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Theorem List for Metamath Proof Explorer - 30101-30200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempadd12N 30101 Commutative/associative law for projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )
 )  ->  ( X  .+  ( Y  .+  Z ) )  =  ( Y  .+  ( X  .+  Z ) ) )
 
Theorempadd4N 30102 Rearrangement of 4 terms in a projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( ( X  .+  Y ) 
 .+  ( Z  .+  W ) )  =  ( ( X  .+  Z )  .+  ( Y 
 .+  W ) ) )
 
Theorempaddidm 30103 Projective subspace sum is idempotent. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 13-Jan-2012.)
 |-  S  =  ( PSubSp `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  B  /\  X  e.  S )  ->  ( X  .+  X )  =  X )
 
TheorempaddclN 30104 The projective sum of two subspaces is a subspace. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
 |-  S  =  ( PSubSp `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  S  /\  Y  e.  S ) 
 ->  ( X  .+  Y )  e.  S )
 
Theorempaddssw1 30105 Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )
 )  ->  ( ( X  C_  Z  /\  Y  C_  Z )  ->  ( X  .+  Y )  C_  ( Z  .+  Z ) ) )
 
Theorempaddssw2 30106 Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )
 )  ->  ( ( X  .+  Y )  C_  Z  ->  ( X  C_  Z  /\  Y  C_  Z ) ) )
 
Theorempaddss 30107 Subset law for projective subspace sum. (unss 3351 analog.) (Contributed by NM, 7-Mar-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  ->  ( ( X  C_  Z  /\  Y  C_  Z )  <->  ( X  .+  Y )  C_  Z ) )
 
Theorempmodlem1 30108* Lemma for pmod1i 30110. (Contributed by NM, 9-Mar-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( Z  e.  S  /\  X  C_  Z  /\  p  e.  Z )  /\  (
 q  e.  X  /\  r  e.  Y  /\  p  .<_  ( q  .\/  r ) ) ) 
 ->  p  e.  ( X  .+  ( Y  i^i  Z ) ) )
 
Theorempmodlem2 30109 Lemma for pmod1i 30110. (Contributed by NM, 9-Mar-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) 
 /\  X  C_  Z )  ->  ( ( X 
 .+  Y )  i^i 
 Z )  C_  ( X  .+  ( Y  i^i  Z ) ) )
 
Theorempmod1i 30110 The modular law holds in a projective subspace. (Contributed by NM, 10-Mar-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  ->  ( X  C_  Z  ->  ( ( X  .+  Y )  i^i 
 Z )  =  ( X  .+  ( Y  i^i  Z ) ) ) )
 
Theorempmod2iN 30111 Dual of the modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )
 )  ->  ( Z  C_  X  ->  ( ( X  i^i  Y )  .+  Z )  =  ( X  i^i  ( Y  .+  Z ) ) ) )
 
TheorempmodN 30112 The modular law for projective subspaces. (Contributed by NM, 26-Mar-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )
 )  ->  ( X  i^i  ( Y  .+  ( X  i^i  Z ) ) )  =  ( ( X  i^i  Y ) 
 .+  ( X  i^i  Z ) ) )
 
Theorempmodl42N 30113 Lemma derived from modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
 |-  S  =  ( PSubSp `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  S  /\  Y  e.  S )  /\  ( Z  e.  S  /\  W  e.  S ) )  ->  ( ( ( X 
 .+  Y )  .+  Z )  i^i  ( ( X  .+  Y ) 
 .+  W ) )  =  ( ( X 
 .+  Y )  .+  ( ( X  .+  Z )  i^i  ( Y 
 .+  W ) ) ) )
 
Theorempmapjoin 30114 The projective map of the join of two lattice elements. Part of Equation 15.5.3 of [MaedaMaeda] p. 63. (Contributed by NM, 27-Jan-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  M  =  ( pmap `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( M `  X )  .+  ( M `
  Y ) ) 
 C_  ( M `  ( X  .\/  Y ) ) )
 
Theorempmapjat1 30115 The projective map of the join of a lattice element and an atom. (Contributed by NM, 28-Jan-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  ( pmap `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A ) 
 ->  ( M `  ( X  .\/  Q ) )  =  ( ( M `
  X )  .+  ( M `  Q ) ) )
 
