HomeHome Metamath Proof Explorer
Theorem List (p. 228 of 315)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21490)
  Hilbert Space Explorer  Hilbert Space Explorer
(21491-23013)
  Users' Mathboxes  Users' Mathboxes
(23014-31421)
 

Theorem List for Metamath Proof Explorer - 22701-22800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremleopsq 22701 The square of a Hermitian operator is positive. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
 |-  ( T  e.  HrmOp  ->  0hop  <_op  ( T  o.  T ) )
 
Theorem0leop 22702 The zero operator is a positive operator. (The literature calls it "positive," even though in some sense it is really "nonnegative.") Part of Example 12.2(i) in [Young] p. 142. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
 |-  0hop  <_op  0hop
 
Theoremidleop 22703 The identity operator is a positive operator. Part of Example 12.2(i) in [Young] p. 142. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
 |-  0hop  <_op  Iop
 
Theoremleopadd 22704 The sum of two positive operators is positive. Exercise 1(i) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
 |-  (
 ( ( T  e.  HrmOp  /\  U  e.  HrmOp )  /\  ( 0hop  <_op  T  /\  0hop  <_op  U ) )  ->  0hop  <_op  ( T  +op  U ) )
 
Theoremleopmuli 22705 The scalar product of a nonnegative real and a positive operator is a positive operator. Exercise 1(ii) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  RR  /\  T  e.  HrmOp ) 
 /\  ( 0  <_  A  /\  0hop  <_op  T ) )  ->  0hop  <_op  ( A  .op  T ) )
 
Theoremleopmul 22706 The scalar product of a positive real and a positive operator is a positive operator. Exercise 1(ii) of [Retherford] p. 49. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  RR  /\  T  e.  HrmOp  /\  0  <  A )  ->  ( 0hop  <_op  T  <->  0hop  <_op  ( A  .op  T ) ) )
 
Theoremleopmul2i 22707 Scalar product applied to operator ordering. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  RR  /\  T  e.  HrmOp  /\  U  e.  HrmOp )  /\  ( 0  <_  A  /\  T  <_op  U )
 )  ->  ( A  .op  T )  <_op  ( A  .op  U ) )
 
Theoremleoptri 22708 The positive operator ordering relation satisfies trichotomy. Exercise 1(iii) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  HrmOp  /\  U  e.  HrmOp )  ->  ( ( T  <_op  U 
 /\  U  <_op  T )  <->  T  =  U )
 )
 
Theoremleoptr 22709 The positive operator ordering relation is transitive. Exercise 1(iv) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
 |-  (
 ( ( S  e.  HrmOp  /\  T  e.  HrmOp  /\  U  e.  HrmOp )  /\  ( S  <_op  T  /\  T  <_op  U ) )  ->  S  <_op  U )
 
Theoremleopnmid 22710 A bounded Hermitian operator is less than or equal to its norm times the identity operator. (Contributed by NM, 11-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  ->  T  <_op  ( ( normop `  T )  .op  Iop  ) )
 
Theoremnmopleid 22711 A nonzero, bounded Hermitian operator divided by its norm is less than or equal to the identity operator. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR  /\  T  =/=  0hop
 )  ->  ( (
 1  /  ( normop `  T ) )  .op  T ) 
 <_op  Iop  )
 
Theoremopsqrlem1 22712* Lemma for opsqri . (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.)
 |-  T  e.  HrmOp   &    |-  ( normop `  T )  e.  RR   &    |-  0hop  <_op  T   &    |-  R  =  ( ( 1  /  ( normop `  T )
 )  .op  T )   &    |-  ( T  =/=  0hop  ->  E. u  e.  HrmOp  ( 0hop  <_op  u  /\  ( u  o.  u )  =  R )
 )   =>    |-  ( T  =/=  0hop  ->  E. v  e.  HrmOp  ( 0hop  <_op 
 v  /\  ( v  o.  v )  =  T ) )
 
Theoremopsqrlem2 22713* Lemma for opsqri .  F `  N is the recursive function An (starting at n=1 instead of 0) of Theorem 9.4-2 of [Kreyszig] p. 476. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
 |-  T  e.  HrmOp   &    |-  S  =  ( x  e.  HrmOp ,  y  e. 
 HrmOp  |->  ( x  +op  ( ( 1  / 
 2 )  .op  ( T  -op  ( x  o.  x ) ) ) ) )   &    |-  F  =  seq  1 ( S ,  ( NN  X.  { 0hop } ) )   =>    |-  ( F `  1
 )  =  0hop
 
