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Theorem List for Metamath Proof Explorer - 32101-32200   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremdihwN 32101* Value of isomorphism H at the fiducial hyperplane . (Contributed by NM, 25-Aug-2014.) (New usage is discouraged.)

Theoremdihmeetlem1N 32102* Isomorphism H of a conjunction. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem5apreN 32103* A conjunction property of isomorphism H. TODO: reduce antecedent size; general review for shorter proof. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem5aN 32104 A conjunction property of isomorphism H. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem2aN 32105* Lemma for isomorphism H of a GLB. (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem2N 32106* The GLB of a set of lattice elements is the same as that of the set with elements of cut down to be under . (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem3N 32107* Isomorphism H of a lattice glb. (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem3aN 32108* Isomorphism H of a lattice glb. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihglblem4 32109* Isomorphism H of a lattice glb. (Contributed by NM, 21-Mar-2014.)

Theoremdihglblem5 32110* Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)

Theoremdihmeetlem2N 32111 Isomorphism H of a conjunction. (Contributed by NM, 22-Mar-2014.) (New usage is discouraged.)

TheoremdihglbcpreN 32112* Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane . (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.)

TheoremdihglbcN 32113* Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane . (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)

TheoremdihmeetcN 32114 Isomorphism H of a lattice meet when the meet is not under the fiducial hyperplane . (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)

TheoremdihmeetbN 32115 Isomorphism H of a lattice meet when one element is under the fiducial hyperplane . (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)

TheoremdihmeetbclemN 32116 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

Theoremdihmeetlem3N 32117 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

Theoremdihmeetlem4preN 32118* Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

Theoremdihmeetlem4N 32119 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

Theoremdihmeetlem5 32120 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 6-Apr-2014.)

Theoremdihmeetlem6 32121 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.)

Theoremdihmeetlem7N 32122 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihjatc1 32123 Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change order of here and down? (Contributed by NM, 6-Apr-2014.)

Theoremdihjatc2N 32124 Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.)

Theoremdihjatc3 32125 Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.)

Theoremdihmeetlem8N 32126 Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change order of here and down? (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem9N 32127 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem10N 32128 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem11N 32129 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem12N 32130 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem13N 32131* Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem14N 32132 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem15N 32133 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem16N 32134 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem17N 32135 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem18N 32136 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem19N 32137 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem20N 32138 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

TheoremdihmeetALTN 32139 Isomorphism H of a lattice meet. This version does not depend on the atomisticity of the constructed vector space. TODO: Delete? (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdih1dimatlem0 32140* Lemma for dih1dimat 32142. (Contributed by NM, 11-Apr-2014.)
LSAtoms                                                               Scalar

Theoremdih1dimatlem 32141* Lemma for dih1dimat 32142. (Contributed by NM, 10-Apr-2014.)
LSAtoms                                                               Scalar

Theoremdih1dimat 32142 Any 1-dimensional subspace is a value of isomorphism H. (Contributed by NM, 11-Apr-2014.)
LSAtoms

Theoremdihlsprn 32143 The span of a vector belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)

TheoremdihlspsnssN 32144 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 32145 The inverse isomorphism H of the span of a singleton is a Hilbert lattice atom. (Contributed by NM, 27-Apr-2014.)

Theoremdihatlat 32146 The isomorphism H of an atom is a 1-dim subspace. (Contributed by NM, 28-Apr-2014.)
LSAtoms

Theoremdihat 32147 There exists at least one atom in the subspaces of vector space H. (Contributed by NM, 12-Aug-2014.)
LSAtoms

TheoremdihpN 32148* The value of isomorphism H at the fiducial atom is determined by the vector (the zero translation ltrnid 30946 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 32149 The reverse isomorphism H of a 1-dim subspace is an atom. (Contributed by NM, 28-Apr-2014.)
LSAtoms

Theoremdihatexv 32150* There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 16-Aug-2014.)

Theoremdihatexv2 32151* There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 17-Aug-2014.)

Theoremdihglblem6 32152* Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)
LSAtoms

Theoremdihglb 32153* Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)

Theoremdihglb2 32154* Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)

Theoremdihmeet 32155 Isomorphism H of a lattice meet. (Contributed by NM, 13-Apr-2014.)

Theoremdihintcl 32156 The intersection of closed subspaces (the range of isomorphism H) is a closed subspace. (Contributed by NM, 14-Apr-2014.)

Theoremdihmeetcl 32157 Closure of closed subspace meet for vector space. (Contributed by NM, 5-Aug-2014.)

