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Theorem List for Metamath Proof Explorer - 29101-29200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremldillaut 29101 A lattice dilation is an automorphism. (Contributed by NM, 20-May-2012.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( LAut `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  D ) 
 ->  F  e.  I )
 
Theoremldil1o 29102 A lattice dilation is a one-to-one onto function. (Contributed by NM, 19-Apr-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  D ) 
 ->  F : B -1-1-onto-> B )
 
Theoremldilval 29103 Value of a lattice dilation under its co-atom. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  D  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( F `  X )  =  X )
 
Theoremidldil 29104 The identity function is a lattice dilation. (Contributed by NM, 18-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   =>    |-  ( ( K  e.  A  /\  W  e.  H )  ->  (  _I  |`  B )  e.  D )
 
Theoremldilcnv 29105 The converse of a lattice dilation is a lattice dilation. (Contributed by NM, 10-May-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D ) 
 ->  `' F  e.  D )
 
Theoremldilco 29106 The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  D  /\  G  e.  D )  ->  ( F  o.  G )  e.  D )
 
Theoremltrnfset 29107* The set of all lattice translations for a lattice  K. (Contributed by NM, 11-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  C  ->  (
 LTrn `  K )  =  ( w  e.  H  |->  { f  e.  ( (
 LDil `  K ) `  w )  |  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  w  /\  -.  q  .<_  w )  ->  ( ( p  .\/  ( f `  p ) )  ./\  w )  =  ( ( q 
 .\/  ( f `  q ) )  ./\  w ) ) } )
 )
 
Theoremltrnset 29108* The set of lattice translations for a fiducial co-atom  W. (Contributed by NM, 11-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( K  e.  B  /\  W  e.  H )  ->  T  =  { f  e.  D  |  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  W 
 /\  -.  q  .<_  W )  ->  ( ( p  .\/  ( f `  p ) )  ./\  W )  =  ( ( q  .\/  ( f `  q ) )  ./\  W ) ) } )
 
Theoremisltrn 29109* The predicate "is a lattice translation". Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( K  e.  B  /\  W  e.  H )  ->  ( F  e.  T  <->  ( F  e.  D  /\  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  ( ( p  .\/  ( F `  p ) )  ./\  W )  =  ( ( q  .\/  ( F `  q ) )  ./\  W )
 ) ) ) )
 
Theoremisltrn2N 29110* The predicate "is a lattice translation". Version of isltrn 29109 that considers only different  p and  q. TODO: Can this eliminate some separate proofs for the 
p  =  q case? (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( K  e.  B  /\  W  e.  H )  ->  ( F  e.  T  <->  ( F  e.  D  /\  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  p  =/=  q )  ->  (
 ( p  .\/  ( F `  p ) ) 
 ./\  W )  =  ( ( q  .\/  ( F `  q ) ) 
 ./\  W ) ) ) ) )
 
Theoremltrnu 29111 Uniqueness property of a lattice translation value for atoms not under the fiducial co-atom  W. Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 20-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( ( K  e.  V  /\  W  e.  H ) 
 /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 ->  ( ( P  .\/  ( F `  P ) )  ./\  W )  =  ( ( Q  .\/  ( F `  Q ) )  ./\  W )
 )
 
Theoremltrnldil 29112 A lattice translation is a lattice dilation. (Contributed by NM, 20-May-2012.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T ) 
 ->  F  e.  D )
 
Theoremltrnlaut 29113 A lattice translation is a lattice automorphism. (Contributed by NM, 20-May-2012.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( LAut `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T ) 
 ->  F  e.  I )
 
Theoremltrn1o 29114 A lattice translation is a one-to-one onto function. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T ) 
 ->  F : B -1-1-onto-> B )
 
Theoremltrncl 29115 Closure of a lattice translation. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T  /\  X  e.  B )  ->  ( F `  X )  e.  B )
 
Theoremltrn11 29116 One-to-one property of a lattice translation. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( ( F `  X )  =  ( F `  Y ) 
 <->  X  =  Y ) )
 
Theoremltrncnvnid 29117 If a translation is different from the identity, so is its converse. (Contributed by NM, 17-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  `' F  =/=  (  _I  |`  B ) )
 
TheoremltrncoidN 29118 Two translations are equal if the composition of one with the converse of the other is the zero translation. This is an analog of vector subtraction. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  ->  ( ( F  o.  `' G )  =  (  _I  |`  B )  <->  F  =  G ) )
 
