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Theorem List for Metamath Proof Explorer - 28801-28900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempcl0bN 28801 The projective subspace closure of the empty subspace. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( K  e.  HL  /\  P  C_  A )  ->  ( ( U `
  P )  =  (/) 
 <->  P  =  (/) ) )
 
TheorempmaplubN 28802 The LUB of a projective map is the projective map's argument. (Contributed by NM, 13-Mar-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( U `  ( M `  X ) )  =  X )
 
TheoremsspmaplubN 28803 A set of atoms is a subset of the projective map of its LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
 |-  U  =  ( lub `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  (
 pmap `  K )   =>    |-  ( ( K  e.  HL  /\  S  C_  A )  ->  S  C_  ( M `  ( U `  S ) ) )
 
Theorem2pmaplubN 28804 Double projective map of an LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
 |-  U  =  ( lub `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  (
 pmap `  K )   =>    |-  ( ( K  e.  HL  /\  S  C_  A )  ->  ( M `  ( U `  ( M `  ( U `
  S ) ) ) )  =  ( M `  ( U `
  S ) ) )
 
TheorempaddunN 28805 The closure of the projective sum of two sets of atoms is the same as the closure of their union. (Closure is actually double polarity, which can be trivially inferred from this theorem using fveq2d 5381.) (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   =>    |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (  ._|_  `  ( S  .+  T ) )  =  (  ._|_  `  ( S  u.  T ) ) )
 
Theorempoldmj1N 28806 DeMorgan's law for polarity of projective sum. (oldmj1 28100 analog.) (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   =>    |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (  ._|_  `  ( S  .+  T ) )  =  (
 (  ._|_  `  S )  i^i  (  ._|_  `  T ) ) )
 
Theorempmapj2N 28807 The projective map of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  M  =  ( pmap `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( M `
  ( X  .\/  Y ) )  =  ( 
 ._|_  `  (  ._|_  `  (
 ( M `  X )  .+  ( M `  Y ) ) ) ) )
 
TheorempmapocjN 28808 The projective map of the orthocomplement of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  F  =  ( pmap `  K )   &    |-  .+  =  ( + P `  K )   &    |-  N  =  ( _|_ P `
  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `
  (  ._|_  `  ( X  .\/  Y ) ) )  =  ( N `
  ( ( F `
  X )  .+  ( F `  Y ) ) ) )
 
TheorempolatN 28809 The polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
 |-  ._|_  =  ( oc `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  (
 pmap `  K )   &    |-  P  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  OL  /\  Q  e.  A )  ->  ( P `  { Q } )  =  ( M `  (  ._|_  `  Q ) ) )
 
Theorem2polatN 28810 Double polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  P  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  ( P `  { Q } ) )  =  { Q } )
 
TheorempnonsingN 28811 The intersection of a set of atoms and its polarity is empty. Definition of nonsingular in [Holland95] p. 214. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  P  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  HL  /\  X  C_  A )  ->  ( X  i^i  ( P `  X ) )  =  (/) )
 
SyntaxcpscN 28812 Extend class notation with set of all closed projective subspaces for a Hilbert lattice.
 class  PSubCl
 
Definitiondf-psubclN 28813* Define set of all closed projective subspaces, which are those sets of atoms that equal their double polarity. Based on definition in [Holland95] p. 223. (Contributed by NM, 23-Jan-2012.)
 |-  PSubCl  =  ( k  e.  _V  |->  { s  |  ( s 
 C_  ( Atoms `  k
 )  /\  ( ( _|_ P `  k ) `
  ( ( _|_
 P `  k ) `  s ) )  =  s ) } )
 
TheorempsubclsetN 28814* The set of closed projective subspaces in a Hilbert lattice. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  ._|_  =  ( _|_ P `  K )   &    |-  C  =  ( PSubCl `  K )   =>    |-  ( K  e.  B  ->  C  =  { s  |  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) }
 )
 
TheoremispsubclN 28815 The predicate "is a closed projective subspace". (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  ._|_  =  ( _|_ P `  K )   &    |-  C  =  ( PSubCl `  K )   =>    |-  ( K  e.  D  ->  ( X  e.  C  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
 
