HomeHome Metamath Proof Explorer
Theorem List (p. 239 of 329)
< 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
(1-22452)
  Hilbert Space Explorer  Hilbert Space Explorer
(22453-23975)
  Users' Mathboxes  Users' Mathboxes
(23976-32860)
 

Theorem List for Metamath Proof Explorer - 23801-23900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlargei 23801* 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 23802 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 23803* 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 23804* 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 23786 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 ) )
 
18.7.2  Godowski's equation
 
Theoremgolem1 23805 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 23806 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 23807 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 23808* 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 23809* 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 23810* 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 23811* 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 23812* 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 ) )
 
18.8  Cover relation, atoms, exchange axiom, and modular symmetry
 
18.8.1  Covers relation; modular pairs
 
Definitiondf-cv 23813* 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 23816 and cvbr2 23817 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 23814* 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 23828 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 23815* 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 23833 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 23816* 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 23817* 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 23818 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 23819 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 23820 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 23821 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 23822 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 23823 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 23824 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 ) )
 
Theoremcvnref 23825 The covers relation is not reflexive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  -.  A  <oH  A )
 
Theoremcvntr 23826 The covers relation is not transitive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( ( A  <oH  B 
 /\  B  <oH  C ) 
 ->  -.  A  <oH  C ) )
 
Theoremspansncv2 23827 Hilbert space has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  ~H )  ->  ( -.  ( span ` 
 { B } )  C_  A  ->  A  <oH  ( A  vH  ( span ` 
 { B } )
 ) ) )
 
Theoremmdbr 23828* Binary relation expressing  <. A ,  B >. is a modular pair. Definition 1.1 of [MaedaMaeda] p. 1. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. x  e.  CH  ( x  C_  B  ->  (
 ( x  vH  A )  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) ) ) )
 
Theoremmdi 23829 Consequence of the modular pair property. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  MH  B  /\  C  C_  B )
 )  ->  ( ( C  vH  A )  i^i 
 B )  =  ( C  vH  ( A  i^i  B ) ) )
 
Theoremmdbr2 23830* Binary relation expressing the modular pair property. This version has a weaker constraint than mdbr 23828. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. x  e.  CH  ( x  C_  B  ->  (
 ( x  vH  A )  i^i  B )  C_  ( x  vH  ( A  i^i  B ) ) ) ) )
 
Theoremmdbr3 23831* Binary relation expressing the modular pair property. This version quantifies an equality instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. x  e.  CH  (
 ( ( x  i^i  B )  vH  A )  i^i  B )  =  ( ( x  i^i  B )  vH  ( A  i^i  B ) ) ) )
 
Theoremmdbr4 23832* Binary relation expressing the modular pair property. This version quantifies an ordering instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. x  e.  CH  (
 ( ( x  i^i  B )  vH  A )  i^i  B )  C_  ( ( x  i^i  B )  vH  ( A  i^i  B ) ) ) )
 
Theoremdmdbr 23833* Binary relation expressing the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH*  B  <->  A. x  e.  CH  ( B  C_  x  ->  (
 ( x  i^i  A )  vH  B )  =  ( x  i^i  ( A  vH  B ) ) ) ) )
 
Theoremdmdmd 23834 The dual modular pair property expressed in terms of the modular pair property, that hold in Hilbert lattices. Remark 29.6 of [MaedaMaeda] p. 130. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH*  B  <->  ( _|_ `  A )  MH  ( _|_ `  B ) ) )
 
Theoremmddmd 23835 The modular pair property expressed in terms of the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  ( _|_ `  A )  MH*  ( _|_ `  B ) ) )
 
Theoremdmdi 23836 Consequence of the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  MH*  B  /\  B  C_  C )
 )  ->  ( ( C  i^i  A )  vH  B )  =  ( C  i^i  ( A  vH  B ) ) )
 
Theoremdmdbr2 23837* Binary relation expressing the dual modular pair property. This version has a weaker constraint than dmdbr 23833. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH*  B  <->  A. x  e.  CH  ( B  C_  x  ->  ( x  i^i  ( A  vH  B ) )  C_  ( ( x  i^i  A )  vH  B ) ) ) )
 
Theoremdmdi2 23838 Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  MH*  B  /\  B  C_  C )
 )  ->  ( C  i^i  ( A  vH  B ) )  C_  ( ( C  i^i  A ) 
 vH  B ) )
 
