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Theorem List for Metamath Proof Explorer - 30101-30200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorematbase 30101 An atom is a member of the lattice base set (i.e. a lattice element). (atelch 22940 analog.) (Contributed by NM, 10-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( P  e.  A  ->  P  e.  B )
 
Theorematssbase 30102 The set of atoms is a subset of the base set. (atssch 22939 analog.) (Contributed by NM, 21-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  A  C_  B
 
Theorem0ltat 30103 An atom is greater than zero. (Contributed by NM, 4-Jul-2012.)
 |-  .0.  =  ( 0. `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  OP  /\  P  e.  A )  ->  .0.  .<  P )
 
Theoremleatb 30104 A poset element less than or equal to an atom equals either zero or the atom. (atss 22942 analog.) (Contributed by NM, 17-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  ( X 
 .<_  P  <->  ( X  =  P  \/  X  =  .0.  ) ) )
 
Theoremleat 30105 A poset element less than or equal to an atom equals either zero or the atom. (Contributed by NM, 15-Oct-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  /\  X  .<_  P )  ->  ( X  =  P  \/  X  =  .0.  )
 )
 
Theoremleat2 30106 A nonzero poset element less than or equal to an atom equals the atom. (Contributed by NM, 6-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  /\  ( X  =/=  .0.  /\  X  .<_  P ) ) 
 ->  X  =  P )
 
Theoremleat3 30107 A poset element less than or equal to an atom is either an atom or zero. (Contributed by NM, 2-Dec-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  /\  X  .<_  P )  ->  ( X  e.  A  \/  X  =  .0.  )
 )
 
Theoremmeetat 30108 The meet of any element with an atom is either the atom or zero. (Contributed by NM, 28-Aug-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A ) 
 ->  ( ( X  ./\  P )  =  P  \/  ( X  ./\  P )  =  .0.  ) )
 
Theoremmeetat2 30109 The meet of any element with an atom is either the atom or zero. (Contributed by NM, 30-Aug-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A ) 
 ->  ( ( X  ./\  P )  e.  A  \/  ( X  ./\  P )  =  .0.  ) )
 
Definitiondf-atl 30110* Define the class of atomic lattices, in which every nonzero element is greater than or equal to an atom. . We also ensure the existence of a lattice zero, since a lattice by itself may not have a zero. (Contributed by NM, 18-Sep-2011.)
 |-  AtLat  =  {
 k  e.  Lat  |  ( ( 0. `  k
 )  e.  ( Base `  k )  /\  A. x  e.  ( Base `  k ) ( x  =/=  ( 0. `  k
 )  ->  E. p  e.  ( Atoms `  k ) p ( le `  k
 ) x ) ) }
 
Theoremisatl 30111* The predicate "is an atomic lattice." Every nonzero element is less than or equal to an atom. (Contributed by NM, 18-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( K  e.  AtLat  <->  ( K  e.  Lat  /\  .0.  e.  B  /\  A. x  e.  B  ( x  =/=  .0.  ->  E. y  e.  A  y  .<_  x ) ) )
 
Theorematllat 30112 An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.)
 |-  ( K  e.  AtLat  ->  K  e.  Lat )
 
Theorematlpos 30113 An atomic lattice is a poset. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  AtLat  ->  K  e.  Poset )
 
TheoremisatliN 30114* Properties that determine an atomic lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
 |-  K  e.  Lat   &    |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  .0.  e.  B   &    |-  (
 ( x  e.  B  /\  x  =/=  .0.  )  ->  E. y  e.  A  y  .<_  x )   =>    |-  K  e.  AtLat
 
Theorematl0cl 30115 An atomic lattice has a zero element. We can use this in place of op0cl 29996 for lattices without orthocomplements. (Contributed by NM, 5-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( K  e.  AtLat  ->  .0.  e.  B )
 
Theorematl0le 30116 Orthoposet zero is less than or equal to any element. (ch0le 22036 analog.) (Contributed by NM, 12-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  AtLat  /\  X  e.  B ) 
 ->  .0.  .<_  X )
 
