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Theorem List for Metamath Proof Explorer - 29701-29800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremllni2 29701 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 29702 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 29703 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 29704 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 29705 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 29706 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 29707* 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 29708 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 29709 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 29710* 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 29711 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 29712 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 29713 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 29714 Special case of 2atmat0 29715 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 29715 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 29716 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 29717 Variation of projective geometry axiom ps-2 29667. (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 29718* 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 29719* 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 29720* 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 29721 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 29722* 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 29723 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 29724* 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 29725* 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 29726 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 29727* 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 29728 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 29729* 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 29730 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 29731 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 29732 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 29733 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 29734 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 29735 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 29736 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 29737 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 29738 The predicate "is a lattice plane" for join of atoms. Version of islpln2a 29737 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 29739 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 29740 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 29741 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 29742 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 29743 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 29744 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 29745 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 29746 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 29747 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 29748 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 29749 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 29750 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 29751 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 29752* 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 29753* 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 29754 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 29755 The join of two different lattice lines in a lattice plane equals the plane (version of 2llnjN 29756 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 29756 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 29757 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 29758 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 29759 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 29760 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 29761* 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 29762* 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 29763* 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 29764 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 29765* 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 29766 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 29767 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 29768* 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 29769* 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 29770 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 )
 
Theoremlvolnle3at 29771 A lattice plane (or lattice line or atom) cannot majorize a lattice volume. (Contributed by NM, 8-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  V  =  ( LVols `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  V )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  -.  X  .<_  ( ( P  .\/  Q )  .\/  R ) )
 
Theoremlvolnleat 29772 An atom cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  V  =  (
 LVols `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  V  /\  P  e.  A )  ->  -.  X  .<_  P )
 
Theoremlvolnlelln 29773 A lattice line cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |-  N  =  ( LLines `  K )   &    |-  V  =  (
 LVols `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  ->  -.  X  .<_  Y )
 
Theoremlvolnlelpln 29774 A lattice plane cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |-  P  =  ( LPlanes `  K )   &    |-  V  =  (
 LVols `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  ->  -.  X  .<_  Y )
 
Theorem3atnelvolN 29775 The join of 3 atoms is not a lattice volume. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  V  =  ( LVols `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
 )  ->  -.  (
 ( P  .\/  Q )  .\/  R )  e.  V )
 
Theorem2atnelvolN 29776 The join of two atoms is not a lattice volume. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  V  =  ( LVols `  K )   =>    |-  (
 ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  -.  ( P  .\/  Q )  e.  V )
 
TheoremlvolneatN 29777 No lattice volume is an atom. (Contributed by NM, 15-Jul-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  V  =  ( LVols `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  V ) 
 ->  -.  X  e.  A )
 
Theoremlvolnelln 29778 No lattice volume is a lattice line. (Contributed by NM, 15-Jul-2012.)
 |-  N  =  ( LLines `  K )   &    |-  V  =  ( LVols `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  V ) 
 ->  -.  X  e.  N )
 
Theoremlvolnelpln 29779 No lattice volume is a lattice plane. (Contributed by NM, 19-Jun-2012.)
 |-  P  =  ( LPlanes `  K )   &    |-  V  =  ( LVols `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  V ) 
 ->  -.  X  e.  P )
 
Theoremlvoln0N 29780 A lattice volume is non-zero. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)
 |-  .0.  =  ( 0. `  K )   &    |-  V  =  ( LVols `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  V )  ->  X  =/=  .0.  )
 
Theoremislvol2aN 29781 The predicate "is a lattice volume". (Contributed by NM, 16-Jul-2012.) (New usage is discouraged.)
 |-  .<_  =  ( 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 )  .\/  S )  e.  V  <->  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R )
 ) ) )
 
Theorem4atlem0a 29782 Lemma for 4at 29802. (Contributed by NM, 10-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A ) 
 /\  ( R  e.  A  /\  S  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R )
 ) )  ->  -.  R  .<_  ( ( P  .\/  Q )  .\/  S )
 )
 
