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Theorem List for Metamath Proof Explorer - 30101-30200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheorempolpmapN 30101 The polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  M  =  ( pmap `  K )   &    |-  P  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( P `  ( M `  X ) )  =  ( M `
  (  ._|_  `  X ) ) )
 
Theorem2polpmapN 30102 Double polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  M  =  ( pmap `  K )   &    |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B )  ->  (  ._|_  `  (  ._|_  `  ( M `  X ) ) )  =  ( M `  X ) )
 
Theorem2polvalN 30103 Value of double polarity. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
 |-  U  =  ( lub `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  (
 pmap `  K )   &    |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  HL  /\  X  C_  A )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  ( M `  ( U `  X ) ) )
 
Theorem2polssN 30104 A set of atoms is a subset of its double polarity. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  C_  (  ._|_  `  (  ._|_  `  X ) ) )
 
Theorem3polN 30105 Triple polarity cancels to a single polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  HL  /\  S  C_  A )  ->  (  ._|_  `  (  ._|_  `  (  ._|_  `  S ) ) )  =  (  ._|_  `  S ) )
 
Theorempolcon3N 30106 Contraposition law for polarity. Remark in [Holland95] p. 223. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  Y )  ->  (  ._|_  `  Y ) 
 C_  (  ._|_  `  X ) )
 
Theorem2polcon4bN 30107 Contraposition law for polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  ->  ( (  ._|_  `  (  ._|_  `  X ) ) 
 C_  (  ._|_  `  (  ._|_  `  Y ) )  <-> 
 (  ._|_  `  Y )  C_  (  ._|_  `  X ) ) )
 
Theorempolcon2N 30108 Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  (  ._|_  `  Y ) )  ->  Y  C_  (  ._|_  `  X ) )
 
Theorempolcon2bN 30109 Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  C_  (  ._|_  `  Y )  <->  Y  C_  (  ._|_  `  X ) ) )
 
Theorempclss2polN 30110 The projective subspace closure is a subset of closed subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  ._|_  =  ( _|_ P `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( K  e.  HL  /\  X  C_  A )  ->  ( U `  X )  C_  (  ._|_  `  (  ._|_  `  X ) ) )
 
Theorempcl0N 30111 The projective subspace closure of the empty subspace. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
 |-  U  =  ( PCl `  K )   =>    |-  ( K  e.  HL  ->  ( U `  (/) )  =  (/) )
 
Theorempcl0bN 30112 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 30113 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 30114 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 30115 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 30116 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 5529.) (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 30117 DeMorgan's law for polarity of projective sum. (oldmj1 29411 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 30118 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 30119 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 30120 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 30121 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 30122 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 30123 Extend class notation with set of all closed projective subspaces for a Hilbert lattice.
 class  PSubCl
 
Definitiondf-psubclN 30124* 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 30125* 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 30126 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 30127 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 30128 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 30129 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 30130 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 30131 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 30132 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 30133 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 30134 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 30135 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 30136* 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 30137 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 30138 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 30139 The projective subspace closure of a finite set of atoms is a closed subspace. Compare the (non-closed) subspace version pclfinN 30089 and also pclcmpatN 30090. (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 30140 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 30141 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 30142 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 30143 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 30144 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 30145 Lemma for osumclN 30156. (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 30146 Lemma for osumclN 30156. (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 30147 Lemma for osumclN 30156. (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 30148 Lemma for osumclN 30156. (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 30149 Lemma for osumclN 30156. (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 30150 Lemma for osumclN 30156. Use atom exchange hlatexch1 29584 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 30151* Lemma for osumclN 30156. (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 30152 Lemma for osumclN 30156. (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 30153 Lemma for osumclN 30156. (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 30154 Lemma for osumclN 30156. Contradict osumcllem9N 30153. (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 30155 Lemma for osumclN 30156. (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 30156 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 30157 For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 30041 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 30158 Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 30142. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 30156. (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 30159 Lemma for pexmidN 30158. 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 30160 Lemma for pexmidN 30158. (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 30161 Lemma for pexmidN 30158. Use atom exchange hlatexch1 29584 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 30162* Lemma for pexmidN 30158. (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 30163 Lemma for pexmidN 30158. (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 30164 Lemma for pexmidN 30158. (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 30165 Lemma for pexmidN 30158. Contradict pexmidlem6N 30164. (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 30166 Lemma for pexmidN 30158. The contradiction of pexmidlem6N 30164 and pexmidlem7N 30165 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 30167 Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 30142. 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 30168 Lemma for pl42N 30172. (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 30169 Lemma for pl42N 30172. (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 30170 Lemma for pl42N 30172. (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 30171 Lemma for pl42N 30172. (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 30172 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 30173 Extend class notation with set of all co-atoms (lattice hyperplanes).
 class  LHyp
 
Syntaxclaut 30174 Extend class notation with set of all lattice automorphisms.
 class  LAut
 
SyntaxcwpointsN 30175 Extend class notation with W points.
 class  WAtoms
 
SyntaxcpautN 30176 Extend class notation with set of all projective automorphisms.
 class  PAut
 
Definitiondf-lhyp 30177* 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 30178* 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 30179* 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 30180* 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 30181* 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 30182 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 30183 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 30184* 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 30185 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 30186 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 30187 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 30188 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 30189 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 30190 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 30191* 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 30192 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 30193 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 30194* 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 30195* 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 30196* Lemma for lhpexle1 30197 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 30197* 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 30198* Lemma for lhpexle2 30199. (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 30199* 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 30200* 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 ) ) )
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