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Theorem List for Metamath Proof Explorer - 32701-32800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorematcvrj0 32701 Two atoms covering the zero subspace are equal. (atcv1 27868 analog.) (Contributed by NM, 29-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A )  /\  X C ( P 
 .\/  Q ) )  ->  ( X  =  .0.  <->  P  =  Q ) )
 
Theoremcvrat2 32702 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 27875 analog.) (Contributed by NM, 30-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A ) 
 /\  ( P  =/=  Q 
 /\  X C ( P  .\/  Q )
 ) )  ->  X  e.  A )
 
TheorematcvrneN 32703 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) 
 /\  P C ( Q  .\/  R )
 )  ->  Q  =/=  R )
 
Theorematcvrj1 32704 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P C ( Q  .\/  R ) )
 
Theorematcvrj2b 32705 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
 )  ->  ( ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) )  <->  P C ( Q 
 .\/  R ) ) )
 
Theorematcvrj2 32706 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P C ( Q  .\/  R ) )
 
TheorematleneN 32707 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  ->  Q  =/=  R )
 
Theorematltcvr 32708 An equivalence of less-than ordering and covers relation. (Contributed by NM, 7-Feb-2012.)
 |-  .<  =  ( lt `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  ( P  .<  ( Q  .\/  R ) 
 <->  P C ( Q 
 .\/  R ) ) )
 
Theorematle 32709* Any non-zero element has an atom under it. (Contributed by NM, 28-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  E. p  e.  A  p  .<_  X )
 
Theorematlt 32710 Two atoms are unequal iff their join is greater than one of them. (Contributed by NM, 6-May-2012.)
 |-  .<  =  ( lt `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P 
 .<  ( P  .\/  Q ) 
 <->  P  =/=  Q ) )
 
Theorematlelt 32711 Transfer less-than relation from one atom to another. (Contributed by NM, 7-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  Q  .<  X ) ) 
 ->  P  .<  X )
 
Theorem2atlt 32712* Given an atom less than an element, there is another atom less than the element. (Contributed by NM, 6-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  ->  E. q  e.  A  ( q  =/= 
 P  /\  q  .<  X ) )
 
TheorematexchcvrN 32713 Atom exchange property. Version of hlatexch2 32669 with covers relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) 
 /\  P  =/=  R )  ->  ( P C ( Q  .\/  R ) 
 ->  Q C ( P 
 .\/  R ) ) )
 
TheorematexchltN 32714 Atom exchange property. Version of hlatexch2 32669 with less-than ordering. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
 |-  .<  =  ( lt `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R ) 
 ->  ( P  .<  ( Q 
 .\/  R )  ->  Q  .<  ( P  .\/  R ) ) )
 
Theoremcvrat3 32715 A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 27884 analog.) (Contributed by NM, 30-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A )
 )  ->  ( ( P  =/=  Q  /\  -.  Q  .<_  X  /\  P  .<_  ( X  .\/  Q ) )  ->  ( X 
 ./\  ( P  .\/  Q ) )  e.  A ) )
 
Theoremcvrat4 32716* A condition implying existence of an atom with the properties shown. Lemma 3.2.20 in [PtakPulmannova] p. 68. Also Lemma 9.2(delta) in [MaedaMaeda] p. 41. (atcvat4i 27885 analog.) (Contributed by NM, 30-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A ) )  ->  ( ( X  =/=  .0.  /\  P  .<_  ( X  .\/  Q ) )  ->  E. r  e.  A  ( r  .<_  X 
 /\  P  .<_  ( Q 
 .\/  r ) ) ) )
 
Theoremcvrat42 32717* Commuted version of cvrat4 32716. (Contributed by NM, 28-Jan-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A ) )  ->  ( ( X  =/=  .0.  /\  P  .<_  ( X  .\/  Q ) )  ->  E. r  e.  A  ( r  .<_  X 
 /\  P  .<_  ( r 
 .\/  Q ) ) ) )
 
Theorem2atjm 32718 The meet of a line (expressed with 2 atoms) and a lattice element. (Contributed by NM, 30-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) 
 ->  ( ( P  .\/  Q )  ./\  X )  =  P )
 