Theorem4atlem0ae 29783 Lemma for 4at 29802. (Contributed by NM, 10-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  .<_  ( P 
 .\/  R ) )
 
Theorem4atlem0be 29784 Lemma for 4at 29802. (Contributed by NM, 10-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  -.  R  .<_  ( P 
 .\/  Q ) )  ->  P  =/=  R )
 
Theorem4atlem3 29785 Lemma for 4at 29802. Break inequality into 4 cases. (Contributed by NM, 8-Jul-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  /\  V  e.  A ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R )
 ) )  ->  (
 ( -.  P  .<_  ( ( T  .\/  U )  .\/  V )  \/ 
 -.  Q  .<_  ( ( T  .\/  U )  .\/  V ) )  \/  ( -.  R  .<_  ( ( T  .\/  U )  .\/  V )  \/ 
 -.  S  .<_  ( ( T  .\/  U )  .\/  V ) ) ) )
 
Theorem4atlem3a 29786 Lemma for 4at 29802. Break inequality into 3 cases. (Contributed by NM, 9-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A ) 
 /\  ( U  e.  A  /\  V  e.  A ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R )
 ) )  ->  ( -.  Q  .<_  ( ( P 
 .\/  U )  .\/  V )  \/  -.  R  .<_  ( ( P  .\/  U )  .\/  V )  \/ 
 -.  S  .<_  ( ( P  .\/  U )  .\/  V ) ) )
 
Theorem4atlem3b 29787 Lemma for 4at 29802. Break inequality into 2 cases. (Contributed by NM, 9-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  V  e.  A )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
 .\/  Q )  /\  -.  S  .<_  ( ( P 
 .\/  Q )  .\/  R ) ) )  ->  ( -.  R  .<_  ( ( P  .\/  Q )  .\/  V )  \/  -.  S  .<_  ( ( P 
 .\/  Q )  .\/  V ) ) )
 
Theorem4atlem4a 29788 Lemma for 4at 29802. Frequently used associative law. (Contributed by NM, 9-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  .\/  S ) )  =  ( P 
 .\/  ( ( Q 
 .\/  R )  .\/  S ) ) )
 
Theorem4atlem4b 29789 Lemma for 4at 29802. Frequently used associative law. (Contributed by NM, 9-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  .\/  S ) )  =  ( Q 
 .\/  ( ( P 
 .\/  R )  .\/  S ) ) )
 
Theorem4atlem4c 29790 Lemma for 4at 29802. Frequently used associative law. (Contributed by NM, 9-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  .\/  S ) )  =  ( R 
 .\/  ( ( P 
 .\/  Q )  .\/  S ) ) )
 
Theorem4atlem4d 29791 Lemma for 4at 29802. Frequently used associative law. (Contributed by NM, 9-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  .\/  S ) )  =  ( S 
 .\/  ( ( P 
 .\/  Q )  .\/  R ) ) )
 
Theorem4atlem9 29792 Lemma for 4at 29802. Substitute  W for 
S. (Contributed by NM, 9-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  ( S  .<_  ( ( P  .\/  Q )  .\/  ( R  .\/  W ) )  <->  ( ( P 
 .\/  Q )  .\/  ( R  .\/  S ) )  =  ( ( P 
 .\/  Q )  .\/  ( R  .\/  W ) ) ) )
 
Theorem4atlem10a 29793 Lemma for 4at 29802. Substitute  V for 
R. (Contributed by NM, 9-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  V  e.  A  /\  W  e.  A )  /\  -.  R  .<_  ( ( P  .\/  Q )  .\/  W ) )  ->  ( R  .<_  ( ( P  .\/  Q )  .\/  ( V  .\/  W ) )  <->  ( ( P 
 .\/  Q )  .\/  ( R  .\/  W ) )  =  ( ( P 
 .\/  Q )  .\/  ( V  .\/  W ) ) ) )
 
