P. 354 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 35301-35400 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	dochsordN 35301	Strict ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dochn0nv 35302	An orthocomplement is nonzero iff the double orthocomplement is not the whole vector space. (Contributed by NM, 1-Jan-2015.)
$/\$

Theorem	dihoml4c 35303	Version of dihoml4 35304 with closed subspaces. (Contributed by NM, 15-Jan-2015.)
$/\$

Theorem	dihoml4 35304	Orthomodular law for constructed vector space H. Lemma 3.3(1) in [Holland95] p. 215. (poml4N 33879 analog.) (Contributed by NM, 15-Jan-2015.)
$/\$

Theorem	dochspss 35305	The span of a set of vectors is included in their double orthocomplement. (Contributed by NM, 26-Jul-2014.)
$/\$

Theorem	dochocsp 35306	The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.)
$/\$

Theorem	dochspocN 35307	The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dochocsn 35308	The double orthocomplement of a singleton is its span. (Contributed by NM, 13-Jan-2015.)
$/\$

Theorem	dochsncom 35309	Swap vectors in an orthocomplement of a singleton. (Contributed by NM, 17-Jun-2015.)
$/\$

Theorem	dochsat 35310	The double orthocomplement of an atom is an atom. (Contributed by NM, 29-Oct-2014.)
LSAtoms $/\$

Theorem	dochshpncl 35311	If a hyperplane is not closed, its closure equals the vector space. (Contributed by NM, 29-Oct-2014.)
LSHyp $/\$

Theorem	dochlkr 35312	Equivalent conditions for the closure of a kernel to be a hyperplane. (Contributed by NM, 29-Oct-2014.)
LFnl LSHyp LKer $/\$ $/\$

Theorem	dochkrshp 35313	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 35314	Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LSHyp LFnl LKer $/\$ $/\$

Theorem	dochkrshp3 35315	Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $/\$

Theorem	dochkrshp4 35316	Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $\/$

Theorem	dochdmj1 35317	De Morgan-like law for subspace orthocomplement. (Contributed by NM, 5-Aug-2014.)
$/\$ $/\$ $/\$

Theorem	dochnoncon 35318	Law of noncontradiction. The intersection of a subspace and its orthocomplement is the zero subspace. (Contributed by NM, 16-Apr-2014.)
$/\$ $/\$

Theorem	dochnel2 35319	A nonzero member of a subspace doesn't belong to the orthocomplement of the subspace. (Contributed by NM, 28-Feb-2015.)
$/\$ $\$

Theorem	dochnel 35320	A nonzero vector doesn't belong to the orthocomplement of its singleton. (Contributed by NM, 27-Oct-2014.)
$/\$ $\$

Syntax	cdjh 35321	Extend class notation with subspace join for vector space.
joinH

Definition	df-djh 35322*	Define (closed) subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theorem	djhffval 35323*	Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theorem	djhfval 35324*	Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhval 35325	Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhval2 35326	Value of subspace join for vector space. (Contributed by NM, 6-Aug-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhcl 35327	Closure of subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhlj 35328	Transfer lattice join to vector space closed subspace join. (Contributed by NM, 19-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $/\$ $.\/$

Theorem	djhljjN 35329	Lattice join in terms of vector space closed subspace join. (Contributed by NM, 17-Aug-2014.) (New usage is discouraged.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhjlj 35330	vector space closed subspace join in terms of lattice join. (Contributed by NM, 9-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhj 35331	vector space closed subspace join in terms of lattice join. (Contributed by NM, 17-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhcom 35332	Subspace join commutes. (Contributed by NM, 8-Aug-2014.)
$.\/$ joinH $/\$ $.\/$ $.\/$

Theorem	djhspss 35333	Subspace span of union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhsumss 35334	Subspace sum is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	dihsumssj 35335	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 35336	Subspace union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.) (New usage is discouraged.)
$.\/$ joinH $/\$ $.\/$