Theoremopsqrlem3 22714* Lemma for opsqri . (Contributed by NM, 22-Aug-2006.) (New usage is discouraged.)
 |-  T  e.  HrmOp   &    |-  S  =  ( x  e.  HrmOp ,  y  e. 
 HrmOp  |->  ( x  +op  ( ( 1  / 
 2 )  .op  ( T  -op  ( x  o.  x ) ) ) ) )   &    |-  F  =  seq  1 ( S ,  ( NN  X.  { 0hop } ) )   =>    |-  ( ( G  e.  HrmOp  /\  H  e.  HrmOp )  ->  ( G S H )  =  ( G  +op  ( ( 1  / 
 2 )  .op  ( T  -op  ( G  o.  G ) ) ) ) )
 
Theoremopsqrlem4 22715* Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
 |-  T  e.  HrmOp   &    |-  S  =  ( x  e.  HrmOp ,  y  e. 
 HrmOp  |->  ( x  +op  ( ( 1  / 
 2 )  .op  ( T  -op  ( x  o.  x ) ) ) ) )   &    |-  F  =  seq  1 ( S ,  ( NN  X.  { 0hop } ) )   =>    |-  F : NN --> HrmOp
 
Theoremopsqrlem5 22716* Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
 |-  T  e.  HrmOp   &    |-  S  =  ( x  e.  HrmOp ,  y  e. 
 HrmOp  |->  ( x  +op  ( ( 1  / 
 2 )  .op  ( T  -op  ( x  o.  x ) ) ) ) )   &    |-  F  =  seq  1 ( S ,  ( NN  X.  { 0hop } ) )   =>    |-  ( N  e.  NN  ->  ( F `  ( N  +  1 )
 )  =  ( ( F `  N ) 
 +op  ( ( 1 
 /  2 )  .op  ( T  -op  ( ( F `  N )  o.  ( F `  N ) ) ) ) ) )
 
Theoremopsqrlem6 22717* Lemma for opsqri . (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
 |-  T  e.  HrmOp   &    |-  S  =  ( x  e.  HrmOp ,  y  e. 
 HrmOp  |->  ( x  +op  ( ( 1  / 
 2 )  .op  ( T  -op  ( x  o.  x ) ) ) ) )   &    |-  F  =  seq  1 ( S ,  ( NN  X.  { 0hop } ) )   &    |-  T  <_op  Iop   =>    |-  ( N  e.  NN  ->  ( F `  N )  <_op  Iop  )
 
17.6.16  Projectors as operators
 
Theorempjhmopi 22718 A projector is a Hermitian operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( proj  h `  H )  e.  HrmOp
 
Theorempjlnopi 22719 A projector is a linear operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( proj  h `  H )  e.  LinOp
 
Theorempjnmopi 22720 The operator norm of a projector on a nonzero closed subspace is one. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( H  =/=  0H  ->  ( normop `  ( proj  h `
  H ) )  =  1 )
 
Theorempjbdlni 22721 A projector is a bounded linear operator. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( proj  h `  H )  e.  BndLinOp
 
Theorempjhmop 22722 A projection is a Hermitian operator. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
 |-  ( H  e.  CH  ->  ( proj  h `  H )  e.  HrmOp )
 
Theoremhmopidmchi 22723 An idempotent Hermitian operator generates a closed subspace. Part of proof of Theorem of [AkhiezerGlazman] p. 64. (Contributed by NM, 21-Apr-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  T  e.  HrmOp   &    |-  ( T  o.  T )  =  T   =>    |- 
 ran  T  e.  CH
 
Theoremhmopidmpji 22724 An idempotent Hermitian operator is a projection operator. Theorem 26.4 of [Halmos] p. 44. (Halmos seems to omit the proof that  H is a closed subspace, which is not trivial as hmopidmchi 22723 shows.) (Contributed by NM, 22-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  T  e.  HrmOp   &    |-  ( T  o.  T )  =  T   =>    |-  T  =  ( proj  h `
  ran  T )
 
