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Theorem List for Metamath Proof Explorer - 28801-28900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremps-2b 28801 Variation of projective geometry axiom ps-2 28797. (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 28802 Lemma for 3at 28809. (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 28803 Lemma for 3at 28809. (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 28804 Lemma for 3at 28809. (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 28805 Lemma for 3at 28809. (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 28806 Lemma for 3at 28809. (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 28807 Lemma for 3at 28809. (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 28808 Lemma for 3at 28809. (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 28809 Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analog of ps-1 28796 for lines and 4at 28932 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 )
 ) )
 
16.24.13  Projective geometries based on Hilbert lattices
 
Syntaxclln 28810 Extend class notation with set of all "lattice lines" (lattice elements which cover an atom) in a Hilbert lattice.
 class  LLines
 
Syntaxclpl 28811 Extend class notation with set of all "lattice planes" (lattice elements which cover a line) in a Hilbert lattice.
 class  LPlanes
 
Syntaxclvol 28812 Extend class notation with set of all 3-dimensional "lattice volumes" (lattice elements which cover a plane) in a Hilbert lattice.
 class  LVols
 
Syntaxclines 28813 Extend class notation with set of all projective lines for a Hilbert lattice.
 class  Lines
 
SyntaxcpointsN 28814 Extend class notation with set of all projective points.
 class  Points
 
Syntaxcpsubsp 28815 Extend class notation with set of all projective subspaces.
 class  PSubSp
 
Syntaxcpmap 28816 Extend class notation with projective map.
 class  pmap
 
Definitiondf-llines 28817* 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 28818* 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 28819* 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 28820* 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 28821* 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 28822* 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 28823* 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 28824* 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 28825* 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 28826* 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 28827 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 28828 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 28829* 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 28830* 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 28831 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 28832 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 28833 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 28834 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 28835 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 28836 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 28837* 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 28838 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 28839 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 28840* 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 28841 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 28842 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 28843 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 28844 Special case of 2atmat0 28845 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 28845 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 28846 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 28847 Variation of projective geometry axiom ps-2 28797. (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 28848* 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 28849* 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 28850* 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 28851 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 28852* 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 28853 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 28854* 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 28855* 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 28856 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 28857* 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 28858 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 28859* 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 28860 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 28861 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 28862 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 28863 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 28864 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 28865 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 28866 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 28867 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 28868 The predicate "is a lattice plane" for join of atoms. Version of islpln2a 28867 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 28869 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 28870 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 28871 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 ) )
 
Theoremlplnri1 28872 Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.)
 |-  .\/  =  ( 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 )
 
Theoremlplnri2N 28873 Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
 |-  .\/  =  ( 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  =/=  S )
 
Theoremlplnri3N 28874 Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
 |-  .\/  =  ( 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  =/=  S )
 
TheoremlplnllnneN 28875 Two lattice lines defined by atoms defining a lattice plane are not equal. (Contributed by NM, 9-Oct-2012.) (New usage is discouraged.)
 |-  .\/  =  ( 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  .\/  S )  =/=  ( R  .\/  S ) )
 
Theoremllncvrlpln2 28876 A lattice line under a lattice plane is covered by it. (Contributed by NM, 24-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |-  C  =  (  <o  `  K )   &    |-  N  =  (
 LLines `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  P ) 
 /\  X  .<_  Y ) 
 ->  X C Y )
 
Theoremllncvrlpln 28877 An element covering a lattice line is a lattice plane and vice-versa. (Contributed by NM, 26-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  N  =  ( LLines `  K )   &    |-  P  =  (
 LPlanes `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y ) 
 ->  ( X  e.  N  <->  Y  e.  P ) )
 
Theorem2lplnmN 28878 If the join of two lattice planes covers one of them, their meet is a lattice line. (Contributed by NM, 30-Jun-2012.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  C  =  (  <o  `  K )   &    |-  N  =  ( LLines `  K )   &    |-  P  =  (
 LPlanes `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
 .\/  Y ) )  ->  ( X  ./\  Y )  e.  N )
 
Theorem2llnmj 28879 The meet of two lattice lines is an atom iff their join is a lattice plane. (Contributed by NM, 27-Jun-2012.)
 |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  ( LLines `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  ->  ( ( X  ./\  Y )  e.  A  <->  ( X  .\/  Y )  e.  P ) )
 
Theorem2atmat 28880 The meet of two intersecting lines (expressed as joins of atoms) is an atom. (Contributed by NM, 21-Nov-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  /\  P  =/=  Q ) 
 /\  ( R  =/=  S 
 /\  -.  R  .<_  ( P  .\/  Q )  /\  S  .<_  ( ( P 
 .\/  Q )  .\/  R ) ) )  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  A )
 
Theoremlplncmp 28881 If two lattice planes are comparable, they are equal. (Contributed by NM, 24-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P ) 
 ->  ( X  .<_  Y  <->  X  =  Y ) )
 
TheoremlplnexatN 28882* Given a lattice line on a lattice plane, there is an atom whose join with the line equals the plane. (Contributed by NM, 29-Jun-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  N  =  ( LLines `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N ) 
 /\  Y  .<_  X ) 
 ->  E. q  e.  A  ( -.  q  .<_  Y  /\  X  =  ( Y  .\/  q ) ) )
 
