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Theorem List for Metamath Proof Explorer - 29501-29600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorematcmp 29501 If two atoms are comparable, they are equal. (atsseq 22927 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 29502 Frequently-used variation of atcmp 29501. (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 29503 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 29504 An element covered by an atom must be zero. (atcveq0 22928 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 29505 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 29506* Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 22940 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 29507 Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 22956 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 29508 The meet of distinct atoms is zero. (atnemeq0 22957 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 29509* 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 22942 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 29510* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 22951 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 29511* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 22943, 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 29512* 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 29513* 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 29514* 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 29515 An atomic lattice with the covering property is an atomic lattice. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  CvLat  ->  K  e.  AtLat )
 
Theoremcvllat 29516 An atomic lattice with the covering property is a lattice. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  CvLat  ->  K  e.  Lat )
 
TheoremcvlposN 29517 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 29518 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 29519 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 29520 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 29521 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 29522 An atomic covering lattice has the exchange property. (atexch 22961 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 29523 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 29524 A version of cvlexchb1 29520 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 29525 A version of cvlexchb2 29521 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 29526 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 29527 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 29528 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 29529 The covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 22935 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 29530 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 22955 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 29531 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 29532 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 29533 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 29534 Two equivalent ways of expressing that  R is a superposition of  P and  Q, which can replace the superposition part of ishlat1 29542,  ( 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 29544. (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 29535 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 29536 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 29537 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 29538 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 29539 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 ) )
 
18.27.8  Hilbert lattices
 
Syntaxchlt 29540 Extend class notation with Hilbert lattices.
 class  HL
 
Definitiondf-hlat 29541* 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 29542* 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 29543* 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 29544* 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 29545* 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 29546 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 29547 A Hilbert lattice is orthomodular. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  OML )
 
Theoremhlclat 29548 A Hilbert lattice is complete. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  CLat )
 
Theoremhlcvl 29549 A Hilbert lattice is an atomic lattice with the covering property. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  HL  ->  K  e.  CvLat )
 
Theoremhlatl 29550 A Hilbert lattice is atomic. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  AtLat )
 
Theoremhlol 29551 A Hilbert lattice is an ortholattice. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  OL )
 
Theoremhlop 29552 A Hilbert lattice is an orthoposet. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  OP )
 
Theoremhllat 29553 A Hilbert lattice is a lattice. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  Lat )
 
Theoremhlomcmat 29554 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 29555 A Hilbert lattice is a poset. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  Poset )
 
Theoremhlatjcl 29556 Closure of join operation. Frequently-used special case of latjcl 14156 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 29557 Commutatitivity of join operation. Frequently-used special case of latjcom 14165 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 29558 Idempotence of join operation. Frequently-used special case of latjcom 14165 for atoms. (Contributed by NM, 15-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  A ) 
 ->  ( X  .\/  X )  =  X )
 
Theoremhlatjass 29559 Lattice join is associative. Frequently-used special case of latjass 14201 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 29560 Swap 1st and 2nd members of lattice join. Frequently-used special case of latj32 14203 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 29561 Swap 2nd and 3rd members of lattice join. Frequently-used special case of latj32 14203 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 29562 Rotate lattice join of 3 classes. Frequently-used special case of latjrot 14206 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 29563 Rearrangement of lattice join of 4 classes. Frequently-used special case of latj4 14207 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 29564 A join's first argument is less than or equal to the join. Special case of latlej1 14166 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 29565 A join's second argument is less than or equal to the join. Special case of latlej2 14167 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 29566* DeMorgan'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 29567* DeMorgan's law for GLB and LUB. Index-set version of glbconN 29566, 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 29568 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 29569 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 29570* 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 29571 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 29572 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 29573 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 29574 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 29575* 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 29576* 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 29577* 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 29578* 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.  ) )
 
Theoremhl0lt1N 29579 Lattice 0 is less than lattice 1 in a Hilbert lattice. (Contributed by NM, 4-Dec-2011.) (New usage is discouraged.)
 |-  .<  =  ( lt `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  .1.  =  ( 1. `  K )   =>    |-  ( K  e.  HL  ->  .0.  .<  .1.  )
 
Theoremhlexch3 29580 A Hilbert lattice has the exchange property. (atexch 22961 analog.) (Contributed by NM, 15-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B ) 
 /\  ( P  ./\  X )  =  .0.  )  ->  ( P  .<_  ( X 
 .\/  Q )  ->  Q  .<_  ( X  .\/  P ) ) )
 
Theoremhlexch4N 29581 A Hilbert lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (Contributed by NM, 15-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B ) 
 /\  ( P  ./\  X )  =  .0.  )  ->  ( P  .<_  ( X 
 .\/  Q )  <->  ( X  .\/  P )  =  ( X 
 .\/  Q ) ) )
 
