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Theorem List for Metamath Proof Explorer - 29101-29200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempmaple 29101 The projective map of a Hilbert lattice preserves ordering. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .<_  Y  <->  ( M `  X )  C_  ( M `
  Y ) ) )
 
Theorempmap11 29102 The projective map of a Hilbert lattice is one-to-one. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( M `  X )  =  ( M `  Y )  <->  X  =  Y ) )
 
Theorempmapat 29103 The projective map of an atom. (Contributed by NM, 25-Jan-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( K  e.  HL  /\  P  e.  A ) 
 ->  ( M `  P )  =  { P } )
 
Theoremelpmapat 29104 Member of the projective map of an atom. (Contributed by NM, 27-Jan-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( K  e.  HL  /\  P  e.  A ) 
 ->  ( X  e.  ( M `  P )  <->  X  =  P ) )
 
Theorempmap0 29105 Value of the projective map of a Hilbert lattice at lattice zero. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 17-Oct-2011.)
 |-  .0.  =  ( 0. `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  ( K  e.  AtLat  ->  ( M `  .0.  )  =  (/) )
 
Theorempmapeq0 29106 A projective map value is zero iff its argument is lattice zero. (Contributed by NM, 27-Jan-2012.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( M `
  X )  =  (/) 
 <->  X  =  .0.  )
 )
 
Theorempmap1N 29107 Value of the projective map of a Hilbert lattice at lattice unit. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.) (New usage is discouraged.)
 |-  .1.  =  ( 1. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  (
 pmap `  K )   =>    |-  ( K  e.  OP  ->  ( M `  .1.  )  =  A )
 
Theorempmapsub 29108 The projective map of a Hilbert lattice maps to projective subspaces. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 17-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( K  e.  Lat  /\  X  e.  B ) 
 ->  ( M `  X )  e.  S )
 
Theorempmapglbx 29109* The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 29110, where we read  S as  S ( i ). Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B  /\  I  =/=  (/) )  ->  ( M `  ( G `
  { y  | 
 E. i  e.  I  y  =  S }
 ) )  =  |^|_ i  e.  I  ( M `
  S ) )
 
Theorempmapglb 29110* The projective map of the GLB of a set of lattice elements  S. Variant of Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  ( M `  ( G `
  S ) )  =  |^|_ x  e.  S  ( M `  x ) )
 
Theorempmapglb2N 29111* The projective map of the GLB of a set of lattice elements  S. Variant of Theorem 15.5.2 of [MaedaMaeda] p. 62. Allows  S  =  (/). (Contributed by NM, 21-Jan-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  (
 pmap `  K )   =>    |-  ( ( K  e.  HL  /\  S  C_  B )  ->  ( M `  ( G `  S ) )  =  ( A  i^i  |^|_ x  e.  S  ( M `  x ) ) )
 
Theorempmapglb2xN 29112* The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 29111, where we read  S as  S ( i ). Extension of Theorem 15.5.2 of [MaedaMaeda] p. 62 that allows  I  =  (/). (Contributed by NM, 21-Jan-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  (
 pmap `  K )   =>    |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B )  ->  ( M `  ( G `  { y  |  E. i  e.  I  y  =  S } ) )  =  ( A  i^i  |^|_ i  e.  I  ( M `  S ) ) )
 
Theorempmapmeet 29113 The projective map of a meet. (Contributed by NM, 25-Jan-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  P  =  ( pmap `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( P `  ( X  ./\  Y ) )  =  ( ( P `
  X )  i^i  ( P `  Y ) ) )
 
Theoremisline2 29114* Definition of line in terms of projective map. (Contributed by NM, 25-Jan-2012.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  ( Lines `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  ( K  e.  Lat  ->  ( X  e.  N  <->  E. p  e.  A  E. q  e.  A  ( p  =/=  q  /\  X  =  ( M `
  ( p  .\/  q ) ) ) ) )
 
Theoremlinepmap 29115 A line described with a projective map. (Contributed by NM, 3-Feb-2012.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  ( Lines `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q ) 
 ->  ( M `  ( P  .\/  Q ) )  e.  N )
 
Theoremisline3 29116* Definition of line in terms of original lattice elements. (Contributed by NM, 29-Apr-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  ( Lines `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B ) 
 ->  ( ( M `  X )  e.  N  <->  E. p  e.  A  E. q  e.  A  ( p  =/=  q  /\  X  =  ( p  .\/  q
 ) ) ) )
 
