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Theorem List for Metamath Proof Explorer - 28901-29000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcvlatexch3 28901 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 28902 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 28903 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 28904 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 28905 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 28906 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 28907 Two equivalent ways of expressing that  R is a superposition of  P and  Q, which can replace the superposition part of ishlat1 28915,  ( 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 28917. (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 28908 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 28909 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 28910 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 28911 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 28912 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 28913 Extend class notation with Hilbert lattices.
 class  HL
 
Definitiondf-hlat 28914* 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 28915* 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 28916* 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 28917* 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 28918* 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 28919 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 28920 A Hilbert lattice is orthomodular. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  OML )
 
Theoremhlclat 28921 A Hilbert lattice is complete. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  CLat )
 
Theoremhlcvl 28922 A Hilbert lattice is an atomic lattice with the covering property. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  HL  ->  K  e.  CvLat )
 
Theoremhlatl 28923 A Hilbert lattice is atomic. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  AtLat )
 
Theoremhlol 28924 A Hilbert lattice is an ortholattice. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  OL )
 
Theoremhlop 28925 A Hilbert lattice is an orthoposet. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  OP )
 
Theoremhllat 28926 A Hilbert lattice is a lattice. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  Lat )
 
Theoremhlomcmat 28927 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 28928 A Hilbert lattice is a poset. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  Poset )
 
Theoremhlatjcl 28929 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 28930 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 28931 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 28932 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 28933 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 28934 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 28935 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 28936 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 28937 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 28938 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 28939* 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 28940* DeMorgan's law for GLB and LUB. Index-set version of glbconN 28939, 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 28941 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 28942 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 28943* 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 28944 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 28945 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 28946 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 28947 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 28948* 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 28949* 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 28950* 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 28951* 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 28952 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 28953 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 28954 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 28955 A version of hlexchb1 28946 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 28956 A version of hlexchb2 28947 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 28957 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 28958 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 28959* 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 28960* 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 28961* 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 28962* 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 28963* 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 28964* 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 28965* 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 28966 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 28967* 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 28968 At least one atom exists. (Contributed by NM, 15-Jul-2012.)
 |-  A  =  ( Atoms `  K )   =>    |-  ( K  e.  HL  ->  A  =/=  (/) )
 
TheoremintnatN 28969 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 28970 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 28971 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 28972 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 28973 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 ) ) )
 
Theoremhlrelat3 28974* The Hilbert lattice is relatively atomic. Stronger version of hlrelat 28964. (Contributed by NM, 2-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  .\/  =  ( join `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  E. p  e.  A  ( X C ( X  .\/  p ) 
 /\  ( X  .\/  p )  .<_  Y ) )
 
Theoremcvrval3 28975* Binary relation expressing  Y covers  X. (Contributed by NM, 16-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <->  E. p  e.  A  ( -.  p  .<_  X  /\  ( X  .\/  p )  =  Y ) ) )
 
Theoremcvrval4N 28976* Binary relation expressing  Y covers  X. (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <->  ( X  .<  Y  /\  E. p  e.  A  ( X  .\/  p )  =  Y ) ) )
 
Theoremcvrval5 28977* Binary relation expressing  X covers  X  ./\  Y. (Contributed by NM, 7-Dec-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( ( X  ./\  Y ) C X  <->  E. p  e.  A  ( -.  p  .<_  Y  /\  ( p  .\/  ( X 
 ./\  Y ) )  =  X ) ) )
 
Theoremcvrp 28978 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 22955 analog.) (Contributed by NM, 18-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  ( ( X  ./\  P )  =  .0.  <->  X C ( X 
 .\/  P ) ) )
 
Theorematcvr1 28979 An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012.)
 |-  .\/  =  ( join `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A ) 
 ->  ( P  =/=  Q  <->  P C ( P  .\/  Q ) ) )
 
Theorematcvr2 28980 An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012.)
 |-  .\/  =  ( join `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A ) 
 ->  ( P  =/=  Q  <->  P C ( Q  .\/  P ) ) )
 
