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Theorem List for Metamath Proof Explorer - 21801-21900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremchss 21801 A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.)
 |-  ( H  e.  CH  ->  H  C_ 
 ~H )
 
Theoremchel 21802 A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  H ) 
 ->  A  e.  ~H )
 
Theoremchssii 21803 A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  H  C_  ~H
 
Theoremcheli 21804 A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( A  e.  H  ->  A  e.  ~H )
 
Theoremchelii 21805 A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  H   =>    |-  A  e.  ~H
 
Theoremchlimi 21806 The limit property of a closed subspace of a Hilbert space. (Contributed by NM, 14-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( ( H  e.  CH 
 /\  F : NN --> H  /\  F  ~~>v  A ) 
 ->  A  e.  H )
 
Theoremhlim0 21807 The zero sequence in Hilbert space converges to the zero vector. (Contributed by NM, 17-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( NN  X.  { 0h }
 )  ~~>v  0h
 
Theoremhlimcaui 21808 If a sequence in Hilbert space subset converges to a limit, it is a Cauchy sequence. (Contributed by NM, 17-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( F  ~~>v  A  ->  F  e.  Cauchy )
 
Theoremhlimf 21809 Function-like behavior of the convergence relation. (Contributed by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ~~>v  : dom  ~~>v  --> ~H
 
Theoremhlimuni 21810 A Hilbert space sequence converges to at most one limit. (Contributed by NM, 19-Aug-1999.) (Revised by Mario Carneiro, 2-May-2015.) (New usage is discouraged.)
 |-  (
 ( F  ~~>v  A  /\  F  ~~>v  B )  ->  A  =  B )
 
Theoremhlimreui 21811* The limit of a Hilbert space sequence is unique. (Contributed by NM, 19-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( E. x  e.  H  F  ~~>v  x  <->  E! x  e.  H  F  ~~>v  x )
 
Theoremhlimeui 21812* The limit of a Hilbert space sequence is unique. (Contributed by NM, 19-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( E. x  F  ~~>v  x  <->  E! x  F  ~~>v  x )
 
Theoremisch3 21813* A Hilbert subspace is closed iff it is complete. A complete subspace is one in which every Cauchy sequence of vectors in the subspace converges to a member of the subspace (Definition of complete subspace in [Beran] p. 96). Remark 3.12 of [Beran] p. 107. (Contributed by NM, 24-Dec-2001.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( H  e.  CH  <->  ( H  e.  SH  /\  A. f  e.  Cauchy  ( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) ) )
 
Theoremchcompl 21814* Completeness of a closed subspace of Hilbert space. (Contributed by NM, 4-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  F  e.  Cauchy  /\  F : NN --> H )  ->  E. x  e.  H  F  ~~>v  x )
 
Theoremhelch 21815 The unit Hilbert lattice element (which is all of Hilbert space) belongs to the Hilbert lattice. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 6-Sep-1999.) (New usage is discouraged.)
 |-  ~H  e.  CH
 
Theoremhelsh 21816 Hilbert space is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  ~H  e.  SH
 
Theoremshsspwh 21817 Subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
 |-  SH  C_ 
 ~P ~H
 
Theoremchsspwh 21818 Closed subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
 |-  CH  C_  ~P ~H
 
Theoremhsn0elch 21819 The zero subspace belongs to the set of closed subspaces of Hilbert space. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
 |-  { 0h }  e.  CH
 
Theoremnorm1 21820 From any nonzero Hilbert space vector, construct a vector whose norm is 1. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  A  =/=  0h )  ->  ( normh `  ( (
 1  /  ( normh `  A ) )  .h  A ) )  =  1 )
 
Theoremnorm1exi 21821* A normalized vector exists in a subspace iff the subspace has a nonzero vector. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)
 |-  H  e.  SH   =>    |-  ( E. x  e.  H  x  =/=  0h  <->  E. y  e.  H  ( normh `  y )  =  1 )
 
Theoremnorm1hex 21822 A normalized vector can exist only iff the Hilbert space has a nonzero vector. (Contributed by NM, 21-Jan-2006.) (New usage is discouraged.)
 |-  ( E. x  e.  ~H  x  =/=  0h  <->  E. y  e.  ~H  ( normh `  y )  =  1 )
 
