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Theorem List for Metamath Proof Explorer - 30701-30800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheorempclvalN 30701* Value of the projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( K  e.  V  /\  X  C_  A )  ->  ( U `  X )  =  |^| { y  e.  S  |  X  C_  y } )
 
TheorempclclN 30702 Closure of the projective subspace closure function. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( K  e.  V  /\  X  C_  A )  ->  ( U `  X )  e.  S )
 
TheoremelpclN 30703* Membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  U  =  ( PCl `  K )   &    |-  Q  e.  _V   =>    |-  ( ( K  e.  V  /\  X  C_  A )  ->  ( Q  e.  ( U `  X )  <->  A. y  e.  S  ( X  C_  y  ->  Q  e.  y )
 ) )
 
TheoremelpcliN 30704 Implication of membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
 |-  S  =  ( PSubSp `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  /\  Q  e.  ( U `  X ) )  ->  Q  e.  Y )
 
TheorempclssN 30705 Ordering is preserved by subspace closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  ( U `  X )  C_  ( U `  Y ) )
 
TheorempclssidN 30706 A set of atoms is included in its projective subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( K  e.  V  /\  X  C_  A )  ->  X  C_  ( U `  X ) )
 
TheorempclidN 30707 The projective subspace closure of a projective subspace is itself. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
 |-  S  =  ( PSubSp `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( K  e.  V  /\  X  e.  S )  ->  ( U `  X )  =  X )
 
TheorempclbtwnN 30708 A projective subspace sandwiched between a set of atoms and the set's projective subspace closure equals the closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
 |-  S  =  ( PSubSp `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( ( K  e.  V  /\  X  e.  S )  /\  ( Y  C_  X  /\  X  C_  ( U `  Y ) ) )  ->  X  =  ( U `  Y ) )
 
TheorempclunN 30709 The projective subspace closure of the union of two sets of atoms equals the closure of their projective sum. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( K  e.  V  /\  X  C_  A  /\  Y  C_  A )  ->  ( U `  ( X  u.  Y ) )  =  ( U `  ( X  .+  Y ) ) )
 
Theorempclun2N 30710 The projective subspace closure of the union of two subspaces equals their projective sum. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
 |-  S  =  ( PSubSp `  K )   &    |-  .+  =  ( + P `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  S  /\  Y  e.  S ) 
 ->  ( U `  ( X  u.  Y ) )  =  ( X  .+  Y ) )
 
TheorempclfinN 30711* The projective subspace closure of a set equals the union of the closures of its finite subsets. Analogous to Lemma 3.3.6 of [PtakPulmannova] p. 72. Compare the closed subspace version pclfinclN 30761. (Contributed by NM, 10-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( K  e.  AtLat  /\  X  C_  A )  ->  ( U `  X )  =  U_ y  e.  ( Fin  i^i  ~P X ) ( U `
  y ) )
 
TheorempclcmpatN 30712* The set of projective subspaces is compactly atomistic: if an atom is in the projective subspace closure of a set of atoms, it also belongs to the projective subspace closure of a finite subset of that set. Analogous to Lemma 3.3.10 of [PtakPulmannova] p. 74. (Contributed by NM, 10-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( K  e.  AtLat  /\  X  C_  A  /\  P  e.  ( U `  X ) )  ->  E. y  e.  Fin  ( y  C_  X  /\  P  e.  ( U `  y ) ) )
 
SyntaxcpolN 30713 Extend class notation with polarity of projective subspace $m$.
 class  _|_ P
 
Definitiondf-polarityN 30714* Define polarity of projective subspace, which is a kind of complement of the subspace. Item 2 in [Holland95] p. 222 bottom. For more generality, we define it for all subsets of atoms, not just projective subspaces. The intersection with  Atoms `  l ensures it is defined when  m  =  (/). (Contributed by NM, 23-Oct-2011.)
 |-  _|_ P  =  ( l  e. 
 _V  |->  ( m  e. 
 ~P ( Atoms `  l
 )  |->  ( ( Atoms `  l )  i^i  |^|_ p  e.  m  ( ( pmap `  l ) `  (
 ( oc `  l
 ) `  p )
 ) ) ) )
 
TheorempolfvalN 30715* The projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)
 |-  ._|_  =  ( oc `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  (
 pmap `  K )   &    |-  P  =  ( _|_ P `  K )   =>    |-  ( K  e.  B  ->  P  =  ( m  e.  ~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p )
 ) ) ) )
 
