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Theorem List for Metamath Proof Explorer - 29701-29800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem2atlt 29701* 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 29702 Atom exchange property. Version of hlatexch2 29658 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 29703 Atom exchange property. Version of hlatexch2 29658 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 29704 A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 22978 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 29705* 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 22979 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 29706* Commuted version of cvrat4 29705. (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 29707 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 29708 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 29709* 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 29710* 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 29711 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 29712 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 29713 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 29714 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 29715 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 29716 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 29717 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 29718* 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 29719* 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 29720 Lemma for 3dim1 29729. (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 29721 Lemma for 3dim1 29729. (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 29722 Lemma for 3dim3 29731. (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 29723 Lemma for 3dim1 29729. (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 29724 Lemma for 3dim1 29729. (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 29725 Lemma for 3dim3 29731. (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 29726 Lemma for 3dim1 29729. (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 29727 Lemma for 3dim1 29729. (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 29728* Lemma for 3dim1 29729. (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 29729* 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 29730* 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 29731* 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 29732* 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 29733* 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 29734 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 29735* 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 29736 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 29737 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 29738 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 29739 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 29740* 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 29741 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 29742 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 29743 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 29744 Variation of projective geometry axiom ps-2 29740. (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 29745 Lemma for 3at 29752. (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 29746 Lemma for 3at 29752. (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 29747 Lemma for 3at 29752. (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 29748 Lemma for 3at 29752. (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 29749 Lemma for 3at 29752. (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 29750 Lemma for 3at 29752. (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 29751 Lemma for 3at 29752. (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 29752 Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analog of ps-1 29739 for lines and 4at 29875 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 )
 ) )
 
18.27.9  Projective geometries based on Hilbert lattices
 
Syntaxclln 29753 Extend class notation with set of all "lattice lines" (lattice elements which cover an atom) in a Hilbert lattice.
 class  LLines
 
Syntaxclpl 29754 Extend class notation with set of all "lattice planes" (lattice elements which cover a line) in a Hilbert lattice.
 class  LPlanes
 
Syntaxclvol 29755 Extend class notation with set of all 3-dimensional "lattice volumes" (lattice elements which cover a plane) in a Hilbert lattice.
 class  LVols
 
Syntaxclines 29756 Extend class notation with set of all projective lines for a Hilbert lattice.
 class  Lines
 
SyntaxcpointsN 29757 Extend class notation with set of all projective points.
 class  Points
 
Syntaxcpsubsp 29758 Extend class notation with set of all projective subspaces.
 class  PSubSp
 
Syntaxcpmap 29759 Extend class notation with projective map.
 class  pmap
 
Definitiondf-llines 29760* 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 29761* 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 29762* 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 29763* 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 29764* 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 29765* 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 29766* 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 29767* 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 29768* 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 29769* 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 29770 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 29771 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 29772* 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 29773* 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 29774 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 29775 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 29776 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 29777 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 29778 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 29779 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 29780* 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 29781 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 29782 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 29783* 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 29784 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 29785 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 29786 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 29787 Special case of 2atmat0 29788 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 29788 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 29789 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 ) ) )
 
Theoremps-2c 29790 Variation of projective geometry axiom ps-2 29740. (Contributed by NM, 3-Jul-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 ) 
 /\  ( ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T )  /\  ( P 
 .\/  R )  =/=  ( S  .\/  T )  /\  ( S  .<_  ( P 
 .\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) ) 
 ->  ( ( P  .\/  R )  ./\  ( S  .\/  T ) )  e.  A )
 
Theoremlplnset 29791* The set of lattice planes in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  N  =  ( LLines `  K )   &    |-  P  =  (
 LPlanes `  K )   =>    |-  ( K  e.  A  ->  P  =  { x  e.  B  |  E. y  e.  N  y C x } )
 
Theoremislpln 29792* The predicate "is a lattice plane". (Contributed by NM, 16-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  N  =  ( LLines `  K )   &    |-  P  =  (
 LPlanes `  K )   =>    |-  ( K  e.  A  ->  ( X  e.  P 
 <->  ( X  e.  B  /\  E. y  e.  N  y C X ) ) )
 
Theoremislpln4 29793* The predicate "is a lattice plane". (Contributed by NM, 17-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  N  =  ( LLines `  K )   &    |-  P  =  (
 LPlanes `  K )   =>    |-  ( ( K  e.  A  /\  X  e.  B )  ->  ( X  e.  P  <->  E. y  e.  N  y C X ) )
 
Theoremlplni 29794 Condition implying a lattice plane. (Contributed by NM, 20-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  N  =  ( LLines `  K )   &    |-  P  =  (
 LPlanes `  K )   =>    |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  N )  /\  X C Y ) 
 ->  Y  e.  P )
 
Theoremislpln3 29795* The predicate "is a lattice plane". (Contributed by NM, 17-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  N  =  ( LLines `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B ) 
 ->  ( X  e.  P  <->  E. y  e.  N  E. p  e.  A  ( -.  p  .<_  y  /\  X  =  ( y  .\/  p ) ) ) )
 
Theoremlplnbase 29796 A lattice plane is a lattice element. (Contributed by NM, 17-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  ( X  e.  P  ->  X  e.  B )
 
Theoremislpln5 29797* The predicate "is a lattice plane" in terms of atoms. (Contributed by NM, 24-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B ) 
 ->  ( X  e.  P  <->  E. p  e.  A  E. q  e.  A  E. r  e.  A  ( p  =/=  q  /\  -.  r  .<_  ( p  .\/  q
 )  /\  X  =  ( ( p  .\/  q )  .\/  r ) ) ) )
 
Theoremislpln2 29798* The predicate "is a lattice plane" in terms of atoms. (Contributed by NM, 25-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  ( K  e.  HL  ->  ( X  e.  P  <->  ( X  e.  B  /\  E. p  e.  A  E. q  e.  A  E. r  e.  A  ( p  =/=  q  /\  -.  r  .<_  ( p  .\/  q
 )  /\  X  =  ( ( p  .\/  q )  .\/  r ) ) ) ) )
 
Theoremlplni2 29799 The join of 3 different atoms is a lattice plane. (Contributed by NM, 4-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  (
 ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q 
 .\/  R ) ) ) 
 ->  ( ( Q  .\/  R )  .\/  S )  e.  P )
 
Theoremlvolex3N 29800* There is an atom outside of a lattice plane i.e. a 3-dimensional lattice volume exists. (Contributed by NM, 28-Jul-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  P  =  (
 LPlanes `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  P )  ->  E. q  e.  A  -.  q  .<_  X )
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