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Theorem List for Metamath Proof Explorer - 22601-22700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempjinvari 22601 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 22602 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 22603 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 22604 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 22605 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 22606 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 22607 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 22608 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 22609 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 22610 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 22611 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 22612 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 22613 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 22614 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 22615 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 22616 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 22617 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 22618 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 22619 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 22620 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 ) ) )
 
15.9.48  States on a Hilbert lattice
 
Definitiondf-st 22621* 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 22622* 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 22623* 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 22624* 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 22625  [ 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 22626 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 22627 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 22628 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 22629 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 22630 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 22631 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 22632 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 22633 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 22634 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 22635 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 22636 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 22637 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 22638 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 ) )
 
Theoremhstpyth 22639 Pythagorean property of a Hilbert-space-valued state for orthogonal vectors  A and  B. (Contributed by NM, 26-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( B  e.  CH  /\  A  C_  ( _|_ `  B ) ) ) 
 ->  ( ( normh `  ( S `  ( A  vH  B ) ) ) ^ 2 )  =  ( ( ( normh `  ( S `  A ) ) ^ 2
 )  +  ( (
 normh `  ( S `  B ) ) ^
 2 ) ) )
 
Theoremhstle 22640 Ordering property of a Hilbert-space-valued state. (Contributed by NM, 26-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( B  e.  CH  /\  A  C_  B )
 )  ->  ( normh `  ( S `  A ) )  <_  ( normh `  ( S `  B ) ) )
 
Theoremhstles 22641 Ordering property of a Hilbert-space-valued state. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( B  e.  CH  /\  A  C_  B )
 )  ->  ( ( normh `  ( S `  A ) )  =  1  ->  ( normh `  ( S `  B ) )  =  1
 ) )
 
Theoremhstoh 22642 A Hilbert-space-valued state orthogonal to the state of the lattice unit is zero. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( S  e.  CHStates  /\  A  e.  CH  /\  (
 ( S `  A )  .ih  ( S `  ~H ) )  =  0 )  ->  ( S `  A )  =  0h )
 
Theoremhst0 22643 A Hilbert-space-valued state is zero at the zero subspace. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
 |-  ( S  e.  CHStates  ->  ( S `
  0H )  =  0h )
 
Theoremsthil 22644 The value of a state at the full Hilbert space. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
 |-  ( S  e.  States  ->  ( S `  ~H )  =  1 )
 
Theoremstj 22645 The value of a state on a join. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
 |-  ( S  e.  States  ->  (
 ( ( A  e.  CH 
 /\  B  e.  CH )  /\  A  C_  ( _|_ `  B ) ) 
 ->  ( S `  ( A  vH  B ) )  =  ( ( S `
  A )  +  ( S `  B ) ) ) )
 
Theoremsto1i 22646 The state of a subspace plus the state of its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( S  e.  States  ->  ( ( S `  A )  +  ( S `  ( _|_ `  A ) ) )  =  1 )
 
Theoremsto2i 22647 The state of the orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( S  e.  States  ->  ( S `  ( _|_ `  A ) )  =  ( 1  -  ( S `  A ) ) )
 
Theoremstge1i 22648 If a state is greater than or equal to 1, it is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( S  e.  States  ->  ( 1  <_  ( S `  A )  <->  ( S `  A )  =  1
 ) )
 
Theoremstle0i 22649 If a state is less than or equal to 0, it is 0. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( S  e.  States  ->  ( ( S `  A )  <_  0  <->  ( S `  A )  =  0
 ) )
 
Theoremstlei 22650 Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( S  e.  States  ->  ( A  C_  B  ->  ( S `  A )  <_  ( S `  B ) ) )
 
Theoremstlesi 22651 Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( S  e.  States  ->  ( A  C_  B  ->  (
 ( S `  A )  =  1  ->  ( S `  B )  =  1 ) ) )
 
Theoremstji1i 22652 Join of components of Sasaki arrow ->1. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( S  e.  States  ->  ( S `  ( ( _|_ `  A )  vH  ( A  i^i  B ) ) )  =  ( ( S `  ( _|_ `  A ) )  +  ( S `  ( A  i^i  B ) ) ) )
 
Theoremstm1i 22653 State of component of unit meet. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( S  e.  States  ->  (
 ( S `  ( A  i^i  B ) )  =  1  ->  ( S `  A )  =  1 ) )
 