Theorempmapjat2 30116 The projective map of the join of an atom with a lattice element. (Contributed by NM, 12-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  ( pmap `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A ) 
 ->  ( M `  ( Q  .\/  X ) )  =  ( ( M `
  Q )  .+  ( M `  X ) ) )
 
Theorempmapjlln1 30117 The projective map of the join of a lattice element and a lattice line (expressed as the join  Q  .\/  R of two atoms). (Contributed by NM, 16-Sep-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  ( pmap `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A ) )  ->  ( M `  ( X  .\/  ( Q  .\/  R ) ) )  =  ( ( M `  X ) 
 .+  ( M `  ( Q  .\/  R ) ) ) )
 
Theoremhlmod1i 30118 A version of the modular law pmod1i 30110 that holds in a Hilbert lattice. (Contributed by NM, 13-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  F  =  ( pmap `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .<_  Z  /\  ( F `  ( X  .\/  Y ) )  =  ( ( F `  X )  .+  ( F `  Y ) ) ) 
 ->  ( ( X  .\/  Y )  ./\  Z )  =  ( X  .\/  ( Y  ./\  Z ) ) ) )
 
Theorematmod1i1 30119 Version of modular law pmod1i 30110 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 11-May-2012.) (Revised by Mario Carneiro, 10-May-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B )  /\  P  .<_  Y )  ->  ( P  .\/  ( X 
 ./\  Y ) )  =  ( ( P  .\/  X )  ./\  Y )
 )
 
Theorematmod1i1m 30120 Version of modular law pmod1i 30110 that holds in a Hilbert lattice, when an element meets an atom. (Contributed by NM, 2-Sep-2012.) (Revised by Mario Carneiro, 10-May-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) 
 /\  ( X  ./\  P )  .<_  Z )  ->  ( ( X  ./\  P )  .\/  ( Y  ./\ 
 Z ) )  =  ( ( ( X 
 ./\  P )  .\/  Y )  ./\  Z ) )
 
Theorematmod1i2 30121 Version of modular law pmod1i 30110 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-2012.) (Revised by Mario Carneiro, 10-May-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  Y )  ->  ( X  .\/  ( P 
 ./\  Y ) )  =  ( ( X  .\/  P )  ./\  Y )
 )
 
Theoremllnmod1i2 30122 Version of modular law pmod1i 30110 that holds in a Hilbert lattice, when one element is a lattice line (expressed as the join  P  .\/  Q). (Contributed by NM, 16-Sep-2012.) (Revised by Mario Carneiro, 10-May-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 /\  ( P  e.  A  /\  Q  e.  A )  /\  X  .<_  Y ) 
 ->  ( X  .\/  (
 ( P  .\/  Q )  ./\  Y ) )  =  ( ( X 
 .\/  ( P  .\/  Q ) )  ./\  Y ) )
 
Theorematmod2i1 30123 Version of modular law pmod2iN 30111 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-2012.) (Revised by Mario Carneiro, 10-May-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B )  /\  P  .<_  X )  ->  ( ( X  ./\  Y )  .\/  P )  =  ( X  ./\  ( Y  .\/  P ) ) )
 
Theorematmod2i2 30124 Version of modular law pmod2iN 30111 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-2012.) (Revised by Mario Carneiro, 10-May-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B )  /\  Y  .<_  X )  ->  ( ( X  ./\  P )  .\/  Y )  =  ( X  ./\  ( P  .\/  Y ) ) )
 
Theoremllnmod2i2 30125 Version of modular law pmod1i 30110 that holds in a Hilbert lattice, when one element is a lattice line (expressed as the join  P  .\/  Q). (Contributed by NM, 16-Sep-2012.) (Revised by Mario Carneiro, 10-May-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 /\  ( P  e.  A  /\  Q  e.  A )  /\  Y  .<_  X ) 
 ->  ( ( X  ./\  ( P  .\/  Q ) )  .\/  Y )  =  ( X  ./\  (
 ( P  .\/  Q )  .\/  Y ) ) )
 
Theorematmod3i1 30126 Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 4-Jun-2012.) (Revised by Mario Carneiro, 10-May-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B )  /\  P  .<_  X )  ->  ( P  .\/  ( X 
 ./\  Y ) )  =  ( X  ./\  ( P  .\/  Y ) ) )
 