Theoremdmdbr3 23839* Binary relation expressing the dual modular pair property. This version quantifies an equality instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH*  B  <->  A. x  e.  CH  (
 ( ( x  vH  B )  i^i  A ) 
 vH  B )  =  ( ( x  vH  B )  i^i  ( A 
 vH  B ) ) ) )
 
Theoremdmdbr4 23840* Binary relation expressing the dual modular pair property. This version quantifies an ordering instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH*  B  <->  A. x  e.  CH  (
 ( x  vH  B )  i^i  ( A  vH  B ) )  C_  ( ( ( x 
 vH  B )  i^i 
 A )  vH  B ) ) )
 
Theoremdmdi4 23841 Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  MH*  B  ->  ( ( C  vH  B )  i^i  ( A  vH  B ) )  C_  ( ( ( C 
 vH  B )  i^i 
 A )  vH  B ) ) )
 
Theoremdmdbr5 23842* Binary relation expressing the dual modular pair property. (Contributed by NM, 15-Jan-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH*  B  <->  A. x  e.  CH  ( x  C_  ( A  vH  B )  ->  x  C_  ( ( ( x 
 vH  B )  i^i 
 A )  vH  B ) ) ) )
 
Theoremmddmd2 23843* Relationship between modular pairs and dual-modular pairs. Lemma 1.2 of [MaedaMaeda] p. 1. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A. x  e.  CH  A  MH  x  <->  A. x  e.  CH  A  MH*  x ) )
 
Theoremmdsl0 23844 A sublattice condition that transfers the modular pair property. Exercise 12 of [Kalmbach] p. 103. Also Lemma 1.5.3 of [MaedaMaeda] p. 2. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH )  /\  ( C  e.  CH 
 /\  D  e.  CH ) )  ->  ( ( ( ( C  C_  A  /\  D  C_  B )  /\  ( A  i^i  B )  =  0H )  /\  A  MH  B ) 
 ->  C  MH  D ) )
 
Theoremssmd1 23845 Ordering implies the modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  A  C_  B )  ->  A  MH  B )
 
Theoremssmd2 23846 Ordering implies the modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  A  C_  B )  ->  B  MH  A )
 
Theoremssdmd1 23847 Ordering implies the dual modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  A  C_  B )  ->  A  MH*  B )
 
Theoremssdmd2 23848 Ordering implies the dual modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  A  C_  B )  ->  ( _|_ `  B )  MH  ( _|_ `  A ) )
 
Theoremdmdsl3 23849 Sublattice mapping for a dual-modular pair. Part of Theorem 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 26-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  MH*  A  /\  A  C_  C  /\  C  C_  ( A  vH  B ) ) ) 
 ->  ( ( C  i^i  B )  vH  A )  =  C )
 
Theoremmdsl3 23850 Sublattice mapping for a modular pair. Part of Theorem 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 26-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  MH  B  /\  ( A  i^i  B )  C_  C  /\  C  C_  B ) )  ->  ( ( C  vH  A )  i^i  B )  =  C )
 
Theoremmdslle1i 23851 Order preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( B  MH*  A  /\  A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) )  ->  ( C  C_  D  <->  ( C  i^i  B )  C_  ( D  i^i  B ) ) )
 
Theoremmdslle2i 23852 Order preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( A  MH  B  /\  ( A  i^i  B )  C_  ( C  i^i  D )  /\  ( C 
 vH  D )  C_  B )  ->  ( C 
 C_  D  <->  ( C  vH  A )  C_  ( D 
 vH  A ) ) )
 
Theoremmdslj1i 23853 Join preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D ) 
 /\  ( C  vH  D )  C_  ( A 
 vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
 B )  =  ( ( C  i^i  B )  vH  ( D  i^i  B ) ) )
 
Theoremmdslj2i 23854 Meet preservation of the reverse mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A  i^i  B )  C_  ( C  i^i  D ) 
 /\  ( C  vH  D )  C_  B ) )  ->  ( ( C  i^i  D )  vH  A )  =  (
 ( C  vH  A )  i^i  ( D  vH  A ) ) )
 
Theoremmdsl1i 23855* If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A. x  e.  CH  (
 ( ( A  i^i  B )  C_  x  /\  x  C_  ( A  vH  B ) )  ->  ( x  C_  B  ->  ( ( x  vH  A )  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) ) )  <->  A  MH  B )
 