Theoremhmopidmch 22725 An idempotent Hermitian operator generates a closed subspace. Part of proof of Theorem of [AkhiezerGlazman] p. 64. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  HrmOp  /\  ( T  o.  T )  =  T )  ->  ran  T  e.  CH )
 
Theoremhmopidmpj 22726 An idempotent Hermitian operator is a projection operator. Theorem 26.4 of [Halmos] p. 44. (Contributed by NM, 22-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  HrmOp  /\  ( T  o.  T )  =  T )  ->  T  =  ( proj  h `
  ran  T )
 )
 
Theorempjsdii 22727 Distributive law for Hilbert space operator sum. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( ( proj  h `
  H )  o.  ( S  +op  T ) )  =  (
 ( ( proj  h `  H )  o.  S )  +op  ( ( proj  h `
  H )  o.  T ) )
 
Theorempjddii 22728 Distributive law for Hilbert space operator difference. (Contributed by NM, 24-Nov-2000.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( ( proj  h `
  H )  o.  ( S  -op  T ) )  =  (
 ( ( proj  h `  H )  o.  S )  -op  ( ( proj  h `
  H )  o.  T ) )
 
Theorempjsdi2i 22729 Chained distributive law for Hilbert space operator difference. (Contributed by NM, 30-Nov-2000.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  R : ~H --> ~H   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( ( R  o.  ( S  +op  T ) )  =  ( ( R  o.  S ) 
 +op  ( R  o.  T ) )  ->  ( ( ( proj  h `
  H )  o.  R )  o.  ( S  +op  T ) )  =  ( ( ( ( proj  h `  H )  o.  R )  o.  S )  +op  (
 ( ( proj  h `  H )  o.  R )  o.  T ) ) )
 
Theorempjcoi 22730 Composition of projections. (Contributed by NM, 16-Aug-2000.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( A  e.  ~H  ->  ( ( ( proj  h `  G )  o.  ( proj  h `  H ) ) `  A )  =  ( ( proj  h `
  G ) `  ( ( proj  h `  H ) `  A ) ) )
 
Theorempjcocli 22731 Closure of composition of projections. (Contributed by NM, 29-Nov-2000.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( A  e.  ~H  ->  ( ( ( proj  h `  G )  o.  ( proj  h `  H ) ) `  A )  e.  G )
 
Theorempjcohcli 22732 Closure of composition of projections. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( A  e.  ~H  ->  ( ( ( proj  h `  G )  o.  ( proj  h `  H ) ) `  A )  e.  ~H )
 
Theorempjadjcoi 22733 Adjoint of composition of projections. Special case of Theorem 3.11(viii) of [Beran] p. 106. (Contributed by NM, 6-Oct-2000.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( ( (
 proj  h `  G )  o.  ( proj  h `  H ) ) `  A )  .ih  B )  =  ( A  .ih  ( ( ( proj  h `
  H )  o.  ( proj  h `  G ) ) `  B ) ) )
 
Theorempjcofni 22734 Functionality of composition of projections. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  (
 ( proj  h `  G )  o.  ( proj  h `  H ) )  Fn 
 ~H
 
Theorempjss1coi 22735 Subset relationship for projections. Theorem 4.5(i)<->(iii) of [Beran] p. 112. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( G  C_  H  <->  ( ( proj  h `
  H )  o.  ( proj  h `  G ) )  =  ( proj  h `  G ) )
 
Theorempjss2coi 22736 Subset relationship for projections. Theorem 4.5(i)<->(ii) of [Beran] p. 112. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( G  C_  H  <->  ( ( proj  h `
  G )  o.  ( proj  h `  H ) )  =  ( proj  h `  G ) )
 
Theorempjssmi 22737 Projection meet property. Remark in [Kalmbach] p. 66. Also Theorem 4.5(i)->(iv) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( A  e.  ~H  ->  ( H  C_  G  ->  ( ( ( proj  h `  G ) `  A )  -h  ( ( proj  h `
  H ) `  A ) )  =  ( ( proj  h `  ( G  i^i  ( _|_ `  H ) ) ) `
  A ) ) )
 