Theorematbtwn 32719 Property of a 3rd atom  R on a line  P  .\/  Q intersecting element  X at  P. (Contributed by NM, 30-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B ) 
 /\  ( P  .<_  X 
 /\  -.  Q  .<_  X 
 /\  R  .<_  ( P 
 .\/  Q ) ) ) 
 ->  ( R  =/=  P  <->  -.  R  .<_  X ) )
 
TheorematbtwnexOLDN 32720* There exists a 3rd atom  r on a line  P  .\/  Q intersecting element  X at  P, such that  r is different from  Q and not in  X. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) ) 
 ->  E. r  e.  A  ( r  =/=  Q  /\  -.  r  .<_  X  /\  r  .<_  ( P  .\/  Q ) ) )
 
Theorematbtwnex 32721* Given atoms  P in  X and  Q not in  X, there exists an atom  r not in  X such that the line  Q  .\/  r intersects  X at  P. (Contributed by NM, 1-Aug-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) ) 
 ->  E. r  e.  A  ( r  =/=  Q  /\  -.  r  .<_  X  /\  P  .<_  ( Q  .\/  r
 ) ) )
 
Theorem3noncolr2 32722 Two ways to express 3 non-colinear atoms (rotated right 2 places). (Contributed by NM, 12-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
 .\/  Q ) ) ) 
 ->  ( Q  =/=  R  /\  -.  P  .<_  ( Q 
 .\/  R ) ) )
 
Theorem3noncolr1N 32723 Two ways to express 3 non-colinear atoms (rotated right 1 place). (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
 .\/  Q ) ) ) 
 ->  ( R  =/=  P  /\  -.  Q  .<_  ( R 
 .\/  P ) ) )
 
Theoremhlatcon3 32724 Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
 .\/  Q ) ) ) 
 ->  -.  P  .<_  ( Q 
 .\/  R ) )
 
Theoremhlatcon2 32725 Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
 .\/  Q ) ) ) 
 ->  -.  P  .<_  ( R 
 .\/  Q ) )
 
Theorem4noncolr3 32726 A way to express 4 non-colinear atoms (rotated right 3 places). (Contributed by NM, 11-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A ) 
 /\  ( P  =/=  Q 
 /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) ) 
 ->  ( Q  =/=  R  /\  -.  S  .<_  ( Q 
 .\/  R )  /\  -.  P  .<_  ( ( Q 
 .\/  R )  .\/  S ) ) )
 
Theorem4noncolr2 32727 A way to express 4 non-colinear atoms (rotated right 2 places). (Contributed by NM, 11-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A ) 
 /\  ( P  =/=  Q 
 /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) ) 
 ->  ( R  =/=  S  /\  -.  P  .<_  ( R 
 .\/  S )  /\  -.  Q  .<_  ( ( R 
 .\/  S )  .\/  P ) ) )
 
Theorem4noncolr1 32728 A way to express 4 non-colinear atoms (rotated right 1 places). (Contributed by NM, 11-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A ) 
 /\  ( P  =/=  Q 
 /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) ) 
 ->  ( S  =/=  P  /\  -.  Q  .<_  ( S 
 .\/  P )  /\  -.  R  .<_  ( ( S 
 .\/  P )  .\/  Q ) ) )
 
Theoremathgt 32729* A Hilbert lattice, whose height is at least 4, has a chain of 4 successively covering atom joins. (Contributed by NM, 3-May-2012.)
 |-  .\/  =  ( join `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( K  e.  HL  ->  E. p  e.  A  E. q  e.  A  ( p C ( p 
 .\/  q )  /\  E. r  e.  A  ( ( p  .\/  q
 ) C ( ( p  .\/  q )  .\/  r )  /\  E. s  e.  A  (
 ( p  .\/  q
 )  .\/  r ) C ( ( ( p  .\/  q )  .\/  r )  .\/  s
 ) ) ) )
 
Theorem3dim0 32730* There exists a 3-dimensional (height-4) element i.e. a volume. (Contributed by NM, 25-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( K  e.  HL  ->  E. p  e.  A  E. q  e.  A  E. r  e.  A  E. s  e.  A  ( p  =/=  q  /\  -.  r  .<_  ( p 
 .\/  q )  /\  -.  s  .<_  ( ( p 
 .\/  q )  .\/  r ) ) )
 
