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Theorem List for Metamath Proof Explorer - 28401-28500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem4noncolr3 28401 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 28402 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 28403 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 28404* 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 28405* 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 28406 Lemma for 3dim1 28415. (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 28407 Lemma for 3dim1 28415. (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 28408 Lemma for 3dim3 28417. (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 28409 Lemma for 3dim1 28415. (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 28410 Lemma for 3dim1 28415. (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 28411 Lemma for 3dim3 28417. (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 28412 Lemma for 3dim1 28415. (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 28413 Lemma for 3dim1 28415. (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 28414* Lemma for 3dim1 28415. (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 28415* 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 28416* 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 28417* 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 28418* 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 28419* 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 28420 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 28421* 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 28422 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 28423 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 28424 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 28425 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 28426* 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 28427 Two atoms are be 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 28428 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 28429 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 28430 Variation of projective geometry axiom ps-2 28426. (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 28431 Lemma for 3at 28438. (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 28432 Lemma for 3at 28438. (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 28433 Lemma for 3at 28438. (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 28434 Lemma for 3at 28438. (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 28435 Lemma for 3at 28438. (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 28436 Lemma for 3at 28438. (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 28437 Lemma for 3at 28438. (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 28438 Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analog of ps-1 28425 for lines and 4at 28561 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 )
 ) )
 
16.22.13  Projective geometries based on Hilbert lattices
 
Syntaxclln 28439 Extend class notation with set of all "lattice lines" (lattice elements which cover an atom) in a Hilbert lattice.
 class  LLines
 
Syntaxclpl 28440 Extend class notation with set of all "lattice planes" (lattice elements which cover a line) in a Hilbert lattice.
 class  LPlanes
 
Syntaxclvol 28441 Extend class notation with set of all 3-dimensional "lattice volumes" (lattice elements which cover a plane) in a Hilbert lattice.
 class  LVols
 
Syntaxclines 28442 Extend class notation with set of all projective lines for a Hilbert lattice.
 class  Lines
 
SyntaxcpointsN 28443 Extend class notation with set of all projective points.
 class  Points
 
Syntaxcpsubsp 28444 Extend class notation with set of all projective subspaces.
 class  PSubSp
 
Syntaxcpmap 28445 Extend class notation with projective map.
 class  pmap
 
Definitiondf-llines 28446* 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 28447* 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 28448* 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 28449* 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 28450* 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 28451* 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 28452* 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 28453* 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 28454* 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 28455* 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 28456 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 28457 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 28458* 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 28459* 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 28460 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 28461 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 28462 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 28463 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 28464 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 28465 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 28466* 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 28467 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 28468 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 28469* 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 28470 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 28471 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 28472 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 28473 Special case of 2atmat0 28474 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 28474 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 28475 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 28476 Variation of projective geometry axiom ps-2 28426. (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 28477* 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 28478* 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 28479* 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 28480 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 28481* 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 28482 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 28483* 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 28484* 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 28485 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 28486* 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 )
 
TheoremllnmlplnN 28487 The intersection of a line with a plane not containing it is an atom. (Contributed by NM, 29-Jun-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  N  =  ( LLines `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  P ) 
 /\  ( -.  X  .<_  Y  /\  ( X 
 ./\  Y )  =/=  .0.  ) )  ->  ( X 
 ./\  Y )  e.  A )
 
Theoremlplnle 28488* Any element greater than 0 and not an atom and not a lattice line majorizes a lattice plane. (Contributed by NM, 28-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  N  =  ( LLines `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A  /\  -.  X  e.  N ) )  ->  E. y  e.  P  y  .<_  X )
 
Theoremlplnnle2at 28489 A lattice lattice line (or atom) cannot majorize a lattice plane. (Contributed by NM, 8-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  (
 ( K  e.  HL  /\  ( X  e.  P  /\  Q  e.  A  /\  R  e.  A )
 )  ->  -.  X  .<_  ( Q  .\/  R )
 )
 
Theoremlplnnleat 28490 A lattice plane cannot majorize an atom. (Contributed by NM, 14-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  P  =  (
 LPlanes `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  ->  -.  X  .<_  Q )
 
Theoremlplnnlelln 28491 A lattice plane is not less than or equal to a lattice line. (Contributed by NM, 14-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |-  N  =  ( LLines `  K )   &    |-  P  =  (
 LPlanes `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  ->  -.  X  .<_  Y )
 
Theorem2atnelpln 28492 The join of two atoms is not a lattice plane. (Contributed by NM, 16-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  (
 ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  -.  ( Q  .\/  R )  e.  P )
 
Theoremlplnneat 28493 No lattice plane is an atom. (Contributed by NM, 15-Jul-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  P ) 
 ->  -.  X  e.  A )
 
Theoremlplnnelln 28494 No lattice plane is a lattice line. (Contributed by NM, 19-Jun-2012.)
 |-  N  =  ( LLines `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  P ) 
 ->  -.  X  e.  N )
 
Theoremlplnn0N 28495 A lattice plane is non-zero. (Contributed by NM, 15-Jul-2012.) (New usage is discouraged.)
 |-  .0.  =  ( 0. `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  P )  ->  X  =/=  .0.  )
 
Theoremislpln2a 28496 The predicate "is a lattice plane" for join of atoms. (Contributed by NM, 16-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 )  e.  P  <->  ( Q  =/=  R 
 /\  -.  S  .<_  ( Q  .\/  R )
 ) ) )
 
Theoremislpln2ah 28497 The predicate "is a lattice plane" for join of atoms. Version of islpln2a 28496 expressed with an abbreviation hypothesis. (Contributed by NM, 30-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  P  =  ( LPlanes `  K )   &    |-  Y  =  ( ( Q  .\/  R )  .\/  S )   =>    |-  (
 ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A )
 )  ->  ( Y  e.  P  <->  ( Q  =/=  R 
 /\  -.  S  .<_  ( Q  .\/  R )
 ) ) )
 
TheoremlplnriaN 28498 Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  P  =  ( LPlanes `  K )   &    |-  Y  =  ( ( Q  .\/  R )  .\/  S )   =>    |-  (
 ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A )  /\  Y  e.  P ) 
 ->  -.  Q  .<_  ( R 
 .\/  S ) )
 
TheoremlplnribN 28499 Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  P  =  ( LPlanes `  K )   &    |-  Y  =  ( ( Q  .\/  R )  .\/  S )   =>    |-  (
 ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A )  /\  Y  e.  P ) 
 ->  -.  R  .<_  ( Q 
 .\/  S ) )
 
Theoremlplnric 28500 Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  P  =  ( LPlanes `  K )   &    |-  Y  =  ( ( Q  .\/  R )  .\/  S )   =>    |-  (
 ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A )  /\  Y  e.  P ) 
 ->  -.  S  .<_  ( Q 
 .\/  R ) )
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