TheorempsubcliN 28816 Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  ._|_  =  ( _|_ P `  K )   &    |-  C  =  ( PSubCl `  K )   =>    |-  ( ( K  e.  D  /\  X  e.  C )  ->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) )
 
Theorempsubcli2N 28817 Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
 |-  ._|_  =  ( _|_ P `  K )   &    |-  C  =  ( PSubCl `  K )   =>    |-  ( ( K  e.  D  /\  X  e.  C )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
 
TheorempsubclsubN 28818 A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
 |-  S  =  ( PSubSp `  K )   &    |-  C  =  ( PSubCl `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  C ) 
 ->  X  e.  S )
 
TheorempsubclssatN 28819 A closed projective subspace is a set of atoms. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  C  =  ( PSubCl `  K )   =>    |-  (
 ( K  e.  D  /\  X  e.  C ) 
 ->  X  C_  A )
 
TheorempmapidclN 28820 Projective map of the LUB of a closed subspace. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
 |-  U  =  ( lub `  K )   &    |-  M  =  ( pmap `  K )   &    |-  C  =  (
 PSubCl `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  C )  ->  ( M `  ( U `  X ) )  =  X )
 
Theorem0psubclN 28821 The empty set is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
 |-  C  =  ( PSubCl `  K )   =>    |-  ( K  e.  HL  ->  (/)  e.  C )
 
Theorem1psubclN 28822 The set of all atoms is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  C  =  ( PSubCl `  K )   =>    |-  ( K  e.  HL  ->  A  e.  C )
 
TheorematpsubclN 28823 A point (singleton of an atom) is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  C  =  ( PSubCl `  K )   =>    |-  (
 ( K  e.  HL  /\  Q  e.  A ) 
 ->  { Q }  e.  C )
 
TheorempmapsubclN 28824 A projective map value is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  M  =  ( pmap `  K )   &    |-  C  =  ( PSubCl `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B ) 
 ->  ( M `  X )  e.  C )
 
Theoremispsubcl2N 28825* Alternate predicate for "is a closed projective subspace". Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  M  =  ( pmap `  K )   &    |-  C  =  ( PSubCl `  K )   =>    |-  ( K  e.  HL  ->  ( X  e.  C  <->  E. y  e.  B  X  =  ( M `  y ) ) )
 
TheorempsubclinN 28826 The intersection of two closed subspaces is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
 |-  C  =  ( PSubCl `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( X  i^i  Y )  e.  C )
 
TheorempaddatclN 28827 The projective sum of a closed subspace and an atom is a closed projective subspace. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |-  C  =  ( PSubCl `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A ) 
 ->  ( X  .+  { Q } )  e.  C )
 
TheorempclfinclN 28828 The projective subspace closure of a finite set of atoms is a closed subspace. Compare the (non-closed) subspace version pclfinN 28778 and also pclcmpatN 28779. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  U  =  ( PCl `  K )   &    |-  S  =  ( PSubCl `  K )   =>    |-  ( ( K  e.  HL  /\  X  C_  A  /\  X  e.  Fin )  ->  ( U `  X )  e.  S )
 
TheoremlinepsubclN 28829 A line is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
 |-  N  =  ( Lines `  K )   &    |-  C  =  ( PSubCl `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  N ) 
 ->  X  e.  C )
 
TheorempolsubclN 28830 A polarity is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  ._|_  =  ( _|_ P `  K )   &    |-  C  =  ( PSubCl `  K )   =>    |-  ( ( K  e.  HL  /\  X  C_  A )  ->  (  ._|_  `  X )  e.  C )
 
Theorempoml4N 28831 Orthomodular law for projective lattices. Lemma 3.3(1) in [Holland95] p. 215. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  ->  ( ( X  C_  Y  /\  (  ._|_  `  (  ._|_  `  Y ) )  =  Y )  ->  ( (  ._|_  `  (
 (  ._|_  `  X )  i^i  Y ) )  i^i 
 Y )  =  ( 
 ._|_  `  (  ._|_  `  X ) ) ) )
 