Theorem3dimlem1 32731 Lemma for 3dim1 32740. (Contributed by NM, 25-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R )  .\/  S )
 )  /\  P  =  Q )  ->  ( P  =/=  R  /\  -.  S  .<_  ( P  .\/  R )  /\  -.  T  .<_  ( ( P  .\/  R )  .\/  S )
 ) )
 
Theorem3dimlem2 32732 Lemma for 3dim1 32740. (Contributed by NM, 25-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q  .\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R )  .\/  S )
 )  /\  ( P  =/=  Q  /\  P  .<_  ( Q  .\/  R )
 ) )  ->  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( ( P  .\/  Q )  .\/  S )
 ) )
 
Theorem3dimlem3a 32733 Lemma for 3dim3 32742. (Contributed by NM, 27-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( -.  T  .<_  ( ( Q 
 .\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q 
 .\/  R )  .\/  S ) ) )  ->  -.  T  .<_  ( ( P 
 .\/  Q )  .\/  R ) )
 
Theorem3dimlem3 32734 Lemma for 3dim1 32740. (Contributed by NM, 25-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A ) 
 /\  ( Q  =/=  R 
 /\  -.  T  .<_  ( ( Q  .\/  R )  .\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R )  .\/  S ) ) )  ->  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( ( P  .\/  Q )  .\/  R ) ) )
 
Theorem3dimlem3OLDN 32735 Lemma for 3dim1 32740. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A ) 
 /\  ( Q  =/=  R 
 /\  -.  T  .<_  ( ( Q  .\/  R )  .\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R )  .\/  S ) ) )  ->  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( ( P  .\/  Q )  .\/  R ) ) )
 
Theorem3dimlem4a 32736 Lemma for 3dim3 32742. (Contributed by NM, 27-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( -.  S  .<_  ( Q  .\/  R )  /\  -.  P  .<_  ( Q  .\/  R )  /\  -.  P  .<_  ( ( Q  .\/  R )  .\/  S ) ) )  ->  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )
 
Theorem3dimlem4 32737 Lemma for 3dim1 32740. (Contributed by NM, 25-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A ) 
 /\  ( Q  =/=  R 
 /\  -.  S  .<_  ( Q  .\/  R )
 ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R ) )  /\  -.  P  .<_  ( ( Q 
 .\/  R )  .\/  S ) )  ->  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R )
 ) )
 
Theorem3dimlem4OLDN 32738 Lemma for 3dim1 32740. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A ) 
 /\  ( Q  =/=  R 
 /\  -.  S  .<_  ( Q  .\/  R )
 ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R ) )  /\  -.  P  .<_  ( ( Q 
 .\/  R )  .\/  S ) )  ->  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R )
 ) )
 
Theorem3dim1lem5 32739* Lemma for 3dim1 32740. (Contributed by NM, 26-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( u  e.  A  /\  v  e.  A  /\  w  e.  A )  /\  ( P  =/=  u  /\  -.  v  .<_  ( P  .\/  u )  /\  -.  w  .<_  ( ( P  .\/  u )  .\/  v )
 ) )  ->  E. q  e.  A  E. r  e.  A  E. s  e.  A  ( P  =/=  q  /\  -.  r  .<_  ( P  .\/  q )  /\  -.  s  .<_  ( ( P  .\/  q )  .\/  r ) ) )
 
Theorem3dim1 32740* Construct a 3-dimensional volume (height-4 element) on top of a given atom  P. (Contributed by NM, 25-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  P  e.  A )  ->  E. q  e.  A  E. r  e.  A  E. s  e.  A  ( P  =/=  q  /\  -.  r  .<_  ( P 
 .\/  q )  /\  -.  s  .<_  ( ( P 
 .\/  q )  .\/  r ) ) )
 
Theorem3dim2 32741* Construct 2 new layers on top of 2 given atoms. (Contributed by NM, 27-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A ) 
 ->  E. r  e.  A  E. s  e.  A  ( -.  r  .<_  ( P 
 .\/  Q )  /\  -.  s  .<_  ( ( P 
 .\/  Q )  .\/  r
 ) ) )
 