Theorematlle0 30117 An element less than or equal to zero equals zero. (chle0 22038 analog.) (Contributed by NM, 21-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  AtLat  /\  X  e.  B ) 
 ->  ( X  .<_  .0.  <->  X  =  .0.  ) )
 
Theorematlltn0 30118 A lattice element greater than zero is nonzero. (Contributed by NM, 1-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  AtLat  /\  X  e.  B ) 
 ->  (  .0.  .<  X  <->  X  =/=  .0.  )
 )
 
Theoremisat3 30119* The predicate "is an atom". (elat2 22936 analog.) (Contributed by NM, 27-Apr-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( K  e.  AtLat  ->  ( P  e.  A  <->  ( P  e.  B  /\  P  =/=  .0.  /\  A. x  e.  B  ( x  .<_  P  ->  ( x  =  P  \/  x  =  .0.  )
 ) ) ) )
 
Theorematn0 30120 An atom is not zero. (atne0 22941 analog.) (Contributed by NM, 5-Nov-2012.)
 |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  AtLat  /\  P  e.  A ) 
 ->  P  =/=  .0.  )
 
Theorematnle0 30121 An atom is not less than or equal to zero. (Contributed by NM, 17-Oct-2011.)
 |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  -.  P  .<_  .0.  )
 
Theorematlen0 30122 A lattice element is nonzero if an atom is under it. (Contributed by NM, 26-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  X  =/=  .0.  )
 
Theorematcmp 30123 If two atoms are comparable, they are equal. (atsseq 22943 analog.) (Contributed by NM, 13-Oct-2011.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .<_  Q  <->  P  =  Q ) )
 
Theorematncmp 30124 Frequently-used variation of atcmp 30123. (Contributed by NM, 29-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( -.  P  .<_  Q  <->  P  =/=  Q ) )
 
Theorematnlt 30125 Two atoms cannot satisfy the less than relation. (Contributed by NM, 7-Feb-2012.)
 |-  .<  =  ( lt `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  -.  P  .<  Q )
 
Theorematcvreq0 30126 An element covered by an atom must be zero. (atcveq0 22944 analog.) (Contributed by NM, 4-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  ->  ( X C P  <->  X  =  .0.  ) )
 
TheorematncvrN 30127 Two atoms cannot satisfy the covering relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
 |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  -.  P C Q )
 
Theorematlex 30128* Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 22956 analog.) (Contributed by NM, 21-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  X  =/=  .0.  )  ->  E. y  e.  A  y  .<_  X )
 
Theorematnle 30129 Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 22972 analog.) (Contributed by NM, 5-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  ( -.  P  .<_  X  <->  ( P  ./\  X )  =  .0.  )
 )
 
Theorematnem0 30130 The meet of distinct atoms is zero. (atnemeq0 22973 analog.) (Contributed by NM, 5-Nov-2012.)
 |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  ( P  ./\  Q )  =  .0.  ) )
 
Theorematlatmstc 30131* An atomic, complete, orthomodular lattice is atomistic i.e. every element is the join of the atoms under it. See remark before Proposition 1 in [Kalmbach] p. 140; also remark in [BeltramettiCassinelli] p. 98. (hatomistici 22958 analog.) (Contributed by NM, 5-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .1.  =  ( lub `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B )  ->  (  .1.  `  { y  e.  A  |  y  .<_  X } )  =  X )
 
Theorematlatle 30132* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 22967 analog.) (Contributed by NM, 5-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  A. p  e.  A  ( p  .<_  X  ->  p 
 .<_  Y ) ) )
 
Theorematlrelat1 30133* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 22959, with  /\ swapped, analog.) (Contributed by NM, 4-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B  /\  Y  e.  B )  ->  ( X 
 .<  Y  ->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
 