Theoremstm1ri 22654 State of component of unit meet. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( S  e.  States  ->  (
 ( S `  ( A  i^i  B ) )  =  1  ->  ( S `  B )  =  1 ) )
 
Theoremstm1addi 22655 Sum of states whose meet is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( S  e.  States  ->  (
 ( S `  ( A  i^i  B ) )  =  1  ->  (
 ( S `  A )  +  ( S `  B ) )  =  2 ) )
 
Theoremstaddi 22656 If the sum of 2 states is 2, then each state is 1. (Contributed by NM, 12-Nov-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( S  e.  States  ->  (
 ( ( S `  A )  +  ( S `  B ) )  =  2  ->  ( S `  A )  =  1 ) )
 
Theoremstm1add3i 22657 Sum of states whose meet is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( S  e.  States  ->  ( ( S `  ( ( A  i^i  B )  i^i  C ) )  =  1  ->  ( ( ( S `
  A )  +  ( S `  B ) )  +  ( S `
  C ) )  =  3 ) )
 
Theoremstadd3i 22658 If the sum of 3 states is 3, then each state is 1. (Contributed by NM, 13-Nov-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( S  e.  States  ->  ( ( ( ( S `  A )  +  ( S `  B ) )  +  ( S `  C ) )  =  3  ->  ( S `  A )  =  1 ) )
 
Theoremst0 22659 The state of the zero subspace. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  ( S  e.  States  ->  ( S `  0H )  =  0 )
 
Theoremstrlem1 22660* Lemma for strong state theorem: if closed subspace  A is not contained in  B, there is a unit vector  u in their difference. (Contributed by NM, 25-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( -.  A  C_  B  ->  E. u  e.  ( A 
 \  B ) (
 normh `  u )  =  1 )
 
Theoremstrlem2 22661* Lemma for strong state theorem. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
 |-  S  =  ( x  e.  CH  |->  ( ( normh `  (
 ( proj  h `  x ) `  u ) ) ^ 2 ) )   =>    |-  ( C  e.  CH  ->  ( S `  C )  =  ( ( normh `  ( ( proj  h `  C ) `  u ) ) ^ 2
 ) )
 
Theoremstrlem3a 22662* Lemma for strong state theorem: the function  S, that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a state. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
 |-  S  =  ( x  e.  CH  |->  ( ( normh `  (
 ( proj  h `  x ) `  u ) ) ^ 2 ) )   =>    |-  ( ( u  e. 
 ~H  /\  ( normh `  u )  =  1 )  ->  S  e.  States )
 
Theoremstrlem3 22663* Lemma for strong state theorem: the function  S, that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a state. This lemma restates the hypotheses in a more convenient form to work with. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
 |-  S  =  ( x  e.  CH  |->  ( ( normh `  (
 ( proj  h `  x ) `  u ) ) ^ 2 ) )   &    |-  ( ph  <->  ( u  e.  ( A  \  B )  /\  ( normh `  u )  =  1 )
 )   &    |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( ph  ->  S  e.  States )
 
Theoremstrlem4 22664* Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
 |-  S  =  ( x  e.  CH  |->  ( ( normh `  (
 ( proj  h `  x ) `  u ) ) ^ 2 ) )   &    |-  ( ph  <->  ( u  e.  ( A  \  B )  /\  ( normh `  u )  =  1 )
 )   &    |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( ph  ->  ( S `  A )  =  1
 )
 
Theoremstrlem5 22665* Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
 |-  S  =  ( x  e.  CH  |->  ( ( normh `  (
 ( proj  h `  x ) `  u ) ) ^ 2 ) )   &    |-  ( ph  <->  ( u  e.  ( A  \  B )  /\  ( normh `  u )  =  1 )
 )   &    |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( ph  ->  ( S `  B )  <  1 )
 
Theoremstrlem6 22666* Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
 |-  S  =  ( x  e.  CH  |->  ( ( normh `  (
 ( proj  h `  x ) `  u ) ) ^ 2 ) )   &    |-  ( ph  <->  ( u  e.  ( A  \  B )  /\  ( normh `  u )  =  1 )
 )   &    |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( ph  ->  -.  ( ( S `  A )  =  1  ->  ( S `  B )  =  1 ) )
 
Theoremstri 22667* Strong state theorem. The states on a Hilbert lattice define an ordering. Remark in [Mayet] p. 370. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A. f  e.  States  ( ( f `  A )  =  1  ->  (
 f `  B )  =  1 )  ->  A  C_  B )
 