Theoremisline4N 29117* Definition of line in terms of original lattice elements. (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  (
 Lines `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B ) 
 ->  ( ( M `  X )  e.  N  <->  E. p  e.  A  p C X ) )
 
Theoremlneq2at 29118 A line equals the join of any two of its distinct points (atoms). (Contributed by NM, 29-Apr-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  N  =  ( Lines `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( P  .<_  X  /\  Q  .<_  X ) ) 
 ->  X  =  ( P 
 .\/  Q ) )
 
TheoremlnatexN 29119* There is an atom in a line different from any other. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  (
 Lines `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) )
 
TheoremlnjatN 29120* Given an atom in a line, there is another atom which when joined equals the line. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  N  =  ( Lines `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A ) 
 /\  ( ( M `
  X )  e.  N  /\  P  .<_  X ) )  ->  E. q  e.  A  ( q  =/= 
 P  /\  X  =  ( P  .\/  q ) ) )
 
TheoremlncvrelatN 29121 A lattice element covered by a line is an atom. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  (
 Lines `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B ) 
 /\  ( ( M `
  X )  e.  N  /\  P C X ) )  ->  P  e.  A )
 
Theoremlncvrat 29122 A line covers the atoms it contains. (Contributed by NM, 30-Apr-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  N  =  ( Lines `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A ) 
 /\  ( ( M `
  X )  e.  N  /\  P  .<_  X ) )  ->  P C X )
 
Theoremlncmp 29123 If two lines are comparable, they are equal. (Contributed by NM, 30-Apr-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  N  =  ( Lines `  K )   &    |-  M  =  (
 pmap `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( ( M `  X )  e.  N  /\  ( M `  Y )  e.  N )
 )  ->  ( X  .<_  Y  <->  X  =  Y ) )
 
Theorem2lnat 29124 Two intersecting lines intersect at an atom. (Contributed by NM, 30-Apr-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  (
 Lines `  K )   &    |-  F  =  ( pmap `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 /\  ( ( F `
  X )  e.  N  /\  ( F `
  Y )  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
 .0.  ) )  ->  ( X  ./\  Y )  e.  A )
 
Theorem2atm2atN 29125 Two joins with a common atom have a nonzero meet. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  ( ( R  .\/  P )  ./\  ( R  .\/  Q ) )  =/=  .0.  )
 
Theorem2llnma1b 29126 Generalization of 2llnma1 29127. (Contributed by NM, 26-Apr-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A )  /\  -.  Q  .<_  ( P 
 .\/  X ) )  ->  ( ( P  .\/  X )  ./\  ( P  .\/  Q ) )  =  P )
 
Theorem2llnma1 29127 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 11-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  -.  R  .<_  ( P 
 .\/  Q ) )  ->  ( ( Q  .\/  P )  ./\  ( Q  .\/  R ) )  =  Q )
 
Theorem2llnma3r 29128 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 30-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  (
 ( P  .\/  R )  ./\  ( Q  .\/  R ) )  =  R )
 
Theorem2llnma2 29129 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 28-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
 .\/  Q ) ) ) 
 ->  ( ( R  .\/  P )  ./\  ( R  .\/  Q ) )  =  R )
 
Theorem2llnma2rN 29130 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 2-May-2013.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
 .\/  Q ) ) ) 
 ->  ( ( P  .\/  R )  ./\  ( Q  .\/  R ) )  =  R )
 
16.24.14  Construction of a vector space from a Hilbert lattice
 
Theoremcdlema1N 29131 A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 29-Apr-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  ( Lines `  K )   &    |-  F  =  ( pmap `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) 
 /\  ( ( R  =/=  P  /\  R  .<_  ( P  .\/  Q ) )  /\  ( P 
 .<_  X  /\  Q  .<_  Y )  /\  ( ( F `  Y )  e.  N  /\  ( X  ./\  Y )  e.  A  /\  -.  Q  .<_  X ) ) ) 
 ->  ( X  .\/  R )  =  ( X  .\/  Y ) )
 
Theoremcdlema2N 29132 A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 9-May-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( ( R  =/=  P 
 /\  R  .<_  ( P 
 .\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  ( R  ./\ 
 X )  =  .0.  )
 
