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Theorem List for Metamath Proof Explorer - 30701-30800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdochocsn 30701 The double orthocomplement of a singleton is its span. (Contributed by NM, 13-Jan-2015.)

Theoremdochsncom 30702 Swap vectors in an orthocomplement of a singleton. (Contributed by NM, 17-Jun-2015.)

Theoremdochsat 30703 The double orthocomplement of an atom is an atom. (Contributed by NM, 29-Oct-2014.)
LSAtoms

Theoremdochshpncl 30704 If a hyperplane is not closed, its closure equals the vector space. (Contributed by NM, 29-Oct-2014.)
LSHyp

Theoremdochlkr 30705 Equivalent conditions for the closure of a kernel to be a hyperplane. (Contributed by NM, 29-Oct-2014.)
LFnl       LSHyp       LKer

Theoremdochkrshp 30706 The closure of a kernel is a hyperplane iff it doesn't contain all vectors. (Contributed by NM, 1-Nov-2014.)
LSHyp       LFnl       LKer

Theoremdochkrshp2 30707 Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LSHyp       LFnl       LKer

Theoremdochkrshp3 30708 Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer

Theoremdochkrshp4 30709 Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer

Theoremdochdmj1 30710 DeMorgan-like law for subspace orthocomplement. (Contributed by NM, 5-Aug-2014.)

Theoremdochnoncon 30711 Law of noncontradiction. The intersection of a subspace and its orthocomplement is the zero subspace. (Contributed by NM, 16-Apr-2014.)

Theoremdochnel2 30712 A nonzero member of a subspace doesn't belong to the orthocomplement of the subspace. (Contributed by NM, 28-Feb-2015.)

Theoremdochnel 30713 A nonzero vector doesn't belong to the orthocomplement of its singleton. (Contributed by NM, 27-Oct-2014.)

Syntaxcdjh 30714 Extend class notation with subspace join for vector space.
joinH

Definitiondf-djh 30715* Define (closed) subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theoremdjhffval 30716* Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theoremdjhfval 30717* Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theoremdjhval 30718 Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theoremdjhval2 30719 Value of subspace join for vector space. (Contributed by NM, 6-Aug-2014.)
joinH

Theoremdjhcl 30720 Closure of subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theoremdjhlj 30721 Transfer lattice join to vector space closed subspace join. (Contributed by NM, 19-Jul-2014.)
joinH

TheoremdjhljjN 30722 Lattice join in terms of vector space closed subspace join. (Contributed by NM, 17-Aug-2014.) (New usage is discouraged.)
joinH

Theoremdjhjlj 30723 vector space closed subspace join in terms of lattice join. (Contributed by NM, 9-Aug-2014.)
joinH

Theoremdjhj 30724 vector space closed subspace join in terms of lattice join. (Contributed by NM, 17-Aug-2014.)
joinH

Theoremdjhcom 30725 Subspace join commutes. (Contributed by NM, 8-Aug-2014.)
joinH

Theoremdjhspss 30726 Subspace span of union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
joinH

Theoremdjhsumss 30727 Subspace sum is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
joinH

Theoremdihsumssj 30728 The subspace sum of two isomorphisms of lattice elements is less than the isomorphism of their lattice join. (Contributed by NM, 23-Sep-2014.)

TheoremdjhunssN 30729 Subspace union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.) (New usage is discouraged.)
joinH

Theoremdochdmm1 30730 DeMorgan-like law for closed subspace orthocomplement. (Contributed by NM, 13-Jan-2015.)
joinH

Theoremdjhexmid 30731 Excluded middle property of vector space closed subspace join. (Contributed by NM, 22-Jul-2014.)
joinH

Theoremdjh01 30732 Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
joinH

Theoremdjh02 30733 Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
joinH

Theoremdjhlsmcl 30734 A closed subspace sum equals subspace join. (shjshseli 21997 analog.) (Contributed by NM, 13-Aug-2014.)
joinH

Theoremdjhcvat42 30735* A covering property. (cvrat42 28763 analog.) (Contributed by NM, 17-Aug-2014.)
joinH

Theoremdihjatb 30736 Isomorphism H of lattice join of two atoms under the fiducial hyperplane. (Contributed by NM, 23-Sep-2014.)