Theoremstrb 22668* Strong state theorem (bidirectional version). (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A. f  e.  States  ( ( f `  A )  =  1  ->  (
 f `  B )  =  1 )  <->  A  C_  B )
 
Theoremhstrlem2 22669* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
 |-  S  =  ( x  e.  CH  |->  ( ( proj  h `  x ) `  u ) )   =>    |-  ( C  e.  CH  ->  ( S `  C )  =  ( ( proj  h `  C ) `
  u ) )
 
Theoremhstrlem3a 22670* Lemma for strong set of CH states theorem: the function  S, that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a state. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
 |-  S  =  ( x  e.  CH  |->  ( ( proj  h `  x ) `  u ) )   =>    |-  ( ( u  e. 
 ~H  /\  ( normh `  u )  =  1 )  ->  S  e.  CHStates )
 
Theoremhstrlem3 22671* Lemma for strong set of CH states theorem: the function  S, that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a state. This lemma restates the hypotheses in a more convenient form to work with. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
 |-  S  =  ( x  e.  CH  |->  ( ( proj  h `  x ) `  u ) )   &    |-  ( ph  <->  ( u  e.  ( A  \  B )  /\  ( normh `  u )  =  1 )
 )   &    |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( ph  ->  S  e.  CHStates )
 
Theoremhstrlem4 22672* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
 |-  S  =  ( x  e.  CH  |->  ( ( proj  h `  x ) `  u ) )   &    |-  ( ph  <->  ( u  e.  ( A  \  B )  /\  ( normh `  u )  =  1 )
 )   &    |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( ph  ->  ( normh `  ( S `  A ) )  =  1 )
 
Theoremhstrlem5 22673* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
 |-  S  =  ( x  e.  CH  |->  ( ( proj  h `  x ) `  u ) )   &    |-  ( ph  <->  ( u  e.  ( A  \  B )  /\  ( normh `  u )  =  1 )
 )   &    |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( ph  ->  ( normh `  ( S `  B ) )  <  1 )
 
Theoremhstrlem6 22674* Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
 |-  S  =  ( x  e.  CH  |->  ( ( proj  h `  x ) `  u ) )   &    |-  ( ph  <->  ( u  e.  ( A  \  B )  /\  ( normh `  u )  =  1 )
 )   &    |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( ph  ->  -.  ( ( normh `  ( S `  A ) )  =  1  ->  ( normh `  ( S `  B ) )  =  1
 ) )
 
Theoremhstri 22675* Hilbert space admits a strong set of Hilbert-space-valued states (CH-states). Theorem in [Mayet3] p. 10. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A. f  e.  CHStates  ( (
 normh `  ( f `  A ) )  =  1  ->  ( normh `  ( f `  B ) )  =  1
 )  ->  A  C_  B )
 
Theoremhstrbi 22676* Strong CH-state theorem (bidirectional version). Theorem in [Mayet3] p. 10 and its converse. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A. f  e.  CHStates  ( (
 normh `  ( f `  A ) )  =  1  ->  ( normh `  ( f `  B ) )  =  1
 ) 
 <->  A  C_  B )
 
Theoremlargei 22677* A Hilbert lattice admits a largei set of states. Remark in [Mayet] p. 370. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( -.  A  =  0H 
 <-> 
 E. f  e.  States  ( f `  A )  =  1 )
 
Theoremjplem1 22678 Lemma for Jauch-Piron theorem. (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( u  e. 
 ~H  /\  ( normh `  u )  =  1 )  ->  ( u  e.  A  <->  ( ( normh `  ( ( proj  h `  A ) `  u ) ) ^ 2
 )  =  1 ) )
 
Theoremjplem2 22679* Lemma for Jauch-Piron theorem. (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)
 |-  S  =  ( x  e.  CH  |->  ( ( normh `  (
 ( proj  h `  x ) `  u ) ) ^ 2 ) )   &    |-  A  e.  CH   =>    |-  ( ( u  e. 
 ~H  /\  ( normh `  u )  =  1 )  ->  ( u  e.  A  <->  ( S `  A )  =  1
 ) )
 