Theorempjssge0i 22738 Theorem 4.5(iv)->(v) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( A  e.  ~H  ->  ( ( ( ( proj  h `
  G ) `  A )  -h  (
 ( proj  h `  H ) `  A ) )  =  ( ( proj  h `
  ( G  i^i  ( _|_ `  H )
 ) ) `  A )  ->  0  <_  (
 ( ( ( proj  h `
  G ) `  A )  -h  (
 ( proj  h `  H ) `  A ) ) 
 .ih  A ) ) )
 
Theorempjdifnormi 22739 Theorem 4.5(v)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( A  e.  ~H  ->  ( 0  <_  ( (
 ( ( proj  h `  G ) `  A )  -h  ( ( proj  h `
  H ) `  A ) )  .ih  A )  <->  ( normh `  (
 ( proj  h `  H ) `  A ) ) 
 <_  ( normh `  ( ( proj  h `  G ) `
  A ) ) ) )
 
Theorempjnormssi 22740* Theorem 4.5(i)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( G  C_  H  <->  A. x  e.  ~H  ( normh `  ( ( proj  h `  G ) `
  x ) ) 
 <_  ( normh `  ( ( proj  h `  H ) `
  x ) ) )
 
Theorempjorthcoi 22741 Composition of projections of orthogonal subspaces. Part (i)->(iia) of Theorem 27.4 of [Halmos] p. 45. (Contributed by NM, 6-Nov-2000.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( G  C_  ( _|_ `  H )  ->  ( ( proj  h `
  G )  o.  ( proj  h `  H ) )  =  0hop )
 
Theorempjscji 22742 The projection of orthogonal subspaces is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( G  C_  ( _|_ `  H )  ->  ( proj  h `  ( G  vH  H ) )  =  ( (
 proj  h `  G ) 
 +op  ( proj  h `  H ) ) )
 
Theorempjssumi 22743 The projection on a subspace sum is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( G  C_  ( _|_ `  H )  ->  ( proj  h `  ( G  +H  H ) )  =  ( (
 proj  h `  G ) 
 +op  ( proj  h `  H ) ) )
 
Theorempjssposi 22744* Projector ordering can be expressed by the subset relationship between their projection subspaces. (i)<->(iii) of Theorem 29.2 of [Halmos] p. 48. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( A. x  e.  ~H  0  <_  ( ( ( ( proj  h `  H )  -op  ( proj  h `  G ) ) `  x )  .ih  x )  <->  G  C_  H )
 
Theorempjordi 22745* The definition of projector ordering in [Halmos] p. 42 is equivalent to the definition of projector ordering in [Beran] p. 110. (We will usually express projector ordering with the even simpler equivalent  G  C_  H; see pjssposi 22744). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( A. x  e.  ~H  0  <_  ( ( ( ( proj  h `  H )  -op  ( proj  h `  G ) ) `  x )  .ih  x )  <-> 
 ( ( proj  h `  G ) " ~H )  C_  ( ( proj  h `
  H ) " ~H ) )
 
Theorempjssdif2i 22746 The projection subspace of the difference between two projectors. Part 2 of Theorem 29.3 of [Halmos] p. 48 (shortened with pjssposi 22744). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( G  C_  H  <->  ( ( proj  h `
  H )  -op  ( proj  h `  G ) )  =  ( proj  h `  ( H  i^i  ( _|_ `  G ) ) ) )
 
Theorempjssdif1i 22747 A necessary and sufficient condition for the difference between two projectors to be a projector. Part 1 of Theorem 29.3 of [Halmos] p. 48 (shortened with pjssposi 22744). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( G  C_  H  <->  ( ( proj  h `
  H )  -op  ( proj  h `  G ) )  e.  ran  proj  h )
 
Theorempjimai 22748 The image of a projection. Lemma 5 in Daniel Lehmann, "A presentation of Quantum Logic based on an and then connective" http://www.arxiv.org/pdf/quant-ph/0701113 p. 20. (Contributed by NM, 20-Jan-2007.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  CH   =>    |-  (
 ( proj  h `  B ) " A )  =  ( ( A  +H  ( _|_ `  B )
 )  i^i  B )
 
Theorempjidmcoi 22749 A projection is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( ( proj  h `  H )  o.  ( proj  h `  H ) )  =  ( proj  h `
  H )
 
Theorempjoccoi 22750 Composition of projections of a subspace and its orthocomplement. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( ( proj  h `  H )  o.  ( proj  h `  ( _|_ `  H ) ) )  =  0hop
 