Theoremhlatexchb1 29582 A version of hlexchb1 29573 for atoms. (Contributed by NM, 15-Nov-2011.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R ) 
 ->  ( P  .<_  ( R 
 .\/  Q )  <->  ( R  .\/  P )  =  ( R 
 .\/  Q ) ) )
 
Theoremhlatexchb2 29583 A version of hlexchb2 29574 for atoms. (Contributed by NM, 7-Feb-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R ) 
 ->  ( P  .<_  ( Q 
 .\/  R )  <->  ( P  .\/  R )  =  ( Q 
 .\/  R ) ) )
 
Theoremhlatexch1 29584 Atom exchange property. (Contributed by NM, 7-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R ) 
 ->  ( P  .<_  ( R 
 .\/  Q )  ->  Q  .<_  ( R  .\/  P ) ) )
 
Theoremhlatexch2 29585 Atom exchange property. (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  =/=  R ) 
 ->  ( P  .<_  ( Q 
 .\/  R )  ->  Q  .<_  ( P  .\/  R ) ) )
 
TheoremhlatmstcOLDN 29586* 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 22942 analog.) (Contributed by NM, 21-Oct-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  U  =  ( lub `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( U `  { y  e.  A  |  y  .<_  X } )  =  X )
 
Theoremhlatle 29587* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 22951 analog.) (Contributed by NM, 4-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .<_  Y  <->  A. p  e.  A  ( p  .<_  X  ->  p 
 .<_  Y ) ) )
 
Theoremhlateq 29588* The equality of two Hilbert lattice elements is determined by the atoms under them. (chrelat4i 22953 analog.) (Contributed by NM, 24-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( A. p  e.  A  ( p  .<_  X  <-> 
 p  .<_  Y )  <->  X  =  Y ) )
 
Theoremhlrelat1 29589* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 22943, with  /\ swapped, analog.) (Contributed by NM, 4-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X 
 .<  Y  ->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
 
Theoremhlrelat5N 29590* An atomistic lattice with 0 is relatively atomic, using the definition in Remark 2 of [Kalmbach] p. 149. (Contributed by NM, 21-Oct-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 /\  X  .<  Y ) 
 ->  E. p  e.  A  ( X  .<  ( X 
 .\/  p )  /\  p  .<_  Y ) )
 
Theoremhlrelat 29591* A Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 22944 analog.) (Contributed by NM, 4-Feb-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 /\  X  .<  Y ) 
 ->  E. p  e.  A  ( X  .<  ( X 
 .\/  p )  /\  ( X  .\/  p ) 
 .<_  Y ) )
 
Theoremhlrelat2 29592* A consequence of relative atomicity. (chrelat2i 22945 analog.) (Contributed by NM, 5-Feb-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( -.  X  .<_  Y  <->  E. p  e.  A  ( p  .<_  X  /\  -.  p  .<_  Y ) ) )
 
TheoremexatleN 29593 A condition for an atom to be less than or equal to a lattice element. Part of proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( R  .<_  X  <->  R  =  P ) )
 
Theoremhl2at 29594* A Hilbert lattice has at least 2 atoms. (Contributed by NM, 5-Dec-2011.)
 |-  A  =  ( Atoms `  K )   =>    |-  ( K  e.  HL  ->  E. p  e.  A  E. q  e.  A  p  =/=  q )
 
Theorematex 29595 At least one atom exists. (Contributed by NM, 15-Jul-2012.)
 |-  A  =  ( Atoms `  K )   =>    |-  ( K  e.  HL  ->  A  =/=  (/) )
 
TheoremintnatN 29596 If the intersection with a non-majorizing element is an atom, the intersecting element is not an atom. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( -.  Y  .<_  X  /\  ( X  ./\  Y )  e.  A ) )  ->  -.  Y  e.  A )
 
Theorem2llnne2N 29597 Condition implying that two intersecting lines are different. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  R  e.  A )  /\  -.  P  .<_  ( R 
 .\/  Q ) )  ->  ( R  .\/  P )  =/=  ( R  .\/  Q ) )
 
Theorem2llnneN 29598 Condition implying that two intersecting lines are different. (Contributed by NM, 29-May-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
 .\/  Q ) ) ) 
 ->  ( R  .\/  P )  =/=  ( R  .\/  Q ) )
 
Theoremcvr1 29599 A Hilbert lattice has the covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 22935 analog.) (Contributed by NM, 17-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  ( -.  P  .<_  X  <->  X C ( X  .\/  P ) ) )
 
Theoremcvr2N 29600 Less-than and covers equivalence in a Hilbert lattice. (chcv2 22936 analog.) (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  ( X  .<  ( X 
 .\/  P )  <->  X C ( X 
 .\/  P ) ) )
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