Theoremmdsl2i 23856* If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 28-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  MH  B  <->  A. x  e.  CH  ( ( ( A  i^i  B )  C_  x  /\  x  C_  B )  ->  ( ( x 
 vH  A )  i^i 
 B )  C_  ( x  vH  ( A  i^i  B ) ) ) )
 
Theoremmdsl2bi 23857* If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 24-Dec-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  MH  B  <->  A. x  e.  CH  ( ( ( A  i^i  B )  C_  x  /\  x  C_  B )  ->  ( ( x 
 vH  A )  i^i 
 B )  =  ( x  vH  ( A  i^i  B ) ) ) )
 
Theoremcvmdi 23858 The covering property implies the modular pair property. Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( A  i^i  B )  <oH  B  ->  A  MH  B )
 
Theoremmdslmd1lem1 23859 Lemma for mdslmd1i 23863. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   &    |-  R  e.  CH   =>    |-  ( ( ( A  MH  B  /\  B  MH* 
 A )  /\  (
 ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) ) )  ->  ( (
 ( R  vH  A )  C_  D  ->  (
 ( ( R  vH  A )  vH  C )  i^i  D )  C_  ( ( R  vH  A )  vH  ( C  i^i  D ) ) )  ->  ( (
 ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) 
 C_  R  /\  R  C_  ( D  i^i  B ) )  ->  ( ( R  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) )  C_  ( R  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) ) ) )
 
Theoremmdslmd1lem2 23860 Lemma for mdslmd1i 23863. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   &    |-  R  e.  CH   =>    |-  ( ( ( A  MH  B  /\  B  MH* 
 A )  /\  (
 ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) ) )  ->  ( (
 ( R  i^i  B )  C_  ( D  i^i  B )  ->  ( (
 ( R  i^i  B )  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
 C_  ( ( R  i^i  B )  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) )  ->  (
 ( ( C  i^i  D )  C_  R  /\  R  C_  D )  ->  ( ( R  vH  C )  i^i  D ) 
 C_  ( R  vH  ( C  i^i  D ) ) ) ) )
 
Theoremmdslmd1lem3 23861* Lemma for mdslmd1i 23863. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( x  e.  CH  /\  ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
 C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) ) ) ) 
 ->  ( ( ( x 
 vH  A )  C_  D  ->  ( ( ( x  vH  A ) 
 vH  C )  i^i 
 D )  C_  (
 ( x  vH  A )  vH  ( C  i^i  D ) ) )  ->  ( ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  x  /\  x  C_  ( D  i^i  B ) ) 
 ->  ( ( x  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
 C_  ( x  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) ) ) )
 
Theoremmdslmd1lem4 23862* Lemma for mdslmd1i 23863. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( x  e.  CH  /\  ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
 C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) ) ) ) 
 ->  ( ( ( x  i^i  B )  C_  ( D  i^i  B ) 
 ->  ( ( ( x  i^i  B )  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
 C_  ( ( x  i^i  B )  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) )  ->  (
 ( ( C  i^i  D )  C_  x  /\  x  C_  D )  ->  ( ( x  vH  C )  i^i  D ) 
 C_  ( x  vH  ( C  i^i  D ) ) ) ) )
 
Theoremmdslmd1i 23863 Preservation of the modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2 (meet version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D ) 
 /\  ( C  vH  D )  C_  ( A 
 vH  B ) ) )  ->  ( C  MH  D  <->  ( C  i^i  B )  MH  ( D  i^i  B ) ) )
 
Theoremmdslmd2i 23864 Preservation of the modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2 (join version). (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A  i^i  B )  C_  ( C  i^i  D ) 
 /\  ( C  vH  D )  C_  B ) )  ->  ( C  MH  D  <->  ( C  vH  A )  MH  ( D  vH  A ) ) )
 
Theoremmdsldmd1i 23865 Preservation of the dual modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D ) 
 /\  ( C  vH  D )  C_  ( A 
 vH  B ) ) )  ->  ( C  MH* 
 D 
 <->  ( C  i^i  B )  MH*  ( D  i^i  B ) ) )
 