Theorempjtoi 22751 Subspace sum of projection and projection of orthocomplement. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( ( proj  h `  H )  +op  ( proj  h `
  ( _|_ `  H ) ) )  =  ( proj  h `  ~H )
 
Theorempjoci 22752 Projection of orthocomplement. First part of Theorem 27.3 of [Halmos] p. 45. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( ( proj  h `  ~H )  -op  ( proj  h `
  H ) )  =  ( proj  h `  ( _|_ `  H )
 )
 
Theorempjidmco 22753 A projection operator is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
 |-  ( H  e.  CH  ->  (
 ( proj  h `  H )  o.  ( proj  h `  H ) )  =  ( proj  h `  H ) )
 
Theoremdfpjop 22754 Definition of projection operator in [Hughes] p. 47, except that we do not need linearity to be explicit by virtue of hmoplin 22514. (Contributed by NM, 24-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  ( T  e.  ran  proj  h  <->  ( T  e.  HrmOp  /\  ( T  o.  T )  =  T )
 )
 
Theorempjhmopidm 22755 Two ways to express the set of all projection operators. (Contributed by NM, 24-Apr-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  ran  proj  h  =  { t  e.  HrmOp  |  ( t  o.  t )  =  t }
 
Theoremelpjidm 22756 A projection operator is idempotent. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
 |-  ( T  e.  ran  proj  h  ->  ( T  o.  T )  =  T )
 
Theoremelpjhmop 22757 A projection operator is Hermitian. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
 |-  ( T  e.  ran  proj  h  ->  T  e.  HrmOp )
 
Theorem0leopj 22758 A projector is a positive operator. (Contributed by NM, 27-Sep-2008.) (New usage is discouraged.)
 |-  ( T  e.  ran  proj  h  ->  0hop  <_op  T )
 
Theorempjadj2 22759 A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
 |-  ( T  e.  ran  proj  h  ->  (
 adjh `  T )  =  T )
 
Theorempjadj3 22760 A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
 |-  ( H  e.  CH  ->  ( adjh `  ( proj  h `  H ) )  =  ( proj  h `  H ) )
 
Theoremelpjch 22761 Reconstruction of the subspace of a projection operator. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
 |-  ( T  e.  ran  proj  h  ->  ( ran  T  e.  CH  /\  T  =  ( proj  h `
  ran  T )
 ) )
 
Theoremelpjrn 22762* Reconstruction of the subspace of a projection operator. (Contributed by NM, 24-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  ( T  e.  ran  proj  h  ->  ran 
 T  =  { x  e.  ~H  |  ( T `
  x )  =  x } )
 
Theorempjinvari 22763 A closed subspace  H with projection  T is invariant under an operator  S iff  S T  =  T S T. Theorem 27.1 of [Halmos] p. 45. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  H  e.  CH   &    |-  T  =  ( proj  h `  H )   =>    |-  ( ( S  o.  T ) : ~H --> H 
 <->  ( S  o.  T )  =  ( T  o.  ( S  o.  T ) ) )
 
Theorempjin1i 22764 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( proj  h `  ( G  i^i  H ) )  =  ( ( proj  h `
  G )  o.  ( proj  h `  ( G  i^i  H ) ) )
 
Theorempjin2i 22765 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  (
 ( ( proj  h `  G )  =  (
 ( proj  h `  G )  o.  ( proj  h `  H ) )  /\  ( proj  h `  H )  =  ( ( proj  h `  H )  o.  ( proj  h `  G ) ) )  <-> 
 ( proj  h `  G )  =  ( proj  h `
  H ) )
 
Theorempjin3i 22766 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
 |-  F  e.  CH   &    |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( ( ( proj  h `
  F )  =  ( ( proj  h `  F )  o.  ( proj  h `  G ) )  /\  ( proj  h `
  F )  =  ( ( proj  h `  F )  o.  ( proj  h `  H ) ) )  <->  ( proj  h `  F )  =  (
 ( proj  h `  F )  o.  ( proj  h `  ( G  i^i  H ) ) ) )
 
Theorempjclem1 22767 Lemma for projection commutation theorem. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( G  C_H  H  ->  (
 ( proj  h `  G )  o.  ( proj  h `  H ) )  =  ( proj  h `  ( G  i^i  H ) ) )
 
