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Theorem List for Metamath Proof Explorer - 34901-35000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdihord5apre 34901 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord5a 34902 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord 34903 The isomorphism H is order-preserving. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)

Theoremdih11 34904 The isomorphism H is one-to-one. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)

Theoremdihf11lem 34905 Functionality of the isomorphism H. (Contributed by NM, 6-Mar-2014.)

Theoremdihf11 34906 The isomorphism H for a lattice is a one-to-one function. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)

Theoremdihfn 34907 Functionality and domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)

Theoremdihdm 34908 Domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)

Theoremdihcl 34909 Closure of isomorphism H. (Contributed by NM, 8-Mar-2014.)

Theoremdihcnvcl 34910 Closure of isomorphism H converse. (Contributed by NM, 8-Mar-2014.)

Theoremdihcnvid1 34911 The converse isomorphism of an isomorphism. (Contributed by NM, 5-Aug-2014.)

Theoremdihcnvid2 34912 The isomorphism of a converse isomorphism. (Contributed by NM, 5-Aug-2014.)

Theoremdihcnvord 34913 Ordering property for converse of isomorphism H. (Contributed by NM, 17-Aug-2014.)

Theoremdihcnv11 34914 The converse of isomorphism H is one-to-one. (Contributed by NM, 17-Aug-2014.)

Theoremdihsslss 34915 The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)

Theoremdihrnlss 34916 The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)

Theoremdihrnss 34917 The isomorphism H maps to a set of vectors. (Contributed by NM, 14-Mar-2014.)

Theoremdihvalrel 34918 The value of isomorphism H is a relation. (Contributed by NM, 9-Mar-2014.)

Theoremdih0 34919 The value of isomorphism H at the lattice zero is the singleton of the zero vector i.e. the zero subspace. (Contributed by NM, 9-Mar-2014.)

Theoremdih0bN 34920 A lattice element is zero iff its isomorphism is the zero subspace. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.)

Theoremdih0vbN 34921 A vector is zero iff its span is the isomorphism of lattice zero. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.)

Theoremdih0cnv 34922 The isomorphism H converse value of the zero subspace is the lattice zero. (Contributed by NM, 19-Jun-2014.)

Theoremdih0rn 34923 The zero subspace belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)

Theoremdih0sb 34924 A subspace is zero iff the converse of its isomorphism is lattice zero. (Contributed by NM, 17-Aug-2014.)

Theoremdih1 34925 The value of isomorphism H at the lattice unit is the set of all vectors. (Contributed by NM, 13-Mar-2014.)

Theoremdih1rn 34926 The full vector space belongs to the range of isomorphism H. (Contributed by NM, 19-Jun-2014.)

Theoremdih1cnv 34927 The isomorphism H converse value of the full vector space is the lattice one. (Contributed by NM, 19-Jun-2014.)

TheoremdihwN 34928* Value of isomorphism H at the fiducial hyperplane . (Contributed by NM, 25-Aug-2014.) (New usage is discouraged.)

Theoremdihmeetlem1N 34929* Isomorphism H of a conjunction. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem5apreN 34930* 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 34931 A conjunction property of isomorphism H. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem2aN 34932* Lemma for isomorphism H of a GLB. (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem2N 34933* 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 34934* Isomorphism H of a lattice glb. (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem3aN 34935* Isomorphism H of a lattice glb. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihglblem4 34936* Isomorphism H of a lattice glb. (Contributed by NM, 21-Mar-2014.)

Theoremdihglblem5 34937* Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)

Theoremdihmeetlem2N 34938 Isomorphism H of a conjunction. (Contributed by NM, 22-Mar-2014.) (New usage is discouraged.)

TheoremdihglbcpreN 34939* 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 34940* 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 34941 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 34942 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 34943 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

Theoremdihmeetlem3N 34944 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

Theoremdihmeetlem4preN 34945* Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

Theoremdihmeetlem4N 34946 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

Theoremdihmeetlem5 34947 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 6-Apr-2014.)

Theoremdihmeetlem6 34948 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.)