Theoremltrnle 29119 Less-than or equal property of a lattice translation. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( X  .<_  Y  <->  ( F `  X )  .<_  ( F `
  Y ) ) )
 
TheoremltrncnvleN 29120 Less-than or equal property of lattice translation converse. (Contributed by NM, 10-May-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( X  .<_  Y  <->  ( `' F `  X )  .<_  ( `' F `  Y ) ) )
 
Theoremltrnm 29121 Lattice translation of a meet. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( F `  ( X  ./\  Y ) )  =  ( ( F `  X ) 
 ./\  ( F `  Y ) ) )
 
Theoremltrnj 29122 Lattice translation of a meet. TODO: change antecedent to  K  e.  HL (Contributed by NM, 25-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( F `  ( X  .\/  Y ) )  =  (
 ( F `  X )  .\/  ( F `  Y ) ) )
 
Theoremltrncvr 29123 Covering property of a lattice translation. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( X C Y  <->  ( F `  X ) C ( F `  Y ) ) )
 
Theoremltrnval1 29124 Value of a lattice translation under its co-atom. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( F `  X )  =  X )
 
Theoremltrnid 29125* A lattice translation is the identity function iff all atoms not under the fiducial co-atom  W are equal to their values. (Contributed by NM, 24-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T ) 
 ->  ( A. p  e.  A  ( -.  p  .<_  W  ->  ( F `  p )  =  p )  <->  F  =  (  _I  |`  B ) ) )
 
Theoremltrnnid 29126* If a lattice translation is not the identity, then there is an atom not under the fiducial co-atom 
W and not equal to its translation. (Contributed by NM, 24-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  ( F `
  p )  =/= 
 p ) )
 
Theoremltrnatb 29127 The lattice translation of an atom is an atom. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B )  ->  ( P  e.  A  <->  ( F `  P )  e.  A ) )
 
Theoremltrncnvatb 29128 The converse of the lattice translation of an atom is an atom. (Contributed by NM, 2-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B )  ->  ( P  e.  A  <->  ( `' F `  P )  e.  A ) )
 
Theoremltrnel 29129 The lattice translation of an atom not under the fiducial co-atom is also an atom not under the fiducial co-atom. Remark below Lemma B in [Crawley] p. 112. (Contributed by NM, 22-May-2012.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
 
Theoremltrnat 29130 The lattice translation of an atom is also an atom. TODO: See if this can shorten some ltrnel 29129 uses. (Contributed by NM, 25-May-2012.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A )  ->  ( F `  P )  e.  A )
 
Theoremltrncnvat 29131 The converse of the lattice translation of an atom is an atom. (Contributed by NM, 9-May-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A )  ->  ( `' F `  P )  e.  A )
 
Theoremltrncnvel 29132 The converse of the lattice translation of an atom not under the fiducial co-atom. (Contributed by NM, 10-May-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( `' F `  P )  e.  A  /\  -.  ( `' F `  P ) 
 .<_  W ) )
 
TheoremltrncoelN 29133 Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel 29129 uses. (Contributed by NM, 1-May-2013.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  (
 ( F `  ( G `  P ) )  e.  A  /\  -.  ( F `  ( G `
  P ) ) 
 .<_  W ) )
 
Theoremltrncoat 29134 Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel 29129, ltrnat 29130 uses. (Contributed by NM, 1-May-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T ) 
 /\  P  e.  A )  ->  ( F `  ( G `  P ) )  e.  A )
 
Theoremltrncoval 29135 Two ways to express value of translation composition. (Contributed by NM, 31-May-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T ) 
 /\  P  e.  A )  ->  ( ( F  o.  G ) `  P )  =  ( F `  ( G `  P ) ) )
 
Theoremltrncnv 29136 The converse of a lattice translation is a lattice translation. (Contributed by NM, 10-May-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T ) 
 ->  `' F  e.  T )
 
Theoremltrn11at 29137 Frequently used one-to-one property of lattice translation atoms. (Contributed by NM, 5-May-2013.)
 |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q ) ) 
 ->  ( F `  P )  =/=  ( F `  Q ) )
 
Theoremltrneq2 29138* The equality of two translations is determined by their equality at atoms. (Contributed by NM, 2-Mar-2014.)
 |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  ->  ( A. p  e.  A  ( F `  p )  =  ( G `  p )  <->  F  =  G ) )
 
