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Theorem List for Metamath Proof Explorer - 29201-29300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempmap11 29201 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 29202 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 29203 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 29204 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 29205 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 29206 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 29207 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 29208* The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 29209, 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 29209* 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 29210* 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 29211* The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 29210, 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 29212 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 29213* 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 29214 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 29215* 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 29216* 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 29217 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 29218* 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 29219* 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 29220 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 29221 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 29222 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 29223 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 29224 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 29225 Generalization of 2llnma1 29226. (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 29226 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 29227 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 29228 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 29229 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 )
 
18.25.14  Construction of a vector space from a Hilbert lattice
 
Theoremcdlema1N 29230 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 29231 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 29232 Lemma for cdlemb 29233. (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 29233* 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 29234 Extend class notation with projective subspace sum.
 class  + P
 
Definitiondf-padd 29235* 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 29236* 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 29237* 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 29238* 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 29239* 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 29240* 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 29241 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 29242 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 29243* 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 29244* 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 29245 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 29246 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 29247 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 29248 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 29249 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 29250 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 29251 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 29252 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 29253 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 29254 A projective subspace sum is a superset of its first summand. (ssun1 3313 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 29255 A projective subspace sum is a superset of its second summand. (ssun2 3314 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 29256 Subset law for projective subspace sum. (unss1 3319 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 29257 Subset law for projective subspace sum. (unss2 3321 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 29258 Subset law for projective subspace sum. (unss12 3322 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 29259 Lemma for paddass 29277. (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 29260 Lemma for paddass 29277. (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 29261* Lemma for paddass 29277. Restate projective space axiom ps-2 28917. (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 29262* Lemma for paddass 29277. Combine paddasslem1 29259, paddasslem2 29260, and paddasslem3 29261. (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 29263 Lemma for paddass 29277. 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 29264 Lemma for paddass 29277. (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 29265 Lemma for paddass 29277. Combine paddasslem5 29263 and paddasslem6 29264. (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 29266 Lemma for paddass 29277. (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 29267 Lemma for paddass 29277. Combine paddasslem7 29265 and paddasslem8 29266. (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 29268 Lemma for paddass 29277. Use paddasslem4 29262 to eliminate  s from paddasslem9 29267. (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 29269 Lemma for paddass 29277. 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 29270 Lemma for paddass 29277. 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 29271 Lemma for paddass 29277. 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 29272 Lemma for paddass 29277. Remove  p  =/=  z,  x  =/=  y, and  -.  r  .<_  ( x  .\/  y ) from antecedent of paddasslem10 29268, using paddasslem11 29269, paddasslem12 29270, and paddasslem13 29271. (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 29273 Lemma for paddass 29277. Use elpaddn0 29239 to eliminate  y and  z from paddasslem14 29272. (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 29274 Lemma for paddass 29277. Use elpaddn0 29239 to eliminate  x and  r from paddasslem15 29273. (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 29275 Lemma for paddass 29277. 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 29276 Lemma for paddass 29277. Combine paddasslem16 29274 and paddasslem17 29275. (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 29277 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 29278 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 29279 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 29280 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 29281 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 29282 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 29283 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 29284 Subset law for projective subspace sum. (unss 3324 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 29285* Lemma for pmod1i 29287. (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 29286 Lemma for pmod1i 29287. (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 29287 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 29288 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 29289 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 29290 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 29291 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 29292 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 29293 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 29294 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 29295 A version of the modular law pmod1i 29287 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 29296 Version of modular law pmod1i 29287 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 29297 Version of modular law pmod1i 29287 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 29298 Version of modular law pmod1i 29287 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 29299 Version of modular law pmod1i 29287 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 29300 Version of modular law pmod2iN 29288 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 ) ) )
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