Theorem	dochdmm1 35337	De Morgan-like law for closed subspace orthocomplement. (Contributed by NM, 13-Jan-2015.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhexmid 35338	Excluded middle property of vector space closed subspace join. (Contributed by NM, 22-Jul-2014.)
$.\/$ joinH $/\$ $/\$ $.\/$

Theorem	djh01 35339	Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djh02 35340	Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhlsmcl 35341	A closed subspace sum equals subspace join. (shjshseli 25017 analog.) (Contributed by NM, 13-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	djhcvat42 35342*	A covering property. (cvrat42 33370 analog.) (Contributed by NM, 17-Aug-2014.)
$.\/$ joinH $/\$ $\$ $\$ $/\$ $.\/$ $\$ $/\$ $.\/$

Theorem	dihjatb 35343	Isomorphism H of lattice join of two atoms under the fiducial hyperplane. (Contributed by NM, 23-Sep-2014.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjatc 35344	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 35345	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjatcclem2 35346	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjatcclem3 35347*	Lemma for dihjatcc 35349. (Contributed by NM, 28-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$

Theorem	dihjatcclem4 35348*	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$

Theorem	dihjatcc 35349	Isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	dihjat 35350	Isomorphism H of lattice join of two atoms. (Contributed by NM, 29-Sep-2014.)
$.\/$ $/\$ $.\/$

Theorem	dihprrnlem1N 35351	Lemma for dihprrn 35353, showing one of 4 cases. (Contributed by NM, 30-Aug-2014.) (New usage is discouraged.)
$/\$

Theorem	dihprrnlem2 35352	Lemma for dihprrn 35353. (Contributed by NM, 29-Sep-2014.)
$/\$

Theorem	dihprrn 35353	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 35354	The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 35353; should we directly use dihjat 35350? (Contributed by NM, 13-Aug-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	dihjat1lem 35355	Subspace sum of a closed subspace and an atom. (pmapjat1 33779 analog.) TODO: merge into dihjat1 35356? (Contributed by NM, 18-Aug-2014.)
$.\/$ joinH $/\$ $\$ $.\/$

Theorem	dihjat1 35356	Subspace sum of a closed subspace and an atom. (pmapjat1 33779 analog.) (Contributed by NM, 1-Oct-2014.)
$.\/$ joinH $/\$ $.\/$

Theorem	dihsmsprn 35357	Subspace sum of a closed subspace and the span of a singleton. (Contributed by NM, 17-Jan-2015.)
$/\$

Theorem	dihjat2 35358	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 35359	Isomorphism H of lattice join with an atom. (Contributed by NM, 25-Apr-2015.)
$.\/$ $/\$ $.\/$

Theorem	dihjat4 35360	Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
$.\/$ LSAtoms $/\$ $.\/$

Theorem	dihjat6 35361	Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
$.\/$ LSAtoms $/\$ $.\/$

Theorem	dihsmsnrn 35362	The subspace sum of two singleton spans is closed. (Contributed by NM, 27-Feb-2015.)
$/\$

Theorem	dihsmatrn 35363	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 35358. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dihjat5N 35364	Transfer lattice join with atom to subspace sum. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
$.\/$ $/\$ $.\/$

Theorem	dvh4dimat 35365*	There is an atom that is outside the subspace sum of 3 others. (Contributed by NM, 25-Apr-2015.)
LSAtoms $/\$

Theorem	dvh3dimatN 35366*	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 35367*	Given an atom, there exists another. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
LSAtoms $/\$

Theorem	dvh1dimat 35368*	There exists an atom. (Contributed by NM, 25-Apr-2015.)
LSAtoms $/\$

Theorem	dvh1dim 35369*	There exists a nonzero vector. (Contributed by NM, 26-Apr-2015.)
$/\$

Theorem	dvh4dimlem 35370*	Lemma for dvh4dimN 35374. (Contributed by NM, 22-May-2015.)
$/\$

Theorem	dvhdimlem 35371*	Lemma for dvh2dim 35372 and dvh3dim 35373. TODO: make this obsolete and use dvh4dimlem 35370 directly? (Contributed by NM, 24-Apr-2015.)
$/\$

