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Theorem List for Metamath Proof Explorer - 28801-28900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempadd01 28801 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 28802 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 28803 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 28804 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 28805 A projective subspace sum is a superset of its first summand. (ssun1 3248 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 28806 A projective subspace sum is a superset of its second summand. (ssun2 3249 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 28807 Subset law for projective subspace sum. (unss1 3254 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 28808 Subset law for projective subspace sum. (unss2 3256 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 28809 Subset law for projective subspace sum. (unss12 3257 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 28810 Lemma for paddass 28828. (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 28811 Lemma for paddass 28828. (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 28812* Lemma for paddass 28828. Restate projective space axiom ps-2 28468. (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 28813* Lemma for paddass 28828. Combine paddasslem1 28810, paddasslem2 28811, and paddasslem3 28812. (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 28814 Lemma for paddass 28828. 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 28815 Lemma for paddass 28828. (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 28816 Lemma for paddass 28828. Combine paddasslem5 28814 and paddasslem6 28815. (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 28817 Lemma for paddass 28828. (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 28818 Lemma for paddass 28828. Combine paddasslem7 28816 and paddasslem8 28817. (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 28819 Lemma for paddass 28828. Use paddasslem4 28813 to eliminate  s from paddasslem9 28818. (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 28820 Lemma for paddass 28828. 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 28821 Lemma for paddass 28828. 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 28822 Lemma for paddass 28828. 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 28823 Lemma for paddass 28828. Remove  p  =/=  z,  x  =/=  y, and  -.  r  .<_  ( x  .\/  y ) from antecedent of paddasslem10 28819, using paddasslem11 28820, paddasslem12 28821, and paddasslem13 28822. (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 28824 Lemma for paddass 28828. Use elpaddn0 28790 to eliminate  y and  z from paddasslem14 28823. (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 28825 Lemma for paddass 28828. Use elpaddn0 28790 to eliminate  x and  r from paddasslem15 28824. (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 28826 Lemma for paddass 28828. 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 28827 Lemma for paddass 28828. Combine paddasslem16 28825 and paddasslem17 28826. (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 28828 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 28829 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 28830 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 28831 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 28832 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 28833 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 28834 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 28835 Subset law for projective subspace sum. (unss 3259 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 28836* Lemma for pmod1i 28838. (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 28837 Lemma for pmod1i 28838. (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 28838 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 28839 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 28840 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 28841 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 28842 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 28843 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 28844 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 28845 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 28846 A version of the modular law pmod1i 28838 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 28847 Version of modular law pmod1i 28838 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 28848 Version of modular law pmod1i 28838 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 28849 Version of modular law pmod1i 28838 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 28850 Version of modular law pmod1i 28838 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 28851 Version of modular law pmod2iN 28839 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 28852 Version of modular law pmod2iN 28839 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 28853 Version of modular law pmod1i 28838 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 28854 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 28855 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 28856 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 28857 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 28858 Lemma for llnexchb2 28859. (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 28859 Line exchange property (compare cvlatexchb2 28326 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 28860 Line exchange property (compare cvlatexch2 28328 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 28861 Lemma for dalaw 28876. Special case of dath2 28727, where  C is replaced by  ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) ). The remaining lemmas will eliminate the conditions on the atoms imposed by dath2 28727. (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 28862 Lemma for dalaw 28876. 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 28863 Lemma for dalaw 28876. First piece of dalawlem5 28865. (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 28864 Lemma for dalaw 28876. Second piece of dalawlem5 28865. (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 28865 Lemma for dalaw 28876. Special case to eliminate the requirement  -.  ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q ) in dalawlem1 28861. (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 28866 Lemma for dalaw 28876. First piece of dalawlem8 28868. (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 28867 Lemma for dalaw 28876. Second piece of dalawlem8 28868. (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 28868 Lemma for dalaw 28876. Special case to eliminate the requirement  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( Q 
.\/  R ) in dalawlem1 28861. (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 28869 Lemma for dalaw 28876. Special case to eliminate the requirement  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) in dalawlem1 28861. (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 28870 Lemma for dalaw 28876. Combine dalawlem5 28865, dalawlem8 28868, 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 28871 Lemma for dalaw 28876. First part of dalawlem13 28873. (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 28872 Lemma for dalaw 28876. Second part of dalawlem13 28873. (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 28873 Lemma for dalaw 28876. Special case to eliminate the requirement  ( ( P  .\/  Q )  .\/  R )  e.  O in dalawlem1 28861. (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 28874 Lemma for dalaw 28876. Combine dalawlem10 28870 and dalawlem13 28873. (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 28875 Lemma for dalaw 28876. Swap variable triples  P Q R and  S T U in dalawlem14 28874, to obtain the elimination of the remaining conditions in dalawlem1 28861. (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 28876 Desargues' law, derived from Desargues' theorem dath 28726 and with no conditions on the atoms. If triples  <. P ,  Q ,  R >. and  <. S ,  T ,  U >. are centrally perspective, i.e.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ), then they are axially perspective. Theorem 13.3 of [Crawley] p. 110. (Contributed by NM, 7-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
 )  ->  ( (
 ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R 
 .\/  U )  ->  (
 ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .\/  (
 ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) ) )
 