Theorem4atlem10b 29794 Lemma for 4at 29802. Substitute  V for 
R (cont.). (Contributed by NM, 10-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  V  e.  A )  /\  ( W  e.  A  /\  -.  R  .<_  ( ( P  .\/  Q )  .\/  W )  /\  -.  S  .<_  ( ( P 
 .\/  Q )  .\/  R ) ) )  /\  ( R  .<_  ( ( P  .\/  Q )  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( P  .\/  Q )  .\/  ( V  .\/  W ) ) ) ) 
 ->  ( ( P  .\/  Q )  .\/  ( R  .\/  S ) )  =  ( ( P  .\/  Q )  .\/  ( V  .\/  W ) ) )
 
Theorem4atlem10 29795 Lemma for 4at 29802. Combine both possible cases. (Contributed by NM, 9-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( ( R  e.  A  /\  S  e.  A )  /\  V  e.  A  /\  W  e.  A ) 
 /\  ( P  =/=  Q 
 /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) ) 
 ->  ( ( R  .\/  S )  .<_  ( ( P 
 .\/  Q )  .\/  ( V  .\/  W ) ) 
 ->  ( ( P  .\/  Q )  .\/  ( R  .\/  S ) )  =  ( ( P  .\/  Q )  .\/  ( V  .\/  W ) ) ) )
 
Theorem4atlem11a 29796 Lemma for 4at 29802. Substitute  U for 
Q. (Contributed by NM, 9-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A )  /\  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) )  ->  ( Q  .<_  ( ( P  .\/  U )  .\/  ( V  .\/  W ) )  <->  ( ( P 
 .\/  Q )  .\/  ( V  .\/  W ) )  =  ( ( P 
 .\/  U )  .\/  ( V  .\/  W ) ) ) )
 
Theorem4atlem11b 29797 Lemma for 4at 29802. Substitute  U for 
Q (cont.). (Contributed by NM, 10-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A ) 
 /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A ) )  /\  ( ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R )
 )  /\  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) ) 
 /\  ( Q  .<_  ( ( P  .\/  U )  .\/  ( V  .\/  W ) )  /\  R  .<_  ( ( P  .\/  U )  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( P 
 .\/  U )  .\/  ( V  .\/  W ) ) ) )  ->  (
 ( P  .\/  Q )  .\/  ( R  .\/  S ) )  =  ( ( P  .\/  U )  .\/  ( V  .\/  W ) ) )
 
Theorem4atlem11 29798 Lemma for 4at 29802. Combine all three possible cases. (Contributed by NM, 10-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A ) 
 /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R )
 ) )  ->  (
 ( Q  .\/  ( R  .\/  S ) ) 
 .<_  ( ( P  .\/  U )  .\/  ( V  .\/  W ) )  ->  ( ( P  .\/  Q )  .\/  ( R  .\/  S ) )  =  ( ( P  .\/  U )  .\/  ( V  .\/  W ) ) ) )
 
Theorem4atlem12a 29799 Lemma for 4at 29802. Substitute  T for 
P. (Contributed by NM, 9-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A )  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )  ->  ( P  .<_  ( ( T  .\/  U )  .\/  ( V  .\/  W ) )  <->  ( ( P 
 .\/  U )  .\/  ( V  .\/  W ) )  =  ( ( T 
 .\/  U )  .\/  ( V  .\/  W ) ) ) )
 
Theorem4atlem12b 29800 Lemma for 4at 29802. Substitute  T for 
P (cont.). (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  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A )
 )  /\  ( ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R )
 )  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) ) 
 /\  ( ( P 
 .<_  ( ( T  .\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T 
 .\/  U )  .\/  ( V  .\/  W ) ) )  /\  ( R 
 .<_  ( ( T  .\/  U )  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
 .\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( ( P  .\/  Q )  .\/  ( R  .\/  S ) )  =  ( ( T  .\/  U )  .\/  ( V  .\/  W ) ) )
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