Theoremdihmeetlem7N 34949 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihjatc1 34950 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 34951 Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.)

Theoremdihjatc3 34952 Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.)

Theoremdihmeetlem8N 34953 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 34954 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem10N 34955 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem11N 34956 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem12N 34957 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem13N 34958* Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem14N 34959 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem15N 34960 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem16N 34961 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem17N 34962 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem18N 34963 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem19N 34964 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihmeetlem20N 34965 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

TheoremdihmeetALTN 34966 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 34967* Lemma for dih1dimat 34969. (Contributed by NM, 11-Apr-2014.)
LSAtoms                                                               Scalar

Theoremdih1dimatlem 34968* Lemma for dih1dimat 34969. (Contributed by NM, 10-Apr-2014.)
LSAtoms                                                               Scalar

Theoremdih1dimat 34969 Any 1-dimensional subspace is a value of isomorphism H. (Contributed by NM, 11-Apr-2014.)
LSAtoms

Theoremdihlsprn 34970 The span of a vector belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)

TheoremdihlspsnssN 34971 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 34972 The inverse isomorphism H of the span of a singleton is a Hilbert lattice atom. (Contributed by NM, 27-Apr-2014.)

Theoremdihatlat 34973 The isomorphism H of an atom is a 1-dim subspace. (Contributed by NM, 28-Apr-2014.)
LSAtoms

Theoremdihat 34974 There exists at least one atom in the subspaces of vector space H. (Contributed by NM, 12-Aug-2014.)
LSAtoms

TheoremdihpN 34975* The value of isomorphism H at the fiducial atom is determined by the vector (the zero translation ltrnid 33771 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 34976 The reverse isomorphism H of a 1-dim subspace is an atom. (Contributed by NM, 28-Apr-2014.)
LSAtoms

Theoremdihatexv 34977* There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 16-Aug-2014.)

Theoremdihatexv2 34978* There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 17-Aug-2014.)

Theoremdihglblem6 34979* Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)
LSAtoms

Theoremdihglb 34980* Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)

Theoremdihglb2 34981* Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)

Theoremdihmeet 34982 Isomorphism H of a lattice meet. (Contributed by NM, 13-Apr-2014.)

Theoremdihintcl 34983 The intersection of closed subspaces (the range of isomorphism H) is a closed subspace. (Contributed by NM, 14-Apr-2014.)

Theoremdihmeetcl 34984 Closure of closed subspace meet for vector space. (Contributed by NM, 5-Aug-2014.)

Theoremdihmeet2 34985 Reverse isomorphism H of a closed subspace intersection. (Contributed by NM, 15-Jan-2015.)

Syntaxcoch 34986 Extend class notation with subspace orthocomplement for vector space.

Definitiondf-doch 34987* 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 34988* Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)

Theoremdochfval 34989* Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)

Theoremdochval 34990* Subspace orthocomplement for vector space. (Contributed by NM, 14-Mar-2014.)

Theoremdochval2 34991* Subspace orthocomplement for vector space. (Contributed by NM, 14-Apr-2014.)

Theoremdochcl 34992 Closure of subspace orthocomplement for vector space. (Contributed by NM, 9-Mar-2014.)

Theoremdochlss 34993 A subspace orthocomplement is a subspace of the vector space. (Contributed by NM, 22-Jul-2014.)

Theoremdochssv 34994 A subspace orthocomplement belongs to the vector space. (Contributed by NM, 22-Jul-2014.)

TheoremdochfN 34995 Domain and codomain of the subspace orthocomplement for the vector space. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)

Theoremdochvalr 34996 Orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)

Theoremdoch0 34997 Orthocomplement of the zero subspace. (Contributed by NM, 19-Jun-2014.)

Theoremdoch1 34998 Orthocomplement of the unit subspace (all vectors). (Contributed by NM, 19-Jun-2014.)

Theoremdochoc0 34999 The zero subspace is closed. (Contributed by NM, 16-Feb-2015.)

Theoremdochoc1 35000 The unit subspace (all vectors) is closed. (Contributed by NM, 16-Feb-2015.)

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