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Theorem List for Metamath Proof Explorer - 28701-28800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremomlspjN 28701 Contraction of a Sasaki projection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B ) 
 /\  X  .<_  Y ) 
 ->  ( ( X  .\/  (  ._|_  `  Y )
 )  ./\  Y )  =  X )
 
18.25.11  Atomic lattices with covering property
 
Syntaxccvr 28702 Extend class notation with covers relation.
 class  <o
 
Syntaxcatm 28703 Extend class notation with atoms.
 class  Atoms
 
Syntaxcal 28704 Extend class notation with atomic lattices.
 class  AtLat
 
Syntaxclc 28705 Extend class notation with lattices with the covering property.
 class  CvLat
 
Definitiondf-covers 28706* Define the covers relation ("is covered by") for posets. " a is covered by  b " means that  a is strictly less than  b and there is nothing in between. See cvrval 28709 for the relation form. (Contributed by NM, 18-Sep-2011.)
 |-  <o  =  ( p  e.  _V  |->  {
 <. a ,  b >.  |  ( ( a  e.  ( Base `  p )  /\  b  e.  ( Base `  p ) ) 
 /\  a ( lt `  p ) b  /\  -. 
 E. z  e.  ( Base `  p ) ( a ( lt `  p ) z  /\  z ( lt `  p ) b ) ) }
 )
 
Definitiondf-ats 28707* Define the class of poset atoms. (Contributed by NM, 18-Sep-2011.)
 |-  Atoms  =  ( p  e.  _V  |->  { a  e.  ( Base `  p )  |  ( 0. `  p ) (  <o  `  p )
 a } )
 
Theoremcvrfval 28708* Value of covers relation "is covered by". (Contributed by NM, 18-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( K  e.  A  ->  C  =  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B )  /\  x  .<  y  /\  -. 
 E. z  e.  B  ( x  .<  z  /\  z  .<  y ) ) } )
 
Theoremcvrval 28709* 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. (cvbr 22823 analog.) (Contributed by NM, 18-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X C Y  <->  ( X  .<  Y  /\  -. 
 E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) ) ) )
 
Theoremcvrlt 28710 The covers relation implies the less-than relation. (cvpss 22826 analog.) (Contributed by NM, 8-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  X  .<  Y )
 
Theoremcvrnbtwn 28711 There is no element between the two arguments of the covers relation. (cvnbtwn 22827 analog.) (Contributed by NM, 18-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) 
 /\  X C Y )  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) )
 
Theoremncvr1 28712 No element covers the lattice unit. (Contributed by NM, 8-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  -.  .1.  C X )
 
TheoremcvrletrN 28713 Property of an element above a covering. (Contributed by NM, 7-Dec-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  C  =  ( 
 <o  `  K )   =>    |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X C Y  /\  Y  .<_  Z )  ->  X  .<  Z ) )
 
Theoremcvrval2 28714* Binary relation expressing  Y covers  X. Definition of covers in [Kalmbach] p. 15. (cvbr2 22824 analog.) (Contributed by NM, 16-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  C  =  ( 
 <o  `  K )   =>    |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <->  ( X  .<  Y 
 /\  A. z  e.  B  ( ( X  .<  z 
 /\  z  .<_  Y ) 
 ->  z  =  Y ) ) ) )
 
Theoremcvrnbtwn2 28715 The covers relation implies no in-betweenness. (cvnbtwn2 22828 analog.) (Contributed by NM, 17-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  C  =  ( 
 <o  `  K )   =>    |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( ( X 
 .<  Z  /\  Z  .<_  Y )  <->  Z  =  Y ) )
 
Theoremcvrnbtwn3 28716 The covers relation implies no in-betweenness. (cvnbtwn3 22829 analog.) (Contributed by NM, 4-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  C  =  ( 
 <o  `  K )   =>    |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( ( X 
 .<_  Z  /\  Z  .<  Y )  <->  X  =  Z ) )
 
Theoremcvrcon3b 28717 Contraposition law for the covers relation. (cvcon3 22825 analog.) (Contributed by NM, 4-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X C Y  <->  ( 
 ._|_  `  Y ) C (  ._|_  `  X ) ) )
 
Theoremcvrle 28718 The covers relation implies the less-than-or-equal relation. (Contributed by NM, 12-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  X  .<_  Y )
 
Theoremcvrnbtwn4 28719 The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 22830 analog.) (Contributed by NM, 18-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y ) 
 ->  ( ( X  .<_  Z 
 /\  Z  .<_  Y )  <-> 
 ( X  =  Z  \/  Z  =  Y ) ) )
 
Theoremcvrnle 28720 The covers relation implies the negation of the reverse less-than-or-equal relation. (Contributed by NM, 18-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  -.  Y  .<_  X )
 
Theoremcvrne 28721 The covers relation implies inequality. (Contributed by NM, 13-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  X  =/=  Y )
 
TheoremcvrnrefN 28722 The covers relation is not reflexive. (cvnref 22832 analog.) (Contributed by NM, 1-Nov-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  A  /\  X  e.  B )  ->  -.  X C X )
 
