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Theorem List for Metamath Proof Explorer - 32201-32300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdochkrshp4 32201 Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer

Theoremdochdmj1 32202 De Morgan-like law for subspace orthocomplement. (Contributed by NM, 5-Aug-2014.)

Theoremdochnoncon 32203 Law of noncontradiction. The intersection of a subspace and its orthocomplement is the zero subspace. (Contributed by NM, 16-Apr-2014.)

Theoremdochnel2 32204 A nonzero member of a subspace doesn't belong to the orthocomplement of the subspace. (Contributed by NM, 28-Feb-2015.)

Theoremdochnel 32205 A nonzero vector doesn't belong to the orthocomplement of its singleton. (Contributed by NM, 27-Oct-2014.)

Syntaxcdjh 32206 Extend class notation with subspace join for vector space.
joinH

Definitiondf-djh 32207* Define (closed) subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theoremdjhffval 32208* Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theoremdjhfval 32209* Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theoremdjhval 32210 Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theoremdjhval2 32211 Value of subspace join for vector space. (Contributed by NM, 6-Aug-2014.)
joinH

Theoremdjhcl 32212 Closure of subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theoremdjhlj 32213 Transfer lattice join to vector space closed subspace join. (Contributed by NM, 19-Jul-2014.)
joinH

TheoremdjhljjN 32214 Lattice join in terms of vector space closed subspace join. (Contributed by NM, 17-Aug-2014.) (New usage is discouraged.)
joinH

Theoremdjhjlj 32215 vector space closed subspace join in terms of lattice join. (Contributed by NM, 9-Aug-2014.)
joinH

Theoremdjhj 32216 vector space closed subspace join in terms of lattice join. (Contributed by NM, 17-Aug-2014.)
joinH

Theoremdjhcom 32217 Subspace join commutes. (Contributed by NM, 8-Aug-2014.)
joinH

Theoremdjhspss 32218 Subspace span of union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
joinH

Theoremdjhsumss 32219 Subspace sum is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
joinH

Theoremdihsumssj 32220 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 32221 Subspace union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.) (New usage is discouraged.)
joinH

Theoremdochdmm1 32222 De Morgan-like law for closed subspace orthocomplement. (Contributed by NM, 13-Jan-2015.)
joinH

Theoremdjhexmid 32223 Excluded middle property of vector space closed subspace join. (Contributed by NM, 22-Jul-2014.)
joinH

Theoremdjh01 32224 Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
joinH

Theoremdjh02 32225 Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
joinH

Theoremdjhlsmcl 32226 A closed subspace sum equals subspace join. (shjshseli 22088 analog.) (Contributed by NM, 13-Aug-2014.)
joinH

Theoremdjhcvat42 32227* A covering property. (cvrat42 30255 analog.) (Contributed by NM, 17-Aug-2014.)
joinH

Theoremdihjatb 32228 Isomorphism H of lattice join of two atoms under the fiducial hyperplane. (Contributed by NM, 23-Sep-2014.)

Theoremdihjatc 32229 Isomorphism H of lattice join of an element under the fiducial hyperplane with atom not under it. (Contributed by NM, 26-Aug-2014.)

Theoremdihjatcclem1 32230 Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)

Theoremdihjatcclem2 32231 Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)

Theoremdihjatcclem3 32232* Lemma for dihjatcc 32234. (Contributed by NM, 28-Sep-2014.)

Theoremdihjatcclem4 32233* Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)

Theoremdihjatcc 32234 Isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)

Theoremdihjat 32235 Isomorphism H of lattice join of two atoms. (Contributed by NM, 29-Sep-2014.)

Theoremdihprrnlem1N 32236 Lemma for dihprrn 32238, showing one of 4 cases. (Contributed by NM, 30-Aug-2014.) (New usage is discouraged.)

Theoremdihprrnlem2 32237 Lemma for dihprrn 32238. (Contributed by NM, 29-Sep-2014.)

Theoremdihprrn 32238 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 32239 The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 32238; should we directly use dihjat 32235? (Contributed by NM, 13-Aug-2014.)
joinH

Theoremdihjat1lem 32240 Subspace sum of a closed subspace and an atom. (pmapjat1 30664 analog.) TODO: merge into dihjat1 32241? (Contributed by NM, 18-Aug-2014.)
joinH

Theoremdihjat1 32241 Subspace sum of a closed subspace and an atom. (pmapjat1 30664 analog.) (Contributed by NM, 1-Oct-2014.)
joinH

Theoremdihsmsprn 32242 Subspace sum of a closed subspace and the span of a singleton. (Contributed by NM, 17-Jan-2015.)

Theoremdihjat2 32243 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 32244 Isomorphism H of lattice join with an atom. (Contributed by NM, 25-Apr-2015.)

Theoremdihjat4 32245 Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
LSAtoms

Theoremdihjat6 32246 Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
LSAtoms

Theoremdihsmsnrn 32247 The subspace sum of two singleton spans is closed. (Contributed by NM, 27-Feb-2015.)

Theoremdihsmatrn 32248 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 32243. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdihjat5N 32249 Transfer lattice join with atom to subspace sum. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)

Theoremdvh4dimat 32250* There is an atom that is outside the subspace sum of 3 others. (Contributed by NM, 25-Apr-2015.)
LSAtoms

Theoremdvh3dimatN 32251* 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 32252* Given an atom, there exists another. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
LSAtoms

Theoremdvh1dimat 32253* There exists an atom. (Contributed by NM, 25-Apr-2015.)
LSAtoms

Theoremdvh1dim 32254* There exists a nonzero vector. (Contributed by NM, 26-Apr-2015.)

Theoremdvh4dimlem 32255* Lemma for dvh4dimN 32259. (Contributed by NM, 22-May-2015.)