Theoremltrneq 29139* The equality of two translations is determined by their equality at atoms not under co-atom  W. (Contributed by NM, 20-Jun-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  ->  ( A. p  e.  A  ( -.  p  .<_  W  ->  ( F `  p )  =  ( G `  p ) )  <->  F  =  G ) )
 
Theoremidltrn 29140 The identity function is a lattice translation. Remark below Lemma B in [Crawley] p. 112. (Contributed by NM, 18-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
 
Theoremltrnmw 29141 Property of lattice translation value. Remark below Lemma B in [Crawley] p. 112. TODO: Can this be used in more places? (Contributed by NM, 20-May-2012.)
 |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  ./\  W )  =  .0.  )
 
TheoremdilfsetN 29142* The mapping from fiducial atom to set of dilations. (Contributed by NM, 30-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  W  =  ( WAtoms `  K )   &    |-  M  =  ( PAut `  K )   &    |-  L  =  ( Dil `  K )   =>    |-  ( K  e.  B  ->  L  =  ( d  e.  A  |->  { f  e.  M  |  A. x  e.  S  ( x  C_  ( W `  d ) 
 ->  ( f `  x )  =  x ) } ) )
 
TheoremdilsetN 29143* The set of dilations for a fiducial atom  D. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  W  =  ( WAtoms `  K )   &    |-  M  =  ( PAut `  K )   &    |-  L  =  ( Dil `  K )   =>    |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( L `  D )  =  {
 f  e.  M  |  A. x  e.  S  ( x  C_  ( W `
  D )  ->  ( f `  x )  =  x ) } )
 
TheoremisdilN 29144* The predicate "is a dilation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  W  =  ( WAtoms `  K )   &    |-  M  =  ( PAut `  K )   &    |-  L  =  ( Dil `  K )   =>    |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( F  e.  ( L `  D )  <-> 
 ( F  e.  M  /\  A. x  e.  S  ( x  C_  ( W `
  D )  ->  ( F `  x )  =  x ) ) ) )
 
TheoremtrnfsetN 29145* The mapping from fiducial atom to set of translations. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  W  =  ( WAtoms `  K )   &    |-  M  =  ( PAut `  K )   &    |-  L  =  ( Dil `  K )   &    |-  T  =  ( Trn `  K )   =>    |-  ( K  e.  C  ->  T  =  ( d  e.  A  |->  { f  e.  ( L `  d
 )  |  A. q  e.  ( W `  d
 ) A. r  e.  ( W `  d ) ( ( q  .+  (
 f `  q )
 )  i^i  (  ._|_  ` 
 { d } )
 )  =  ( ( r  .+  ( f `
  r ) )  i^i  (  ._|_  `  { d } ) ) }
 ) )
 
TheoremtrnsetN 29146* The set of translations for a fiducial atom  D. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  W  =  ( WAtoms `  K )   &    |-  M  =  ( PAut `  K )   &    |-  L  =  ( Dil `  K )   &    |-  T  =  ( Trn `  K )   =>    |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( T `  D )  =  {
 f  e.  ( L `
  D )  | 
 A. q  e.  ( W `  D ) A. r  e.  ( W `  D ) ( ( q  .+  ( f `
  q ) )  i^i  (  ._|_  `  { D } ) )  =  ( ( r  .+  ( f `  r
 ) )  i^i  (  ._|_  `  { D }
 ) ) } )
 
TheoremistrnN 29147* The predicate "is a translation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  W  =  ( WAtoms `  K )   &    |-  M  =  ( PAut `  K )   &    |-  L  =  ( Dil `  K )   &    |-  T  =  ( Trn `  K )   =>    |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( F  e.  ( T `  D )  <-> 
 ( F  e.  ( L `  D )  /\  A. q  e.  ( W `
  D ) A. r  e.  ( W `  D ) ( ( q  .+  ( F `
  q ) )  i^i  (  ._|_  `  { D } ) )  =  ( ( r  .+  ( F `  r ) )  i^i  (  ._|_  ` 
 { D } )
 ) ) ) )
 
Syntaxctrl 29148 Extend class notation with set of all traces of lattice translations.
 class  trL
 