Theorem	dvh2dim 35372*	There is a vector that is outside the span of another. (Contributed by NM, 25-Apr-2015.)
$/\$

Theorem	dvh3dim 35373*	There is a vector that is outside the span of 2 others. (Contributed by NM, 24-Apr-2015.)
$/\$

Theorem	dvh4dimN 35374*	There is a vector that is outside the span of 3 others. (Contributed by NM, 22-May-2015.) (New usage is discouraged.)
$/\$

Theorem	dvh3dim2 35375*	There is a vector that is outside of 2 spans with a common vector. (Contributed by NM, 13-May-2015.)
$/\$ $/\$

Theorem	dvh3dim3N 35376*	There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 35375 everywhere. If this one is needed, make dvh3dim2 35375 into a lemma. (Contributed by NM, 21-May-2015.) (New usage is discouraged.)
$/\$ $/\$

Theorem	dochsnnz 35377	The orthocomplement of a singleton is nonzero. (Contributed by NM, 13-Jun-2015.)
$/\$

Theorem	dochsatshp 35378	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 35379	The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 29-Oct-2014.)
LSAtoms LSHyp $/\$

Theorem	dochsnshp 35380	The orthocomplement of a nonzero singleton is a hyperplane. (Contributed by NM, 3-Jan-2015.)
LSHyp $/\$ $\$

Theorem	dochshpsat 35381	A hyperplane is closed iff its orthocomplement is an atom. (Contributed by NM, 29-Oct-2014.)
LSAtoms LSHyp $/\$

Theorem	dochkrsat 35382	The orthocomplement of a kernel is an atom iff it is nonzero. (Contributed by NM, 1-Nov-2014.)
LSAtoms LFnl LKer $/\$

Theorem	dochkrsat2 35383	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 35384	The orthocomplement of a kernel is either an atom or zero. (Contributed by NM, 29-Jan-2015.)
LSAtoms LFnl LKer $/\$ $\/$

Theorem	dochkrsm 35385	The subspace sum of a closed subspace and a kernel orthocomplement is closed. (djhlsmcl 35341 can be used to convert sum to join.) (Contributed by NM, 29-Jan-2015.)
LFnl LKer $/\$

Theorem	dochexmidat 35386	Special case of excluded middle for the singleton of a vector. (Contributed by NM, 27-Oct-2014.)
$/\$ $\$

Theorem	dochexmidlem1 35387	Lemma for dochexmid 35395. Holland's proof implicitly requires , which we prove here. (Contributed by NM, 14-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem2 35388	Lemma for dochexmid 35395. (Contributed by NM, 14-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem3 35389	Lemma for dochexmid 35395. Use atom exchange lsatexch1 32973 to swap and . (Contributed by NM, 14-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem4 35390	Lemma for dochexmid 35395. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem5 35391	Lemma for dochexmid 35395. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem6 35392	Lemma for dochexmid 35395. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem7 35393	Lemma for dochexmid 35395. Contradict dochexmidlem6 35392. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem8 35394	Lemma for dochexmid 35395. The contradiction of dochexmidlem6 35392 and dochexmidlem7 35393 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 35395	Excluded middle law for closed subspaces, which is equivalent to (and derived from) the orthomodular law dihoml4 35304. 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 33904 analog.) (Contributed by NM, 15-Jan-2015.)
$/\$

Theorem	dochsnkrlem1 35396	Lemma for dochsnkr 35399. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $\$

Theorem	dochsnkrlem2 35397	Lemma for dochsnkr 35399. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $\$ LSAtoms

Theorem	dochsnkrlem3 35398	Lemma for dochsnkr 35399. (Contributed by NM, 2-Jan-2015.)
LFnl LKer $/\$ $\$

Theorem	dochsnkr 35399	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 35400*	Kernel of the explicit functional determined by a nonzero vector . Compare the more general lshpkr 33044. (Contributed by NM, 27-Oct-2014.)
LKer Scalar $/\$ $\$

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