Theoremdvhdimlem 32256* Lemma for dvh2dim 32257 and dvh3dim 32258. TODO: make this obsolete and use dvh4dimlem 32255 directly? (Contributed by NM, 24-Apr-2015.)

Theoremdvh2dim 32257* There is a vector that is outside the span of another. (Contributed by NM, 25-Apr-2015.)

Theoremdvh3dim 32258* There is a vector that is outside the span of 2 others. (Contributed by NM, 24-Apr-2015.)

Theoremdvh4dimN 32259* There is a vector that is outside the span of 3 others. (Contributed by NM, 22-May-2015.) (New usage is discouraged.)

Theoremdvh3dim2 32260* There is a vector that is outside of 2 spans with a common vector. (Contributed by NM, 13-May-2015.)

Theoremdvh3dim3N 32261* There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 32260 everywhere. If this one is needed, make dvh3dim2 32260 into a lemma. (Contributed by NM, 21-May-2015.) (New usage is discouraged.)

Theoremdochsnnz 32262 The orthocomplement of a singleton is nonzero. (Contributed by NM, 13-Jun-2015.)

Theoremdochsatshp 32263 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 32264 The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 29-Oct-2014.)
LSAtoms       LSHyp

Theoremdochsnshp 32265 The orthocomplement of a nonzero singleton is a hyperplane. (Contributed by NM, 3-Jan-2015.)
LSHyp

Theoremdochshpsat 32266 A hyperplane is closed iff its orthocomplement is an atom. (Contributed by NM, 29-Oct-2014.)
LSAtoms       LSHyp

Theoremdochkrsat 32267 The orthocomplement of a kernel is an atom iff it is nonzero. (Contributed by NM, 1-Nov-2014.)
LSAtoms       LFnl       LKer

Theoremdochkrsat2 32268 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 32269 The orthocomplement of a kernel is either an atom or zero. (Contributed by NM, 29-Jan-2015.)
LSAtoms       LFnl       LKer

Theoremdochkrsm 32270 The subspace sum of a closed subspace and a kernel orthocomplement is closed. (djhlsmcl 32226 can be used to convert sum to join.) (Contributed by NM, 29-Jan-2015.)
LFnl       LKer

Theoremdochexmidat 32271 Special case of excluded middle for the singleton of a vector. (Contributed by NM, 27-Oct-2014.)

Theoremdochexmidlem1 32272 Lemma for dochexmid 32280. Holland's proof implicitly requires , which we prove here. (Contributed by NM, 14-Jan-2015.)
LSAtoms

Theoremdochexmidlem2 32273 Lemma for dochexmid 32280. (Contributed by NM, 14-Jan-2015.)
LSAtoms

Theoremdochexmidlem3 32274 Lemma for dochexmid 32280. Use atom exchange lsatexch1 29858 to swap and . (Contributed by NM, 14-Jan-2015.)
LSAtoms

Theoremdochexmidlem4 32275 Lemma for dochexmid 32280. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdochexmidlem5 32276 Lemma for dochexmid 32280. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdochexmidlem6 32277 Lemma for dochexmid 32280. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdochexmidlem7 32278 Lemma for dochexmid 32280. Contradict dochexmidlem6 32277. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdochexmidlem8 32279 Lemma for dochexmid 32280. The contradiction of dochexmidlem6 32277 and dochexmidlem7 32278 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 32280 Excluded middle law for closed subspaces, which is equivalent to (and derived from) the orthomodular law dihoml4 32189. 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 30789 analog.) (Contributed by NM, 15-Jan-2015.)

Theoremdochsnkrlem1 32281 Lemma for dochsnkr 32284. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer

Theoremdochsnkrlem2 32282 Lemma for dochsnkr 32284. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer                            LSAtoms

Theoremdochsnkrlem3 32283 Lemma for dochsnkr 32284. (Contributed by NM, 2-Jan-2015.)
LFnl       LKer

Theoremdochsnkr 32284 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 32285* Kernel of the explicit functional determined by a nonzero vector . Compare the more general lshpkr 29929. (Contributed by NM, 27-Oct-2014.)
LKer       Scalar

Theoremdochsnkr2cl 32286* The determining functional belongs to the atom formed by the orthocomplement of the kernel. (Contributed by NM, 4-Jan-2015.)
LKer       Scalar

Theoremdochflcl 32287* Closure of the explicit functional determined by a nonzero vector . Compare the more general lshpkrcl 29928. (Contributed by NM, 27-Oct-2014.)
LFnl       Scalar

Theoremdochfl1 32288* The value of the explicit functional is 1 at the that determines it. (Contributed by NM, 27-Oct-2014.)
Scalar

Theoremdochfln0 32289 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 32290* 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 29882. (Contributed by NM, 2-Jan-2015.)
Scalar                     LFnl       LKer

Theoremdochkr1OLDN 32291* 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 29882. (Contributed by NM, 2-Jan-2015.) (New usage is discouraged.)
Scalar                     LFnl       LKer

18.27.12  Construction of involution and inner product from a Hilbert lattice

SyntaxclpoN 32292 Extend class notation with all polarities of a left module or left vector space.
LPol

Definitiondf-lpolN 32293* 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

TheoremlpolsetN 32294* 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

TheoremislpolN 32295* The predicate "is a polarity". (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)
LSAtoms       LSHyp       LPol

TheoremislpoldN 32296* Properties that determine a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LSAtoms       LSHyp       LPol

TheoremlpolfN 32297 Functionality of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LPol

TheoremlpolvN 32298 The polarity of the whole space is the zero subspace. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LPol

TheoremlpolconN 32299 Contraposition property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LPol

TheoremlpolsatN 32300 The polarity of an atomic subspace is a hyperplane. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LSAtoms       LSHyp       LPol

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