Theoremdihjatc 30737 Isomorphism H of lattice join of a element under the fiducial hyperplane with atom not under it. (Contributed by NM, 26-Aug-2014.)

Theoremdihjatcclem1 30738 Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)

Theoremdihjatcclem2 30739 Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)

Theoremdihjatcclem3 30740* Lemma for dihjatcc 30742. (Contributed by NM, 28-Sep-2014.)

Theoremdihjatcclem4 30741* Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)

Theoremdihjatcc 30742 Isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)

Theoremdihjat 30743 Isomorphism H of lattice join of two atoms. (Contributed by NM, 29-Sep-2014.)

Theoremdihprrnlem1N 30744 Lemma for dihprrn 30746, showing one of 4 cases. (Contributed by NM, 30-Aug-2014.) (New usage is discouraged.)

Theoremdihprrnlem2 30745 Lemma for dihprrn 30746. (Contributed by NM, 29-Sep-2014.)

Theoremdihprrn 30746 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.)

Theoremdjhlsmat 30747 The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 30746; should we directly use dihjat 30743? (Contributed by NM, 13-Aug-2014.)
joinH

Theoremdihjat1lem 30748 Subspace sum of a closed subspace and an atom. (pmapjat1 29172 analog.) TODO: merge into dihjat1 30749? (Contributed by NM, 18-Aug-2014.)
joinH

Theoremdihjat1 30749 Subspace sum of a closed subspace and an atom. (pmapjat1 29172 analog.) (Contributed by NM, 1-Oct-2014.)
joinH

Theoremdihsmsprn 30750 Subspace sum of a closed subspace and the span of a singleton. (Contributed by NM, 17-Jan-2015.)

Theoremdihjat2 30751 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

Theoremdihjat3 30752 Isomorphism H of lattice join with an atom. (Contributed by NM, 25-Apr-2015.)

Theoremdihjat4 30753 Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
LSAtoms

Theoremdihjat6 30754 Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
LSAtoms

Theoremdihsmsnrn 30755 The subspace sum of two singleton spans is closed. (Contributed by NM, 27-Feb-2015.)

Theoremdihsmatrn 30756 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 30751. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdihjat5N 30757 Transfer lattice join with atom to subspace sum. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)

Theoremdvh4dimat 30758* There is an atom that is outside the subspace sum of 3 others. (Contributed by NM, 25-Apr-2015.)
LSAtoms

Theoremdvh3dimatN 30759* There is an atom that is outside the subspace sum of 2 others. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
LSAtoms

Theoremdvh2dimatN 30760* Given an atom, there exists another. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
LSAtoms

Theoremdvh1dimat 30761* There exists an atom. (Contributed by NM, 25-Apr-2015.)
LSAtoms

Theoremdvh1dim 30762* There exists a nonzero vector. (Contributed by NM, 26-Apr-2015.)

Theoremdvh4dimlem 30763* Lemma for dvh4dimN 30767. (Contributed by NM, 22-May-2015.)

Theoremdvhdimlem 30764* Lemma for dvh2dim 30765 and dvh3dim 30766. TODO: make this obsolete and use dvh4dimlem 30763 directly? (Contributed by NM, 24-Apr-2015.)

Theoremdvh2dim 30765* There is a vector that is outside the span of another. (Contributed by NM, 25-Apr-2015.)

Theoremdvh3dim 30766* There is a vector that is outside the span of 2 others. (Contributed by NM, 24-Apr-2015.)

Theoremdvh4dimN 30767* There is a vector that is outside the span of 3 others. (Contributed by NM, 22-May-2015.) (New usage is discouraged.)

Theoremdvh3dim2 30768* There is a vector that is outside of 2 spans with a common vector. (Contributed by NM, 13-May-2015.)

Theoremdvh3dim3N 30769* There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 30768 everywhere. If this one is needed, make dvh3dim2 30768 into a lemma. (Contributed by NM, 21-May-2015.) (New usage is discouraged.)

Theoremdochsnnz 30770 The orthocomplement of a singleton is nonzero. (Contributed by NM, 13-Jun-2015.)