SyntaxcpclN 28877 Extend class notation with projective subspace closure.
 class  PCl
 
Definitiondf-pclN 28878* Projective subspace closure, which is the smallest projective subspace containing an arbitrary set of atoms. The subspace closure of the union of a set of projective subspaces is their supremum in  PSubSp. Related to an analogous definition of closure used in Lemma 3.1.4 of [PtakPulmannova] p. 68. (Note that this closure is not necessarily one of the closed projective subspaces  PSubCl of df-psubclN 28925.) (Contributed by NM, 7-Sep-2013.)
 |-  PCl  =  ( k  e.  _V  |->  ( x  e.  ~P ( Atoms `  k )  |-> 
 |^| { y  e.  ( PSubSp `
  k )  |  x  C_  y }
 ) )
 
TheorempclfvalN 28879* The projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( K  e.  V  ->  U  =  ( x  e.  ~P A  |->  |^| { y  e.  S  |  x  C_  y } )
 )
 
TheorempclvalN 28880* Value of the projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( K  e.  V  /\  X  C_  A )  ->  ( U `  X )  =  |^| { y  e.  S  |  X  C_  y } )
 
TheorempclclN 28881 Closure of the projective subspace closure function. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( K  e.  V  /\  X  C_  A )  ->  ( U `  X )  e.  S )
 
TheoremelpclN 28882* Membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  U  =  ( PCl `  K )   &    |-  Q  e.  _V   =>    |-  ( ( K  e.  V  /\  X  C_  A )  ->  ( Q  e.  ( U `  X )  <->  A. y  e.  S  ( X  C_  y  ->  Q  e.  y )
 ) )
 
TheoremelpcliN 28883 Implication of membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
 |-  S  =  ( PSubSp `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  /\  Q  e.  ( U `  X ) )  ->  Q  e.  Y )
 
TheorempclssN 28884 Ordering is preserved by subspace closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  ( U `  X )  C_  ( U `  Y ) )
 
TheorempclssidN 28885 A set of atoms is included in its projective subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( K  e.  V  /\  X  C_  A )  ->  X  C_  ( U `  X ) )
 
TheorempclidN 28886 The projective subspace closure of a projective subspace is itself. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
 |-  S  =  ( PSubSp `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( K  e.  V  /\  X  e.  S )  ->  ( U `  X )  =  X )
 
TheorempclbtwnN 28887 A projective subspace sandwiched between a set of atoms and the set's projective subspace closure equals the closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
 |-  S  =  ( PSubSp `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( ( K  e.  V  /\  X  e.  S )  /\  ( Y  C_  X  /\  X  C_  ( U `  Y ) ) )  ->  X  =  ( U `  Y ) )
 
TheorempclunN 28888 The projective subspace closure of the union of two sets of atoms equals the closure of their projective sum. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  .+  =  ( + P `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( K  e.  V  /\  X  C_  A  /\  Y  C_  A )  ->  ( U `  ( X  u.  Y ) )  =  ( U `  ( X  .+  Y ) ) )
 