Theorempoml5N 28832 Orthomodular law for projective lattices. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  (  ._|_  `  Y ) )  ->  ( (  ._|_  `  (
 (  ._|_  `  X )  i^i  (  ._|_  `  Y ) ) )  i^i  (  ._|_  `  Y ) )  =  (  ._|_  `  (  ._|_  `  X ) ) )
 
Theorempoml6N 28833 Orthomodular law for projective lattices. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
 |-  C  =  ( PSubCl `  K )   &    |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  Y )  ->  ( ( 
 ._|_  `  ( (  ._|_  `  X )  i^i  Y ) )  i^i  Y )  =  X )
 
Theoremosumcllem1N 28834 Lemma for osumclN 28845. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  C  =  ( PSubCl `  K )   &    |-  M  =  ( X  .+  { p } )   &    |-  U  =  ( 
 ._|_  `  (  ._|_  `  ( X  .+  Y ) ) )   =>    |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  ( U  i^i  M )  =  M )
 
Theoremosumcllem2N 28835 Lemma for osumclN 28845. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  C  =  ( PSubCl `  K )   &    |-  M  =  ( X  .+  { p } )   &    |-  U  =  ( 
 ._|_  `  (  ._|_  `  ( X  .+  Y ) ) )   =>    |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  X  C_  ( U  i^i  M ) )
 
Theoremosumcllem3N 28836 Lemma for osumclN 28845. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  C  =  ( PSubCl `  K )   &    |-  M  =  ( X  .+  { p } )   &    |-  U  =  ( 
 ._|_  `  (  ._|_  `  ( X  .+  Y ) ) )   =>    |-  ( ( K  e.  HL  /\  Y  e.  C  /\  X  C_  (  ._|_  `  Y ) )  ->  ( (  ._|_  `  X )  i^i  U )  =  Y )
 
Theoremosumcllem4N 28837 Lemma for osumclN 28845. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  C  =  ( PSubCl `  K )   &    |-  M  =  ( X  .+  { p } )   &    |-  U  =  ( 
 ._|_  `  (  ._|_  `  ( X  .+  Y ) ) )   =>    |-  ( ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  (  ._|_  `  Y )
 )  /\  ( r  e.  X  /\  q  e.  Y ) )  ->  q  =/=  r )
 
Theoremosumcllem5N 28838 Lemma for osumclN 28845. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  C  =  ( PSubCl `  K )   &    |-  M  =  ( X  .+  { p } )   &    |-  U  =  ( 
 ._|_  `  (  ._|_  `  ( X  .+  Y ) ) )   =>    |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  ( r  e.  X  /\  q  e.  Y  /\  p  .<_  ( r  .\/  q )
 ) )  ->  p  e.  ( X  .+  Y ) )
 
Theoremosumcllem6N 28839 Lemma for osumclN 28845. Use atom exchange hlatexch1 28273 to swap  p and  q. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  C  =  ( PSubCl `  K )   &    |-  M  =  ( X  .+  { p } )   &    |-  U  =  ( 
 ._|_  `  (  ._|_  `  ( X  .+  Y ) ) )   =>    |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X 
 C_  (  ._|_  `  Y )  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  Y  /\  q  .<_  ( r  .\/  p )
 ) )  ->  p  e.  ( X  .+  Y ) )
 
Theoremosumcllem7N 28840* Lemma for osumclN 28845. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  C  =  ( PSubCl `  K )   &    |-  M  =  ( X  .+  { p } )   &    |-  U  =  ( 
 ._|_  `  (  ._|_  `  ( X  .+  Y ) ) )   =>    |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X 
 C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M ) )  ->  p  e.  ( X  .+  Y ) )
 
Theoremosumcllem8N 28841 Lemma for osumclN 28845. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  C  =  ( PSubCl `  K )   &    |-  M  =  ( X  .+  { p } )   &    |-  U  =  ( 
 ._|_  `  (  ._|_  `  ( X  .+  Y ) ) )   =>    |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X 
 C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  -.  p  e.  ( X  .+  Y ) )  ->  ( Y  i^i  M )  =  (/) )
 
