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Theorem List for Metamath Proof Explorer - 22901-23000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempjeq 22901* Equality with a projection. (Contributed by NM, 20-Jan-2007.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremaxpjcl 22902 Closure of a projection in its subspace. If we consider this together with axpjpj 22922 to be axioms, the need for the ax-hcompl 22704 can often be avoided for the kinds of theorems we are interested in here. An interesting project is to see how far we can go by using them in place of it. In particular, we can prove the orthomodular law pjomli 22937.) (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theorempjhcl 22903 Closure of a projection in Hilbert space. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)

18.5  Properties of Hilbert subspaces

18.5.1  Orthomodular law

Theoremomlsilem 22904 Lemma for orthomodular law in the Hilbert lattice. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)

Theoremomlsii 22905 Subspace inference form of orthomodular law in the Hilbert lattice. (Contributed by NM, 14-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremomlsi 22906 Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of [Kalmbach] p. 22. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)

Theoremococi 22907 Complement of complement of a closed subspace of Hilbert space. Theorem 3.7(ii) of [Beran] p. 102. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)

Theoremococ 22908 Complement of complement of a closed subspace of Hilbert space. Theorem 3.7(ii) of [Beran] p. 102. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)

Theoremdfch2 22909 Alternate definition of the Hilbert lattice. (Contributed by NM, 8-Aug-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremococin 22910* The double complement is the smallest closed subspace containing a subset of Hilbert space. Remark 3.12(B) of [Beran] p. 107. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.)

Theoremhsupval2 22911* Alternate definition of supremum of a subset of the Hilbert lattice. Definition of supremum in Proposition 1 of [Kalmbach] p. 65. We actually define it on any collection of Hilbert space subsets, not just the Hilbert lattice , to allow more general theorems. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)

Theoremchsupval2 22912* The value of the supremum of a set of closed subspaces of Hilbert space. Definition of supremum in Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)

Theoremsshjval2 22913* Value of join in the set of closed subspaces of Hilbert space . (Contributed by NM, 1-Nov-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremchsupid 22914* A subspace is the supremum of all smaller subspaces. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)

Theoremchsupsn 22915 Value of supremum of subset of on a singleton. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)

Theoremshlub 22916 Hilbert lattice join is the least upper bound (among Hilbert lattice elements) of two subspaces. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)

Theoremshlubi 22917 Hilbert lattice join is the least upper bound (among Hilbert lattice elements) of two subspaces. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)

18.5.2  Projectors (cont.)

Theorempjhtheu2 22918* Uniqueness of for the projection theorem. (Contributed by NM, 6-Nov-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theorempjcli 22919 Closure of a projection in its subspace. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)

Theorempjhcli 22920 Closure of a projection in Hilbert space. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)

Theorempjpjpre 22921 Decomposition of a vector into projections. This formulation of axpjpj 22922 avoids pjhth 22895. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremaxpjpj 22922 Decomposition of a vector into projections. See comment in axpjcl 22902. (Contributed by NM, 26-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theorempjclii 22923 Closure of a projection in its subspace. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)

Theorempjhclii 22924 Closure of a projection in Hilbert space. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)

Theorempjpj0i 22925 Decomposition of a vector into projections. (Contributed by NM, 26-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theorempjpji 22926 Decomposition of a vector into projections. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)

Theorempjpjhth 22927* Projection Theorem: Any Hilbert space vector can be decomposed into a member of a closed subspace and a member of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part). (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)

Theorempjpjhthi 22928* Projection Theorem: Any Hilbert space vector can be decomposed into a member of a closed subspace and a member of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part). (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)

Theorempjop 22929 Orthocomplement projection in terms of projection. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)

Theorempjpo 22930 Projection in terms of orthocomplement projection. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)

Theorempjopi 22931 Orthocomplement projection in terms of projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)

Theorempjpoi 22932 Projection in terms of orthocomplement projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)

Theorempjoc1i 22933 Projection of a vector in the orthocomplement of the projection subspace. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)

Theorempjchi 22934 Projection of a vector in the projection subspace. Lemma 4.4(ii) of [Beran] p. 111. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)

Theorempjoccl 22935 The part of a vector that belongs to the orthocomplemented space. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)

Theorempjoc1 22936 Projection of a vector in the orthocomplement of the projection subspace. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)