17.4.3  Orthocomplements
 
Definitiondf-oc 21823* Define orthogonal complement of a subset (usually a subspace) of Hilbert space. The orthogonal complement is the set of all vectors orthogonal to all vectors in the subset. See ocval 21851 and chocvali 21870 for its value. Textbooks usually denote this unary operation with the symbol  _|_ as a small superscript, although Mittelstaedt uses the symbol as a prefix operation. Here we define a function (prefix operation)  _|_ rather than introducing a new syntactical form. This lets us take advantage of the theorems about functions that we already have proved under set theory. Definition of [Mittelstaedt] p. 9. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
 |-  _|_  =  ( x  e.  ~P ~H  |->  { y  e.  ~H  |  A. z  e.  x  ( y  .ih  z )  =  0 } )
 
Definitiondf-ch0 21824 Define the zero for closed subspaces of Hilbert space. See h0elch 21826 for closure law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  0H  =  { 0h }
 
Theoremelch0 21825 Membership in zero for closed subspaces of Hilbert space. (Contributed by NM, 6-Apr-2001.) (New usage is discouraged.)
 |-  ( A  e.  0H  <->  A  =  0h )
 
Theoremh0elch 21826 The zero subspace is a closed subspace. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  0H  e.  CH
 
Theoremh0elsh 21827 The zero subspace is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  0H  e.  SH
 
Theoremhhssva 21828 The vector addition operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   =>    |-  (  +h  |`  ( H  X.  H ) )  =  ( +v `  W )
 
Theoremhhsssm 21829 The scalar multiplication operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   =>    |-  (  .h  |`  ( CC 
 X.  H ) )  =  ( .s OLD `  W )
 
Theoremhhssnm 21830 The norm operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   =>    |-  ( normh  |`  H )  =  ( normCV `  W )
 
Theoremhhssabloi 21831 Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  H  e.  SH   =>    |-  (  +h  |`  ( H  X.  H ) )  e.  AbelOp
 
Theoremhhssablo 21832 Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)
 |-  ( H  e.  SH  ->  (  +h  |`  ( H  X.  H ) )  e.  AbelOp )
 
Theoremhhssnv 21833 Normed complex vector space property of a subspace. (Contributed by NM, 26-Mar-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  H  e.  SH   =>    |-  W  e.  NrmCVec
 
Theoremhhssnvt 21834 Normed complex vector space property of a subspace. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   =>    |-  ( H  e.  SH  ->  W  e.  NrmCVec )
 
Theoremhhsst 21835 A member of  SH is a subspace. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) ) >. ,  ( normh  |`  H )
 >.   =>    |-  ( H  e.  SH  ->  W  e.  ( SubSp `  U ) )
 
Theoremhhshsslem1 21836 Lemma for hhsssh 21838. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) ) >. ,  ( normh  |`  H )
 >.   &    |-  W  e.  ( SubSp `  U )   &    |-  H  C_  ~H   =>    |-  H  =  (
 BaseSet `  W )
 
Theoremhhshsslem2 21837 Lemma for hhsssh 21838. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) ) >. ,  ( normh  |`  H )
 >.   &    |-  W  e.  ( SubSp `  U )   &    |-  H  C_  ~H   =>    |-  H  e.  SH
 
Theoremhhsssh 21838 The predicate " H is a subspace of Hilbert space." (Contributed by NM, 25-Mar-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) ) >. ,  ( normh  |`  H )
 >.   =>    |-  ( H  e.  SH  <->  ( W  e.  ( SubSp `  U )  /\  H  C_ 
 ~H ) )
 
Theoremhhsssh2 21839 The predicate " H is a subspace of Hilbert space." (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   =>    |-  ( H  e.  SH  <->  ( W  e.  NrmCVec  /\  H  C_  ~H ) )
 
Theoremhhssba 21840 The base set of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  H  e.  SH   =>    |-  H  =  (
 BaseSet `  W )
 
Theoremhhssvs 21841 The vector subtraction operation on a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  H  e.  SH   =>    |-  (  -h  |`  ( H  X.  H ) )  =  ( -v `  W )
 
Theoremhhssvsf 21842 Mapping of the vector subtraction operation on a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  H  e.  SH   =>    |-  (  -h  |`  ( H  X.  H ) ) : ( H  X.  H ) --> H
 
Theoremhhssph 21843 Inner product space property of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  H  e.  SH   =>    |-  W  e.  CPreHil OLD
 