Theoremosumcllem9N 28842 Lemma for osumclN 28845. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  C  =  ( PSubCl `  K )   &    |-  M  =  ( X  .+  { p } )   &    |-  U  =  ( 
 ._|_  `  (  ._|_  `  ( X  .+  Y ) ) )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  ( X 
 C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  U )  /\  -.  p  e.  ( X  .+  Y ) )  ->  M  =  X )
 
Theoremosumcllem10N 28843 Lemma for osumclN 28845. Contradict osumcllem9N 28842. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  C  =  ( PSubCl `  K )   &    |-  M  =  ( X  .+  { p } )   &    |-  U  =  ( 
 ._|_  `  (  ._|_  `  ( X  .+  Y ) ) )   =>    |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  M  =/=  X )
 
Theoremosumcllem11N 28844 Lemma for osumclN 28845. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
 |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  C  =  ( PSubCl `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C ) 
 /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/) ) ) 
 ->  ( X  .+  Y )  =  (  ._|_  `  (  ._|_  `  ( X 
 .+  Y ) ) ) )
 
TheoremosumclN 28845 Closure of orthogonal sum. If  X and  Y are orthogonal closed projective subspaces, then their sum is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
 |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  C  =  ( PSubCl `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C ) 
 /\  X  C_  (  ._|_  `  Y ) ) 
 ->  ( X  .+  Y )  e.  C )
 
TheorempmapojoinN 28846 For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 28730 where only one direction holds. (Contributed by NM, 11-Apr-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  M  =  (
 pmap `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  ( 
 ._|_  `  Y ) ) 
 ->  ( M `  ( X  .\/  Y ) )  =  ( ( M `
  X )  .+  ( M `  Y ) ) )
 
TheorempexmidN 28847 Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 28831. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 28845. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   =>    |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  ( X  .+  (  ._|_  `  X )
 )  =  A )
 
Theorempexmidlem1N 28848 Lemma for pexmidN 28847. Holland's proof implicitly requires  q  =/=  r, which we prove here. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  M  =  ( X  .+  { p } )   =>    |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
 r  e.  X  /\  q  e.  (  ._|_  `  X ) ) ) 
 ->  q  =/=  r
 )
 
Theorempexmidlem2N 28849 Lemma for pexmidN 28847. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  M  =  ( X  .+  { p } )   =>    |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  (
 r  e.  X  /\  q  e.  (  ._|_  `  X )  /\  p  .<_  ( r  .\/  q
 ) ) )  ->  p  e.  ( X  .+  (  ._|_  `  X ) ) )
 
Theorempexmidlem3N 28850 Lemma for pexmidN 28847. Use atom exchange hlatexch1 28273 to swap  p and  q. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  M  =  ( X  .+  { p } )   =>    |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  (
 r  e.  X  /\  q  e.  (  ._|_  `  X ) )  /\  q  .<_  ( r  .\/  p ) )  ->  p  e.  ( X  .+  (  ._|_  `  X ) ) )
 
Theorempexmidlem4N 28851* Lemma for pexmidN 28847. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  M  =  ( X  .+  { p } )   =>    |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X )  i^i  M ) ) )  ->  p  e.  ( X  .+  (  ._|_  `  X ) ) )
 
Theorempexmidlem5N 28852 Lemma for pexmidN 28847. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  M  =  ( X  .+  { p } )   =>    |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (
 (  ._|_  `  X )  i^i  M )  =  (/) )
 
Theorempexmidlem6N 28853 Lemma for pexmidN 28847. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  M  =  ( X  .+  { p } )   =>    |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  (
 (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  M  =  X )
 
Theorempexmidlem7N 28854 Lemma for pexmidN 28847. Contradict pexmidlem6N 28853. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  M  =  ( X  .+  { p } )   =>    |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  (
 (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  M  =/=  X )
 
Theorempexmidlem8N 28855 Lemma for pexmidN 28847. The contradiction of pexmidlem6N 28853 and pexmidlem7N 28854 shows that there can be no atom  p that is not in  X  .+  (  ._|_  `  X ), which is therefore the whole atom space. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   =>    |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/= 
 (/) ) )  ->  ( X  .+  (  ._|_  `  X ) )  =  A )
 