Theoremmdslmd3i 23866 Modular pair conditions that imply the modular pair property in a sublattice. Lemma 1.5.1 of [MaedaMaeda] p. 2. (Contributed by NM, 23-Dec-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( ( A  MH  B  /\  ( A  i^i  B )  MH  C ) 
 /\  ( ( A  i^i  C )  C_  D  /\  D  C_  A ) )  ->  D  MH  ( B  i^i  C ) )
 
Theoremmdslmd4i 23867 Modular pair condition that implies the modular pair property in a sublattice. Lemma 1.5.2 of [MaedaMaeda] p. 2. (Contributed by NM, 24-Dec-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( A  MH  B  /\  ( ( A  i^i  B )  C_  C  /\  C  C_  A )  /\  ( ( A  i^i  B )  C_  D  /\  D  C_  B ) ) 
 ->  C  MH  D )
 
Theoremcsmdsymi 23868* Cross-symmetry implies M-symmetry. Theorem 1.9.1 of [MaedaMaeda] p. 3. (Contributed by NM, 24-Dec-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( A. c  e.  CH  ( c  MH  B  ->  B  MH*  c )  /\  A  MH  B ) 
 ->  B  MH  A )
 
Theoremmdexchi 23869 An exchange lemma for modular pairs. Lemma 1.6 of [MaedaMaeda] p. 2. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  MH  B  /\  C  MH  ( A  vH  B )  /\  ( C  i^i  ( A 
 vH  B ) ) 
 C_  A )  ->  ( ( C  vH  A )  MH  B  /\  ( ( C  vH  A )  i^i  B )  =  ( A  i^i  B ) ) )
 
Theoremcvmd 23870 The covering property implies the modular pair property. Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  ( A  i^i  B ) 
 <oH  B )  ->  A  MH  B )
 
Theoremcvdmd 23871 The covering property implies the dual modular pair property. Lemma 7.5.2 of [MaedaMaeda] p. 31. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  B  <oH  ( A  vH  B ) )  ->  A  MH*  B )
 
18.8.2  Atoms
 
Definitiondf-at 23872 Define the set of atoms in a Hilbert lattice. An atom is a nonzero element of a lattice such that anything less than it is zero, i.e. it is the smallest nonzero element of the lattice. Definition of atom in [Kalmbach] p. 15. See ela 23873 and elat2 23874 for membership relations. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
 |- HAtoms  =  { x  e.  CH  |  0H  <oH  x }
 
Theoremela 23873 Atoms in a Hilbert lattice are the elements that cover the zero subspace. Definition of atom in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e. HAtoms  <->  ( A  e.  CH 
 /\  0H  <oH  A ) )
 
Theoremelat2 23874* Expanded membership relation for the set of atoms, i.e. the predicate "is an atom (of the Hilbert lattice)." An atom is a nonzero element of a lattice such that anything less than it is zero, i.e. it is the smallest nonzero element of the lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e. HAtoms  <->  ( A  e.  CH 
 /\  ( A  =/=  0H 
 /\  A. x  e.  CH  ( x  C_  A  ->  ( x  =  A  \/  x  =  0H )
 ) ) ) )
 
Theoremelatcv0 23875 A Hilbert lattice element is an atom iff it covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  e. HAtoms  <->  0H  <oH  A ) )
 
Theorematcv0 23876 An atom covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e. HAtoms  ->  0H  <oH  A )
 
Theorematssch 23877 Atoms are a subset of the Hilbert lattice. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
 |- HAtoms  C_  CH
 
Theorematelch 23878 An atom is a Hilbert lattice element. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e. HAtoms  ->  A  e.  CH )
 
Theorematne0 23879 An atom is not the Hilbert lattice zero. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
 |-  ( A  e. HAtoms  ->  A  =/=  0H )
 
Theorematss 23880 A lattice element smaller than an atom is either the atom or zero. (Contributed by NM, 25-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  ( A  C_  B  ->  ( A  =  B  \/  A  =  0H )
 ) )
 
Theorematsseq 23881 Two atoms in a subset relationship are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e. HAtoms  /\  B  e. HAtoms )  ->  ( A  C_  B  <->  A  =  B ) )
 
Theorematcveq0 23882 A Hilbert lattice element covered by an atom must be the zero subspace. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  ( A  <oH  B  <->  A  =  0H ) )
 
Theoremh1da 23883 A 1-dimensional subspace is an atom. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  A  =/=  0h )  ->  ( _|_ `  ( _|_ `  { A }
 ) )  e. HAtoms )
 