Theoremhhssims 21844 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  H  e.  SH   &    |-  D  =  ( ( normh  o.  -h  )  |`  ( H  X.  H ) )   =>    |-  D  =  (
 IndMet `  W )
 
Theoremhhssims2 21845 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  D  =  ( IndMet `  W )   &    |-  H  e.  SH   =>    |-  D  =  ( ( normh  o.  -h  )  |`  ( H  X.  H ) )
 
Theoremhhssmet 21846 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  D  =  ( IndMet `  W )   &    |-  H  e.  SH   =>    |-  D  e.  ( Met `  H )
 
Theoremhhssmetdval 21847 Value of the distance function of the metric space of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  D  =  ( IndMet `  W )   &    |-  H  e.  SH   =>    |-  (
 ( A  e.  H  /\  B  e.  H ) 
 ->  ( A D B )  =  ( normh `  ( A  -h  B ) ) )
 
Theoremhhsscms 21848 The induced metric of a closed subspace is complete. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  D  =  ( IndMet `  W )   &    |-  H  e.  CH   =>    |-  D  e.  ( CMet `  H )
 
Theoremhhssbn 21849 Banach space property of a closed subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  H  e.  CH   =>    |-  W  e.  CBan
 
Theoremhhsshl 21850 Hilbert space property of a closed subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  H  e.  CH   =>    |-  W  e.  CHil OLD
 
Theoremocval 21851* Value of orthogonal complement of a subset of Hilbert space. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ( H  C_  ~H  ->  ( _|_ `  H )  =  { x  e.  ~H  |  A. y  e.  H  ( x  .ih  y )  =  0 } )
 
Theoremocel 21852* Membership in orthogonal complement of H subset. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
 |-  ( H  C_  ~H  ->  ( A  e.  ( _|_ `  H )  <->  ( A  e.  ~H 
 /\  A. x  e.  H  ( A  .ih  x )  =  0 ) ) )
 
Theoremshocel 21853* Membership in orthogonal complement of H subspace. (Contributed by NM, 9-Oct-1999.) (New usage is discouraged.)
 |-  ( H  e.  SH  ->  ( A  e.  ( _|_ `  H )  <->  ( A  e.  ~H 
 /\  A. x  e.  H  ( A  .ih  x )  =  0 ) ) )
 
Theoremocsh 21854 The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
 |-  ( A  C_  ~H  ->  ( _|_ `  A )  e. 
 SH )
 
Theoremshocsh 21855 The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  SH  ->  ( _|_ `  A )  e.  SH )
 
Theoremocss 21856 An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
 |-  ( A  C_  ~H  ->  ( _|_ `  A )  C_  ~H )
 
Theoremshocss 21857 An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  SH  ->  ( _|_ `  A )  C_ 
 ~H )
 
Theoremoccon 21858 Contraposition law for orthogonal complement. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.)
 |-  (
 ( A  C_  ~H  /\  B  C_  ~H )  ->  ( A  C_  B  ->  ( _|_ `  B )  C_  ( _|_ `  A ) ) )
 
Theoremoccon2 21859 Double contraposition for orthogonal complement. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
 |-  (
 ( A  C_  ~H  /\  B  C_  ~H )  ->  ( A  C_  B  ->  ( _|_ `  ( _|_ `  A ) )  C_  ( _|_ `  ( _|_ `  B ) ) ) )
 
Theoremoccon2i 21860 Double contraposition for orthogonal complement. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
 |-  A  C_ 
 ~H   &    |-  B  C_  ~H   =>    |-  ( A  C_  B  ->  ( _|_ `  ( _|_ `  A ) ) 
 C_  ( _|_ `  ( _|_ `  B ) ) )
 
Theoremoc0 21861 The zero vector belongs to an orthogonal complement of a Hilbert subspace. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
 |-  ( H  e.  SH  ->  0h  e.  ( _|_ `  H ) )
 
Theoremocorth 21862 Members of a subset and its complement are orthogonal. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
 |-  ( H  C_  ~H  ->  (
 ( A  e.  H  /\  B  e.  ( _|_ `  H ) )  ->  ( A  .ih  B )  =  0 ) )
 
Theoremshocorth 21863 Members of a subspace and its complement are orthogonal. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
 |-  ( H  e.  SH  ->  ( ( A  e.  H  /\  B  e.  ( _|_ `  H ) )  ->  ( A  .ih  B )  =  0 ) )
 