Theorempjomli 22937 Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of [Kalmbach] p. 22. Derived using projections; compare omlsi 22906. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)

Theorempjoml 22938 Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of [Kalmbach] p. 22. Derived using projections; compare omlsi 22906. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)

Theorempjococi 22939 Proof of orthocomplement theorem using projections. Compare ococ 22908. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)

Theorempjoc2i 22940 Projection of a vector in the orthocomplement of the projection subspace. Lemma 4.4(iii) of [Beran] p. 111. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)

Theorempjoc2 22941 Projection of a vector in the orthocomplement of the projection subspace. Lemma 4.4(iii) of [Beran] p. 111. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

18.5.3  Hilbert lattice operations

Theoremsh0le 22942 The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

Theoremch0le 22943 The zero subspace is the smallest member of . (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)

Theoremshle0 22944 No subspace is smaller than the zero subspace. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)

Theoremchle0 22945 No Hilbert lattice element is smaller than zero. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)

Theoremchnlen0 22946 A Hilbert lattice element that is not a subset of another is nonzero. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)

Theoremch0pss 22947 The zero subspace is a proper subset of nonzero Hilbert lattice elements. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)

Theoremorthin 22948 The intersection of orthogonal subspaces is the zero subspace. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)

Theoremssjo 22949 The lattice join of a subset with its orthocomplement is the whole space. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremshne0i 22950* A nonzero subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)

Theoremshs0i 22951 Hilbert subspace sum with the zero subspace. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)

Theoremshs00i 22952 Two subspaces are zero iff their join is zero. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)

Theoremch0lei 22953 The closed subspace zero is the smallest member of . (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchle0i 22954 No Hilbert closed subspace is smaller than zero. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)

Theoremchne0i 22955* A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)

Theoremchocini 22956 Intersection of a closed subspace and its orthocomplement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)

Theoremchj0i 22957 Join with lattice zero in . (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchm1i 22958 Meet with lattice one in . (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremchjcli 22959 Closure of join. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)

Theoremchsleji 22960 Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)

Theoremchseli 22961* Membership in subspace sum. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremchincli 22962 Closure of Hilbert lattice intersection. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchsscon3i 22963 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchsscon1i 22964 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchsscon2i 22965 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchcon2i 22966 Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)

Theoremchcon1i 22967 Hilbert lattice contraposition law. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)

Theoremchcon3i 22968 Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)

Theoremchunssji 22969 Union is smaller than join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchjcomi 22970 Commutative law for join in . (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)

Theoremchub1i 22971 join is an upper bound of two elements. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchub2i 22972 join is an upper bound of two elements. (Contributed by NM, 5-Nov-2000.) (New usage is discouraged.)

Theoremchlubi 22973 Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)

Theoremchlubii 22974 Hilbert lattice join is the least upper bound of two elements (one direction of chlubi 22973). (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchlej1i 22975 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremchlej2i 22976 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremchlej12i 22977 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremchlejb1i 22978 Hilbert lattice ordering in terms of join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchdmm1i 22979 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmm2i 22980 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmm3i 22981 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmm4i 22982 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj1i 22983 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj2i 22984 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj3i 22985 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj4i 22986 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchnlei 22987 Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)

Theoremchjassi 22988 Associative law for Hilbert lattice join. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)

Theoremchj00i 22989 Two Hilbert lattice elements are zero iff their join is zero. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)

Theoremchjoi 22990 The join of a closed subspace and its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremchj1i 22991 Join with Hilbert lattice unit. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)

Theoremchm0i 22992 Meet with Hilbert lattice zero. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)

Theoremchm0 22993 Meet with Hilbert lattice zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)

Theoremshjshsi 22994 Hilbert lattice join equals the double orthocomplement of subspace sum. (Contributed by NM, 27-Nov-2004.) (New usage is discouraged.)

Theoremshjshseli 22995 A closed subspace sum equals Hilbert lattice join. Part of Lemma 31.1.5 of [MaedaMaeda] p. 136. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.)

Theoremchne0 22996* A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)

Theoremchocin 22997 Intersection of a closed subspace and its orthocomplement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)

Theoremchssoc 22998 A closed subspace less than its orthocomplement is zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)

Theoremchj0 22999 Join with Hilbert lattice zero. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Theoremchslej 23000 Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 12-Jul-2004.) (New usage is discouraged.)

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