Theorem3dim3 32742* Construct a new layer on top of 3 given atoms. (Contributed by NM, 27-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  E. s  e.  A  -.  s  .<_  ( ( P  .\/  Q )  .\/  R ) )
 
Theorem2dim 32743* Generate a height-3 element (2-dimensional plane) from an atom. (Contributed by NM, 3-May-2012.)
 |-  .\/  =  ( join `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  P  e.  A )  ->  E. q  e.  A  E. r  e.  A  ( P C ( P 
 .\/  q )  /\  ( P  .\/  q ) C ( ( P 
 .\/  q )  .\/  r ) ) )
 
Theorem1dimN 32744* An atom is covered by a height-2 element (1-dimensional line). (Contributed by NM, 3-May-2012.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  P  e.  A )  ->  E. q  e.  A  P C ( P  .\/  q ) )
 
Theorem1cvrco 32745 The orthocomplement of an element covered by 1 is an atom. (Contributed by NM, 7-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .1.  =  ( 1. `  K )   &    |- 
 ._|_  =  ( oc `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B ) 
 ->  ( X C  .1.  <->  (  ._|_  `  X )  e.  A ) )
 
Theorem1cvratex 32746* There exists an atom less than an element covered by 1. (Contributed by NM, 7-May-2012.) (Revised by Mario Carneiro, 13-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 .1.  =  ( 1. `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  ->  E. p  e.  A  p  .<  X )
 
Theorem1cvratlt 32747 An atom less than or equal to an element covered by 1 is less than the element. (Contributed by NM, 7-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1.  /\  P  .<_  X ) ) 
 ->  P  .<  X )
 
Theorem1cvrjat 32748 An element covered by the lattice unit, when joined with an atom not under it, equals the lattice unit. (Contributed by NM, 30-Apr-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1.  /\ 
 -.  P  .<_  X ) )  ->  ( X  .\/  P )  =  .1.  )
 
Theorem1cvrat 32749 Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 30-Apr-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  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 ) ) 
 ->  ( ( P  .\/  Q )  ./\  X )  e.  A )
 
Theoremps-1 32750 The join of two atoms  R  .\/  S (specifying a projective geometry line) is determined uniquely by any two atoms (specifying two points) less than or equal to that join. Part of Lemma 16.4 of [MaedaMaeda] p. 69, showing projective space postulate PS1 in [MaedaMaeda] p. 67. (Contributed by NM, 15-Nov-2011.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  ( ( P  .\/  Q )  .<_  ( R  .\/  S )  <->  ( P  .\/  Q )  =  ( R  .\/  S ) ) )
 
Theoremps-2 32751* Lattice analog for the projective geometry axiom, "if a line intersects two sides of a triangle at different points then it also intersects the third side." Projective space condition PS2 in [MaedaMaeda] p. 68 and part of Theorem 16.4 in [MaedaMaeda] p. 69. (Contributed by NM, 1-Dec-2011.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( -.  P  .<_  ( Q 
 .\/  R )  /\  S  =/=  T )  /\  ( S  .<_  ( P  .\/  Q )  /\  T  .<_  ( Q  .\/  R )
 ) ) )  ->  E. u  e.  A  ( u  .<_  ( P 
 .\/  R )  /\  u  .<_  ( S  .\/  T ) ) )
 
Theorem2atjlej 32752 Two atoms are different if their join majorizes the join of two different atoms. (Contributed by NM, 4-Jun-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q ) 
 .<_  ( R  .\/  S ) ) )  ->  R  =/=  S )
 
Theoremhlatexch3N 32753 Rearrange join of atoms in an equality. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( Q  =/=  R  /\  ( P  .\/  Q )  =  ( P  .\/  R ) ) ) 
 ->  ( P  .\/  Q )  =  ( Q  .\/  R ) )
 
Theoremhlatexch4 32754 Exchange 2 atoms. (Contributed by NM, 13-May-2013.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A ) 
 /\  ( R  e.  A  /\  S  e.  A )  /\  ( P  =/=  R 
 /\  Q  =/=  S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) ) 
 ->  ( P  .\/  R )  =  ( Q  .\/  S ) )
 