Theoremjpi 22680* The function  S, that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a Jauch-Piron state. Remark in [Mayet] p. 370. (See strlem3a 22662 for the proof that  S is a state.) (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)
 |-  S  =  ( x  e.  CH  |->  ( ( normh `  (
 ( proj  h `  x ) `  u ) ) ^ 2 ) )   &    |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( u  e.  ~H  /\  ( normh `  u )  =  1 )  ->  ( ( ( S `
  A )  =  1  /\  ( S `
  B )  =  1 )  <->  ( S `  ( A  i^i  B ) )  =  1 ) )
 
15.9.49  Godowski's equation
 
Theoremgolem1 22681 Lemma for Godowski's equation. (Contributed by NM, 10-Nov-2002.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  F  =  ( ( _|_ `  A )  vH  ( A  i^i  B ) )   &    |-  G  =  ( ( _|_ `  B )  vH  ( B  i^i  C ) )   &    |-  H  =  ( ( _|_ `  C )  vH  ( C  i^i  A ) )   &    |-  D  =  ( ( _|_ `  B )  vH  ( B  i^i  A ) )   &    |-  R  =  ( ( _|_ `  C )  vH  ( C  i^i  B ) )   &    |-  S  =  ( ( _|_ `  A )  vH  ( A  i^i  C ) )   =>    |-  ( f  e.  States  ->  ( ( ( f `
  F )  +  ( f `  G ) )  +  (
 f `  H )
 )  =  ( ( ( f `  D )  +  ( f `  R ) )  +  ( f `  S ) ) )
 
Theoremgolem2 22682 Lemma for Godowski's equation. (Contributed by NM, 13-Nov-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  F  =  ( ( _|_ `  A )  vH  ( A  i^i  B ) )   &    |-  G  =  ( ( _|_ `  B )  vH  ( B  i^i  C ) )   &    |-  H  =  ( ( _|_ `  C )  vH  ( C  i^i  A ) )   &    |-  D  =  ( ( _|_ `  B )  vH  ( B  i^i  A ) )   &    |-  R  =  ( ( _|_ `  C )  vH  ( C  i^i  B ) )   &    |-  S  =  ( ( _|_ `  A )  vH  ( A  i^i  C ) )   =>    |-  ( f  e.  States  ->  ( ( f `  ( ( F  i^i  G )  i^i  H ) )  =  1  ->  ( f `  D )  =  1 )
 )
 
Theoremgoeqi 22683 Godowski's equation, shown here as a variant equivalent to Equation SF of [Godowski] p. 730. (Contributed by NM, 10-Nov-2002.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  F  =  ( ( _|_ `  A )  vH  ( A  i^i  B ) )   &    |-  G  =  ( ( _|_ `  B )  vH  ( B  i^i  C ) )   &    |-  H  =  ( ( _|_ `  C )  vH  ( C  i^i  A ) )   &    |-  D  =  ( ( _|_ `  B )  vH  ( B  i^i  A ) )   =>    |-  ( ( F  i^i  G )  i^i  H ) 
 C_  D
 
Theoremstcltr1i 22684* Property of a strong classical state. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( S  e.  States  /\ 
 A. x  e.  CH  A. y  e.  CH  (
 ( ( S `  x )  =  1  ->  ( S `  y
 )  =  1 ) 
 ->  x  C_  y ) ) )   &    |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( ph  ->  (
 ( ( S `  A )  =  1  ->  ( S `  B )  =  1 )  ->  A  C_  B )
 )
 
Theoremstcltr2i 22685* Property of a strong classical state. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( S  e.  States  /\ 
 A. x  e.  CH  A. y  e.  CH  (
 ( ( S `  x )  =  1  ->  ( S `  y
 )  =  1 ) 
 ->  x  C_  y ) ) )   &    |-  A  e.  CH   =>    |-  ( ph  ->  ( ( S `
  A )  =  1  ->  A  =  ~H ) )
 
Theoremstcltrlem1 22686* Lemma for strong classical state theorem. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( S  e.  States  /\ 
 A. x  e.  CH  A. y  e.  CH  (
 ( ( S `  x )  =  1  ->  ( S `  y
 )  =  1 ) 
 ->  x  C_  y ) ) )   &    |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( ph  ->  (
 ( S `  B )  =  1  ->  ( S `  ( ( _|_ `  A )  vH  ( A  i^i  B ) ) )  =  1 ) )
 