TheorempexmidALTN 28856 Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 28831. Lemma 3.3(2) in [Holland95] p. 215. In our proof, we use the variables  X,  M,  p,  q,  r in place of Hollands' l, m, P, Q, L respectively. TODO: should we make this obsolete? (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   =>    |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  ( X  .+  (  ._|_  `  X )
 )  =  A )
 
Theorempl42lem1N 28857 Lemma for pl42N 28861. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  F  =  ( pmap `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B ) )  ->  ( ( X  .<_  ( 
 ._|_  `  Y )  /\  Z  .<_  (  ._|_  `  W ) )  ->  ( F `
  ( ( ( ( X  .\/  Y )  ./\  Z )  .\/  W )  ./\  V )
 )  =  ( ( ( ( ( F `
  X )  .+  ( F `  Y ) )  i^i  ( F `
  Z ) ) 
 .+  ( F `  W ) )  i^i  ( F `  V ) ) ) )
 
Theorempl42lem2N 28858 Lemma for pl42N 28861. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  F  =  ( pmap `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B ) )  ->  ( ( ( F `
  X )  .+  ( F `  Y ) )  .+  ( ( ( F `  X )  .+  ( F `  W ) )  i^i  ( ( F `  Y )  .+  ( F `
  V ) ) ) )  C_  ( F `  ( ( X 
 .\/  Y )  .\/  (
 ( X  .\/  W )  ./\  ( Y  .\/  V ) ) ) ) )
 
Theorempl42lem3N 28859 Lemma for pl42N 28861. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  F  =  ( pmap `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B ) )  ->  ( ( ( ( ( F `  X )  .+  ( F `  Y ) )  i^i  ( F `  Z ) )  .+  ( F `
  W ) )  i^i  ( F `  V ) )  C_  ( ( ( ( F `  X ) 
 .+  ( F `  Y ) )  .+  ( F `  W ) )  i^i  ( ( ( F `  X )  .+  ( F `  Y ) )  .+  ( F `  V ) ) ) )
 
Theorempl42lem4N 28860 Lemma for pl42N 28861. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  F  =  ( pmap `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B ) )  ->  ( ( X  .<_  ( 
 ._|_  `  Y )  /\  Z  .<_  (  ._|_  `  W ) )  ->  ( F `
  ( ( ( ( X  .\/  Y )  ./\  Z )  .\/  W )  ./\  V )
 )  C_  ( F `  ( ( X  .\/  Y )  .\/  ( ( X  .\/  W )  ./\  ( Y  .\/  V ) ) ) ) ) )
 
Theorempl42N 28861 Law holding in a Hilbert lattice that fails in orthomodular lattice L42 (Figure 7 in [MegPav2000] p. 2366). (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B ) )  ->  ( ( X  .<_  ( 
 ._|_  `  Y )  /\  Z  .<_  (  ._|_  `  W ) )  ->  ( ( ( ( X  .\/  Y )  ./\  Z )  .\/  W )  ./\  V ) 
 .<_  ( ( X  .\/  Y )  .\/  ( ( X  .\/  W )  ./\  ( Y  .\/  V ) ) ) ) )
 
Syntaxclh 28862 Extend class notation with set of all co-atoms (lattice hyperplanes).
 class  LHyp
 
Syntaxclaut 28863 Extend class notation with set of all lattice automorphisms.
 class  LAut
 
SyntaxcwpointsN 28864 Extend class notation with W points.
 class  WAtoms
 
SyntaxcpautN 28865 Extend class notation with set of all projective automorphisms.
 class  PAut
 
Definitiondf-lhyp 28866* Define the set of lattice hyperplanes, which are all lattice elements covered by 1 (i.e. all co-atoms). We call them "hyperplanes" instead of "co-atoms" in analogy with projective geometry hyperplanes. (Contributed by NM, 11-May-2012.)
 |-  LHyp  =  ( k  e.  _V  |->  { x  e.  ( Base `  k )  |  x (  <o  `  k )
 ( 1. `  k
 ) } )
 