Theorempjclem2 22768 Lemma for projection commutation theorem. (Contributed by NM, 17-Nov-2000.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( G  C_H  H  ->  (
 ( proj  h `  G )  o.  ( proj  h `  H ) )  =  ( ( proj  h `  H )  o.  ( proj  h `  G ) ) )
 
Theorempjclem3 22769 Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  (
 ( ( proj  h `  G )  o.  ( proj  h `  H ) )  =  ( (
 proj  h `  H )  o.  ( proj  h `  G ) )  ->  ( ( proj  h `  G )  o.  ( proj  h `  ( _|_ `  H ) ) )  =  ( ( proj  h `
  ( _|_ `  H ) )  o.  ( proj  h `  G ) ) )
 
Theorempjclem4a 22770 Lemma for projection commutation theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( A  e.  ( G  i^i  H )  ->  (
 ( ( proj  h `  G )  o.  ( proj  h `  H ) ) `  A )  =  A )
 
Theorempjclem4 22771 Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  (
 ( ( proj  h `  G )  o.  ( proj  h `  H ) )  =  ( (
 proj  h `  H )  o.  ( proj  h `  G ) )  ->  ( ( proj  h `  G )  o.  ( proj  h `  H ) )  =  ( proj  h `
  ( G  i^i  H ) ) )
 
Theorempjci 22772 Two subspaces commute iff their projections commute. Lemma 4 of [Kalmbach] p. 67. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( G  C_H  H  <->  ( ( proj  h `
  G )  o.  ( proj  h `  H ) )  =  (
 ( proj  h `  H )  o.  ( proj  h `  G ) ) )
 
Theorempjcmul1i 22773 A necessary and sufficient condition for the product of two projectors to be a projector is that the projectors commute. Part 1 of Theorem 1 of [AkhiezerGlazman] p. 65. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  (
 ( ( proj  h `  G )  o.  ( proj  h `  H ) )  =  ( (
 proj  h `  H )  o.  ( proj  h `  G ) )  <->  ( ( proj  h `
  G )  o.  ( proj  h `  H ) )  e.  ran  proj  h )
 
Theorempjcmul2i 22774 The projection subspace of the difference between two projectors. Part 2 of Theorem 1 of [AkhiezerGlazman] p. 65. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  (
 ( ( proj  h `  G )  o.  ( proj  h `  H ) )  =  ( (
 proj  h `  H )  o.  ( proj  h `  G ) )  <->  ( ( proj  h `
  G )  o.  ( proj  h `  H ) )  =  ( proj  h `  ( G  i^i  H ) ) )
 
Theorempjcohocli 22775 Closure of composition of projection and Hilbert space operator. (Contributed by NM, 3-Dec-2000.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  T : ~H --> ~H   =>    |-  ( A  e.  ~H  ->  ( ( ( proj  h `
  H )  o.  T ) `  A )  e.  H )
 
Theorempjadj2coi 22776 Adjoint of double composition of projections. Generalization of special case of Theorem 3.11(viii) of [Beran] p. 106. (Contributed by NM, 1-Dec-2000.) (New usage is discouraged.)
 |-  F  e.  CH   &    |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( ( A  e.  ~H 
 /\  B  e.  ~H )  ->  ( ( ( ( ( proj  h `  F )  o.  ( proj  h `  G ) )  o.  ( proj  h `
  H ) ) `
  A )  .ih  B )  =  ( A 
 .ih  ( ( ( ( proj  h `  H )  o.  ( proj  h `  G ) )  o.  ( proj  h `  F ) ) `  B ) ) )
 
Theorempj2cocli 22777 Closure of double composition of projections. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
 |-  F  e.  CH   &    |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( A  e.  ~H  ->  ( ( ( (
 proj  h `  F )  o.  ( proj  h `  G ) )  o.  ( proj  h `  H ) ) `  A )  e.  F )
 
Theorempj3lem1 22778 Lemma for projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
 |-  F  e.  CH   &    |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( A  e.  (
 ( F  i^i  G )  i^i  H )  ->  ( ( ( (
 proj  h `  F )  o.  ( proj  h `  G ) )  o.  ( proj  h `  H ) ) `  A )  =  A )
 