TheorempolvalN 30716* Value of the projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)
 |-  ._|_  =  ( oc `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  (
 pmap `  K )   &    |-  P  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  B  /\  X  C_  A )  ->  ( P `  X )  =  ( A  i^i  |^|_ p  e.  X  ( M `  (  ._|_  `  p ) ) ) )
 
Theorempolval2N 30717 Alternate expression for value of the projective subspace polarity function. Equation for polarity in [Holland95] p. 223. (Contributed by NM, 22-Jan-2012.) (New usage is discouraged.)
 |-  U  =  ( lub `  K )   &    |- 
 ._|_  =  ( oc `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  M  =  ( pmap `  K )   &    |-  P  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  HL  /\  X  C_  A )  ->  ( P `  X )  =  ( M `  (  ._|_  `  ( U `  X ) ) ) )
 
TheorempolsubN 30718 The polarity of a set of atoms is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  HL  /\  X  C_  A )  ->  (  ._|_  `  X )  e.  S )
 
TheorempolssatN 30719 The polarity of a set of atoms is a set of atoms. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  HL  /\  X  C_  A )  ->  (  ._|_  `  X )  C_  A )
 
Theorempol0N 30720 The polarity of the empty projective subspace is the whole space. (Contributed by NM, 29-Oct-2011.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( K  e.  B  ->  (  ._|_  `  (/) )  =  A )
 
Theorempol1N 30721 The polarity of the whole projective subspace is the empty space. Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( K  e.  HL  ->  (  ._|_  `  A )  =  (/) )
 
Theorem2pol0N 30722 The closed subspace closure of the empty set. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
 |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( K  e.  HL  ->  (  ._|_  `  (  ._|_  `  (/) ) )  =  (/) )
 
TheorempolpmapN 30723 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 30724 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 30725 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 30726 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 30727 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 30728 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 30729 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 30730 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 30731 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 30732 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 30733 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 30734 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 30735 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 30736 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 30737 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 30738 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 5545.) (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 30739 De Morgan's law for polarity of projective sum. (oldmj1 30033 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 30740 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 30741 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 30742 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 30743 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 30744 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 30745 Extend class notation with set of all closed projective subspaces for a Hilbert lattice.
 class  PSubCl
 
Definitiondf-psubclN 30746* 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 30747* 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 30748 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 30749 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 30750 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 30751 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 30752 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 30753 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 30754 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 30755 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 30756 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 30757 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 30758* 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 30759 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 30760 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 30761 The projective subspace closure of a finite set of atoms is a closed subspace. Compare the (non-closed) subspace version pclfinN 30711 and also pclcmpatN 30712. (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 30762 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 30763 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 30764 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 30765 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 30766 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 30767 Lemma for osumclN 30778. (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 30768 Lemma for osumclN 30778. (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 30769 Lemma for osumclN 30778. (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 30770 Lemma for osumclN 30778. (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 30771 Lemma for osumclN 30778. (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 30772 Lemma for osumclN 30778. Use atom exchange hlatexch1 30206 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 30773* Lemma for osumclN 30778. (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 30774 Lemma for osumclN 30778. (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 30775 Lemma for osumclN 30778. (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 30776 Lemma for osumclN 30778. Contradict osumcllem9N 30775. (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 30777 Lemma for osumclN 30778. (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 30778 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 30779 For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 30663 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 30780 Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 30764. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 30778. (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 30781 Lemma for pexmidN 30780. 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 30782 Lemma for pexmidN 30780. (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 30783 Lemma for pexmidN 30780. Use atom exchange hlatexch1 30206 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 30784* Lemma for pexmidN 30780. (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 30785 Lemma for pexmidN 30780. (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 30786 Lemma for pexmidN 30780. (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 30787 Lemma for pexmidN 30780. Contradict pexmidlem6N 30786. (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 30788 Lemma for pexmidN 30780. The contradiction of pexmidlem6N 30786 and pexmidlem7N 30787 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 30789 Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 30764. 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 30790 Lemma for pl42N 30794. (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 30791 Lemma for pl42N 30794. (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 30792 Lemma for pl42N 30794. (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 30793 Lemma for pl42N 30794. (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 30794 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 30795 Extend class notation with set of all co-atoms (lattice hyperplanes).
 class  LHyp
 
Syntaxclaut 30796 Extend class notation with set of all lattice automorphisms.
 class  LAut
 
SyntaxcwpointsN 30797 Extend class notation with W points.
 class  WAtoms
 
SyntaxcpautN 30798 Extend class notation with set of all projective automorphisms.
 class  PAut
 
Definitiondf-lhyp 30799* 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 30800* 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 ) ) ) } )
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