Theorempclun2N 28889 The projective subspace closure of the union of two subspaces equals their projective sum. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
 |-  S  =  ( PSubSp `  K )   &    |-  .+  =  ( + P `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  S  /\  Y  e.  S ) 
 ->  ( U `  ( X  u.  Y ) )  =  ( X  .+  Y ) )
 
TheorempclfinN 28890* The projective subspace closure of a set equals the union of the closures of its finite subsets. Analogous to Lemma 3.3.6 of [PtakPulmannova] p. 72. Compare the closed subspace version pclfinclN 28940. (Contributed by NM, 10-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( K  e.  AtLat  /\  X  C_  A )  ->  ( U `  X )  =  U_ y  e.  ( Fin  i^i  ~P X ) ( U `
  y ) )
 
TheorempclcmpatN 28891* The set of projective subspaces is compactly atomistic: if an atom is in the projective subspace closure of a set of atoms, it also belongs to the projective subspace closure of a finite subset of that set. Analogous to Lemma 3.3.10 of [PtakPulmannova] p. 74. (Contributed by NM, 10-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  U  =  ( PCl `  K )   =>    |-  ( ( K  e.  AtLat  /\  X  C_  A  /\  P  e.  ( U `  X ) )  ->  E. y  e.  Fin  ( y  C_  X  /\  P  e.  ( U `  y ) ) )
 
SyntaxcpolN 28892 Extend class notation with polarity of projective subspace $m$.
 class  _|_ P
 
Definitiondf-polarityN 28893* Define polarity of projective subspace, which is a kind of complement of the subspace. Item 2 in [Holland95] p. 222 bottom. For more generality, we define it for all subsets of atoms, not just projective subspaces. The intersection with  Atoms `  l ensures it is defined when  m  =  (/). (Contributed by NM, 23-Oct-2011.)
 |-  _|_ P  =  ( l  e. 
 _V  |->  ( m  e. 
 ~P ( Atoms `  l
 )  |->  ( ( Atoms `  l )  i^i  |^|_ p  e.  m  ( ( pmap `  l ) `  (
 ( oc `  l
 ) `  p )
 ) ) ) )
 
TheorempolfvalN 28894* The projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)
 |-  ._|_  =  ( oc `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  (
 pmap `  K )   &    |-  P  =  ( _|_ P `  K )   =>    |-  ( K  e.  B  ->  P  =  ( m  e.  ~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p )
 ) ) ) )
 
TheorempolvalN 28895* Value of the projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)
 |-  ._|_  =  ( oc `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  (
 pmap `  K )   &    |-  P  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  B  /\  X  C_  A )  ->  ( P `  X )  =  ( A  i^i  |^|_ p  e.  X  ( M `  (  ._|_  `  p ) ) ) )
 
Theorempolval2N 28896 Alternate expression for value of the projective subspace polarity function. Equation for polarity in [Holland95] p. 223. (Contributed by NM, 22-Jan-2012.) (New usage is discouraged.)
 |-  U  =  ( lub `  K )   &    |- 
 ._|_  =  ( oc `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  M  =  ( pmap `  K )   &    |-  P  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  HL  /\  X  C_  A )  ->  ( P `  X )  =  ( M `  (  ._|_  `  ( U `  X ) ) ) )
 
TheorempolsubN 28897 The polarity of a set of atoms is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  HL  /\  X  C_  A )  ->  (  ._|_  `  X )  e.  S )
 
TheorempolssatN 28898 The polarity of a set of atoms is a set of atoms. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( ( K  e.  HL  /\  X  C_  A )  ->  (  ._|_  `  X )  C_  A )
 
Theorempol0N 28899 The polarity of the empty projective subspace is the whole space. (Contributed by NM, 29-Oct-2011.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( K  e.  B  ->  (  ._|_  `  (/) )  =  A )
 
Theorempol1N 28900 The polarity of the whole projective subspace is the empty space. Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  ._|_  =  ( _|_ P `  K )   =>    |-  ( K  e.  HL  ->  (  ._|_  `  A )  =  (/) )
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