Definitiondf-trl 29149* Define trace of a lattice translation. (Contributed by NM, 20-May-2012.)
 |-  trL  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( f  e.  ( (
 LTrn `  k ) `  w )  |->  ( iota_ x  e.  ( Base `  k
 ) A. p  e.  ( Atoms `  k ) ( -.  p ( le `  k ) w  ->  x  =  ( ( p ( join `  k
 ) ( f `  p ) ) (
 meet `  k ) w ) ) ) ) ) )
 
Theoremtrlfset 29150* The set of all traces of lattice translations for a lattice  K. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  C  ->  ( trL `  K )  =  ( w  e.  H  |->  ( f  e.  (
 ( LTrn `  K ) `  w )  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  ( f `  p ) )  ./\  w ) ) ) ) ) )
 
Theoremtrlset 29151* The set of traces of lattice translations for a fiducial co-atom  W. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( K  e.  C  /\  W  e.  H )  ->  R  =  ( f  e.  T  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p 
 .\/  ( f `  p ) )  ./\  W ) ) ) ) )
 
Theoremtrlval 29152* The value of the trace of a lattice translation. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T ) 
 ->  ( R `  F )  =  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  ( F `  p ) )  ./\  W )
 ) ) )
 
Theoremtrlval2 29153 The value of the trace of a lattice translation, given any atom  P not under the fiducial co-atom  W. Note: this requires only the weaker assumption  K  e.  Lat; we use  K  e.  HL for convenience. (Contributed by NM, 20-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  =  ( ( P  .\/  ( F `  P ) ) 
 ./\  W ) )
 
Theoremtrlcl 29154 Closure of the trace of a lattice translation. (Contributed by NM, 22-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T ) 
 ->  ( R `  F )  e.  B )
 
Theoremtrlcnv 29155 The trace of the converse of a lattice translation. (Contributed by NM, 10-May-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T ) 
 ->  ( R `  `' F )  =  ( R `  F ) )
 
Theoremtrljat1 29156 The value of a translation of an atom  P not under the fiducial co-atom  W, joined with trace. Equation above Lemma C in [Crawley] p. 112. Todo: shorten with atmod3i1 28854? (Contributed by NM, 22-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  F ) )  =  ( P  .\/  ( F `  P ) ) )
 
Theoremtrljat2 29157 The value of a translation of an atom  P not under the fiducial co-atom  W, joined with trace. Equation above Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  .\/  ( R `  F ) )  =  ( P 
 .\/  ( F `  P ) ) )
 
Theoremtrljat3 29158 The value of a translation of an atom  P not under the fiducial co-atom  W, joined with trace. Equation above Lemma C in [Crawley] p. 112. (Contributed by NM, 22-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  F ) )  =  (
 ( F `  P )  .\/  ( R `  F ) ) )
 
Theoremtrlat 29159 If an atom differs from its translation, the trace is an atom. Equation above Lemma C in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  e.  A )
 
Theoremtrl0 29160 If an atom not under the fiducial co-atom  W equals its lattice translation, the trace of the translation is zero. (Contributed by NM, 24-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( R `
  F )  =  .0.  )
 
Theoremtrlator0 29161 The trace of a lattice translation is an atom or zero. (Contributed by NM, 5-May-2013.)
 |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T ) 
 ->  ( ( R `  F )  e.  A  \/  ( R `  F )  =  .0.  )
 )
 
Theoremtrlatn0 29162 The trace of a lattice translation is an atom iff it is nonzero. (Contributed by NM, 14-Jun-2013.)
 |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T ) 
 ->  ( ( R `  F )  e.  A  <->  ( R `  F )  =/=  .0.  ) )
 
Theoremtrlnidat 29163 The trace of a lattice translation other than the identity is an atom. Remark above Lemma C in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( R `  F )  e.  A )
 
Theoremltrnnidn 29164 If a lattice translation is not the identity, then the translation of any atom not under the fiducial co-atom  W is different from the atom. Remark above Lemma C in [Crawley] p. 112. (Contributed by NM, 24-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  P )  =/=  P )
 
Theoremltrnideq 29165 Property of the identity lattice translation. (Contributed by NM, 27-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F  =  (  _I  |`  B )  <-> 
 ( F `  P )  =  P )
 )
 
Theoremtrlid0 29166 The trace of the identity translation is zero. (Contributed by NM, 11-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( R `  (  _I  |`  B ) )  =  .0.  )
 