Theoremspansna 23884 The span of the singleton of a vector is an atom. (Contributed by NM, 18-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  A  =/=  0h )  ->  ( span `  { A }
 )  e. HAtoms )
 
Theoremsh1dle 23885 A 1-dimensional subspace is less than or equal to any subspace containing its generating vector. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  A ) 
 ->  ( _|_ `  ( _|_ `  { B }
 ) )  C_  A )
 
Theoremch1dle 23886 A 1-dimensional subspace is less than or equal to any member of  CH containing its generating vector. (Contributed by NM, 30-May-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  A ) 
 ->  ( _|_ `  ( _|_ `  { B }
 ) )  C_  A )
 
Theorematom1d 23887* The 1-dimensional subspaces of Hilbert space are its atoms. Part of Remark 10.3.5 of [BeltramettiCassinelli] p. 107. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e. HAtoms  <->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( span ` 
 { x } )
 ) )
 
18.8.3  Superposition principle
 
Theoremsuperpos 23888* Superposition Principle. If  A and  B are distinct atoms, there exists a third atom, distinct from  A and  B, that is the superposition of  A and  B. Definition 3.4-3(a) in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e. HAtoms  /\  B  e. HAtoms  /\  A  =/=  B )  ->  E. x  e. HAtoms  ( x  =/=  A  /\  x  =/=  B  /\  x  C_  ( A  vH  B ) ) )
 
18.8.4  Atoms, exchange and covering properties, atomicity
 
Theoremchcv1 23889 The Hilbert lattice has the covering property. Proposition 1(ii) of [Kalmbach] p. 140 (and its converse). (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  ( -.  B  C_  A  <->  A  <oH  ( A 
 vH  B ) ) )
 
Theoremchcv2 23890 The Hilbert lattice has the covering property. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  ( A  C.  ( A  vH  B )  <->  A  <oH  ( A 
 vH  B ) ) )
 
Theoremchjatom 23891 The join of a closed subspace and an atom equals their subspace sum. Special case of remark in [Kalmbach] p. 65, stating that if  A or  B is finite dimensional, then this equality holds. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  ( A  +H  B )  =  ( A  vH  B ) )
 
Theoremshatomici 23892* The lattice of Hilbert subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
 |-  A  e.  SH   =>    |-  ( A  =/=  0H  ->  E. x  e. HAtoms  x  C_  A )
 
Theoremhatomici 23893* The Hilbert lattice is atomic, i.e. any nonzero element is greater than or equal to some atom. Remark in [Kalmbach] p. 140. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( A  =/=  0H  ->  E. x  e. HAtoms  x  C_  A )
 
Theoremhatomic 23894* A Hilbert lattice is atomic, i.e. any nonzero element is greater than or equal to some atom. Remark in [Kalmbach] p. 140. Also Definition 3.4-2 in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  A  =/=  0H )  ->  E. x  e. HAtoms  x  C_  A )
 
Theoremshatomistici 23895* The lattice of Hilbert subspaces is atomistic, i.e. any element is the supremum of its atoms. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.)
 |-  A  e.  SH   =>    |-  A  =  ( span ` 
 U. { x  e. HAtoms  |  x  C_  A }
 )
 
Theoremhatomistici 23896*  CH is atomistic, i.e. any element is the supremum of its atoms. Remark in [Kalmbach] p. 140. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  A  =  (  \/H  ` 
 { x  e. HAtoms  |  x  C_  A } )
 
Theoremchpssati 23897* Two Hilbert lattice elements in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C.  B  ->  E. x  e. HAtoms  ( x  C_  B  /\  -.  x  C_  A ) )
 
Theoremchrelati 23898* The Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C.  B  ->  E. x  e. HAtoms  ( A  C.  ( A  vH  x )  /\  ( A  vH  x ) 
 C_  B ) )
 
Theoremchrelat2i 23899* A consequence of relative atomicity. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( -.  A  C_  B  <->  E. x  e. HAtoms  ( x  C_  A  /\  -.  x  C_  B ) )
 
Theoremcvati 23900* If a Hilbert lattice element covers another, it equals the other joined with some atom. This is a consequence of the relative atomicity of Hilbert space. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  <oH  B  ->  E. x  e. HAtoms  ( A  vH  x )  =  B )
    < 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 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32860
  Copyright terms: Public domain < Previous  Next >