Theoremps-2b 32755 Variation of projective geometry axiom ps-2 32751. (Contributed by NM, 3-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T 
 /\  ( S  .<_  ( P  .\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) ) 
 ->  ( ( P  .\/  R )  ./\  ( S  .\/  T ) )  =/= 
 .0.  )
 
Theorem3atlem1 32756 Lemma for 3at 32763. (Contributed by NM, 22-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  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 )
 )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  P  .<_  ( T  .\/  U )  /\  -.  Q  .<_  ( P  .\/  U )
 )  /\  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S 
 .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T )  .\/  U ) )
 
Theorem3atlem2 32757 Lemma for 3at 32763. (Contributed by NM, 22-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  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 )
 )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/=  U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U ) )  /\  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T )  .\/  U )
 )
 
Theorem3atlem3 32758 Lemma for 3at 32763. (Contributed by NM, 23-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  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 )
 )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  U 
 /\  -.  Q  .<_  ( P  .\/  U )
 )  /\  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S 
 .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T )  .\/  U ) )
 
Theorem3atlem4 32759 Lemma for 3at 32763. (Contributed by NM, 23-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q )  /\  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  R ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T )  .\/  R )
 )
 
Theorem3atlem5 32760 Lemma for 3at 32763. (Contributed by NM, 23-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  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 )
 )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q 
 /\  -.  Q  .<_  ( P  .\/  U )
 )  /\  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S 
 .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T )  .\/  U ) )
 
Theorem3atlem6 32761 Lemma for 3at 32763. (Contributed by NM, 23-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  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 )
 )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q 
 /\  Q  .<_  ( P 
 .\/  U ) )  /\  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U )
 )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T )  .\/  U ) )
 
Theorem3atlem7 32762 Lemma for 3at 32763. (Contributed by NM, 23-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  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 )
 )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q )  /\  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T )  .\/  U )
 )
 
Theorem3at 32763 Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analog of ps-1 32750 for lines and 4at 32886 for volumes. I could not find this proof in the literature on projective geometry (where it is either given as an axiom or stated as an unproved fact), but it is similar to Theorem 15 of Veblen, "The Foundations of Geometry" (1911), p. 18, which uses different axioms. This proof was written before I became aware of Veblen's, and it is possible that a shorter proof could be obtained by using Veblen's proof for hints. (Contributed by NM, 23-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  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 )
 )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q ) )  ->  (
 ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U )  <->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T )  .\/  U )
 ) )
 
21.21.12  Projective geometries based on Hilbert lattices
 
Syntaxclln 32764 Extend class notation with set of all "lattice lines" (lattice elements which cover an atom) in a Hilbert lattice.
 class  LLines
 
Syntaxclpl 32765 Extend class notation with set of all "lattice planes" (lattice elements which cover a line) in a Hilbert lattice.
 class  LPlanes
 
Syntaxclvol 32766 Extend class notation with set of all 3-dimensional "lattice volumes" (lattice elements which cover a plane) in a Hilbert lattice.
 class  LVols
 
Syntaxclines 32767 Extend class notation with set of all projective lines for a Hilbert lattice.
 class  Lines
 
SyntaxcpointsN 32768 Extend class notation with set of all projective points.
 class  Points
 
Syntaxcpsubsp 32769 Extend class notation with set of all projective subspaces.
 class  PSubSp
 
Syntaxcpmap 32770 Extend class notation with projective map.
 class  pmap
 
Definitiondf-llines 32771* Define the set of all "lattice lines" (lattice elements which cover an atom) in a Hilbert lattice  k, in other words all elements of height 2 (or lattice dimension 2 or projective dimension 1). (Contributed by NM, 16-Jun-2012.)
 |-  LLines  =  ( k  e.  _V  |->  { x  e.  ( Base `  k )  |  E. p  e.  ( Atoms `  k ) p ( 
 <o  `  k ) x } )
 
Definitiondf-lplanes 32772* Define the set of all "lattice planes" (lattice elements which cover a line) in a Hilbert lattice  k, in other words all elements of height 3 (or lattice dimension 3 or projective dimension 2). (Contributed by NM, 16-Jun-2012.)
 |-  LPlanes  =  ( k  e.  _V  |->  { x  e.  ( Base `  k )  |  E. p  e.  ( LLines `  k ) p ( 
 <o  `  k ) x } )
 
