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Theorem List for Metamath Proof Explorer - 34601-34700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem2llnma1 34601 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 34602 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 34603 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 34604 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 )
 
21.30.10  Construction of a vector space from a Hilbert lattice
 
Theoremcdlema1N 34605 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 34606 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 34607 Lemma for cdlemb 34608. (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 34608* 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 34609 Extend class notation with projective subspace sum.
 class  +P
 
Definitiondf-padd 34610* 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 34611* 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 34612* 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 34613* 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 34614* Member of projective subspace sum of nonempty 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 34615* Projective subspace sum operation value for nonempty 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 34616 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 34617 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 34618* 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 34619* 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 34620 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 34621 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 34622 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 34623 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 34624 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 34625 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 34626 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 34627 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 34628 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 34629 A projective subspace sum is a superset of its first summand. (ssun1 3667 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 34630 A projective subspace sum is a superset of its second summand. (ssun2 3668 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 34631 Subset law for projective subspace sum. (unss1 3673 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 34632 Subset law for projective subspace sum. (unss2 3675 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 34633 Subset law for projective subspace sum. (unss12 3676 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 34634 Lemma for paddass 34652. (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 34635 Lemma for paddass 34652. (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 34636* Lemma for paddass 34652. Restate projective space axiom ps-2 34292. (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 34637* Lemma for paddass 34652. Combine paddasslem1 34634, paddasslem2 34635, and paddasslem3 34636. (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 34638 Lemma for paddass 34652. 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 34639 Lemma for paddass 34652. (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 34640 Lemma for paddass 34652. Combine paddasslem5 34638 and paddasslem6 34639. (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 34641 Lemma for paddass 34652. (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 34642 Lemma for paddass 34652. Combine paddasslem7 34640 and paddasslem8 34641. (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 34643 Lemma for paddass 34652. Use paddasslem4 34637 to eliminate  s from paddasslem9 34642. (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 34644 Lemma for paddass 34652. 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 34645 Lemma for paddass 34652. 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 34646 Lemma for paddass 34652. 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 34647 Lemma for paddass 34652. Remove  p  =/=  z,  x  =/=  y, and  -.  r  .<_  ( x  .\/  y ) from antecedent of paddasslem10 34643, using paddasslem11 34644, paddasslem12 34645, and paddasslem13 34646. (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 34648 Lemma for paddass 34652. Use elpaddn0 34614 to eliminate  y and  z from paddasslem14 34647. (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 34649 Lemma for paddass 34652. Use elpaddn0 34614 to eliminate  x and  r from paddasslem15 34648. (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 34650 Lemma for paddass 34652. 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 34651 Lemma for paddass 34652. Combine paddasslem16 34649 and paddasslem17 34650. (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 34652 Projective subspace sum is associative. Equation 16.2.1 of [MaedaMaeda] p. 68. In our version, the subspaces do not have to be nonempty. (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 34653 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 34654 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 34655 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 34656 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 34657 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 34658 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 34659 Subset law for projective subspace sum. (unss 3678 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 34660* Lemma for pmod1i 34662. (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 34661 Lemma for pmod1i 34662. (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 34662 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 34663 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 34664 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 34665 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 34666 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 34667 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 34668 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 34669 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 34670 A version of the modular law pmod1i 34662 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 34671 Version of modular law pmod1i 34662 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 34672 Version of modular law pmod1i 34662 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 34673 Version of modular law pmod1i 34662 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 34674 Version of modular law pmod1i 34662 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 34675 Version of modular law pmod2iN 34663 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 34676 Version of modular law pmod2iN 34663 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 34677 Version of modular law pmod1i 34662 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 34678 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 34679 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 34680 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 34681 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 34682 Lemma for llnexchb2 34683. (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 34683 Line exchange property (compare cvlatexchb2 34150 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 34684 Line exchange property (compare cvlatexch2 34152 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 34685 Lemma for dalaw 34700. Special case of dath2 34551, where  C is replaced by  ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) ). The remaining lemmas will eliminate the conditions on the atoms imposed by dath2 34551. (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 34686 Lemma for dalaw 34700. 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 34687 Lemma for dalaw 34700. First piece of dalawlem5 34689. (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 34688 Lemma for dalaw 34700. Second piece of dalawlem5 34689. (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 34689 Lemma for dalaw 34700. Special case to eliminate the requirement  -.  ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q ) in dalawlem1 34685. (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 34690 Lemma for dalaw 34700. First piece of dalawlem8 34692. (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 34691 Lemma for dalaw 34700. Second piece of dalawlem8 34692. (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 34692 Lemma for dalaw 34700. Special case to eliminate the requirement  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( Q 
.\/  R ) in dalawlem1 34685. (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 34693 Lemma for dalaw 34700. Special case to eliminate the requirement  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) in dalawlem1 34685. (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 34694 Lemma for dalaw 34700. Combine dalawlem5 34689, dalawlem8 34692, 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 34695 Lemma for dalaw 34700. First part of dalawlem13 34697. (Contributed by NM, 17-Sep-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  P  .<_  ( Q 
 .\/  R )  /\  (
 ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R 
 .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
 )  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
 
Theoremdalawlem12 34696 Lemma for dalaw 34700. Second part of dalawlem13 34697. (Contributed by NM, 17-Sep-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  Q  =  R  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U )
 )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
 
Theoremdalawlem13 34697 Lemma for dalaw 34700. Special case to eliminate the requirement  ( ( P  .\/  Q )  .\/  R )  e.  O in dalawlem1 34685. (Contributed by NM, 6-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  O  =  ( LPlanes `  K )   =>    |-  (
 ( ( K  e.  HL  /\  -.  ( ( P  .\/  Q )  .\/  R )  e.  O  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U )
 )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
 
Theoremdalawlem14 34698 Lemma for dalaw 34700. Combine dalawlem10 34694 and dalawlem13 34697. (Contributed by NM, 6-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  O  =  ( LPlanes `  K )   =>    |-  (
 ( ( K  e.  HL  /\  -.  ( ( ( P  .\/  Q )  .\/  R )  e.  O  /\  ( -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  -.  ( ( P 
 .\/  S )  ./\  ( Q  .\/  T ) ) 
 .<_  ( Q  .\/  R )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R 
 .\/  P ) ) ) 
 /\  ( ( P 
 .\/  S )  ./\  ( Q  .\/  T ) ) 
 .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
 
Theoremdalawlem15 34699 Lemma for dalaw 34700. Swap variable triples  P Q R and  S T U in dalawlem14 34698, to obtain the elimination of the remaining conditions in dalawlem1 34685. (Contributed by NM, 6-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  O  =  ( LPlanes `  K )   =>    |-  (
 ( ( K  e.  HL  /\  -.  ( ( ( S  .\/  T )  .\/  U )  e.  O  /\  ( -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( S  .\/  T )  /\  -.  ( ( P 
 .\/  S )  ./\  ( Q  .\/  T ) ) 
 .<_  ( T  .\/  U )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( U 
 .\/  S ) ) ) 
 /\  ( ( P 
 .\/  S )  ./\  ( Q  .\/  T ) ) 
 .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
 
Theoremdalaw 34700 Desargues' law, derived from Desargues' theorem dath 34550 and with no conditions on the atoms. If triples  <. P ,  Q ,  R >. and  <. S ,  T ,  U >. are centrally perspective, i.e.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ), then they are axially perspective. Theorem 13.3 of [Crawley] p. 110. (Contributed by NM, 7-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
 )  ->  ( (
 ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R 
 .\/  U )  ->  (
 ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .\/  (
 ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) ) )
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