Theoremcvrcmp 28723 If two lattice elements that cover a third are comparable, then they are equal. (Contributed by NM, 6-Feb-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( Z C X  /\  Z C Y ) )  ->  ( X  .<_  Y  <->  X  =  Y ) )
 
Theoremcvrcmp2 28724 If two lattice elements covered by a third are comparable, then they are equal. (Contributed by NM, 20-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) 
 /\  ( X C Z  /\  Y C Z ) )  ->  ( X 
 .<_  Y  <->  X  =  Y ) )
 
Theorempats 28725* The set of atoms in a poset. (Contributed by NM, 18-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( K  e.  D  ->  A  =  { x  e.  B  |  .0.  C x } )
 
Theoremisat 28726 The predicate "is an atom". (ela 22880 analog.) (Contributed by NM, 18-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( K  e.  D  ->  ( P  e.  A 
 <->  ( P  e.  B  /\  .0.  C P ) ) )
 
Theoremisat2 28727 The predicate "is an atom". (elatcv0 22882 analog.) (Contributed by NM, 18-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  D  /\  P  e.  B )  ->  ( P  e.  A  <->  .0.  C P ) )
 
Theorematcvr0 28728 An atom covers zero. (atcv0 22883 analog.) (Contributed by NM, 4-Nov-2011.)
 |-  .0.  =  ( 0. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  D  /\  P  e.  A )  ->  .0.  C P )
 
Theorematbase 28729 An atom is a member of the lattice base set (i.e. a lattice element). (atelch 22885 analog.) (Contributed by NM, 10-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( P  e.  A  ->  P  e.  B )
 
Theorematssbase 28730 The set of atoms is a subset of the base set. (atssch 22884 analog.) (Contributed by NM, 21-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  A  C_  B
 
Theorem0ltat 28731 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 28732 A poset element less than or equal to an atom equals either zero or the atom. (atss 22887 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 28733 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 28734 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 28735 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 28736 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 28737 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 28738* 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 28739* 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 28740 An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.)
 |-  ( K  e.  AtLat  ->  K  e.  Lat )
 
Theorematlpos 28741 An atomic lattice is a poset. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  AtLat  ->  K  e.  Poset )
 
TheoremisatliN 28742* 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 28743 An atomic lattice has a zero element. We can use this in place of op0cl 28624 for lattices without orthocomplements. (Contributed by NM, 5-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( K  e.  AtLat  ->  .0.  e.  B )
 
Theorematl0le 28744 Orthoposet zero is less than or equal to any element. (ch0le 21981 analog.) (Contributed by NM, 12-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  AtLat  /\  X  e.  B ) 
 ->  .0.  .<_  X )
 
Theorematlle0 28745 An element less than or equal to zero equals zero. (chle0 21983 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 28746 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 28747* The predicate "is an atom". (elat2 22881 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 28748 An atom is not zero. (atne0 22886 analog.) (Contributed by NM, 5-Nov-2012.)
 |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  AtLat  /\  P  e.  A ) 
 ->  P  =/=  .0.  )
 
Theorematnle0 28749 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 28750 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 28751 If two atoms are comparable, they are equal. (atsseq 22888 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 28752 Frequently-used variation of atcmp 28751. (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 28753 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 28754 An element covered by an atom must be zero. (atcveq0 22889 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 28755 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 28756* Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 22901 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 28757 Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 22917 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 28758 The meet of distinct atoms is zero. (atnemeq0 22918 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 28759* 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 22903 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 28760* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 22912 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 28761* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 22904, 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 28762* 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 28763* 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 28764* 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 28765 An atomic lattice with the covering property is an atomic lattice. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  CvLat  ->  K  e.  AtLat )
 
Theoremcvllat 28766 An atomic lattice with the covering property is a lattice. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  CvLat  ->  K  e.  Lat )
 
TheoremcvlposN 28767 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 28768 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 28769 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 28770 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 28771 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 28772 An atomic covering lattice has the exchange property. (atexch 22922 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 28773 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 28774 A version of cvlexchb1 28770 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 28775 A version of cvlexchb2 28771 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 28776 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 28777 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 28778 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 28779 The covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 22896 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 28780 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 22916 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 28781 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 28782 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 28783 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 28784 Two equivalent ways of expressing that  R is a superposition of  P and  Q, which can replace the superposition part of ishlat1 28792,  ( 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 28794. (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 28785 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 28786 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 28787 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 28788 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 28789 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 ) )
 
18.25.12  Hilbert lattices
 
Syntaxchlt 28790 Extend class notation with Hilbert lattices.
 class  HL
 
Definitiondf-hlat 28791* 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 28792* 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 28793* 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 28794* 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 28795* 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 28796 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 28797 A Hilbert lattice is orthomodular. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  OML )
 
Theoremhlclat 28798 A Hilbert lattice is complete. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  CLat )
 
Theoremhlcvl 28799 A Hilbert lattice is an atomic lattice with the covering property. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  HL  ->  K  e.  CvLat )
 
Theoremhlatl 28800 A Hilbert lattice is atomic. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  AtLat )
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