Definitiondf-cvlat 30134* Define the class of atomic lattices with the covering property. (This is actually the exchange property, but they are equivalent. The literature usually uses the covering property terminology.) (Contributed by NM, 5-Nov-2012.)
 |-  CvLat  =  {
 k  e.  AtLat  |  A. a  e.  ( Atoms `  k ) A. b  e.  ( Atoms `  k ) A. c  e.  ( Base `  k ) ( ( -.  a ( le `  k ) c  /\  a ( le `  k ) ( c ( join `  k ) b ) )  ->  b ( le `  k ) ( c ( join `  k
 ) a ) ) }
 
Theoremiscvlat 30135* The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( K  e.  CvLat  <->  ( K  e.  AtLat  /\  A. p  e.  A  A. q  e.  A  A. x  e.  B  ( ( -.  p  .<_  x  /\  p  .<_  ( x  .\/  q
 ) )  ->  q  .<_  ( x  .\/  p ) ) ) )
 
Theoremiscvlat2N 30136* The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( K  e.  CvLat  <->  ( K  e.  AtLat  /\  A. p  e.  A  A. q  e.  A  A. x  e.  B  ( ( ( p  ./\  x )  =  .0.  /\  p  .<_  ( x  .\/  q )
 )  ->  q  .<_  ( x  .\/  p )
 ) ) )
 
Theoremcvlatl 30137 An atomic lattice with the covering property is an atomic lattice. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  CvLat  ->  K  e.  AtLat )
 
Theoremcvllat 30138 An atomic lattice with the covering property is a lattice. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  CvLat  ->  K  e.  Lat )
 
TheoremcvlposN 30139 An atomic lattice with the covering property is a poset. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
 |-  ( K  e.  CvLat  ->  K  e.  Poset )
 
Theoremcvlexch1 30140 An atomic covering lattice has the exchange property. (Contributed by NM, 6-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  ->  Q  .<_  ( X 
 .\/  P ) ) )
 
Theoremcvlexch2 30141 An atomic covering lattice has the exchange property. (Contributed by NM, 6-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( Q  .\/  X )  ->  Q  .<_  ( P 
 .\/  X ) ) )
 
Theoremcvlexchb1 30142 An atomic covering lattice has the exchange property. (Contributed by NM, 16-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q ) 
 <->  ( X  .\/  P )  =  ( X  .\/  Q ) ) )
 
Theoremcvlexchb2 30143 An atomic covering lattice has the exchange property. (Contributed by NM, 22-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( Q  .\/  X ) 
 <->  ( P  .\/  X )  =  ( Q  .\/  X ) ) )
 
Theoremcvlexch3 30144 An atomic covering lattice has the exchange property. (atexch 22977 analog.) (Contributed by NM, 5-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  ( P  ./\  X )  =  .0.  )  ->  ( P  .<_  ( X 
 .\/  Q )  ->  Q  .<_  ( X  .\/  P ) ) )
 
Theoremcvlexch4N 30145 An atomic covering lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  ( P  ./\  X )  =  .0.  )  ->  ( P  .<_  ( X 
 .\/  Q )  <->  ( X  .\/  P )  =  ( X 
 .\/  Q ) ) )
 
Theoremcvlatexchb1 30146 A version of cvlexchb1 30142 for atoms. (Contributed by NM, 5-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R )  ->  ( P  .<_  ( R  .\/  Q )  <->  ( R  .\/  P )  =  ( R  .\/  Q ) ) )
 
Theoremcvlatexchb2 30147 A version of cvlexchb2 30143 for atoms. (Contributed by NM, 5-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R )  ->  ( P  .<_  ( Q  .\/  R )  <->  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
 
Theoremcvlatexch1 30148 Atom exchange property. (Contributed by NM, 5-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R )  ->  ( P  .<_  ( R  .\/  Q )  ->  Q  .<_  ( R  .\/  P ) ) )
 