TheoremlplnexllnN 28883* Given an atom on a lattice plane, there is a lattice line whose join with the atom equals the plane. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  N  =  ( LLines `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A ) 
 /\  Q  .<_  X ) 
 ->  E. y  e.  N  ( -.  Q  .<_  y  /\  X  =  ( y  .\/  Q ) ) )
 
Theoremlplnnlt 28884 Two lattice planes cannot satisfy the less than relation. (Contributed by NM, 7-Jul-2012.)
 |-  .<  =  ( lt `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P ) 
 ->  -.  X  .<  Y )
 
Theorem2llnjaN 28885 The join of two different lattice lines in a lattice plane equals the plane (version of 2llnjN 28886 in terms of atoms). (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  N  =  ( LLines `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T ) ) 
 /\  ( ( Q 
 .\/  R )  .<_  W  /\  ( S  .\/  T ) 
 .<_  W  /\  ( Q 
 .\/  R )  =/=  ( S  .\/  T ) ) )  ->  ( ( Q  .\/  R )  .\/  ( S  .\/  T ) )  =  W )
 
Theorem2llnjN 28886 The join of two different lattice lines in a lattice plane equals the plane. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  N  =  (
 LLines `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  (
 ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  ->  ( X  .\/  Y )  =  W )
 
Theorem2llnm2N 28887 The meet of two different lattice lines in a lattice plane is an atom. (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  (
 LLines `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  (
 ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  ->  ( X  ./\  Y )  e.  A )
 
Theorem2llnm3N 28888 Two lattice lines in a lattice plane always meet. (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  N  =  (
 LLines `  K )   &    |-  P  =  ( LPlanes `  K )   =>    |-  (
 ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P )  /\  ( X  .<_  W  /\  Y  .<_  W ) ) 
 ->  ( X  ./\  Y )  =/=  .0.  )
 
Theorem2llnm4 28889 Two lattice lines that majorize the same atom always meet. (Contributed by NM, 20-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  N  =  ( LLines `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N )  /\  ( P  .<_  X  /\  P  .<_  Y ) ) 
 ->  ( X  ./\  Y )  =/=  .0.  )
 
Theorem2llnmeqat 28890 An atom equals the intersection of two majorizing lines. (Contributed by NM, 3-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  (
 LLines `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A )  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y ) ) )  ->  P  =  ( X  ./\ 
 Y ) )
 
Theoremlvolset 28891* The set of 3-dim lattice volumes in a Hilbert lattice. (Contributed by NM, 1-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  P  =  ( LPlanes `  K )   &    |-  V  =  (
 LVols `  K )   =>    |-  ( K  e.  A  ->  V  =  { x  e.  B  |  E. y  e.  P  y C x } )
 
Theoremislvol 28892* The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  P  =  ( LPlanes `  K )   &    |-  V  =  (
 LVols `  K )   =>    |-  ( K  e.  A  ->  ( X  e.  V 
 <->  ( X  e.  B  /\  E. y  e.  P  y C X ) ) )
 
Theoremislvol4 28893* The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  P  =  ( LPlanes `  K )   &    |-  V  =  (
 LVols `  K )   =>    |-  ( ( K  e.  A  /\  X  e.  B )  ->  ( X  e.  V  <->  E. y  e.  P  y C X ) )
 
Theoremlvoli 28894 Condition implying a 3-dim lattice volume. (Contributed by NM, 1-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  P  =  ( LPlanes `  K )   &    |-  V  =  (
 LVols `  K )   =>    |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  P )  /\  X C Y ) 
 ->  Y  e.  V )
 
Theoremislvol3 28895* The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  P  =  ( LPlanes `  K )   &    |-  V  =  ( LVols `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B ) 
 ->  ( X  e.  V  <->  E. y  e.  P  E. p  e.  A  ( -.  p  .<_  y  /\  X  =  ( y  .\/  p ) ) ) )
 
Theoremlvoli3 28896 Condition implying a 3-dim lattice volume. (Contributed by NM, 2-Aug-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  P  =  ( LPlanes `  K )   &    |-  V  =  ( LVols `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A ) 
 /\  -.  Q  .<_  X )  ->  ( X  .\/  Q )  e.  V )
 
Theoremlvolbase 28897 A 3-dim lattice volume is a lattice element. (Contributed by NM, 1-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  V  =  ( LVols `  K )   =>    |-  ( X  e.  V  ->  X  e.  B )
 
Theoremislvol5 28898* The predicate "is a 3-dim lattice volume" in terms of atoms. (Contributed by NM, 1-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  V  =  ( LVols `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B ) 
 ->  ( X  e.  V  <->  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 ) )  /\  X  =  ( ( ( p 
 .\/  q )  .\/  r )  .\/  s ) ) ) )
 
Theoremislvol2 28899* The predicate "is a 3-dim lattice volume" in terms of atoms. (Contributed by NM, 1-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  V  =  ( LVols `  K )   =>    |-  ( K  e.  HL  ->  ( X  e.  V  <->  ( X  e.  B  /\  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 ) )  /\  X  =  ( ( ( p 
 .\/  q )  .\/  r )  .\/  s ) ) ) ) )
 
Theoremlvoli2 28900 The join of 4 different atoms is a lattice volume. (Contributed by NM, 8-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  V  =  ( LVols `  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 ) ) ) 
 ->  ( ( ( P 
 .\/  Q )  .\/  R )  .\/  S )  e.  V )
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