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Theorem List for Metamath Proof Explorer - 29201-29300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdalawlem12 29201 Lemma for dalaw 29205. Second part of dalawlem13 29202. (Contributed by NM, 17-Sep-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  Q  =  R  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U )
 )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
 
Theoremdalawlem13 29202 Lemma for dalaw 29205. Special case to eliminate the requirement  ( ( P  .\/  Q )  .\/  R )  e.  O in dalawlem1 29190. (Contributed by NM, 6-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  O  =  ( LPlanes `  K )   =>    |-  (
 ( ( K  e.  HL  /\  -.  ( ( P  .\/  Q )  .\/  R )  e.  O  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U )
 )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
 
Theoremdalawlem14 29203 Lemma for dalaw 29205. Combine dalawlem10 29199 and dalawlem13 29202. (Contributed by NM, 6-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  O  =  ( LPlanes `  K )   =>    |-  (
 ( ( K  e.  HL  /\  -.  ( ( ( P  .\/  Q )  .\/  R )  e.  O  /\  ( -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  -.  ( ( P 
 .\/  S )  ./\  ( Q  .\/  T ) ) 
 .<_  ( Q  .\/  R )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R 
 .\/  P ) ) ) 
 /\  ( ( P 
 .\/  S )  ./\  ( Q  .\/  T ) ) 
 .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
 
Theoremdalawlem15 29204 Lemma for dalaw 29205. Swap variable triples  P Q R and  S T U in dalawlem14 29203, to obtain the elimination of the remaining conditions in dalawlem1 29190. (Contributed by NM, 6-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  O  =  ( LPlanes `  K )   =>    |-  (
 ( ( K  e.  HL  /\  -.  ( ( ( S  .\/  T )  .\/  U )  e.  O  /\  ( -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( S  .\/  T )  /\  -.  ( ( P 
 .\/  S )  ./\  ( Q  .\/  T ) ) 
 .<_  ( T  .\/  U )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( U 
 .\/  S ) ) ) 
 /\  ( ( P 
 .\/  S )  ./\  ( Q  .\/  T ) ) 
 .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
 
Theoremdalaw 29205 Desargues' law, derived from Desargues' theorem dath 29055 and with no conditions on the atoms. If triples  <. P ,  Q ,  R >. and  <. S ,  T ,  U >. are centrally perspective, i.e.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ), then they are axially perspective. Theorem 13.3 of [Crawley] p. 110. (Contributed by NM, 7-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
 )  ->  ( (
 ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R 
 .\/  U )  ->  (
 ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .\/  (
 ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) ) )
 
SyntaxcpclN 29206 Extend class notation with projective subspace closure.
 class  PCl
 
Definitiondf-pclN 29207* Projective subspace closure, which is the smallest projective subspace containing an arbitrary set of atoms. The subspace closure of the union of a set of projective subspaces is their supremum in  PSubSp. Related to an analogous definition of closure used in Lemma 3.1.4 of [PtakPulmannova] p. 68. (Note that this closure is not necessarily one of the closed projective subspaces  PSubCl of df-psubclN 29254.) (Contributed by NM, 7-Sep-2013.)
 |-  PCl  =  ( k  e.  _V  |->  ( x  e.  ~P ( Atoms `  k )  |-> 
 |^| { y  e.  ( PSubSp `
  k )  |  x  C_  y }
 ) )
 
TheorempclfvalN 29208* 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  ->  U  =  ( x  e.  ~P A  |->  |^| { y  e.  S  |  x  C_  y } )
 )
 
TheorempclvalN 29209* 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 29210 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 29211* 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 29212 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 29213 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 29214 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 29215 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 29216 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 29217 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 29218 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 29219* 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 29269. (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 29220* 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 29221 Extend class notation with polarity of projective subspace $m$.
 class  _|_ P
 
Definitiondf-polarityN 29222* 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 29223* 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 29224* 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 29225 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 29226 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 29227 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 29228 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 29229 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 29230 The closed subspace closure of the empty set. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
 |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( K  e.  HL  ->  (  ._|_  `  (  ._|_  `  (/) ) )  =  (/) )
 
TheorempolpmapN 29231 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 29232 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 29233 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 29234 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 29235 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 29236 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 29237 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 29238 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 29239 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 29240 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 29241 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 29242 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 29243 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 29244 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 29245 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 29246 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 5427.) (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 29247 DeMorgan's law for polarity of projective sum. (oldmj1 28541 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 29248 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 29249 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 29250 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 29251 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 29252 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 29253 Extend class notation with set of all closed projective subspaces for a Hilbert lattice.
 class  PSubCl
 
Definitiondf-psubclN 29254* 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 29255* 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 29256 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 29257 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 29258 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 29259 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 29260 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 29261 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 29262 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 29263 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 29264 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 29265 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 29266* 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 29267 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 29268 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 29269 The projective subspace closure of a finite set of atoms is a closed subspace. Compare the (non-closed) subspace version pclfinN 29219 and also pclcmpatN 29220. (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 29270 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 29271 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 29272 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 29273 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 29274 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 29275 Lemma for osumclN 29286. (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 29276 Lemma for osumclN 29286. (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 29277 Lemma for osumclN 29286. (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 29278 Lemma for osumclN 29286. (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 29279 Lemma for osumclN 29286. (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 29280 Lemma for osumclN 29286. Use atom exchange hlatexch1 28714 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 29281* Lemma for osumclN 29286. (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 29282 Lemma for osumclN 29286. (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 29283 Lemma for osumclN 29286. (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 29284 Lemma for osumclN 29286. Contradict osumcllem9N 29283. (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 29285 Lemma for osumclN 29286. (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 29286 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 29287 For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 29171 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 29288 Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 29272. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 29286. (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 29289 Lemma for pexmidN 29288. 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 29290 Lemma for pexmidN 29288. (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 29291 Lemma for pexmidN 29288. Use atom exchange hlatexch1 28714 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 29292* Lemma for pexmidN 29288. (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 29293 Lemma for pexmidN 29288. (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 29294 Lemma for pexmidN 29288. (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 29295 Lemma for pexmidN 29288. Contradict pexmidlem6N 29294. (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 29296 Lemma for pexmidN 29288. The contradiction of pexmidlem6N 29294 and pexmidlem7N 29295 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 29297 Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 29272. 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 29298 Lemma for pl42N 29302. (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 29299 Lemma for pl42N 29302. (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 29300 Lemma for pl42N 29302. (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 ) ) ) )
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