Theoremcvlatexch2 30149 Atom exchange property. (Contributed by NM, 5-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R )  ->  ( P  .<_  ( Q  .\/  R )  ->  Q  .<_  ( P  .\/  R ) ) )
 
Theoremcvlatexch3 30150 Atom exchange property. (Contributed by NM, 29-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q 
 /\  P  =/=  R ) )  ->  ( P 
 .<_  ( Q  .\/  R )  ->  ( P  .\/  Q )  =  ( P 
 .\/  R ) ) )
 
Theoremcvlcvr1 30151 The covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 22951 analog.) (Contributed by NM, 5-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  ( -.  P  .<_  X  <->  X C ( X 
 .\/  P ) ) )
 
Theoremcvlcvrp 30152 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 22971 analog.) (Contributed by NM, 5-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  ( ( X  ./\  P )  =  .0.  <->  X C ( X 
 .\/  P ) ) )
 
Theoremcvlatcvr1 30153 An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.)
 |-  .\/  =  ( join `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  P C ( P  .\/  Q ) ) )
 
Theoremcvlatcvr2 30154 An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.)
 |-  .\/  =  ( join `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  P C ( Q  .\/  P ) ) )
 
Theoremcvlsupr2 30155 Two equivalent ways of expressing that  R is a superposition of  P and  Q. (Contributed by NM, 5-Nov-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  Q ) 
 ->  ( ( P  .\/  R )  =  ( Q 
 .\/  R )  <->  ( R  =/=  P 
 /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) ) )
 
Theoremcvlsupr3 30156 Two equivalent ways of expressing that  R is a superposition of  P and  Q, which can replace the superposition part of ishlat1 30164,  ( x  =/=  y  ->  E. z  e.  A ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x  .\/  y ) )  ), with the simpler  E. z  e.  A ( x  .\/  z )  =  ( y  .\/  z ) as shown in ishlat3N 30166. (Contributed by NM, 5-Nov-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
 )  ->  ( ( P  .\/  R )  =  ( Q  .\/  R ) 
 <->  ( P  =/=  Q  ->  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) ) ) )
 
Theoremcvlsupr4 30157 Consequence of superposition condition  ( P  .\/  R )  =  ( Q  .\/  R ). (Contributed by NM, 9-Nov-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R )  =  ( Q  .\/  R ) ) ) 
 ->  R  .<_  ( P  .\/  Q ) )
 
Theoremcvlsupr5 30158 Consequence of superposition condition  ( P  .\/  R )  =  ( Q  .\/  R ). (Contributed by NM, 9-Nov-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  (
 ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R )  =  ( Q  .\/  R ) ) ) 
 ->  R  =/=  P )
 
Theoremcvlsupr6 30159 Consequence of superposition condition  ( P  .\/  R )  =  ( Q  .\/  R ). (Contributed by NM, 9-Nov-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  (
 ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R )  =  ( Q  .\/  R ) ) ) 
 ->  R  =/=  Q )
 
Theoremcvlsupr7 30160 Consequence of superposition condition  ( P  .\/  R )  =  ( Q  .\/  R ). (Contributed by NM, 24-Nov-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  (
 ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R )  =  ( Q  .\/  R ) ) ) 
 ->  ( P  .\/  Q )  =  ( R  .\/  Q ) )
 
Theoremcvlsupr8 30161 Consequence of superposition condition  ( P  .\/  R )  =  ( Q  .\/  R ). (Contributed by NM, 24-Nov-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  (
 ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R )  =  ( Q  .\/  R ) ) ) 
 ->  ( P  .\/  Q )  =  ( P  .\/  R ) )
 