Definitiondf-lvols 32773* Define the set of all 3-dimensional "lattice volumes" (lattice elements which cover a plane) in a Hilbert lattice  k, in other words all elements of height 4 (or lattice dimension 4 or projective dimension 3). (Contributed by NM, 1-Jul-2012.)
 |-  LVols  =  ( k  e.  _V  |->  { x  e.  ( Base `  k )  |  E. p  e.  ( LPlanes `  k ) p ( 
 <o  `  k ) x } )
 
Definitiondf-lines 32774* Define set of all projective lines for a Hilbert lattice (actually in any set at all, for simplicity). The join of two distinct atoms equals a line. Definition of lines in item 1 of [Holland95] p. 222. (Contributed by NM, 19-Sep-2011.)
 |-  Lines  =  ( k  e.  _V  |->  { s  |  E. q  e.  ( Atoms `  k ) E. r  e.  ( Atoms `  k ) ( q  =/=  r  /\  s  =  { p  e.  ( Atoms `  k )  |  p ( le `  k
 ) ( q (
 join `  k ) r ) } ) }
 )
 
Definitiondf-pointsN 32775* Define set of all projective points in a Hilbert lattice (actually in any set at all, for simplicity). A projective point is the singleton of a lattice atom. Definition 15.1 of [MaedaMaeda] p. 61. Note that item 1 in [Holland95] p. 222 defines a point as the atom itself, but this leads to a complicated subspace ordering that may be either membership or inclusion based on its arguments. (Contributed by NM, 2-Oct-2011.)
 |-  Points  =  ( k  e.  _V  |->  { q  |  E. p  e.  ( Atoms `  k )
 q  =  { p } } )
 
Definitiondf-psubsp 32776* Define set of all projective subspaces. Based on definition of subspace in [Holland95] p. 212. (Contributed by NM, 2-Oct-2011.)
 |-  PSubSp  =  ( k  e.  _V  |->  { s  |  ( s 
 C_  ( Atoms `  k
 )  /\  A. p  e.  s  A. q  e.  s  A. r  e.  ( Atoms `  k )
 ( r ( le `  k ) ( p ( join `  k )
 q )  ->  r  e.  s ) ) }
 )
 
Definitiondf-pmap 32777* Define projective map for  k at  a. Definition in Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 2-Oct-2011.)
 |-  pmap  =  ( k  e.  _V  |->  ( a  e.  ( Base `  k )  |->  { p  e.  ( Atoms `  k )  |  p ( le `  k ) a } ) )
 
Theoremllnset 32778* The set of lattice lines in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  (
 LLines `  K )   =>    |-  ( K  e.  D  ->  N  =  { x  e.  B  |  E. p  e.  A  p C x } )
 
Theoremislln 32779* The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  (
 LLines `  K )   =>    |-  ( K  e.  D  ->  ( X  e.  N 
 <->  ( X  e.  B  /\  E. p  e.  A  p C X ) ) )
 
Theoremislln4 32780* The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  (
 LLines `  K )   =>    |-  ( ( K  e.  D  /\  X  e.  B )  ->  ( X  e.  N  <->  E. p  e.  A  p C X ) )
 
Theoremllni 32781 Condition implying a lattice line. (Contributed by NM, 17-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  (
 LLines `  K )   =>    |-  ( ( ( K  e.  D  /\  X  e.  B  /\  P  e.  A )  /\  P C X ) 
 ->  X  e.  N )
 
Theoremllnbase 32782 A lattice line is a lattice element. (Contributed by NM, 16-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  N  =  ( LLines `  K )   =>    |-  ( X  e.  N  ->  X  e.  B )
 
Theoremislln3 32783* The predicate "is a lattice line". (Contributed by NM, 17-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  ( LLines `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B ) 
 ->  ( X  e.  N  <->  E. p  e.  A  E. q  e.  A  ( p  =/=  q  /\  X  =  ( p  .\/  q
 ) ) ) )
 
Theoremislln2 32784* The predicate "is a lattice line". (Contributed by NM, 23-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  ( LLines `  K )   =>    |-  ( K  e.  HL  ->  ( X  e.  N  <->  ( X  e.  B  /\  E. p  e.  A  E. q  e.  A  ( p  =/=  q  /\  X  =  ( p  .\/  q ) ) ) ) )
 
