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

Theoremdih1dimatlem 32201* Lemma for dih1dimat 32202. (Contributed by NM, 10-Apr-2014.)
LSAtoms                                                               Scalar

Theoremdih1dimat 32202 Any 1-dimensional subspace is a value of isomorphism H. (Contributed by NM, 11-Apr-2014.)
LSAtoms

Theoremdihlsprn 32203 The span of a vector belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)

TheoremdihlspsnssN 32204 A subspace included in a 1-dim subspace belongs to the range of isomorphism H. (Contributed by NM, 26-Apr-2014.) (New usage is discouraged.)

Theoremdihlspsnat 32205 The inverse isomorphism H of the span of a singleton is a Hilbert lattice atom. (Contributed by NM, 27-Apr-2014.)

Theoremdihatlat 32206 The isomorphism H of an atom is a 1-dim subspace. (Contributed by NM, 28-Apr-2014.)
LSAtoms

Theoremdihat 32207 There exists at least one atom in the subspaces of vector space H. (Contributed by NM, 12-Aug-2014.)
LSAtoms

TheoremdihpN 32208* The value of isomorphism H at the fiducial atom is determined by the vector (the zero translation ltrnid 31006 and a nonzero member of the endomorphism ring). In particular, can be replaced with the ring unit . (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.)

Theoremdihlatat 32209 The reverse isomorphism H of a 1-dim subspace is an atom. (Contributed by NM, 28-Apr-2014.)
LSAtoms

Theoremdihatexv 32210* There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 16-Aug-2014.)

Theoremdihatexv2 32211* There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 17-Aug-2014.)

Theoremdihglblem6 32212* Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)
LSAtoms

Theoremdihglb 32213* Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)

Theoremdihglb2 32214* Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)

Theoremdihmeet 32215 Isomorphism H of a lattice meet. (Contributed by NM, 13-Apr-2014.)

Theoremdihintcl 32216 The intersection of closed subspaces (the range of isomorphism H) is a closed subspace. (Contributed by NM, 14-Apr-2014.)

Theoremdihmeetcl 32217 Closure of closed subspace meet for vector space. (Contributed by NM, 5-Aug-2014.)

Theoremdihmeet2 32218 Reverse isomorphism H of a closed subspace intersection. (Contributed by NM, 15-Jan-2015.)

Syntaxcoch 32219 Extend class notation with subspace orthocomplement for vector space.

Definitiondf-doch 32220* Define subspace orthocomplement for vector space. Temporarily, we are using the range of the isomorphism instead of the set of closed subspaces. Later, when inner product is introduced, we will show that these are the same. (Contributed by NM, 14-Mar-2014.)

Theoremdochffval 32221* Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)

Theoremdochfval 32222* Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)

Theoremdochval 32223* Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)

Theoremdochval2 32224* Subspace orthocomplement for vector space. (Contributed by NM, 14-Apr-2014.)

Theoremdochcl 32225 Closure of subspace orthocomplement for vector space. (Contributed by NM, 9-Mar-2014.)

Theoremdochlss 32226 A subspace orthocomplement is a subspace of the vector space. (Contributed by NM, 22-Jul-2014.)

Theoremdochssv 32227 A subspace orthocomplement belongs to the vector space. (Contributed by NM, 22-Jul-2014.)

TheoremdochfN 32228 Domain and codomain of the subspace orthocomplement for the vector space. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)

Theoremdochvalr 32229 Orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)

Theoremdoch0 32230 Orthocomplement of the zero subspace. (Contributed by NM, 19-Jun-2014.)

Theoremdoch1 32231 Orthocomplement of the unit subspace (all vectors). (Contributed by NM, 19-Jun-2014.)

Theoremdochoc0 32232 The zero subspace is closed. (Contributed by NM, 16-Feb-2015.)

Theoremdochoc1 32233 The unit subspace (all vectors) is closed. (Contributed by NM, 16-Feb-2015.)

Theoremdochvalr2 32234 Orthocomplement of a closed subspace. (Contributed by NM, 21-Jul-2014.)

Theoremdochvalr3 32235 Orthocomplement of a closed subspace. (Contributed by NM, 15-Jan-2015.)

Theoremdoch2val2 32236* Double orthocomplement for vector space. (Contributed by NM, 26-Jul-2014.)

Theoremdochss 32237 Subset law for orthocomplement. (Contributed by NM, 16-Apr-2014.)

Theoremdochocss 32238 Double negative law for orthocomplement of an arbitrary set of vectors. (Contributed by NM, 16-Apr-2014.)

Theoremdochoc 32239 Double negative law for orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)

Theoremdochsscl 32240 If a set of vectors is included in a closed set, so is its closure. (Contributed by NM, 17-Jun-2015.)

Theoremdochoccl 32241 A set of vectors is closed iff it equals its double orthocomplent. (Contributed by NM, 1-Jan-2015.)

Theoremdochord 32242 Ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.)

Theoremdochord2N 32243 Ordering law for orthocomplement. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.)

Theoremdochord3 32244 Ordering law for orthocomplement. (Contributed by NM, 9-Mar-2015.)

Theoremdoch11 32245 Orthocomplement is one-to-one. (Contributed by NM, 12-Aug-2014.)

TheoremdochsordN 32246 Strict ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.) (New usage is discouraged.)

Theoremdochn0nv 32247 An orthocomplement is nonzero iff the double orthocomplement is not the whole vector space. (Contributed by NM, 1-Jan-2015.)

Theoremdihoml4c 32248 Version of dihoml4 32249 with closed subspaces. (Contributed by NM, 15-Jan-2015.)