18.27.9  Hilbert lattices
 
Syntaxchlt 30162 Extend class notation with Hilbert lattices.
 class  HL
 
Definitiondf-hlat 30163* Define the class of Hilbert lattices, which are complete, atomic lattices satisfying the superposition principle and minimum height. (Contributed by NM, 5-Nov-2012.)
 |-  HL  =  { l  e.  (
 ( OML  i^i  CLat )  i^i  CvLat )  |  (
 A. a  e.  ( Atoms `  l ) A. b  e.  ( Atoms `  l ) ( a  =/=  b  ->  E. c  e.  ( Atoms `  l )
 ( c  =/=  a  /\  c  =/=  b  /\  c ( le `  l
 ) ( a (
 join `  l ) b ) ) )  /\  E. a  e.  ( Base `  l ) E. b  e.  ( Base `  l ) E. c  e.  ( Base `  l ) ( ( ( 0. `  l
 ) ( lt `  l
 ) a  /\  a
 ( lt `  l
 ) b )  /\  ( b ( lt `  l ) c  /\  c ( lt `  l
 ) ( 1. `  l
 ) ) ) ) }
 
Theoremishlat1 30164* The predicate "is a Hilbert lattice," which is orthomodular ( K  e.  OML), complete ( K  e.  CLat), atomic and satisfying the exchange (or covering) property ( K  e.  CvLat), satisfies the superposition principle, and has a minimum height of 4. (Contributed by NM, 5-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( K  e.  HL 
 <->  ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  ( A. x  e.  A  A. y  e.  A  ( x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
 .\/  y ) ) )  /\  E. x  e.  B  E. y  e.  B  E. z  e.  B  ( (  .0. 
 .<  x  /\  x  .<  y )  /\  ( y 
 .<  z  /\  z  .<  .1.  ) ) ) ) )
 
Theoremishlat2 30165* The predicate "is a Hilbert lattice". Here we replace  K  e. 
CvLat with the weaker  K  e.  AtLat and show the exchange property explicitly. (Contributed by NM, 5-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( K  e.  HL 
 <->  ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  ( A. x  e.  A  A. y  e.  A  ( ( x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
 .\/  y ) ) )  /\  A. z  e.  B  ( ( -.  x  .<_  z  /\  x  .<_  ( z  .\/  y
 ) )  ->  y  .<_  ( z  .\/  x ) ) )  /\  E. x  e.  B  E. y  e.  B  E. z  e.  B  ( (  .0. 
 .<  x  /\  x  .<  y )  /\  ( y 
 .<  z  /\  z  .<  .1.  ) ) ) ) )
 
Theoremishlat3N 30166* The predicate "is a Hilbert lattice". Note that the superposition principle is expressed in the compact form  E. z  e.  A ( x  .\/  z )  =  ( y  .\/  z ). The exchange property and atomicity are provided by  K  e.  CvLat, and "minimum height 4" is shown explicitly. (Contributed by NM, 8-Nov-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( K  e.  HL 
 <->  ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  ( A. x  e.  A  A. y  e.  A  E. z  e.  A  ( x  .\/  z )  =  ( y  .\/  z
 )  /\  E. x  e.  B  E. y  e.  B  E. z  e.  B  ( (  .0. 
 .<  x  /\  x  .<  y )  /\  ( y 
 .<  z  /\  z  .<  .1.  ) ) ) ) )
 
TheoremishlatiN 30167* Properties that determine a Hilbert lattice. (Contributed by NM, 13-Nov-2011.) (New usage is discouraged.)
 |-  K  e.  OML   &    |-  K  e.  CLat   &    |-  K  e.  AtLat   &    |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .<  =  ( lt `  K )   &    |-  .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  A. x  e.  A  A. y  e.  A  ( ( x  =/=  y  ->  E. z  e.  A  ( z  =/= 
 x  /\  z  =/=  y  /\  z  .<_  ( x 
 .\/  y ) ) )  /\  A. z  e.  B  ( ( -.  x  .<_  z  /\  x  .<_  ( z  .\/  y
 ) )  ->  y  .<_  ( z  .\/  x ) ) )   &    |-  E. x  e.  B  E. y  e.  B  E. z  e.  B  ( (  .0. 
 .<  x  /\  x  .<  y )  /\  ( y 
 .<  z  /\  z  .<  .1.  ) )   =>    |-  K  e.  HL
 