Theoremtrlnidatb 29167 A lattice translation is not the identity iff its trace is an atom. TODO: Can proofs be reorganized so this goes with trlnidat 29163? Why do both this and ltrnideq 29165 need trlnidat 29163? (Contributed by NM, 4-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T ) 
 ->  ( F  =/=  (  _I  |`  B )  <->  ( R `  F )  e.  A ) )
 
Theoremtrlid0b 29168 A lattice translation is the identity iff its trace is zero. (Contributed by NM, 14-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T ) 
 ->  ( F  =  (  _I  |`  B )  <->  ( R `  F )  =  .0.  ) )
 
Theoremtrlnid 29169 Different translations with the same trace cannot be the identity. (Contributed by NM, 26-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T ) 
 /\  ( F  =/=  G 
 /\  ( R `  F )  =  ( R `  G ) ) )  ->  F  =/=  (  _I  |`  B )
 )
 
Theoremltrn2ateq 29170 Property of the equality of a lattice translation with its value. (Contributed by NM, 27-May-2012.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( ( F `  P )  =  P  <->  ( F `  Q )  =  Q ) )
 
Theoremltrnateq 29171 If any atom (under  W) is not equal to its translation, so is any other atom. (Contributed by NM, 6-May-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `  P )  =  P )  ->  ( F `  Q )  =  Q )
 
Theoremltrnatneq 29172 If any atom (under  W) is not equal to its translation, so is any other atom. TODO:  -.  P  .<_  W isn't needed to prove this. Will removing it shorten (and not lengthen) proofs using it? (Contributed by NM, 6-May-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `  P )  =/= 
 P )  ->  ( F `  Q )  =/= 
 Q )
 
Theoremltrnatlw 29173 If the value of an atom equals the atom in a non-identity translation, the atom is under the fiducial hyperplane. (Contributed by NM, 15-May-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  Q  e.  A )  /\  ( ( F `
  P )  =/= 
 P  /\  ( F `  Q )  =  Q ) )  ->  Q  .<_  W )
 
Theoremtrlle 29174 The trace of a lattice translation is less than the fiducial co-atom  W.. (Contributed by NM, 25-May-2012.)
 |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T ) 
 ->  ( R `  F )  .<_  W )
 
Theoremtrlne 29175 The trace of a lattice translation is not equal to any atom not under the fiducial co-atom  W. Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  =/=  ( R `  F ) )
 
Theoremtrlnle 29176 The atom not under the fiducial co-atom  W is not less than the trace of a lattice translation. Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 26-May-2012.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  ( R `  F ) )
 
Theoremtrlval3 29177 The value of the trace of a lattice translation in terms of 2 atoms. TODO: Try to shorten proof. (Contributed by NM, 3-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  .\/  ( F `
  P ) )  =/=  ( Q  .\/  ( F `  Q ) ) ) )  ->  ( R `  F )  =  ( ( P 
 .\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `
  Q ) ) ) )
 
Theoremtrlval4 29178 The value of the trace of a lattice translation in terms of 2 atoms. (Contributed by NM, 3-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F ) 
 .<_  ( P  .\/  Q ) ) )  ->  ( R `  F )  =  ( ( P 
 .\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `
  Q ) ) ) )
 
Theoremtrlval5 29179 The value of the trace of a lattice translation in terms of itself. (Contributed by NM, 19-Jul-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  =  ( ( P  .\/  ( R `  F ) ) 
 ./\  W ) )
 
Theoremarglem1N 29180 Lemma for Desargues' law. Theorem 13.3 of [Crawley] p. 110, 3rd and 4th lines from bottom. In these lemmas,  P,  Q,  R,  S,  T,  U,  C,  D,  E,  F, and  G represent Crawley's a0, a1, a2, b0, b1, b2, c, z0, z1, z2, and p respectively. (Contributed by NM, 28-Jun-2012.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  F  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )   &    |-  G  =  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )   =>    |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T ) )  /\  G  e.  A )  ->  F  e.  A )
 
Theoremcdlemc1 29181 Part of proof of Lemma C in [Crawley] p. 112. TODO: shorten with atmod3i1 28854? (Contributed by NM, 29-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( ( P  .\/  X )  ./\  W )
 )  =  ( P 
 .\/  X ) )
 
Theoremcdlemc2 29182 Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( F `  Q )  .<_  ( ( F `  P ) 
 .\/  ( ( P 
 .\/  Q )  ./\  W ) ) )
 