Theoremllni2 32785 The join of two different atoms is a lattice line. (Contributed by NM, 26-Jun-2012.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  ( LLines `  K )   =>    |-  (
 ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A ) 
 /\  P  =/=  Q )  ->  ( P  .\/  Q )  e.  N )
 
Theoremllnnleat 32786 An atom cannot majorize a lattice line. (Contributed by NM, 8-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  (
 LLines `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  -.  X  .<_  P )
 
Theoremllnneat 32787 A lattice line is not an atom. (Contributed by NM, 19-Jun-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  N  =  ( LLines `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  N ) 
 ->  -.  X  e.  A )
 
Theorem2atneat 32788 The join of two distinct atoms is not an atom. (Contributed by NM, 12-Oct-2012.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q ) ) 
 ->  -.  ( P  .\/  Q )  e.  A )
 
Theoremllnn0 32789 A lattice line is non-zero. (Contributed by NM, 15-Jul-2012.)
 |-  .0.  =  ( 0. `  K )   &    |-  N  =  ( LLines `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  N )  ->  X  =/=  .0.  )
 
Theoremislln2a 32790 The predicate "is a lattice line" in terms of atoms. (Contributed by NM, 15-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  ( LLines `  K )   =>    |-  (
 ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( ( P  .\/  Q )  e.  N  <->  P  =/=  Q ) )
 
Theoremllnle 32791* Any element greater than 0 and not an atom majorizes a lattice line. (Contributed by NM, 28-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  N  =  ( LLines `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A ) )  ->  E. y  e.  N  y  .<_  X )
 
Theorematcvrlln2 32792 An atom under a line is covered by it. (Contributed by NM, 2-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  N  =  ( LLines `  K )   =>    |-  (
 ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N ) 
 /\  P  .<_  X ) 
 ->  P C X )
 
Theorematcvrlln 32793 An element covering an atom is a lattice line and vice-versa. (Contributed by NM, 18-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  (
 LLines `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y ) 
 ->  ( X  e.  A  <->  Y  e.  N ) )
 
TheoremllnexatN 32794* Given an atom on a line, there is another atom whose join equals the line. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  N  =  ( LLines `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A ) 
 /\  P  .<_  X ) 
 ->  E. q  e.  A  ( P  =/=  q  /\  X  =  ( P 
 .\/  q ) ) )
 
Theoremllncmp 32795 If two lattice lines are comparable, they are equal. (Contributed by NM, 19-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |-  N  =  ( LLines `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N ) 
 ->  ( X  .<_  Y  <->  X  =  Y ) )
 
Theoremllnnlt 32796 Two lattice lines cannot satisfy the less than relation. (Contributed by NM, 26-Jun-2012.)
 |-  .<  =  ( lt `  K )   &    |-  N  =  ( LLines `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N ) 
 ->  -.  X  .<  Y )
 
Theorem2llnmat 32797 Two intersecting lines intersect at an atom. (Contributed by NM, 30-Apr-2012.)
 |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  (
 LLines `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/=  .0.  ) ) 
 ->  ( X  ./\  Y )  e.  A )
 
Theorem2at0mat0 32798 Special case of 2atmat0 32799 where one atom could be zero. (Contributed by NM, 30-May-2013.)
 |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  ->  ( ( ( P 
 .\/  Q )  ./\  ( R  .\/  S ) )  e.  A  \/  (
 ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =  .0.  ) )
 
Theorem2atmat0 32799 The meet of two unequal lines (expressed as joins of atoms) is an atom or zero. (Contributed by NM, 2-Dec-2012.)
 |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P 
 .\/  Q )  =/=  ( R  .\/  S ) ) )  ->  ( (
 ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =  .0.  ) )
 
Theorem2atm 32800 An atom majorized by two different atom joins (which could be atoms or lines) is equal to their intersection. (Contributed by NM, 30-Jun-2013.)
 |-  .<_  =  ( 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 ) 
 /\  ( T  .<_  ( P  .\/  Q )  /\  T  .<_  ( R  .\/  S )  /\  ( P 
 .\/  Q )  =/=  ( R  .\/  S ) ) )  ->  T  =  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) ) )
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