Theoremhlomcmcv 30168 A Hilbert lattice is orthomodular, complete, and has the covering (exchange) property. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat ) )
 
Theoremhloml 30169 A Hilbert lattice is orthomodular. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  OML )
 
Theoremhlclat 30170 A Hilbert lattice is complete. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  CLat )
 
Theoremhlcvl 30171 A Hilbert lattice is an atomic lattice with the covering property. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  HL  ->  K  e.  CvLat )
 
Theoremhlatl 30172 A Hilbert lattice is atomic. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  AtLat )
 
Theoremhlol 30173 A Hilbert lattice is an ortholattice. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  OL )
 
Theoremhlop 30174 A Hilbert lattice is an orthoposet. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  OP )
 
Theoremhllat 30175 A Hilbert lattice is a lattice. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  Lat )
 
Theoremhlomcmat 30176 A Hilbert lattice is orthomodular, complete, and atomic. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat ) )
 
Theoremhlpos 30177 A Hilbert lattice is a poset. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  Poset )
 
Theoremhlatjcl 30178 Closure of join operation. Frequently-used special case of latjcl 14172 for atoms. (Contributed by NM, 15-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  A  /\  Y  e.  A )  ->  ( X  .\/  Y )  e.  B )
 
Theoremhlatjcom 30179 Commutatitivity of join operation. Frequently-used special case of latjcom 14181 for atoms. (Contributed by NM, 15-Jun-2012.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  A  /\  Y  e.  A )  ->  ( X  .\/  Y )  =  ( Y  .\/  X ) )
 
Theoremhlatjidm 30180 Idempotence of join operation. Frequently-used special case of latjcom 14181 for atoms. (Contributed by NM, 15-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  A ) 
 ->  ( X  .\/  X )  =  X )
 
Theoremhlatjass 30181 Lattice join is associative. Frequently-used special case of latjass 14217 for atoms. (Contributed by NM, 27-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
 )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( P 
 .\/  ( Q  .\/  R ) ) )
 
Theoremhlatj12 30182 Swap 1st and 2nd members of lattice join. Frequently-used special case of latj32 14219 for atoms. (Contributed by NM, 4-Jun-2012.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
 )  ->  ( P  .\/  ( Q  .\/  R ) )  =  ( Q  .\/  ( P  .\/  R ) ) )
 
Theoremhlatj32 30183 Swap 2nd and 3rd members of lattice join. Frequently-used special case of latj32 14219 for atoms. (Contributed by NM, 21-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
 )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( P  .\/  R )  .\/  Q ) )
 
Theoremhlatjrot 30184 Rotate lattice join of 3 classes. Frequently-used special case of latjrot 14222 for atoms. (Contributed by NM, 2-Aug-2012.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
 )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( R  .\/  P )  .\/  Q ) )
 
Theoremhlatj4 30185 Rearrangement of lattice join of 4 classes. Frequently-used special case of latj4 14223 for atoms. (Contributed by NM, 9-Aug-2012.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A ) 
 /\  ( R  e.  A  /\  S  e.  A ) )  ->  ( ( P  .\/  Q )  .\/  ( R  .\/  S ) )  =  (
 ( P  .\/  R )  .\/  ( Q  .\/  S ) ) )
 
Theoremhlatlej1 30186 A join's first argument is less than or equal to the join. Special case of latlej1 14182 to show an atom is on a line. (Contributed by NM, 15-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q )
 )
 
Theoremhlatlej2 30187 A join's second argument is less than or equal to the join. Special case of latlej2 14183 to show an atom is on a line. (Contributed by NM, 15-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  Q  .<_  ( P  .\/  Q )
 )
 
TheoremglbconN 30188* De Morgan's law for GLB and LUB. This holds in any complete ortholattice, although we assume  HL for convenience. (Contributed by NM, 17-Jan-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   &    |-  G  =  ( glb `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  HL  /\  S  C_  B )  ->  ( G `  S )  =  (  ._|_  `  ( U `
  { x  e.  B  |  (  ._|_  `  x )  e.  S } ) ) )
 
