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Theorem List for Metamath Proof Explorer - 29401-29500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem1psubclN 29401 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 29402 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 29403 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 29404* 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 29405 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 29406 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 29407 The projective subspace closure of a finite set of atoms is a closed subspace. Compare the (non-closed) subspace version pclfinN 29357 and also pclcmpatN 29358. (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 29408 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 29409 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 29410 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 29411 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 29412 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 29413 Lemma for osumclN 29424. (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 29414 Lemma for osumclN 29424. (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 29415 Lemma for osumclN 29424. (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 29416 Lemma for osumclN 29424. (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 29417 Lemma for osumclN 29424. (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 29418 Lemma for osumclN 29424. Use atom exchange hlatexch1 28852 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 29419* Lemma for osumclN 29424. (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 29420 Lemma for osumclN 29424. (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 29421 Lemma for osumclN 29424. (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 29422 Lemma for osumclN 29424. Contradict osumcllem9N 29421. (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 29423 Lemma for osumclN 29424. (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 29424 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 29425 For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 29309 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 29426 Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 29410. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 29424. (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 29427 Lemma for pexmidN 29426. 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 29428 Lemma for pexmidN 29426. (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 29429 Lemma for pexmidN 29426. Use atom exchange hlatexch1 28852 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 29430* Lemma for pexmidN 29426. (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 29431 Lemma for pexmidN 29426. (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 29432 Lemma for pexmidN 29426. (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 29433 Lemma for pexmidN 29426. Contradict pexmidlem6N 29432. (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 29434 Lemma for pexmidN 29426. The contradiction of pexmidlem6N 29432 and pexmidlem7N 29433 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 29435 Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 29410. 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 29436 Lemma for pl42N 29440. (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 29437 Lemma for pl42N 29440. (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 29438 Lemma for pl42N 29440. (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 29439 Lemma for pl42N 29440. (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 29440 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 29441 Extend class notation with set of all co-atoms (lattice hyperplanes).
 class  LHyp
 
Syntaxclaut 29442 Extend class notation with set of all lattice automorphisms.
 class  LAut
 
SyntaxcwpointsN 29443 Extend class notation with W points.
 class  WAtoms
 
SyntaxcpautN 29444 Extend class notation with set of all projective automorphisms.
 class  PAut
 
Definitiondf-lhyp 29445* 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 29446* 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 29447* 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 29448* 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 29449* 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 29450 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 29451 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 29452* 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 29453 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 29454 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 29455 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 29456 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 29457 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 29458 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 29459* 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 29460 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 29461 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 29462* 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 29463* 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 29464* Lemma for lhpexle1 29465 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 29465* 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 29466* Lemma for lhpexle2 29467. (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 29467* 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 29468* 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 29469* 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 29470* 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 29471 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 29472 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 29473 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 29474 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 29475 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 29476 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 29477 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 29478 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 29479 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.  )
 
Theoremlhpmcvr 29480 The meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 7-Dec-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  C  =  (  <o  `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  ( X  ./\ 
 W ) C X )
 
Theoremlhpmcvr2 29481* Alternate way to express that the meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 9-Apr-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  ( p 
 .\/  ( X  ./\  W ) )  =  X ) )
 
Theoremlhpmcvr3 29482 Specialization of lhpmcvr2 29481. TODO: Use this to simplify many uses of  ( P  .\/  ( X  ./\  W ) )  =  X to become  P  .<_  X. (Contributed by NM, 6-Apr-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) 
 ->  ( P  .<_  X  <->  ( P  .\/  ( X  ./\  W ) )  =  X ) )
 
Theoremlhpmcvr4N 29483 Specialization of lhpmcvr2 29481. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y )  .<_  W 
 /\  P  .<_  X ) )  ->  -.  P  .<_  Y )
 
Theoremlhpmcvr5N 29484* Specialization of lhpmcvr2 29481. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X  ./\  Y )  .<_  W ) )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  ( p 
 .\/  ( X  ./\  W ) )  =  X ) )
 
