged.)
Scalar LFnl LKer $/\$

21.30.11 Construction of involution and inner product from a Hilbert lattice

Syntax	clpoN 36678	Extend class notation with all polarities of a left module or left vector space.
LPol

Definition	df-lpolN 36679*	Define the set of all polarities of a left module or left vector space. A polarity is a kind of complementation operation on a subspace. The double polarity of a subspace is a closure operation. Based on Definition 3.2 of [Holland95] p. 214 for projective geometry polarities. For convenience, we open up the domain to include all vector subsets and not just subspaces, but any more restricted polarity can be converted to this one by taking the span of its argument. (Contributed by NM, 24-Nov-2014.)
LPol $/\$ $/\$ $/\$ $/\$ LSAtoms LSHyp $/\$

Theorem	lpolsetN 36680*	The set of polarities of a left module or left vector space. (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)
LSAtoms LSHyp LPol $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	islpolN 36681*	The predicate "is a polarity". (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)
LSAtoms LSHyp LPol $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	islpoldN 36682*	Properties that determine a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LSAtoms LSHyp LPol $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	lpolfN 36683	Functionality of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LPol

Theorem	lpolvN 36684	The polarity of the whole space is the zero subspace. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LPol

Theorem	lpolconN 36685	Contraposition property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LPol

Theorem	lpolsatN 36686	The polarity of an atomic subspace is a hyperplane. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LSAtoms LSHyp LPol

Theorem	lpolpolsatN 36687	Property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LSAtoms LPol

Theorem	dochpolN 36688	The subspace orthocomplement for the vector space is a polarity. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)
LPol $/\$

Theorem	lcfl1lem 36689*	Property of a functional with a closed kernel. (Contributed by NM, 28-Dec-2014.)
$/\$

Theorem	lcfl1 36690*	Property of a functional with a closed kernel. (Contributed by NM, 31-Dec-2014.)


Theorem	lcfl2 36691*	Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $\/$

Theorem	lcfl3 36692*	Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
LSAtoms LFnl LKer $/\$ $\/$

Theorem	lcfl4N 36693*	Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.) (New usage is discouraged.)
LSHyp LFnl LKer $/\$ $\/$

Theorem	lcfl5 36694*	Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$

Theorem	lcfl5a 36695	Property of a functional with a closed kernel. TODO: Make lcfl5 36694 etc. obsolete and rewrite w/out hypothesis? (Contributed by NM, 29-Jan-2015.)
LFnl LKer $/\$

Theorem	lcfl6lem 36696*	Lemma for lcfl6 36698. A functional (whose kernel is closed by dochsnkr 36670) is comletely determined by a vector in the orthocomplement in its kernel at which the functional value is 1. Note that the $\$ in the hypothesis is redundant by the last hypothesis but allows easier use of other theorems. (Contributed by NM, 3-Jan-2015.)
Scalar LFnl LKer $/\$ $\$

Theorem	lcfl7lem 36697*	Lemma for lcfl7N 36699. If two functionals and are equal, they are determined by the same vector. (Contributed by NM, 4-Jan-2015.)
Scalar LFnl LKer $/\$ $\$ $\$

Theorem	lcfl6 36698*	Property of a functional with a closed kernel. Note that means the functional is zero by lkr0f 34292. (Contributed by NM, 3-Jan-2015.)
Scalar LFnl LKer $/\$ $\/$ $\$

Theorem	lcfl7N 36699*	Property of a functional with a closed kernel. Every nonzero functional is determined by a unique nonzero vector. Note that means the functional is zero by lkr0f 34292. (Contributed by NM, 4-Jan-2015.) (New usage is discouraged.)
Scalar LFnl LKer $/\$ $\/$ $\$

Theorem	lcfl8 36700*	Property of a functional with a closed kernel. (Contributed by NM, 17-Jan-2015.)
LFnl LKer $/\$

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