Theoremococss 21864 Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
 |-  ( A  C_  ~H  ->  A  C_  ( _|_ `  ( _|_ `  A ) ) )
 
Theoremshococss 21865 Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  SH  ->  A 
 C_  ( _|_ `  ( _|_ `  A ) ) )
 
Theoremshorth 21866 Members of orthogonal subspaces are orthogonal. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
 |-  ( H  e.  SH  ->  ( G  C_  ( _|_ `  H )  ->  (
 ( A  e.  G  /\  B  e.  H ) 
 ->  ( A  .ih  B )  =  0 )
 ) )
 
Theoremocin 21867 Intersection of a Hilbert subspace and its complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  SH  ->  ( A  i^i  ( _|_ `  A ) )  =  0H )
 
Theoremoccon3 21868 Hilbert lattice contraposition law. (Contributed by Mario Carneiro, 18-May-2014.) (New usage is discouraged.)
 |-  (
 ( A  C_  ~H  /\  B  C_  ~H )  ->  ( A  C_  ( _|_ `  B )  <->  B  C_  ( _|_ `  A ) ) )
 
Theoremocnel 21869 A nonzero vector in the complement of a subspace does not belong to the subspace. (Contributed by NM, 10-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( H  e.  SH  /\  A  e.  ( _|_ `  H )  /\  A  =/=  0h )  ->  -.  A  e.  H )
 
Theoremchocvali 21870* Value of the orthogonal complement of a Hilbert lattice element. The orthogonal complement of  A is the set of vectors that are orthogonal to all vectors in  A. (Contributed by NM, 8-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( _|_ `  A )  =  { x  e.  ~H  |  A. y  e.  A  ( x  .ih  y )  =  0 }
 
Theoremshuni 21871 Two subspaces with trivial intersection have a unique decomposition of the elements of the subspace sum. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  H  e.  SH )   &    |-  ( ph  ->  K  e.  SH )   &    |-  ( ph  ->  ( H  i^i  K )  =  0H )   &    |-  ( ph  ->  A  e.  H )   &    |-  ( ph  ->  B  e.  K )   &    |-  ( ph  ->  C  e.  H )   &    |-  ( ph  ->  D  e.  K )   &    |-  ( ph  ->  ( A  +h  B )  =  ( C  +h  D ) )   =>    |-  ( ph  ->  ( A  =  C  /\  B  =  D )
 )
 
Theoremchocunii 21872 Lemma for uniqueness part of Projection Theorem. Theorem 3.7(i) of [Beran] p. 102 (uniqueness part). (Contributed by NM, 23-Oct-1999.) (Proof shortened by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( ( ( A  e.  H  /\  B  e.  ( _|_ `  H ) )  /\  ( C  e.  H  /\  D  e.  ( _|_ `  H ) ) )  ->  ( ( R  =  ( A  +h  B ) 
 /\  R  =  ( C  +h  D ) )  ->  ( A  =  C  /\  B  =  D ) ) )
 
Theorempjhthmo 21873* Projection Theorem, uniqueness part. Any two disjoint subspaces yield a unique decomposition of vectors into each subspace. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH  /\  ( A  i^i  B )  =  0H )  ->  E* x ( x  e.  A  /\  E. y  e.  B  C  =  ( x  +h  y ) ) )
 
Theoremoccllem 21874 Lemma for occl 21875. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  A  C_  ~H )   &    |-  ( ph  ->  F  e.  Cauchy )   &    |-  ( ph  ->  F : NN
 --> ( _|_ `  A ) )   &    |-  ( ph  ->  B  e.  A )   =>    |-  ( ph  ->  ( (  ~~>v  `  F )  .ih  B )  =  0 )
 
Theoremoccl 21875 Closure of complement of Hilbert subset. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 8-Aug-2000.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( A  C_  ~H  ->  ( _|_ `  A )  e. 
 CH )
 
Theoremshoccl 21876 Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 13-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  SH  ->  ( _|_ `  A )  e.  CH )
 
Theoremchoccl 21877 Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( _|_ `  A )  e. 
 CH )
 
Theoremchoccli 21878 Closure of  CH orthocomplement. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( _|_ `  A )  e.  CH
 