Theoremstcltrlem2 22687* Lemma for strong classical state theorem. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( S  e.  States  /\ 
 A. x  e.  CH  A. y  e.  CH  (
 ( ( S `  x )  =  1  ->  ( S `  y
 )  =  1 ) 
 ->  x  C_  y ) ) )   &    |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( ph  ->  B  C_  ( ( _|_ `  A )  vH  ( A  i^i  B ) ) )
 
Theoremstcltrthi 22688* Theorem for classically strong set of states. If there exists a "classically strong set of states" on lattice  CH (or actually any ortholattice, which would have an identical proof), then any two elements of the lattice commute, i.e., the lattice is distributive. (Proof due to Mladen Pavicic.) Theorem 3.3 of [MegPav2000] p. 2344. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  E. s  e.  States  A. x  e.  CH  A. y  e.  CH  (
 ( ( s `  x )  =  1  ->  ( s `  y
 )  =  1 ) 
 ->  x  C_  y )   =>    |-  B  C_  ( ( _|_ `  A )  vH  ( A  i^i  B ) )
 
15.9.50  Covers relation; modular pairs
 
Definitiondf-cv 22689* Define the covers relation (on the Hilbert lattice). Definition 3.2.18 of [PtakPulmannova] p. 68, whose notation we use. Ptak/Pulmannova's notation  A  <oH  B is read " B covers  A " or " A is covered by  B " , and it means that  B is larger than  A and there is nothing in between. See cvbr 22692 and cvbr2 22693 for membership relations. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
 |-  <oH  =  { <. x ,  y >.  |  ( ( x  e.  CH  /\  y  e. 
 CH )  /\  ( x  C.  y  /\  -.  E. z  e.  CH  ( x  C.  z  /\  z  C.  y ) ) ) }
 
Definitiondf-md 22690* Define the modular pair relation (on the Hilbert lattice). Definition 1.1 of [MaedaMaeda] p. 1, who use the notation (x,y)M for "the ordered pair <x,y> is a modular pair." See mdbr 22704 for membership relation. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
 |-  MH  =  { <. x ,  y >.  |  ( ( x  e.  CH  /\  y  e. 
 CH )  /\  A. z  e.  CH  ( z 
 C_  y  ->  (
 ( z  vH  x )  i^i  y )  =  ( z  vH  ( x  i^i  y ) ) ) ) }
 
Definitiondf-dmd 22691* Define the dual modular pair relation (on the Hilbert lattice). Definition 1.1 of [MaedaMaeda] p. 1, who use the notation (x,y)M* for "the ordered pair <x,y> is a dual modular pair." See dmdbr 22709 for membership relation. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  MH*  =  { <. x ,  y >.  |  ( ( x  e.  CH  /\  y  e. 
 CH )  /\  A. z  e.  CH  ( y 
 C_  z  ->  (
 ( z  i^i  x )  vH  y )  =  ( z  i^i  ( x  vH  y ) ) ) ) }
 
Theoremcvbr 22692* Binary relation expressing  B covers  A, which means that  B is larger than  A and there is nothing in between. Definition 3.2.18 of [PtakPulmannova] p. 68. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  <->  ( A  C.  B  /\  -.  E. x  e.  CH  ( A  C.  x  /\  x  C.  B ) ) ) )
 
Theoremcvbr2 22693* Binary relation expressing  B covers  A. Definition of covers in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  <->  ( A  C.  B  /\  A. x  e. 
 CH  ( ( A 
 C.  x  /\  x  C_  B )  ->  x  =  B ) ) ) )
 
Theoremcvcon3 22694 Contraposition law for the covers relation. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  <->  ( _|_ `  B )  <oH  ( _|_ `  A ) ) )
 
Theoremcvpss 22695 The covers relation implies proper subset. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  ->  A 
 C.  B ) )
 
Theoremcvnbtwn 22696 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  -.  ( A  C.  C  /\  C  C.  B ) ) )
 
Theoremcvnbtwn2 22697 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  ( ( A  C.  C  /\  C  C_  B )  ->  C  =  B ) ) )
 
Theoremcvnbtwn3 22698 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  ( ( A  C_  C  /\  C  C.  B ) 
 ->  C  =  A ) ) )
 
Theoremcvnbtwn4 22699 The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  ( ( A  C_  C  /\  C  C_  B )  ->  ( C  =  A  \/  C  =  B ) ) ) )
 
Theoremcvnsym 22700 The covers relation is not symmetric. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  ->  -.  B  <oH  A ) )
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