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Theorem List for Metamath Proof Explorer - 28201-28300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-cvlat 28201* 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 28202* 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 28203* 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 28204 An atomic lattice with the covering property is an atomic lattice. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  CvLat  ->  K  e.  AtLat )
 
Theoremcvllat 28205 An atomic lattice with the covering property is a lattice. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  CvLat  ->  K  e.  Lat )
 
TheoremcvlposN 28206 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 28207 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 28208 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 28209 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 28210 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 28211 An atomic covering lattice has the exchange property. (atexch 22791 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 28212 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 28213 A version of cvlexchb1 28209 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 28214 A version of cvlexchb2 28210 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 28215 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 28216 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 28217 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 28218 The covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 22765 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 28219 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 22785 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 28220 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 28221 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 28222 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 28223 Two equivalent ways of expressing that  R is a superposition of  P and  Q, which can replace the superposition part of ishlat1 28231,  ( 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 28233. (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 28224 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 28225 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 28226 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 28227 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 28228 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 ) )
 
16.22.8  Hilbert lattices
 
Syntaxchlt 28229 Extend class notation with Hilbert lattices.
 class  HL
 
Definitiondf-hlat 28230* 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 28231* 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 28232* 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 28233* 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 28234* 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 28235 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 28236 A Hilbert lattice is orthomodular. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  OML )
 
Theoremhlclat 28237 A Hilbert lattice is complete. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  CLat )
 
Theoremhlcvl 28238 A Hilbert lattice is an atomic lattice with the covering property. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  HL  ->  K  e.  CvLat )
 
Theoremhlatl 28239 A Hilbert lattice is atomic. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  AtLat )
 
Theoremhlol 28240 A Hilbert lattice is an ortholattice. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  OL )
 
Theoremhlop 28241 A Hilbert lattice is an orthoposet. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  OP )
 
Theoremhllat 28242 A Hilbert lattice is a lattice. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  Lat )
 
Theoremhlomcmat 28243 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 28244 A Hilbert lattice is a poset. (Contributed by NM, 20-Oct-2011.)
 |-  ( K  e.  HL  ->  K  e.  Poset )
 
Theoremhlatjcl 28245 Closure of join operation. Frequently-used special case of latjcl 14000 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 28246 Commutatitivity of join operation. Frequently-used special case of latjcom 14009 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 28247 Idempotence of join operation. Frequently-used special case of latjcom 14009 for atoms. (Contributed by NM, 15-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  A ) 
 ->  ( X  .\/  X )  =  X )
 
Theoremhlatjass 28248 Lattice join is associative. Frequently-used special case of latjass 14045 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 28249 Swap 1st and 2nd members of lattice join. Frequently-used special case of latj32 14047 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 28250 Swap 2nd and 3rd members of lattice join. Frequently-used special case of latj32 14047 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 28251 Rotate lattice join of 3 classes. Frequently-used special case of latjrot 14050 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 28252 Rearrangement of lattice join of 4 classes. Frequently-used special case of latj4 14051 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 28253 A join's first argument is less than or equal to the join. Special case of latlej1 14010 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 28254 A join's second argument is less than or equal to the join. Special case of latlej2 14011 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 28255* 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 28256* DeMorgan's law for GLB and LUB. Index-set version of glbconN 28255, 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 28257 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 28258 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 28259* 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 28260 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 28261 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 28262 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 28263 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 28264* 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 28265* 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 28266* 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 28267* 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 28268 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 28269 A Hilbert lattice has the exchange property. (atexch 22791 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 28270 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 28271 A version of hlexchb1 28262 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 28272 A version of hlexchb2 28263 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 28273 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 28274 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 28275* 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 22772 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 28276* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 22781 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 28277* The equality of two Hilbert lattice elements is determined by the atoms under them. (chrelat4i 22783 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 28278* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 22773, 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 28279* 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 28280* A Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 22774 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 28281* A consequence of relative atomicity. (chrelat2i 22775 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 28282 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 28283* 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 28284 At least one atom exists. (Contributed by NM, 15-Jul-2012.)
 |-  A  =  ( Atoms `  K )   =>    |-  ( K  e.  HL  ->  A  =/=  (/) )
 
TheoremintnatN 28285 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 28286 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 28287 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 28288 A Hilbert lattice has the covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 22765 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 28289 Less-than and covers equivalence in a Hilbert lattice. (chcv2 22766 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 28290* The Hilbert lattice is relatively atomic. Stronger version of hlrelat 28280. (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 28291* 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 28292* 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 28293* 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 28294 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 22785 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 28295 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 28296 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 28297 Lemma for cvrexch 28298. (cvexchlem 22778 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 28298 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 22779 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 28299 Lemma for cvrat 28300. (atcvatlem 22795 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 28300 A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 22796 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 ) )
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