Theorematmod3i2 30127 Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 10-Jun-2012.) (Revised by Mario Carneiro, 10-May-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  Y )  ->  ( X  .\/  ( Y 
 ./\  P ) )  =  ( Y  ./\  ( X  .\/  P ) ) )
 
Theorematmod4i1 30128 Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 10-Jun-2012.) (Revised by Mario Carneiro, 10-May-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B )  /\  P  .<_  Y )  ->  ( ( X  ./\  Y )  .\/  P )  =  ( ( X  .\/  P )  ./\  Y )
 )
 
Theorematmod4i2 30129 Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 4-Jun-2012.) (Revised by Mario Carneiro, 10-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  Y )  ->  ( ( P  ./\  Y )  .\/  X )  =  ( ( P  .\/  X )  ./\  Y )
 )
 
Theoremllnexchb2lem 30130 Lemma for llnexchb2 30131. (Contributed by NM, 17-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  ( LLines `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N ) 
 /\  ( P  e.  A  /\  Q  e.  A  /\  -.  P  .<_  X ) 
 /\  ( X  ./\  Y )  e.  A ) 
 ->  ( ( X  ./\  Y )  .<_  ( P  .\/  Q )  <->  ( X  ./\  Y )  =  ( X 
 ./\  ( P  .\/  Q ) ) ) )
 
Theoremllnexchb2 30131 Line exchange property (compare cvlatexchb2 29598 for atoms). (Contributed by NM, 17-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  ( LLines `  K )   =>    |-  (
 ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  Z  e.  N )  /\  ( ( X  ./\  Y )  e.  A  /\  X  =/=  Z ) ) 
 ->  ( ( X  ./\  Y )  .<_  Z  <->  ( X  ./\  Y )  =  ( X 
 ./\  Z ) ) )
 
Theoremllnexch2N 30132 Line exchange property (compare cvlatexch2 29600 for atoms). (Contributed by NM, 18-Nov-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  ( LLines `  K )   =>    |-  (
 ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  Z  e.  N )  /\  ( ( X  ./\  Y )  e.  A  /\  X  =/=  Z ) ) 
 ->  ( ( X  ./\  Y )  .<_  Z  ->  ( X  ./\  Z )  .<_  Y ) )
 
Theoremdalawlem1 30133 Lemma for dalaw 30148. Special case of dath2 29999, where  C is replaced by  ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) ). The remaining lemmas will eliminate the conditions on the atoms imposed by dath2 29999. (Contributed by NM, 6-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  O  =  ( LPlanes `  K )   =>    |-  (
 ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) 
 /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( ( ( P  .\/  Q )  .\/  R )  e.  O  /\  ( ( S  .\/  T )  .\/  U )  e.  O )  /\  ( ( -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  -.  ( ( P 
 .\/  S )  ./\  ( Q  .\/  T ) ) 
 .<_  ( Q  .\/  R )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R 
 .\/  P ) )  /\  ( -.  ( ( 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  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
 
Theoremdalawlem2 30134 Lemma for dalaw 30148. Utility lemma that breaks  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) ) into a join of two pieces. (Contributed by NM, 6-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A ) 
 /\  ( S  e.  A  /\  T  e.  A ) )  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( ( P  .\/  Q )  .\/  T )  ./\ 
 S )  .\/  (
 ( ( P  .\/  Q )  .\/  S )  ./\ 
 T ) ) )
 
Theoremdalawlem3 30135 Lemma for dalaw 30148. First piece of dalawlem5 30137. (Contributed by NM, 4-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  ( ( P 
 .\/  S )  ./\  ( Q  .\/  T ) ) 
 .<_  ( P  .\/  Q )  /\  ( ( P 
 .\/  S )  ./\  ( Q  .\/  T ) ) 
 .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  ->  ( ( ( Q 
 .\/  T )  .\/  P )  ./\  S )  .<_  ( ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
 
Theoremdalawlem4 30136 Lemma for dalaw 30148. Second piece of dalawlem5 30137. (Contributed by NM, 4-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  ( ( P 
 .\/  S )  ./\  ( Q  .\/  T ) ) 
 .<_  ( P  .\/  Q )  /\  ( ( 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 
 .\/  S )  .\/  Q )  ./\  T )  .<_  ( ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
 