Theoremcdlemc3 29183 Part of proof of Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( ( F `  P )  .<_  ( Q 
 .\/  ( R `  F ) )  ->  Q  .<_  ( P  .\/  ( F `  P ) ) ) )
 
Theoremcdlemc4 29184 Part of proof of Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  ->  ( Q  .\/  ( R `  F ) )  =/=  ( ( F `  P )  .\/  ( ( P  .\/  Q )  ./\ 
 W ) ) )
 
Theoremcdlemc5 29185 Lemma for cdlemc 29187. (Contributed by NM, 26-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( -.  Q  .<_  ( P  .\/  ( F `  P ) )  /\  ( F `
  P )  =/= 
 P ) )  ->  ( F `  Q )  =  ( ( Q 
 .\/  ( R `  F ) )  ./\  ( ( F `  P )  .\/  ( ( P  .\/  Q )  ./\ 
 W ) ) ) )
 
Theoremcdlemc6 29186 Lemma for cdlemc 29187. (Contributed by NM, 26-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `  P )  =  P )  ->  ( F `  Q )  =  ( ( Q  .\/  ( R `  F ) )  ./\  ( ( F `  P )  .\/  ( ( P  .\/  Q )  ./\  W )
 ) ) )
 
Theoremcdlemc 29187 Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  ->  ( F `  Q )  =  ( ( Q  .\/  ( R `  F ) )  ./\  ( ( F `  P )  .\/  ( ( P  .\/  Q )  ./\  W )
 ) ) )
 
Theoremcdlemd1 29188 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  ( R  e.  A  /\  P  =/=  Q  /\  -.  R  .<_  ( P 
 .\/  Q ) ) ) )  ->  R  =  ( ( P  .\/  ( ( P  .\/  R )  ./\  W )
 )  ./\  ( Q  .\/  ( ( Q  .\/  R )  ./\  W )
 ) ) )
 
Theoremcdlemd2 29189 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  ( P  =/=  Q 
 /\  -.  R  .<_  ( P  .\/  Q )
 ) )  /\  (
 ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  R )  =  ( G `  R ) )
 
Theoremcdlemd3 29190 Part of proof of Lemma D in [Crawley] p. 113. The  R  =/=  P requirement is not mentioned in their proof. (Contributed by NM, 29-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  ( P  =/=  Q 
 /\  R  .<_  ( P 
 .\/  Q )  /\  R  =/=  P ) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q ) ) )  ->  -.  R  .<_  ( P  .\/  S ) )
 
Theoremcdlemd4 29191 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 30-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  ( P  =/=  Q 
 /\  R  .<_  ( P 
 .\/  Q )  /\  R  =/=  P ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) ) 
 ->  ( F `  R )  =  ( G `  R ) )
 
Theoremcdlemd5 29192 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 30-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  P  =/=  Q )  /\  ( ( F `
  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  R )  =  ( G `  R ) )
 
Theoremcdlemd6 29193 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 31-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) ) 
 /\  ( F `  P )  =  ( G `  P ) ) 
 ->  ( F `  Q )  =  ( G `  Q ) )
 
Theoremcdlemd7 29194 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) ) )  ->  ( F `  R )  =  ( G `  R ) )
 
Theoremcdlemd8 29195 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `
  P )  =  P ) )  ->  ( F `  R )  =  ( G `  R ) )
 
Theoremcdlemd9 29196 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 2-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `
  P )  =  ( G `  P ) )  ->  ( F `
  R )  =  ( G `  R ) )
 
Theoremcdlemd 29197 If two translations agree at any atom not under the fiducial co-atom  W, then they are equal. Lemma D in [Crawley] p. 113. (Contributed by NM, 2-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  F  e.  T  /\  G  e.  T ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `
  P )  =  ( G `  P ) )  ->  F  =  G )
 
Theoremltrneq3 29198 Two translations agree at any atom not under the fiducial co-atom  W iff they are equal. (Contributed by NM, 25-Jul-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  (
 ( F `  P )  =  ( G `  P )  <->  F  =  G ) )
 
Theoremcdleme00a 29199 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  -.  R  .<_  ( P 
 .\/  Q ) )  ->  R  =/=  P )
 
Theoremcdleme0aa 29200 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  B  =  ( Base `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A ) 
 ->  U  e.  B )
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