Definitiondf-laut 28867* Define set of lattice autoisomorphisms. (Contributed by NM, 11-May-2012.)
 |-  LAut  =  ( k  e.  _V  |->  { f  |  ( f : ( Base `  k
 )
 -1-1-onto-> ( Base `  k )  /\  A. x  e.  ( Base `  k ) A. y  e.  ( Base `  k ) ( x ( le `  k
 ) y  <->  ( f `  x ) ( le `  k ) ( f `
  y ) ) ) } )
 
Definitiondf-watsN 28868* Define W-atoms corresponding to an arbitrary "fiducial (i.e. reference) atom"  d. These are all atoms not in the polarity of  { d } ), which is the hyperplane determined by  d. Definition of set W in [Crawley] p. 111. (Contributed by NM, 26-Jan-2012.)
 |-  WAtoms  =  ( k  e.  _V  |->  ( d  e.  ( Atoms `  k )  |->  ( (
 Atoms `  k )  \  ( ( _|_ P `  k ) `  { d } ) ) ) )
 
Definitiondf-pautN 28869* Define set of all projective automorphisms. This is the intended definition of automorphism in [Crawley] p. 112. (Contributed by NM, 26-Jan-2012.)
 |-  PAut  =  ( k  e.  _V  |->  { f  |  ( f : ( PSubSp `  k
 )
 -1-1-onto-> ( PSubSp `  k )  /\  A. x  e.  ( PSubSp `
  k ) A. y  e.  ( PSubSp `  k ) ( x 
 C_  y  <->  ( f `  x )  C_  ( f `
  y ) ) ) } )
 
TheoremwatfvalN 28870* The W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  P  =  ( _|_ P `  K )   &    |-  W  =  (
 WAtoms `  K )   =>    |-  ( K  e.  B  ->  W  =  ( d  e.  A  |->  ( A  \  ( ( _|_ P `  K ) `  { d }
 ) ) ) )
 
TheoremwatvalN 28871 Value of the W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  P  =  ( _|_ P `  K )   &    |-  W  =  (
 WAtoms `  K )   =>    |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( W `  D )  =  ( A  \  (
 ( _|_ P `  K ) `  { D }
 ) ) )
 
TheoremiswatN 28872 The predicate "is a W atom" (corresponding to fiducial atom  D). (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  P  =  ( _|_ P `  K )   &    |-  W  =  (
 WAtoms `  K )   =>    |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( P  e.  ( W `  D )  <->  ( P  e.  A  /\  -.  P  e.  ( ( _|_ P `  K ) `  { D } ) ) ) )
 
Theoremlhpset 28873* The set of co-atoms (lattice hyperplanes). (Contributed by NM, 11-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( K  e.  A  ->  H  =  { w  e.  B  |  w C  .1.  } )
 
Theoremislhp 28874 The predicate "is a co-atom (lattice hyperplane)." (Contributed by NM, 11-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( K  e.  A  ->  ( W  e.  H 
 <->  ( W  e.  B  /\  W C  .1.  )
 ) )
 
Theoremislhp2 28875 The predicate "is a co-atom (lattice hyperplane)." (Contributed by NM, 18-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  A  /\  W  e.  B )  ->  ( W  e.  H  <->  W C  .1.  )
 )
 
Theoremlhpbase 28876 A co-atom is a member of the lattice base set (i.e. a lattice element). (Contributed by NM, 18-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  ( W  e.  H  ->  W  e.  B )
 
Theoremlhp1cvr 28877 The lattice unit covers a co-atom (lattice hyperplane). (Contributed by NM, 18-May-2012.)
 |-  .1.  =  ( 1. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  A  /\  W  e.  H )  ->  W C  .1.  )
 
Theoremlhplt 28878 An atom under a co-atom is strictly less than it. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  P  .<_  W ) )  ->  P  .<  W )
 
Theoremlhp2lt 28879 The join of two atoms under a co-atom is strictly less than it. (Contributed by NM, 8-Jul-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  P  .<_  W ) 
 /\  ( Q  e.  A  /\  Q  .<_  W ) )  ->  ( P  .\/  Q )  .<  W )
 
Theoremlhpexlt 28880* There exists an atom less than a co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
 |-  .<  =  ( lt `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  p  .<  W )
 