Theoremdochsatshp 30771 The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 27-Jul-2014.) (Revised by Mario Carneiro, 1-Oct-2014.)
LSAtoms       LSHyp

Theoremdochsatshpb 30772 The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 29-Oct-2014.)
LSAtoms       LSHyp

Theoremdochsnshp 30773 The orthocomplement of a nonzero singleton is a hyperplane. (Contributed by NM, 3-Jan-2015.)
LSHyp

Theoremdochshpsat 30774 A hyperplane is closed iff its orthocomplement is an atom. (Contributed by NM, 29-Oct-2014.)
LSAtoms       LSHyp

Theoremdochkrsat 30775 The orthocomplement of a kernel is an atom iff it is nonzero. (Contributed by NM, 1-Nov-2014.)
LSAtoms       LFnl       LKer

Theoremdochkrsat2 30776 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

Theoremdochsat0 30777 The orthocomplement of a kernel is either an atom or zero. (Contributed by NM, 29-Jan-2015.)
LSAtoms       LFnl       LKer

Theoremdochkrsm 30778 The subspace sum of a closed subspace and a kernel orthocomplement is closed. (djhlsmcl 30734 can be used to convert sum to join.) (Contributed by NM, 29-Jan-2015.)
LFnl       LKer

Theoremdochexmidat 30779 Special case of excluded middle for the singleton of a vector. (Contributed by NM, 27-Oct-2014.)

Theoremdochexmidlem1 30780 Lemma for dochexmid 30788. Holland's proof implicitly requires , which we prove here. (Contributed by NM, 14-Jan-2015.)
LSAtoms

Theoremdochexmidlem2 30781 Lemma for dochexmid 30788. (Contributed by NM, 14-Jan-2015.)
LSAtoms

Theoremdochexmidlem3 30782 Lemma for dochexmid 30788. Use atom exchange lsatexch1 28366 to swap and . (Contributed by NM, 14-Jan-2015.)
LSAtoms

Theoremdochexmidlem4 30783 Lemma for dochexmid 30788. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdochexmidlem5 30784 Lemma for dochexmid 30788. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdochexmidlem6 30785 Lemma for dochexmid 30788. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdochexmidlem7 30786 Lemma for dochexmid 30788. Contradict dochexmidlem6 30785. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdochexmidlem8 30787 Lemma for dochexmid 30788. The contradiction of dochexmidlem6 30785 and dochexmidlem7 30786 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

Theoremdochexmid 30788 Excluded middle law for closed subspaces, which is equivalent to (and derived from) the orthomodular law dihoml4 30697. 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 29297 analog.) (Contributed by NM, 15-Jan-2015.)

Theoremdochsnkrlem1 30789 Lemma for dochsnkr 30792. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer

Theoremdochsnkrlem2 30790 Lemma for dochsnkr 30792. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer                            LSAtoms

Theoremdochsnkrlem3 30791 Lemma for dochsnkr 30792. (Contributed by NM, 2-Jan-2015.)
LFnl       LKer

Theoremdochsnkr 30792 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

Theoremdochsnkr2 30793* Kernel of the explicit functional determined by a nonzero vector . Compare the more general lshpkr 28437. (Contributed by NM, 27-Oct-2014.)
LKer       Scalar

Theoremdochsnkr2cl 30794* The determining functional belongs to the atom formed by the orthocomplement of the kernel. (Contributed by NM, 4-Jan-2015.)
LKer       Scalar

Theoremdochflcl 30795* Closure of the explicit functional determined by a nonzero vector . Compare the more general lshpkrcl 28436. (Contributed by NM, 27-Oct-2014.)
LFnl       Scalar

Theoremdochfl1 30796* The value of the explicit functional is 1 at the that determines it. (Contributed by NM, 27-Oct-2014.)
Scalar

Theoremdochfln0 30797 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

Theoremdochkr1 30798* 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 28390. (Contributed by NM, 2-Jan-2015.)
Scalar                     LFnl       LKer

Theoremdochkr1OLDN 30799* 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 28390. (Contributed by NM, 2-Jan-2015.) (New usage is discouraged.)
Scalar                     LFnl       LKer

16.24.15  Construction of involution and inner product from a Hilbert lattice

SyntaxclpoN 30800 Extend class notation with all polarities of a left module or left vector space.
LPol

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