Theoremdihoml4 32249 Orthomodular law for constructed vector space H. Lemma 3.3(1) in [Holland95] p. 215. (poml4N 30824 analog.) (Contributed by NM, 15-Jan-2015.)

Theoremdochspss 32250 The span of a set of vectors is included in their double orthocomplement. (Contributed by NM, 26-Jul-2014.)

Theoremdochocsp 32251 The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.)

TheoremdochspocN 32252 The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.) (New usage is discouraged.)

Theoremdochocsn 32253 The double orthocomplement of a singleton is its span. (Contributed by NM, 13-Jan-2015.)

Theoremdochsncom 32254 Swap vectors in an orthocomplement of a singleton. (Contributed by NM, 17-Jun-2015.)

Theoremdochsat 32255 The double orthocomplement of an atom is an atom. (Contributed by NM, 29-Oct-2014.)
LSAtoms

Theoremdochshpncl 32256 If a hyperplane is not closed, its closure equals the vector space. (Contributed by NM, 29-Oct-2014.)
LSHyp

Theoremdochlkr 32257 Equivalent conditions for the closure of a kernel to be a hyperplane. (Contributed by NM, 29-Oct-2014.)
LFnl       LSHyp       LKer

Theoremdochkrshp 32258 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 32259 Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LSHyp       LFnl       LKer

Theoremdochkrshp3 32260 Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer

Theoremdochkrshp4 32261 Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer

Theoremdochdmj1 32262 De Morgan-like law for subspace orthocomplement. (Contributed by NM, 5-Aug-2014.)

Theoremdochnoncon 32263 Law of noncontradiction. The intersection of a subspace and its orthocomplement is the zero subspace. (Contributed by NM, 16-Apr-2014.)

Theoremdochnel2 32264 A nonzero member of a subspace doesn't belong to the orthocomplement of the subspace. (Contributed by NM, 28-Feb-2015.)

Theoremdochnel 32265 A nonzero vector doesn't belong to the orthocomplement of its singleton. (Contributed by NM, 27-Oct-2014.)

Syntaxcdjh 32266 Extend class notation with subspace join for vector space.
joinH

Definitiondf-djh 32267* Define (closed) subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theoremdjhffval 32268* Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theoremdjhfval 32269* Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theoremdjhval 32270 Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theoremdjhval2 32271 Value of subspace join for vector space. (Contributed by NM, 6-Aug-2014.)
joinH

Theoremdjhcl 32272 Closure of subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theoremdjhlj 32273 Transfer lattice join to vector space closed subspace join. (Contributed by NM, 19-Jul-2014.)
joinH

TheoremdjhljjN 32274 Lattice join in terms of vector space closed subspace join. (Contributed by NM, 17-Aug-2014.) (New usage is discouraged.)
joinH

Theoremdjhjlj 32275 vector space closed subspace join in terms of lattice join. (Contributed by NM, 9-Aug-2014.)
joinH

Theoremdjhj 32276 vector space closed subspace join in terms of lattice join. (Contributed by NM, 17-Aug-2014.)
joinH

Theoremdjhcom 32277 Subspace join commutes. (Contributed by NM, 8-Aug-2014.)
joinH

Theoremdjhspss 32278 Subspace span of union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
joinH

Theoremdjhsumss 32279 Subspace sum is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
joinH

Theoremdihsumssj 32280 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 32281 Subspace union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.) (New usage is discouraged.)
joinH

Theoremdochdmm1 32282 De Morgan-like law for closed subspace orthocomplement. (Contributed by NM, 13-Jan-2015.)
joinH

Theoremdjhexmid 32283 Excluded middle property of vector space closed subspace join. (Contributed by NM, 22-Jul-2014.)
joinH

Theoremdjh01 32284 Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
joinH

Theoremdjh02 32285 Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
joinH

Theoremdjhlsmcl 32286 A closed subspace sum equals subspace join. (shjshseli 23000 analog.) (Contributed by NM, 13-Aug-2014.)
joinH

Theoremdjhcvat42 32287* A covering property. (cvrat42 30315 analog.) (Contributed by NM, 17-Aug-2014.)
joinH

Theoremdihjatb 32288 Isomorphism H of lattice join of two atoms under the fiducial hyperplane. (Contributed by NM, 23-Sep-2014.)

Theoremdihjatc 32289 Isomorphism H of lattice join of an element under the fiducial hyperplane with atom not under it. (Contributed by NM, 26-Aug-2014.)

Theoremdihjatcclem1 32290 Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)

Theoremdihjatcclem2 32291 Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)

Theoremdihjatcclem3 32292* Lemma for dihjatcc 32294. (Contributed by NM, 28-Sep-2014.)

Theoremdihjatcclem4 32293* Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)

Theoremdihjatcc 32294 Isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)

Theoremdihjat 32295 Isomorphism H of lattice join of two atoms. (Contributed by NM, 29-Sep-2014.)

Theoremdihprrnlem1N 32296 Lemma for dihprrn 32298, showing one of 4 cases. (Contributed by NM, 30-Aug-2014.) (New usage is discouraged.)

Theoremdihprrnlem2 32297 Lemma for dihprrn 32298. (Contributed by NM, 29-Sep-2014.)

Theoremdihprrn 32298 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 32299 The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 32298; should we directly use dihjat 32295? (Contributed by NM, 13-Aug-2014.)
joinH

Theoremdihjat1lem 32300 Subspace sum of a closed subspace and an atom. (pmapjat1 30724 analog.) TODO: merge into dihjat1 32301? (Contributed by NM, 18-Aug-2014.)
joinH

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