17.4.4  Subspace sum, span, lattice join, lattice supremum
 
Definitiondf-shs 21879* Define subspace sum in  SH. See shsval 21883, shsval2i 21958, and shsval3i 21959 for its value. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
 |-  +H  =  ( x  e.  SH ,  y  e.  SH  |->  (  +h  " ( x  X.  y ) ) )
 
Definitiondf-span 21880* Define the linear span of a subset of Hilbert space. Definition of span in [Schechter] p. 276. See spanval 21904 for its value. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  span  =  ( x  e.  ~P ~H  |->  |^| { y  e. 
 SH  |  x  C_  y } )
 
Definitiondf-chj 21881* Define Hilbert lattice join. See chjval 21923 for its value and chjcl 21928 for its closure law. Note that we define it over all Hilbert space subsets to allow proving more general theorems. Even for general subsets the join belongs to  CH; see sshjcl 21926. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.)
 |-  vH  =  ( x  e.  ~P ~H ,  y  e.  ~P ~H  |->  ( _|_ `  ( _|_ `  ( x  u.  y ) ) ) )
 
Definitiondf-chsup 21882 Define the supremum of a set of Hilbert lattice elements. See chsupval2 21981 for its value. We actually define the supremum for an arbitrary collection of Hilbert space subsets, not just elements of the Hilbert lattice  CH, to allow more general theorems. Even for general subsets the supremum still a Hilbert lattice element; see hsupcl 21910. (Contributed by NM, 9-Dec-2003.) (New usage is discouraged.)
 |-  \/H  =  ( x  e.  ~P ~P ~H  |->  ( _|_ `  ( _|_ `  U. x ) ) )
 
Theoremshsval 21883 Value of subspace sum of two Hilbert space subspaces. Definition of subspace sum in [Kalmbach] p. 65. (Contributed by NM, 16-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B )  =  (  +h  " ( A  X.  B ) ) )
 
Theoremshsss 21884 The subspace sum is a subset of Hilbert space. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B )  C_  ~H )
 
Theoremshsel 21885* Membership in the subspace sum of two Hilbert subspaces. (Contributed by NM, 14-Dec-2004.) (Revised by Mario Carneiro, 29-Jan-2014.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( C  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  C  =  ( x  +h  y ) ) )
 
Theoremshsel3 21886* Membership in the subspace sum of two Hilbert subspaces, using vector subtraction. (Contributed by NM, 20-Jan-2007.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( C  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  C  =  ( x  -h  y ) ) )
 
Theoremshseli 21887* Membership in subspace sum. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( C  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  C  =  ( x  +h  y ) )
 
Theoremshscli 21888 Closure of subspace sum. (Contributed by NM, 15-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( A  +H  B )  e. 
 SH
 
Theoremshscl 21889 Closure of subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B )  e.  SH )
 
Theoremshscom 21890 Commutative law for subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B )  =  ( B  +H  A ) )
 
Theoremshsva 21891 Vector sum belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( ( C  e.  A  /\  D  e.  B )  ->  ( C  +h  D )  e.  ( A  +H  B ) ) )
 
Theoremshsel1 21892 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( C  e.  A  ->  C  e.  ( A  +H  B ) ) )
 
Theoremshsel2 21893 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( C  e.  B  ->  C  e.  ( A  +H  B ) ) )
 
Theoremshsvs 21894 Vector subtraction belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( ( C  e.  A  /\  D  e.  B )  ->  ( C  -h  D )  e.  ( A  +H  B ) ) )
 
Theoremshsub1 21895 Subspace sum is an upper bound of its arguments. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  A  C_  ( A  +H  B ) )
 
Theoremshsub2 21896 Subspace sum is an upper bound of its arguments. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  A  C_  ( B  +H  A ) )
 
Theoremchoc0 21897 The orthocomplement of the zero subspace is the unit subspace. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  ( _|_ `  0H )  =  ~H
 
Theoremchoc1 21898 The orthocomplement of the unit subspace is the zero subspace. Does not require Axiom of Choice. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  ( _|_ `  ~H )  =  0H
 
Theoremchocnul 21899 Orthogonal complement of the empty set. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.)
 |-  ( _|_ `  (/) )  =  ~H
 
Theoremshintcli 21900 Closure of intersection of a non-empty subset of  SH. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
 |-  ( A  C_  SH  /\  A  =/= 
 (/) )   =>    |- 
 |^| A  e.  SH
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