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Theorem List for Metamath Proof Explorer - 21801-21900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempjhthlem2 21801* Lemma for pjhth 21802. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theorempjhth 21802 Projection Theorem: Any Hilbert space vector can be decomposed uniquely 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, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)

Theorempjhtheu 21803* Projection Theorem: Any Hilbert space vector can be decomposed uniquely into a member of a closed subspace and a member of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102. See pjhtheu2 21825 for the uniqueness of . (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)

15.9.20  Projectors

Definitiondf-pjh 21804* Define the projection function on a Hilbert space, as a mapping from the Hilbert lattice to a function on Hilbert space. Every closed subspace is associated with a unique projection function. Remark in [Kalmbach] p. 66, adopted as a definition. is the projection of vector onto closed subspace . Note that the range of is the set of all projection operators, so means that is a projection operator. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)

Theorempjhfval 21805* The value of the projection map. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theorempjhval 21806* Value of a projection. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theorempjpreeq 21807* Equality with a projection. This version of pjeq 21808 does not assume the Axiom of Choice via pjhth 21802. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theorempjeq 21808* Equality with a projection. (Contributed by NM, 20-Jan-2007.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremaxpjcl 21809 Closure of a projection in its subspace. If we consider this together with axpjpj 21829 to be axioms, the need for the ax-hcompl 21611 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 21844.) (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theorempjhcl 21810 Closure of a projection in Hilbert space. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)

15.9.21  Orthomodular law

Theoremomlsilem 21811 Lemma for orthomodular law in the Hilbert lattice. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)

Theoremomlsii 21812 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 21813 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 21814 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 21815 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 21816 Alternate definition of the Hilbert lattice. (Contributed by NM, 8-Aug-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremococin 21817* 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 21818* 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 21819* 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 21820* 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 21821* A subspace is the supremum of all smaller subspaces. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)

Theoremchsupsn 21822 Value of supremum of subset of on a singleton. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)

Theoremshlub 21823 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 21824 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.)

15.9.22  Projectors (cont.)

Theorempjhtheu2 21825* Uniqueness of for the projection theorem. (Contributed by NM, 6-Nov-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theorempjcli 21826 Closure of a projection in its subspace. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)

Theorempjhcli 21827 Closure of a projection in Hilbert space. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)

Theorempjpjpre 21828 Decomposition of a vector into projections. This formulation of axpjpj 21829 avoids pjhth 21802. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremaxpjpj 21829 Decomposition of a vector into projections. See comment in axpjcl 21809. (Contributed by NM, 26-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theorempjclii 21830 Closure of a projection in its subspace. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)

Theorempjhclii 21831 Closure of a projection in Hilbert space. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)

Theorempjpj0i 21832 Decomposition of a vector into projections. (Contributed by NM, 26-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theorempjpji 21833 Decomposition of a vector into projections. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)

Theorempjpjhth 21834* 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 21835* 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 21836 Orthocomplement projection in terms of projection. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)

Theorempjpo 21837 Projection in terms of orthocomplement projection. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)

Theorempjopi 21838 Orthocomplement projection in terms of projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)

Theorempjpoi 21839 Projection in terms of orthocomplement projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)

Theorempjoc1i 21840 Projection of a vector in the orthocomplement of the projection subspace. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)

Theorempjchi 21841 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 21842 The part of a vector that belongs to the orthocomplemented space. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)

Theorempjoc1 21843 Projection of a vector in the orthocomplement of the projection subspace. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)

Theorempjomli 21844 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 21813. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)

Theorempjoml 21845 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 21813. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)

Theorempjococi 21846 Proof of orthocomplement theorem using projections. Compare ococ 21815. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)

Theorempjoc2i 21847 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 21848 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.)

15.9.23  Hilbert lattice operations

Theoremsh0le 21849 The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

Theoremch0le 21850 The zero subspace is the smallest member of . (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)

Theoremshle0 21851 No subspace is smaller than the zero subspace. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)

Theoremchle0 21852 No Hilbert lattice element is smaller than zero. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)

Theoremchnlen0 21853 A Hilbert lattice element that is not a subset of another is nonzero. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)

Theoremch0pss 21854 The zero subspace is a proper subset of nonzero Hilbert lattice elements. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)

Theoremorthin 21855 The intersection of orthogonal subspaces is the zero subspace. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)

Theoremssjo 21856 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 21857* A nonzero subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)

Theoremshs0i 21858 Hilbert subspace sum with the zero subspace. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)

Theoremshs00i 21859 Two subspaces are zero iff their join is zero. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)

Theoremch0lei 21860 The closed subspace zero is the smallest member of . (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchle0i 21861 No Hilbert closed subspace is smaller than zero. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)

Theoremchne0i 21862* A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)

Theoremchocini 21863 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 21864 Join with lattice zero in . (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchm1i 21865 Meet with lattice one in . (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremchjcli 21866 Closure of join. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)

Theoremchsleji 21867 Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)

Theoremchseli 21868* Membership in subspace sum. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremchincli 21869 Closure of Hilbert lattice intersection. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchsscon3i 21870 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchsscon1i 21871 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchsscon2i 21872 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchcon2i 21873 Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)

Theoremchcon1i 21874 Hilbert lattice contraposition law. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)

Theoremchcon3i 21875 Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)

Theoremchunssji 21876 Union is smaller than join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchjcomi 21877 Commutative law for join in . (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)

Theoremchub1i 21878 join is an upper bound of two elements. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchub2i 21879 join is an upper bound of two elements. (Contributed by NM, 5-Nov-2000.) (New usage is discouraged.)

Theoremchlubi 21880 Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)

Theoremchlubii 21881 Hilbert lattice join is the least upper bound of two elements (one direction of chlubi 21880). (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchlej1i 21882 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremchlej2i 21883 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremchlej12i 21884 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremchlejb1i 21885 Hilbert lattice ordering in terms of join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchdmm1i 21886 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmm2i 21887 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmm3i 21888 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmm4i 21889 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj1i 21890 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj2i 21891 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj3i 21892 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj4i 21893 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchnlei 21894 Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)

Theoremchjassi 21895 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 21896 Two Hilbert lattice elements are zero iff their join is zero. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)

Theoremchjoi 21897 The join of a closed subspace and its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremchj1i 21898 Join with Hilbert lattice unit. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)

Theoremchm0i 21899 Meet with Hilbert lattice zero. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)

Theoremchm0 21900 Meet with Hilbert lattice zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)

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