Theoremdalawlem5 30137 Lemma for dalaw 30148. Special case to eliminate the requirement  -.  ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q ) in dalawlem1 30133. (Contributed by NM, 4-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  ( ( P 
 .\/  S )  ./\  ( Q  .\/  T ) ) 
 .<_  ( P  .\/  Q )  /\  ( ( 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 ) ) ) )
 
Theoremdalawlem6 30138 Lemma for dalaw 30148. First piece of dalawlem8 30140. (Contributed by NM, 6-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  ( ( P 
 .\/  S )  ./\  ( Q  .\/  T ) ) 
 .<_  ( 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 )  .\/  T )  ./\  S )  .<_  ( ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
 
Theoremdalawlem7 30139 Lemma for dalaw 30148. Second piece of dalawlem8 30140. (Contributed by NM, 6-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  ( ( P 
 .\/  S )  ./\  ( Q  .\/  T ) ) 
 .<_  ( 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 ) ) ) )
 
Theoremdalawlem8 30140 Lemma for dalaw 30148. Special case to eliminate the requirement  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( Q 
.\/  R ) in dalawlem1 30133. (Contributed by NM, 6-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  ( ( P 
 .\/  S )  ./\  ( Q  .\/  T ) ) 
 .<_  ( 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 ) ) ) )
 
Theoremdalawlem9 30141 Lemma for dalaw 30148. Special case to eliminate the requirement  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) in dalawlem1 30133. (Contributed by NM, 6-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  ( ( 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 ) ) ) )
 
Theoremdalawlem10 30142 Lemma for dalaw 30148. Combine dalawlem5 30137, dalawlem8 30140, and dalawlem9 . (Contributed by NM, 6-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  -.  ( -.  ( ( 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 ) ) ) )
 
Theoremdalawlem11 30143 Lemma for dalaw 30148. First part of dalawlem13 30145. (Contributed by NM, 17-Sep-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  P  .<_  ( 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 ) ) ) )
 
Theoremdalawlem12 30144 Lemma for dalaw 30148. Second part of dalawlem13 30145. (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 30145 Lemma for dalaw 30148. Special case to eliminate the requirement  ( ( P  .\/  Q )  .\/  R )  e.  O in dalawlem1 30133. (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 30146 Lemma for dalaw 30148. Combine dalawlem10 30142 and dalawlem13 30145. (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 30147 Lemma for dalaw 30148. Swap variable triples  P Q R and  S T U in dalawlem14 30146, to obtain the elimination of the remaining conditions in dalawlem1 30133. (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 30148 Desargues' law, derived from Desargues' theorem dath 29998 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 30149 Extend class notation with projective subspace closure.
 class  PCl
 
Definitiondf-pclN 30150* 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 30197.) (Contributed by NM, 7-Sep-2013.)
 |-  PCl  =  ( k  e.  _V  |->  ( x  e.  ~P ( Atoms `  k )  |-> 
 |^| { y  e.  ( PSubSp `
  k )  |  x  C_  y }
 ) )
 
TheorempclfvalN 30151* 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 30152* 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 30153 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 30154* 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 30155 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 30156 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 30157 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 30158 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 30159 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 30160 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 30161 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 30162* 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 30212. (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 30163* 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 30164 Extend class notation with polarity of projective subspace $m$.
 class  _|_ P
 
Definitiondf-polarityN 30165* 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 30166* 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 30167* 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 30168 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 30169 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 30170 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 30171 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 30172 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 30173 The closed subspace closure of the empty set. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
 |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( K  e.  HL  ->  (  ._|_  `  (  ._|_  `  (/) ) )  =  (/) )
 
TheorempolpmapN 30174 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 30175 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 30176 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 30177 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 30178 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 30179 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 30180 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 30181 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 30182 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 30183 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 30184 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 30185 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 30186 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 30187 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 30188 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 30189 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 5531.) (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 30190 De Morgan's law for polarity of projective sum. (oldmj1 29484 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 30191 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 30192 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 30193 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 30194 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 30195 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 30196 Extend class notation with set of all closed projective subspaces for a Hilbert lattice.
 class  PSubCl
 
Definitiondf-psubclN 30197* 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 30198* 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 30199 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 30200 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 ) )
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