Theorempj3si 22779 Stronger projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
 |-  F  e.  CH   &    |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( ( ( ( ( proj  h `  F )  o.  ( proj  h `  G ) )  o.  ( proj  h `  H ) )  =  (
 ( ( proj  h `  H )  o.  ( proj  h `  G ) )  o.  ( proj  h `
  F ) ) 
 /\  ran  ( (
 ( proj  h `  F )  o.  ( proj  h `  G ) )  o.  ( proj  h `  H ) )  C_  G ) 
 ->  ( ( ( proj  h `
  F )  o.  ( proj  h `  G ) )  o.  ( proj  h `  H ) )  =  ( proj  h `
  ( ( F  i^i  G )  i^i 
 H ) ) )
 
Theorempj3i 22780 Projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
 |-  F  e.  CH   &    |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( ( ( ( ( proj  h `  F )  o.  ( proj  h `  G ) )  o.  ( proj  h `  H ) )  =  (
 ( ( proj  h `  H )  o.  ( proj  h `  G ) )  o.  ( proj  h `
  F ) ) 
 /\  ( ( (
 proj  h `  F )  o.  ( proj  h `  G ) )  o.  ( proj  h `  H ) )  =  (
 ( ( proj  h `  G )  o.  ( proj  h `  F ) )  o.  ( proj  h `
  H ) ) )  ->  ( (
 ( proj  h `  F )  o.  ( proj  h `  G ) )  o.  ( proj  h `  H ) )  =  ( proj  h `  ( ( F  i^i  G )  i^i  H ) ) )
 
Theorempj3cor1i 22781 Projection triplet corollary. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
 |-  F  e.  CH   &    |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( ( ( ( ( proj  h `  F )  o.  ( proj  h `  G ) )  o.  ( proj  h `  H ) )  =  (
 ( ( proj  h `  H )  o.  ( proj  h `  G ) )  o.  ( proj  h `
  F ) ) 
 /\  ( ( (
 proj  h `  F )  o.  ( proj  h `  G ) )  o.  ( proj  h `  H ) )  =  (
 ( ( proj  h `  G )  o.  ( proj  h `  F ) )  o.  ( proj  h `
  H ) ) )  ->  ( (
 ( proj  h `  F )  o.  ( proj  h `  G ) )  o.  ( proj  h `  H ) )  =  (
 ( ( proj  h `  H )  o.  ( proj  h `  F ) )  o.  ( proj  h `
  G ) ) )
 
Theorempjs14i 22782 Theorem S-14 of Watanabe, p. 486. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( A  e.  ~H  ->  (
 normh `  ( ( (
 proj  h `  H )  o.  ( proj  h `  G ) ) `  A ) )  <_  ( normh `  ( ( proj  h `  G ) `
  A ) ) )
 
17.7  States on an Hilbert lattice and Godowski's equation
 
17.7.1  States on a Hilbert lattice
 
Definitiondf-st 22783* Define the set of states on a Hilbert lattice. Definition of [Kalmbach] p. 266. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
 |-  States  =  {
 f  e.  ( ( 0 [,] 1 ) 
 ^m  CH )  |  ( ( f `  ~H )  =  1  /\  A. x  e.  CH  A. y  e.  CH  ( x  C_  ( _|_ `  y )  ->  ( f `  ( x  vH  y ) )  =  ( ( f `
  x )  +  ( f `  y
 ) ) ) ) }
 
Definitiondf-hst 22784* Define the set of complex Hilbert-space-valued states on a Hilbert lattice. Definition of CH-states in [Mayet3] p. 9. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
 |-  CHStates  =  {
 f  e.  ( ~H 
 ^m  CH )  |  ( ( normh `  ( f `  ~H ) )  =  1  /\  A. x  e.  CH  A. y  e. 
 CH  ( x  C_  ( _|_ `  y )  ->  ( ( ( f `
  x )  .ih  ( f `  y
 ) )  =  0 
 /\  ( f `  ( x  vH  y ) )  =  ( ( f `  x )  +h  ( f `  y ) ) ) ) ) }
 
Theoremisst 22785* Property of a state. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ( S  e.  States  <->  ( S : CH
 --> ( 0 [,] 1
 )  /\  ( S ` 
 ~H )  =  1 
 /\  A. x  e.  CH  A. y  e.  CH  ( x  C_  ( _|_ `  y
 )  ->  ( S `  ( x  vH  y
 ) )  =  ( ( S `  x )  +  ( S `  y ) ) ) ) )
 