Theoremlhpmcvr6N 29485* Specialization of lhpmcvr2 29481. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X  ./\  Y )  .<_  W ) )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  p  .<_  X ) )
 
Theoremlhpm0atN 29486 If the meet of a lattice hyperplane with a nonzero element is zero, the element is an atom. (Contributed by NM, 28-Apr-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/=  .0.  /\  ( X  ./\  W )  =  .0.  ) ) 
 ->  X  e.  A )
 
Theoremlhpmat 29487 An element covered by the lattice unit, when conjoined with an atom not under it, equals the lattice zero. (Contributed by NM, 6-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  ./\  W )  =  .0.  )
 
Theoremlhpmatb 29488 An element covered by the lattice unit, when conjoined with an atom, equals zero iff the atom is not under it. (Contributed by NM, 15-Jun-2013.)
 |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  ->  ( -.  P  .<_  W  <->  ( P  ./\  W )  =  .0.  )
 )
 
Theoremlhp2at0 29489 Join and meet with different atoms under co-atom  W. (Contributed by NM, 15-Jun-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) ) 
 ->  ( ( P  .\/  U )  ./\  V )  =  .0.  )
 
Theoremlhp2atnle 29490 Inequality for 2 different atoms under co-atom  W. (Contributed by NM, 17-Jun-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) ) 
 ->  -.  V  .<_  ( P 
 .\/  U ) )
 
Theoremlhp2atne 29491 Inequality for joins with 2 different atoms under co-atom  W. (Contributed by NM, 22-Jul-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( ( U  e.  A  /\  U  .<_  W ) 
 /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =/=  V )  ->  ( P  .\/  U )  =/=  ( Q  .\/  V ) )
 
Theoremlhp2at0nle 29492 Inequality for 2 different atoms (or an atom and zero) under co-atom  W. (Contributed by NM, 28-Jul-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V )  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) ) 
 ->  -.  V  .<_  ( P 
 .\/  U ) )
 
Theoremlhp2at0ne 29493 Inequality for joins with 2 different atoms (or an atom and zero) under co-atom  W. (Contributed by NM, 28-Jul-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) ) 
 /\  U  =/=  V )  ->  ( P  .\/  U )  =/=  ( Q 
 .\/  V ) )
 
Theoremlhpelim 29494 Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat 29487 to see if this can be used to shorten them. (Contributed by NM, 27-Apr-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B )  ->  ( ( P  .\/  ( X  ./\ 
 W ) )  ./\  W )  =  ( X 
 ./\  W ) )
 
Theoremlhpmod2i2 29495 Modular law for hyperplanes analogous to atmod2i2 29319 for atoms. (Contributed by NM, 9-Feb-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  Y  .<_  X ) 
 ->  ( ( X  ./\  W )  .\/  Y )  =  ( X  ./\  ( W  .\/  Y ) ) )
 
Theoremlhpmod6i1 29496 Modular law for hyperplanes analogous to complement of atmod2i1 29318 for atoms. (Contributed by NM, 1-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  X  .<_  W ) 
 ->  ( X  .\/  ( Y  ./\  W ) )  =  ( ( X 
 .\/  Y )  ./\  W ) )
 
Theoremlhprelat3N 29497* The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 28869. (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  C  =  (  <o  `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  E. w  e.  H  ( X  .<_  ( Y  ./\  w )  /\  ( Y  ./\  w ) C Y ) )
 
Theoremcdlemb2 29498* Given two atoms not under the fiducial (reference) co-atom  W, there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 30-May-2012.)
 |-  .<_  =  ( le `  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 )  ->  E. r  e.  A  ( -.  r  .<_  W  /\  -.  r  .<_  ( P  .\/  Q ) ) )
 
Theoremlhple 29499 Property of a lattice element under a co-atom. (Contributed by NM, 28-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( ( P  .\/  X )  ./\  W )  =  X )
 
Theoremlhpat 29500 Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  ( ( P  .\/  Q )  ./\  W )  e.  A )
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