Theoremcdlemblem 29133 Lemma for cdlemb 29134. (Contributed by NM, 8-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .<  =  ( lt `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  V  =  ( ( P  .\/  Q )  ./\ 
 X )   =>    |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q ) 
 /\  ( X C  .1.  /\  -.  P  .<_  X 
 /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  ( u  =/=  V  /\  u  .<  X ) ) 
 /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u )
 ) ) )  ->  ( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q ) ) )
 
Theoremcdlemb 29134* Given two atoms not less than or equal to an element covered by 1, there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 8-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q ) 
 /\  ( X C  .1.  /\  -.  P  .<_  X 
 /\  -.  Q  .<_  X ) )  ->  E. r  e.  A  ( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q ) ) )
 
Syntaxcpadd 29135 Extend class notation with projective subspace sum.
 class  + P
 
Definitiondf-padd 29136* Define projective sum of two subspaces (or more generally two sets of atoms), which is the union of all lines generated by pairs of atoms from each subspace. Lemma 16.2 of [MaedaMaeda] p. 68. For convenience, our definition is generalized to apply to empty sets. (Contributed by NM, 29-Dec-2011.)
 |-  + P  =  ( l  e.  _V  |->  ( m  e. 
 ~P ( Atoms `  l
 ) ,  n  e. 
 ~P ( Atoms `  l
 )  |->  ( ( m  u.  n )  u. 
 { p  e.  ( Atoms `  l )  | 
 E. q  e.  m  E. r  e.  n  p ( le `  l
 ) ( q (
 join `  l ) r ) } ) ) )
 
Theorempaddfval 29137* Projective subspace sum operation. (Contributed by NM, 29-Dec-2011.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( K  e.  B  ->  .+  =  ( m  e.  ~P A ,  n  e.  ~P A  |->  ( ( m  u.  n )  u.  { p  e.  A  |  E. q  e.  m  E. r  e.  n  p  .<_  ( q  .\/  r ) } ) ) )
 
Theorempaddval 29138* Projective subspace sum operation value. (Contributed by NM, 29-Dec-2011.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  B  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  =  ( ( X  u.  Y )  u. 
 { p  e.  A  |  E. q  e.  X  E. r  e.  Y  p  .<_  ( q  .\/  r ) } )
 )
 
Theoremelpadd 29139* Member of a projective subspace sum. (Contributed by NM, 29-Dec-2011.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  B  /\  X  C_  A  /\  Y  C_  A )  ->  ( S  e.  ( X  .+  Y )  <->  ( ( S  e.  X  \/  S  e.  Y )  \/  ( S  e.  A  /\  E. q  e.  X  E. r  e.  Y  S  .<_  ( q  .\/  r
 ) ) ) ) )
 
Theoremelpaddn0 29140* Member of projective subspace sum of non-empty sets. (Contributed by NM, 3-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) ) 
 ->  ( S  e.  ( X  .+  Y )  <->  ( S  e.  A  /\  E. q  e.  X  E. r  e.  Y  S  .<_  ( q 
 .\/  r ) ) ) )
 
Theorempaddvaln0N 29141* Projective subspace sum operation value for non-empty sets. (Contributed by NM, 27-Jan-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) ) 
 ->  ( X  .+  Y )  =  { p  e.  A  |  E. q  e.  X  E. r  e.  Y  p  .<_  ( q 
 .\/  r ) }
 )
 
Theoremelpaddri 29142 Condition implying membership in a projective subspace sum. (Contributed by NM, 8-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( Q  e.  X  /\  R  e.  Y )  /\  ( S  e.  A  /\  S  .<_  ( Q 
 .\/  R ) ) ) 
 ->  S  e.  ( X 
 .+  Y ) )
 
TheoremelpaddatriN 29143 Condition implying membership in a projective subspace sum with a point. (Contributed by NM, 1-Feb-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  ( R  e.  X  /\  S  e.  A  /\  S  .<_  ( R  .\/  Q ) ) )  ->  S  e.  ( X  .+ 
 { Q } )
 )
 
Theoremelpaddat 29144* Membership in a projective subspace sum with a point. (Contributed by NM, 29-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  ( S  e.  ( X  .+  { Q }
 ) 
 <->  ( S  e.  A  /\  E. p  e.  X  S  .<_  ( p  .\/  Q ) ) ) )
 