Theoremishst 22786* Property of a complex Hilbert-space-valued state. Definition of CH-states in [Mayet3] p. 9. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
 |-  ( S  e.  CHStates  <->  ( S : CH
 --> ~H  /\  ( normh `  ( S `  ~H ) )  =  1  /\  A. x  e.  CH  A. y  e.  CH  ( x  C_  ( _|_ `  y
 )  ->  ( (
 ( S `  x )  .ih  ( S `  y ) )  =  0  /\  ( S `
  ( x  vH  y ) )  =  ( ( S `  x )  +h  ( S `  y ) ) ) ) ) )
 
Theoremsticl 22787  [ 0 ,  1 ] closure of the value of a state. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ( S  e.  States  ->  ( A  e.  CH  ->  ( S `  A )  e.  ( 0 [,] 1
 ) ) )
 
Theoremstcl 22788 Real closure of the value of a state. (Contributed by NM, 24-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ( S  e.  States  ->  ( A  e.  CH  ->  ( S `  A )  e. 
 RR ) )
 
Theoremhstcl 22789 Closure of the value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( S  e.  CHStates  /\  A  e.  CH )  ->  ( S `  A )  e.  ~H )
 
Theoremhst1a 22790 Unit value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
 |-  ( S  e.  CHStates  ->  ( normh `  ( S `  ~H ) )  =  1
 )
 
Theoremhstel2 22791 Properties of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( B  e.  CH  /\  A  C_  ( _|_ `  B ) ) ) 
 ->  ( ( ( S `
  A )  .ih  ( S `  B ) )  =  0  /\  ( S `  ( A 
 vH  B ) )  =  ( ( S `
  A )  +h  ( S `  B ) ) ) )
 
Theoremhstorth 22792 Orthogonality property of a Hilbert-space-valued state. This is a key feature distinguishing it from a real-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( B  e.  CH  /\  A  C_  ( _|_ `  B ) ) ) 
 ->  ( ( S `  A )  .ih  ( S `
  B ) )  =  0 )
 
Theoremhstosum 22793 Orthogonal sum property of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( B  e.  CH  /\  A  C_  ( _|_ `  B ) ) ) 
 ->  ( S `  ( A  vH  B ) )  =  ( ( S `
  A )  +h  ( S `  B ) ) )
 
Theoremhstoc 22794 Sum of a Hilbert-space-valued state of a lattice element and its orthocomplement. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( S  e.  CHStates  /\  A  e.  CH )  ->  ( ( S `  A )  +h  ( S `  ( _|_ `  A ) ) )  =  ( S `  ~H ) )
 
Theoremhstnmoc 22795 Sum of norms of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( S  e.  CHStates  /\  A  e.  CH )  ->  ( ( ( normh `  ( S `  A ) ) ^ 2
 )  +  ( (
 normh `  ( S `  ( _|_ `  A )
 ) ) ^ 2
 ) )  =  1 )
 
Theoremstge0 22796 The value of a state is nonnegative. (Contributed by NM, 24-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ( S  e.  States  ->  ( A  e.  CH  ->  0  <_  ( S `  A ) ) )
 
Theoremstle1 22797 The value of a state is less than or equal to one. (Contributed by NM, 24-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ( S  e.  States  ->  ( A  e.  CH  ->  ( S `  A )  <_ 
 1 ) )
 
Theoremhstle1 22798 The norm of the value of a Hilbert-space-valued state is less than or equal to one. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( S  e.  CHStates  /\  A  e.  CH )  ->  ( normh `  ( S `  A ) )  <_ 
 1 )
 
Theoremhst1h 22799 The norm of a Hilbert-space-valued state equals one iff the state value equals the state value of the lattice unit. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( S  e.  CHStates  /\  A  e.  CH )  ->  ( ( normh `  ( S `  A ) )  =  1  <->  ( S `  A )  =  ( S `  ~H ) ) )
 
Theoremhst0h 22800 The norm of a Hilbert-space-valued state equals zero iff the state value equals zero. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( S  e.  CHStates  /\  A  e.  CH )  ->  ( ( normh `  ( S `  A ) )  =  0  <->  ( S `  A )  =  0h ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31421
  Copyright terms: Public domain < Previous  Next >