Theoremcvrexchlem 28981 Lemma for cvrexch 28982. (cvexchlem 22948 analog.) (Contributed by NM, 18-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( ( X  ./\  Y ) C Y  ->  X C ( X  .\/  Y ) ) )
 
Theoremcvrexch 28982 A Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of [Kalmbach] p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (cvexchi 22949 analog.) (Contributed by NM, 18-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( ( X  ./\  Y ) C Y  <->  X C ( X 
 .\/  Y ) ) )
 
Theoremcvratlem 28983 Lemma for cvrat 28984. (atcvatlem 22965 analog.) (Contributed by NM, 22-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A )
 )  /\  ( X  =/=  .0.  /\  X  .<  ( P  .\/  Q )
 ) )  ->  ( -.  P ( le `  K ) X  ->  X  e.  A ) )
 
Theoremcvrat 28984 A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 22966 analog.) (Contributed by NM, 22-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A ) )  ->  ( ( X  =/=  .0.  /\  X  .<  ( P  .\/  Q ) )  ->  X  e.  A ) )
 
Theoremltltncvr 28985 A chained strong ordering is not a covers relation. (Contributed by NM, 18-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .<  Y  /\  Y  .<  Z )  ->  -.  X C Z ) )
 
Theoremltcvrntr 28986 Non-transitive condition for the covers relation. (Contributed by NM, 18-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .<  Y  /\  Y C Z )  ->  -.  X C Z ) )
 
Theoremcvrntr 28987 The covers relation is not transitive. (cvntr 22872 analog.) (Contributed by NM, 18-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X C Y  /\  Y C Z )  ->  -.  X C Z ) )
 
Theorematcvr0eq 28988 The covers relation is not transitive. (atcv0eq 22959 analog.) (Contributed by NM, 29-Nov-2011.)
 |-  .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  (  .0. 
 C ( P  .\/  Q )  <->  P  =  Q ) )
 
Theoremlnnat 28989 A line (the join of two distinct atoms) is not an atom. (Contributed by NM, 14-Jun-2012.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  -.  ( P  .\/  Q )  e.  A )
 )
 
Theorematcvrj0 28990 Two atoms covering the zero subspace are equal. (atcv1 22960 analog.) (Contributed by NM, 29-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A )  /\  X C ( P 
 .\/  Q ) )  ->  ( X  =  .0.  <->  P  =  Q ) )
 
Theoremcvrat2 28991 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 22967 analog.) (Contributed by NM, 30-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A ) 
 /\  ( P  =/=  Q 
 /\  X C ( P  .\/  Q )
 ) )  ->  X  e.  A )
 
TheorematcvrneN 28992 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) 
 /\  P C ( Q  .\/  R )
 )  ->  Q  =/=  R )
 
Theorematcvrj1 28993 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P C ( Q  .\/  R ) )
 
Theorematcvrj2b 28994 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
 )  ->  ( ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) )  <->  P C ( Q 
 .\/  R ) ) )
 
Theorematcvrj2 28995 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P C ( Q  .\/  R ) )
 
TheorematleneN 28996 Inequality derived from atom condition. (Contributed by NM, 7-Feb-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  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  ->  Q  =/=  R )
 
Theorematltcvr 28997 An equivalence of less-than ordering and covers relation. (Contributed by NM, 7-Feb-2012.)
 |-  .<  =  ( lt `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  ( P  .<  ( Q  .\/  R ) 
 <->  P C ( Q 
 .\/  R ) ) )
 
Theorematle 28998* Any non-zero element has an atom under it. (Contributed by NM, 28-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  E. p  e.  A  p  .<_  X )
 
Theorematlt 28999 Two atoms are unequal iff their join is greater than one of them. (Contributed by NM, 6-May-2012.)
 |-  .<  =  ( lt `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P 
 .<  ( P  .\/  Q ) 
 <->  P  =/=  Q ) )
 
Theorematlelt 29000 Transfer less-than relation from one atom to another. (Contributed by NM, 7-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  Q  .<  X ) ) 
 ->  P  .<  X )
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