TheoremelpaddatiN 29145* Consequence of membership in a projective subspace sum with a point. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  ( X  =/=  (/)  /\  R  e.  ( X 
 .+  { Q } )
 ) )  ->  E. p  e.  X  R  .<_  ( p 
 .\/  Q ) )
 
Theoremelpadd2at 29146 Membership in a projective subspace sum of two points. (Contributed by NM, 29-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  ( S  e.  ( { Q }  .+  { R } )  <->  ( S  e.  A  /\  S  .<_  ( Q 
 .\/  R ) ) ) )
 
Theoremelpadd2at2 29147 Membership in a projective subspace sum of two points. (Contributed by NM, 8-Mar-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  Lat  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A )
 )  ->  ( S  e.  ( { Q }  .+  { R } )  <->  S 
 .<_  ( Q  .\/  R ) ) )
 
TheorempaddunssN 29148 Projective subspace sum includes the set union of its arguments. (Contributed by NM, 12-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  B  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  u.  Y )  C_  ( X  .+  Y ) )
 
Theoremelpadd0 29149 Member of projective subspace sum with at least one empty set.. (Contributed by NM, 29-Dec-2011.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( K  e.  B  /\  X  C_  A  /\  Y  C_  A )  /\  -.  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  ->  ( S  e.  ( X  .+  Y )  <->  ( S  e.  X  \/  S  e.  Y ) ) )
 
Theorempaddval0 29150 Projective subspace sum with at least one empty set. (Contributed by NM, 11-Jan-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( K  e.  B  /\  X  C_  A  /\  Y  C_  A )  /\  -.  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  ->  ( X  .+  Y )  =  ( X  u.  Y ) )
 
Theorempadd01 29151 Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  B  /\  X  C_  A )  ->  ( X  .+  (/) )  =  X )
 
Theorempadd02 29152 Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  B  /\  X  C_  A )  ->  ( (/)  .+  X )  =  X )
 
Theorempaddcom 29153 Projective subspace sum commutes. (Contributed by NM, 3-Jan-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
 
Theorempaddssat 29154 A projective subspace sum is a set of atoms. (Contributed by NM, 3-Jan-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  B  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  C_  A )
 
Theoremsspadd1 29155 A projective subspace sum is a superset of its first summand. (ssun1 3299 analog.) (Contributed by NM, 3-Jan-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  B  /\  X  C_  A  /\  Y  C_  A )  ->  X  C_  ( X  .+  Y ) )
 
Theoremsspadd2 29156 A projective subspace sum is a superset of its second summand. (ssun2 3300 analog.) (Contributed by NM, 3-Jan-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  B  /\  X  C_  A  /\  Y  C_  A )  ->  X  C_  ( Y  .+  X ) )
 
Theorempaddss1 29157 Subset law for projective subspace sum. (unss1 3305 analog.) (Contributed by NM, 7-Mar-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  ->  ( X  C_  Y  ->  ( X  .+  Z )  C_  ( Y  .+  Z ) ) )
 
Theorempaddss2 29158 Subset law for projective subspace sum. (unss2 3307 analog.) (Contributed by NM, 7-Mar-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  ->  ( X  C_  Y  ->  ( Z  .+  X )  C_  ( Z  .+  Y ) ) )
 
Theorempaddss12 29159 Subset law for projective subspace sum. (unss12 3308 analog.) (Contributed by NM, 7-Mar-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  ->  ( ( X  C_  Y  /\  Z  C_  W )  ->  ( X  .+  Z )  C_  ( Y 
 .+  W ) ) )
 
Theorempaddasslem1 29160 Lemma for paddass 29178. (Contributed by NM, 8-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  ( x  e.  A  /\  r  e.  A  /\  y  e.  A )  /\  x  =/=  y
 )  /\  -.  r  .<_  ( x  .\/  y
 ) )  ->  -.  x  .<_  ( r  .\/  y
 ) )
 
Theorempaddasslem2 29161 Lemma for paddass 29178. (Contributed by NM, 8-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  r  e.  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  /\  ( -.  r  .<_  ( x  .\/  y
 )  /\  r  .<_  ( y  .\/  z )
 ) )  ->  z  .<_  ( r  .\/  y
 ) )
 
