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Theorem List for Metamath Proof Explorer - 29101-29200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlncvrat 29101 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 29102 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 29103 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 29104 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 29105 Generalization of 2llnma1 29106. (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 29106 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 29107 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 29108 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 29109 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 29110 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 29111 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 29112 Lemma for cdlemb 29113. (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 29113* 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 29114 Extend class notation with projective subspace sum.
 class  + P
 
Definitiondf-padd 29115* 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 29116* 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 29117* 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 29118* 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 29119* 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 29120* 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 29121 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 29122 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 29123* 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 29124* 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 29125 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 29126 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 29127 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 29128 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 29129 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 29130 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 29131 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 29132 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 29133 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 29134 A projective subspace sum is a superset of its first summand. (ssun1 3280 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 29135 A projective subspace sum is a superset of its second summand. (ssun2 3281 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 29136 Subset law for projective subspace sum. (unss1 3286 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 29137 Subset law for projective subspace sum. (unss2 3288 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 29138 Subset law for projective subspace sum. (unss12 3289 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 29139 Lemma for paddass 29157. (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 29140 Lemma for paddass 29157. (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 29141* Lemma for paddass 29157. Restate projective space axiom ps-2 28797. (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 29142* Lemma for paddass 29157. Combine paddasslem1 29139, paddasslem2 29140, and paddasslem3 29141. (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 29143 Lemma for paddass 29157. 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 29144 Lemma for paddass 29157. (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 29145 Lemma for paddass 29157. Combine paddasslem5 29143 and paddasslem6 29144. (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 29146 Lemma for paddass 29157. (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 29147 Lemma for paddass 29157. Combine paddasslem7 29145 and paddasslem8 29146. (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 29148 Lemma for paddass 29157. Use paddasslem4 29142 to eliminate  s from paddasslem9 29147. (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 29149 Lemma for paddass 29157. 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 29150 Lemma for paddass 29157. 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 29151 Lemma for paddass 29157. 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 29152 Lemma for paddass 29157. Remove  p  =/=  z,  x  =/=  y, and  -.  r  .<_  ( x  .\/  y ) from antecedent of paddasslem10 29148, using paddasslem11 29149, paddasslem12 29150, and paddasslem13 29151. (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 29153 Lemma for paddass 29157. Use elpaddn0 29119 to eliminate  y and  z from paddasslem14 29152. (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 29154 Lemma for paddass 29157. Use elpaddn0 29119 to eliminate  x and  r from paddasslem15 29153. (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 29155 Lemma for paddass 29157. 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 29156 Lemma for paddass 29157. Combine paddasslem16 29154 and paddasslem17 29155. (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 29157 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 29158 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 29159 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 29160 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 29161 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 29162 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 29163 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 29164 Subset law for projective subspace sum. (unss 3291 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 29165* Lemma for pmod1i 29167. (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 29166 Lemma for pmod1i 29167. (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 29167 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 29168 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 29169 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 29170 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 29171 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 29172 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 29173 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 29174 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 29175 A version of the modular law pmod1i 29167 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 29176 Version of modular law pmod1i 29167 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 29177 Version of modular law pmod1i 29167 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 29178 Version of modular law pmod1i 29167 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 29179 Version of modular law pmod1i 29167 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 29180 Version of modular law pmod2iN 29168 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 29181 Version of modular law pmod2iN 29168 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 29182 Version of modular law pmod1i 29167 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 29183 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 29184 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 29185 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 29186 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 29187 Lemma for llnexchb2 29188. (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 29188 Line exchange property (compare cvlatexchb2 28655 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 29189 Line exchange property (compare cvlatexch2 28657 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 29190 Lemma for dalaw 29205. Special case of dath2 29056, where  C is replaced by  ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) ). The remaining lemmas will eliminate the conditions on the atoms imposed by dath2 29056. (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 29191 Lemma for dalaw 29205. 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 29192 Lemma for dalaw 29205. First piece of dalawlem5 29194. (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 29193 Lemma for dalaw 29205. Second piece of dalawlem5 29194. (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 29194 Lemma for dalaw 29205. Special case to eliminate the requirement  -.  ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q ) in dalawlem1 29190. (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 29195 Lemma for dalaw 29205. First piece of dalawlem8 29197. (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 29196 Lemma for dalaw 29205. Second piece of dalawlem8 29197. (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 29197 Lemma for dalaw 29205. Special case to eliminate the requirement  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( Q 
.\/  R ) in dalawlem1 29190. (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 29198 Lemma for dalaw 29205. Special case to eliminate the requirement  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) in dalawlem1 29190. (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 29199 Lemma for dalaw 29205. Combine dalawlem5 29194, dalawlem8 29197, 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 29200 Lemma for dalaw 29205. First part of dalawlem13 29202. (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 ) ) ) )
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