Theoremlhp0lt 28881 A co-atom is greater than zero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
 |-  .<  =  ( lt `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  .<  W )
 
Theoremlhpn0 28882 A co-atom is nonzero. TODO: is this needed? (Contributed by NM, 26-Apr-2013.)
 |-  .0.  =  ( 0. `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W  =/=  .0.  )
 
Theoremlhpexle 28883* There exists an atom under a co-atom. (Contributed by NM, 26-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  p  .<_  W )
 
Theoremlhpexnle 28884* There exists an atom not under a co-atom. (Contributed by NM, 12-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  -.  p  .<_  W )
 
Theoremlhpexle1lem 28885* Lemma for lhpexle1 28886 and others that eliminates restrictions on  X. (Contributed by NM, 24-Jul-2013.)
 |-  ( ph  ->  E. p  e.  A  ( p  .<_  W  /\  ps ) )   &    |-  ( ( ph  /\  ( X  e.  A  /\  X  .<_  W ) ) 
 ->  E. p  e.  A  ( p  .<_  W  /\  ps 
 /\  p  =/=  X ) )   =>    |-  ( ph  ->  E. p  e.  A  ( p  .<_  W 
 /\  ps  /\  p  =/= 
 X ) )
 
Theoremlhpexle1 28886* There exists an atom under a co-atom different from any given element. (Contributed by NM, 24-Jul-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W 
 /\  p  =/=  X ) )
 
Theoremlhpexle2lem 28887* Lemma for lhpexle2 28888. (Contributed by NM, 19-Jun-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) ) 
 ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y ) )
 
Theoremlhpexle2 28888* There exists atom under a co-atom different from any two other elements. (Contributed by NM, 24-Jul-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W 
 /\  p  =/=  X  /\  p  =/=  Y ) )
 
Theoremlhpexle3lem 28889* There exists atom under a co-atom different from any 3 other atoms. TODO: study if adant*,simp* usage can be improved. (Contributed by NM, 9-Jul-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  Y  e.  A  /\  Z  e.  A )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  Z  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  =/=  Z ) ) )
 
Theoremlhpexle3 28890* There exists atom under a co-atom different from any three other elements. (Contributed by NM, 24-Jul-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W 
 /\  ( p  =/= 
 X  /\  p  =/=  Y 
 /\  p  =/=  Z ) ) )
 
Theoremlhpex2leN 28891* There exist at least two different atoms under a co-atom. This allows us to create a line under the co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  E. q  e.  A  ( p  .<_  W 
 /\  q  .<_  W  /\  p  =/=  q ) )
 
Theoremlhpoc 28892 The orthocomplement of a co-atom (lattice hyperplane) is an atom. (Contributed by NM, 18-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  B )  ->  ( W  e.  H  <->  (  ._|_  `  W )  e.  A )
 )
 
Theoremlhpoc2N 28893 The orthocomplement of an atom is a co-atom (lattice hyperplane). (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  B )  ->  ( W  e.  A  <->  (  ._|_  `  W )  e.  H )
 )
 
Theoremlhpocnle 28894 The orthocomplement of a co-atom is not under it. (Contributed by NM, 22-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 ._|_  =  ( oc `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  -.  (  ._|_  `  W )  .<_  W )
 
Theoremlhpocat 28895 The orthocomplement of a co-atom is an atom. (Contributed by NM, 9-Feb-2013.)
 |-  ._|_  =  ( oc `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  ._|_  `  W )  e.  A )
 
Theoremlhpocnel 28896 The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 25-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 ._|_  =  ( oc `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  ( (  ._|_  `  W )  e.  A  /\  -.  (  ._|_  `  W ) 
 .<_  W ) )
 
Theoremlhpocnel2 28897 The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 20-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
 
Theoremlhpjat1 28898 The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 18-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( W  .\/  P )  =  .1.  )
 
Theoremlhpjat2 28899 The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 4-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  W )  =  .1.  )
 
Theoremlhpj1 28900 The join of a co-atom (hyperplane) and an element not under it is the lattice unit. (Contributed by NM, 7-Dec-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) 
 ->  ( W  .\/  X )  =  .1.  )
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