Theorempaddasslem3 29162* Lemma for paddass 29178. Restate projective space axiom ps-2 28818. (Contributed by NM, 8-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( x  e.  A  /\  r  e.  A  /\  y  e.  A )  /\  ( p  e.  A  /\  z  e.  A ) )  ->  ( ( ( -.  x  .<_  ( r  .\/  y )  /\  p  =/=  z
 )  /\  ( p  .<_  ( x  .\/  r
 )  /\  z  .<_  ( r  .\/  y )
 ) )  ->  E. s  e.  A  ( s  .<_  ( x  .\/  y )  /\  s  .<_  ( p 
 .\/  z ) ) ) )
 
Theorempaddasslem4 29163* Lemma for paddass 29178. Combine paddasslem1 29160, paddasslem2 29161, and paddasslem3 29162. (Contributed by NM, 8-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  r  e.  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  /\  ( p  =/=  z  /\  x  =/=  y  /\  -.  r  .<_  ( x  .\/  y
 ) ) )  /\  ( p  .<_  ( x 
 .\/  r )  /\  r  .<_  ( y  .\/  z ) ) ) 
 ->  E. s  e.  A  ( s  .<_  ( x 
 .\/  y )  /\  s  .<_  ( p  .\/  z ) ) )
 
Theorempaddasslem5 29164 Lemma for paddass 29178. Show  s  =/=  z by contradiction. (Contributed by NM, 8-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  r  e.  A  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  /\  ( -.  r  .<_  ( x  .\/  y )  /\  r  .<_  ( y  .\/  z )  /\  s  .<_  ( x 
 .\/  y ) ) )  ->  s  =/=  z )
 
Theorempaddasslem6 29165 Lemma for paddass 29178. (Contributed by NM, 8-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  ( p  e.  A  /\  s  e.  A )  /\  z  e.  A )  /\  ( s  =/=  z  /\  s  .<_  ( p  .\/  z )
 ) )  ->  p  .<_  ( s  .\/  z
 ) )
 
Theorempaddasslem7 29166 Lemma for paddass 29178. Combine paddasslem5 29164 and paddasslem6 29165. (Contributed by NM, 9-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  ( p  e.  A  /\  r  e.  A  /\  s  e.  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  /\  ( ( -.  r  .<_  ( x  .\/  y
 )  /\  r  .<_  ( y  .\/  z )  /\  s  .<_  ( x 
 .\/  y ) ) 
 /\  s  .<_  ( p 
 .\/  z ) ) )  ->  p  .<_  ( s  .\/  z )
 )
 
Theorempaddasslem8 29167 Lemma for paddass 29178. (Contributed by NM, 8-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  ( p  e.  A  /\  s  e.  A ) )  /\  ( ( x  e.  X  /\  y  e.  Y  /\  z  e.  Z )  /\  s  .<_  ( x  .\/  y
 )  /\  p  .<_  ( s  .\/  z )
 ) )  ->  p  e.  ( ( X  .+  Y )  .+  Z ) )
 
Theorempaddasslem9 29168 Lemma for paddass 29178. Combine paddasslem7 29166 and paddasslem8 29167. (Contributed by NM, 9-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  ( p  e.  A  /\  r  e.  A ) )  /\  ( ( x  e.  X  /\  y  e.  Y  /\  z  e.  Z )  /\  ( -.  r  .<_  ( x 
 .\/  y )  /\  r  .<_  ( y  .\/  z ) )  /\  ( s  e.  A  /\  s  .<_  ( x 
 .\/  y )  /\  s  .<_  ( p  .\/  z ) ) ) )  ->  p  e.  ( ( X  .+  Y )  .+  Z ) )
 
Theorempaddasslem10 29169 Lemma for paddass 29178. Use paddasslem4 29163 to eliminate  s from paddasslem9 29168. (Contributed by NM, 9-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  p  =/=  z  /\  x  =/=  y )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  ( p  e.  A  /\  r  e.  A ) )  /\  ( ( x  e.  X  /\  y  e.  Y  /\  z  e.  Z )  /\  ( -.  r  .<_  ( x 
 .\/  y )  /\  p  .<_  ( x  .\/  r )  /\  r  .<_  ( y  .\/  z )
 ) ) )  ->  p  e.  ( ( X  .+  Y )  .+  Z ) )
 