TheoremglbconxN 30189* De Morgan's law for GLB and LUB. Index-set version of glbconN 30188, where we read  S as  S (
i ). (Contributed by NM, 17-Jan-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   &    |-  G  =  ( glb `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B )  ->  ( G `  { x  |  E. i  e.  I  x  =  S }
 )  =  (  ._|_  `  ( U `  { x  |  E. i  e.  I  x  =  (  ._|_  `  S ) } )
 ) )
 
Theorematnlej1 30190 If an atom is not less than or equal to the join of two others, it is not equal to either. (This also holds for non-atoms, but in this form it is convenient.) (Contributed by NM, 8-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  -.  P  .<_  ( Q 
 .\/  R ) )  ->  P  =/=  Q )
 
Theorematnlej2 30191 If an atom is not less than or equal to the join of two others, it is not equal to either. (This also holds for non-atoms, but in this form it is convenient.) (Contributed by NM, 8-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  -.  P  .<_  ( Q 
 .\/  R ) )  ->  P  =/=  R )
 
Theoremhlsuprexch 30192* A Hilbert lattice has the superposition and exchange properties. (Contributed by NM, 13-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( ( P  =/=  Q  ->  E. z  e.  A  ( z  =/=  P  /\  z  =/=  Q  /\  z  .<_  ( P  .\/  Q ) ) )  /\  A. z  e.  B  ( ( -.  P  .<_  z 
 /\  P  .<_  ( z 
 .\/  Q ) )  ->  Q  .<_  ( z  .\/  P ) ) ) )
 
Theoremhlexch1 30193 A Hilbert lattice has the exchange property. (Contributed by NM, 13-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X ) 
 ->  ( P  .<_  ( X 
 .\/  Q )  ->  Q  .<_  ( X  .\/  P ) ) )
 
Theoremhlexch2 30194 A Hilbert lattice has the exchange property. (Contributed by NM, 6-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X ) 
 ->  ( P  .<_  ( Q 
 .\/  X )  ->  Q  .<_  ( P  .\/  X ) ) )
 
Theoremhlexchb1 30195 A Hilbert lattice has the exchange property. (Contributed by NM, 16-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X ) 
 ->  ( P  .<_  ( X 
 .\/  Q )  <->  ( X  .\/  P )  =  ( X 
 .\/  Q ) ) )
 
Theoremhlexchb2 30196 A Hilbert lattice has the exchange property. (Contributed by NM, 22-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X ) 
 ->  ( P  .<_  ( Q 
 .\/  X )  <->  ( P  .\/  X )  =  ( Q 
 .\/  X ) ) )
 
Theoremhlsupr 30197* A Hilbert lattice has the superposition property. Theorem 13.2 in [Crawley] p. 107. (Contributed by NM, 30-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q ) 
 ->  E. r  e.  A  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P  .\/  Q ) ) )
 
Theoremhlsupr2 30198* A Hilbert lattice has the superposition property. (Contributed by NM, 25-Nov-2012.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  E. r  e.  A  ( P  .\/  r )  =  ( Q  .\/  r ) )
 
Theoremhlhgt4 30199* A Hilbert lattice has a height of at least 4. (Contributed by NM, 4-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  .1.  =  ( 1. `  K )   =>    |-  ( K  e.  HL  ->  E. x  e.  B  E. y  e.  B  E. z  e.  B  ( (  .0.  .<  x  /\  x  .<  y )  /\  ( y  .<  z  /\  z  .<  .1.  ) )
 )
 
Theoremhlhgt2 30200* A Hilbert lattice has a height of at least 2. (Contributed by NM, 4-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  .1.  =  ( 1. `  K )   =>    |-  ( K  e.  HL  ->  E. x  e.  B  (  .0.  .<  x  /\  x  .<  .1.  ) )
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