Theoremdihmeet2 32158 Reverse isomorphism H of a closed subspace intersection. (Contributed by NM, 15-Jan-2015.)

Syntaxcoch 32159 Extend class notation with subspace orthocomplement for vector space.

Definitiondf-doch 32160* 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 32161* Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)

Theoremdochfval 32162* Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)

Theoremdochval 32163* Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)

Theoremdochval2 32164* Subspace orthocomplement for vector space. (Contributed by NM, 14-Apr-2014.)

Theoremdochcl 32165 Closure of subspace orthocomplement for vector space. (Contributed by NM, 9-Mar-2014.)

Theoremdochlss 32166 A subspace orthocomplement is a subspace of the vector space. (Contributed by NM, 22-Jul-2014.)

Theoremdochssv 32167 A subspace orthocomplement belongs to the vector space. (Contributed by NM, 22-Jul-2014.)

TheoremdochfN 32168 Domain and codomain of the subspace orthocomplement for the vector space. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)

Theoremdochvalr 32169 Orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)

Theoremdoch0 32170 Orthocomplement of the zero subspace. (Contributed by NM, 19-Jun-2014.)

Theoremdoch1 32171 Orthocomplement of the unit subspace (all vectors). (Contributed by NM, 19-Jun-2014.)

Theoremdochoc0 32172 The zero subspace is closed. (Contributed by NM, 16-Feb-2015.)

Theoremdochoc1 32173 The unit subspace (all vectors) is closed. (Contributed by NM, 16-Feb-2015.)

Theoremdochvalr2 32174 Orthocomplement of a closed subspace. (Contributed by NM, 21-Jul-2014.)

Theoremdochvalr3 32175 Orthocomplement of a closed subspace. (Contributed by NM, 15-Jan-2015.)

Theoremdoch2val2 32176* Double orthocomplement for vector space. (Contributed by NM, 26-Jul-2014.)

Theoremdochss 32177 Subset law for orthocomplement. (Contributed by NM, 16-Apr-2014.)

Theoremdochocss 32178 Double negative law for orthocomplement of an arbitrary set of vectors. (Contributed by NM, 16-Apr-2014.)

Theoremdochoc 32179 Double negative law for orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)

Theoremdochsscl 32180 If a set of vectors is included in a closed set, so is its closure. (Contributed by NM, 17-Jun-2015.)

Theoremdochoccl 32181 A set of vectors is closed iff it equals its double orthocomplent. (Contributed by NM, 1-Jan-2015.)

Theoremdochord 32182 Ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.)

Theoremdochord2N 32183 Ordering law for orthocomplement. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.)

Theoremdochord3 32184 Ordering law for orthocomplement. (Contributed by NM, 9-Mar-2015.)

Theoremdoch11 32185 Orthocomplement is one-to-one. (Contributed by NM, 12-Aug-2014.)

TheoremdochsordN 32186 Strict ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.) (New usage is discouraged.)

Theoremdochn0nv 32187 An orthocomplement is nonzero iff the double orthocomplement is not the whole vector space. (Contributed by NM, 1-Jan-2015.)

Theoremdihoml4c 32188 Version of dihoml4 32189 with closed subspaces. (Contributed by NM, 15-Jan-2015.)

Theoremdihoml4 32189 Orthomodular law for constructed vector space H. Lemma 3.3(1) in [Holland95] p. 215. (poml4N 30764 analog.) (Contributed by NM, 15-Jan-2015.)

Theoremdochspss 32190 The span of a set of vectors is included in their double orthocomplement. (Contributed by NM, 26-Jul-2014.)

Theoremdochocsp 32191 The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.)

TheoremdochspocN 32192 The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.) (New usage is discouraged.)

Theoremdochocsn 32193 The double orthocomplement of a singleton is its span. (Contributed by NM, 13-Jan-2015.)

Theoremdochsncom 32194 Swap vectors in an orthocomplement of a singleton. (Contributed by NM, 17-Jun-2015.)

Theoremdochsat 32195 The double orthocomplement of an atom is an atom. (Contributed by NM, 29-Oct-2014.)
LSAtoms

Theoremdochshpncl 32196 If a hyperplane is not closed, its closure equals the vector space. (Contributed by NM, 29-Oct-2014.)
LSHyp

Theoremdochlkr 32197 Equivalent conditions for the closure of a kernel to be a hyperplane. (Contributed by NM, 29-Oct-2014.)
LFnl       LSHyp       LKer

Theoremdochkrshp 32198 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 32199 Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LSHyp       LFnl       LKer

Theoremdochkrshp3 32200 Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer

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