Theorempaddasslem11 29170 Lemma for paddass 29178. The case when  p  =  z. (Contributed by NM, 11-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  p  =  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) ) 
 /\  z  e.  Z )  ->  p  e.  (
 ( X  .+  Y )  .+  Z ) )
 
Theorempaddasslem12 29171 Lemma for paddass 29178. The case when  x  =  y. (Contributed by NM, 11-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  x  =  y )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  ( p  e.  A  /\  r  e.  A ) )  /\  ( ( y  e.  Y  /\  z  e.  Z )  /\  ( p  .<_  ( x 
 .\/  r )  /\  r  .<_  ( y  .\/  z ) ) ) )  ->  p  e.  ( ( X  .+  Y )  .+  Z ) )
 
Theorempaddasslem13 29172 Lemma for paddass 29178. The case when  r 
.<_  ( x  .\/  y
). (Unlike the proof in Maeda and Maeda, we don't need  x  =/=  y.) (Contributed by NM, 11-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  ( p  e.  A  /\  r  e.  A )
 )  /\  ( ( x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
 .\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  p  e.  ( ( X  .+  Y )  .+  Z ) )
 
Theorempaddasslem14 29173 Lemma for paddass 29178. Remove  p  =/=  z,  x  =/=  y, and  -.  r  .<_  ( x  .\/  y ) from antecedent of paddasslem10 29169, using paddasslem11 29170, paddasslem12 29171, and paddasslem13 29172. (Contributed by NM, 11-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  ( p  e.  A  /\  r  e.  A ) )  /\  ( ( x  e.  X  /\  y  e.  Y  /\  z  e.  Z )  /\  ( p  .<_  ( x  .\/  r )  /\  r  .<_  ( y  .\/  z )
 ) ) )  ->  p  e.  ( ( X  .+  Y )  .+  Z ) )
 
Theorempaddasslem15 29174 Lemma for paddass 29178. Use elpaddn0 29140 to eliminate  y and  z from paddasslem14 29173. (Contributed by NM, 11-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) )  /\  ( p  e.  A  /\  ( x  e.  X  /\  r  e.  ( Y  .+  Z ) ) 
 /\  p  .<_  ( x 
 .\/  r ) ) )  ->  p  e.  ( ( X  .+  Y )  .+  Z ) )
 
Theorempaddasslem16 29175 Lemma for paddass 29178. Use elpaddn0 29140 to eliminate  x and  r from paddasslem15 29174. (Contributed by NM, 11-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  ( ( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  ( X  .+  ( Y 
 .+  Z ) ) 
 C_  ( ( X 
 .+  Y )  .+  Z ) )
 
Theorempaddasslem17 29176 Lemma for paddass 29178. The case when at least one sum argument is empty. (Contributed by NM, 12-Jan-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  -.  ( ( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/= 
 (/) ) ) ) 
 ->  ( X  .+  ( Y  .+  Z ) ) 
 C_  ( ( X 
 .+  Y )  .+  Z ) )
 
Theorempaddasslem18 29177 Lemma for paddass 29178. Combine paddasslem16 29175 and paddasslem17 29176. (Contributed by NM, 12-Jan-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )
 )  ->  ( X  .+  ( Y  .+  Z ) )  C_  ( ( X  .+  Y ) 
 .+  Z ) )
 
Theorempaddass 29178 Projective subspace sum is associative. Equation 16.2.1 of [MaedaMaeda] p. 68. In our version, the subspaces do not have to be non-empty. (Contributed by NM, 29-Dec-2011.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )
 )  ->  ( ( X  .+  Y )  .+  Z )  =  ( X  .+  ( Y  .+  Z ) ) )
 
Theorempadd12N 29179 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 29180 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 29181 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 29182 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 29183 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 29184 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 29185 Subset law for projective subspace sum. (unss 3310 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 29186* Lemma for pmod1i 29188. (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 29187 Lemma for pmod1i 29188. (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 29188 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 29189 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 29190 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 29191 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 29192 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 29193 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 29194 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 29195 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 29196 A version of the modular law pmod1i 29188 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 29197 Version of modular law pmod1i 29188 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 29198 Version of modular law pmod1i 29188 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 29199 Version of modular law pmod1i 29188 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 29200 Version of modular law pmod1i 29188 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 ) )
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