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Theorem List for Metamath Proof Explorer - 29601-29700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltrncvr 29601 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 29602 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 29603* 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 29604* 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 29605 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 29606 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 29607 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 29608 The lattice translation of an atom is also an atom. TODO: See if this can shorten some ltrnel 29607 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 29609 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 29610 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 29611 Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel 29607 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 29612 Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel 29607, ltrnat 29608 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 29613 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 29614 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 29615 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 29616* 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 29617* 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 29618 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 29619 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 29620* 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 29621* 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 29622* 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 29623* 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 29624* 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 29625* 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 29626 Extend class notation with set of all traces of lattice translations.
 class  trL
 
Definitiondf-trl 29627* 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 29628* 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 29629* 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 29630* 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 29631 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 29632 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 29633 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 29634 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 29332? (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 29635 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 29636 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 29637 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 29638 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 29639 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 29640 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 29641 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 29642 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 29643 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 29644 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 29645 A lattice translation is not the identity iff its trace is an atom. TODO: Can proofs be reorganized so this goes with trlnidat 29641? Why do both this and ltrnideq 29643 need trlnidat 29641? (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 29646 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 29647 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 29648 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 29649 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 29650 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 29651 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 29652 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 29653 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 29654 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 29655 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 29656 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 29657 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 29658 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 29659 Part of proof of Lemma C in [Crawley] p. 112. TODO: shorten with atmod3i1 29332? (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 29660 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 29661 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 29662 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 29663 Lemma for cdlemc 29665. (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 29664 Lemma for cdlemc 29665. (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 29665 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 29666 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 29667 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 29668 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 29669 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 29670 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 29671 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 29672 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 29673 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 29674 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 29675 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 29676 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 29677 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 29678 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 )
 
Theoremcdleme0a 29679 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 12-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A )
 
Theoremcdleme0b 29680 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  U  =/=  P )
 
Theoremcdleme0c 29681 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 12-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  U  =/=  R )
 
Theoremcdleme0cp 29682 Part of proof of Lemma E in [Crawley] p. 113. TODO: Reformat as in cdlemg3a 30065- swap consequent equality; make antecedent use df-3an 936. (Contributed by NM, 13-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
 )  ->  ( P  .\/  U )  =  ( P  .\/  Q )
 )
 
Theoremcdleme0cq 29683 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( Q  .\/  U )  =  ( P  .\/  Q ) )
 
Theoremcdleme0dN 29684 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  V  =  ( ( P  .\/  R )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  P  =/=  R ) )  ->  V  e.  A )
 
Theoremcdleme0e 29685 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  V  =  ( ( P  .\/  R )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  U  =/=  V )
 
Theoremcdleme0fN 29686 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  V  =  ( ( P  .\/  R )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  R  e.  A ) )  ->  V  =/=  P )
 
Theoremcdleme0gN 29687 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  V  =  ( ( P  .\/  R )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  R  e.  A )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  V  =/=  Q )
 
Theoremcdlemeulpq 29688 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 5-Dec-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) )  ->  U  .<_  ( P  .\/  Q )
 )
 
Theoremcdleme01N 29689 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  ->  ( ( U  =/=  P  /\  U  =/=  Q  /\  U  .<_  ( P  .\/  Q )
 )  /\  U  .<_  W ) )
 
Theoremcdleme02N 29690 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  ->  ( ( P  .\/  U )  =  ( Q  .\/  U )  /\  U  .<_  W ) )
 
Theoremcdleme0ex1N 29691* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q ) 
 ->  E. u  e.  A  ( u  .<_  ( P 
 .\/  Q )  /\  u  .<_  W ) )
 
Theoremcdleme0ex2N 29692* Part of proof of Lemma E in [Crawley] p. 113. Note that  ( P  .\/  u )  =  ( Q  .\/  u ) is a shorter way to express  u  =/=  P  /\  u  =/=  Q  /\  u  .<_  ( P 
.\/  Q ). (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  ->  E. u  e.  A  ( ( P 
 .\/  u )  =  ( Q  .\/  u )  /\  u  .<_  W ) )
 
Theoremcdleme0moN 29693* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q )  /\  E* r ( r  e.  A  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( R  =  P  \/  R  =  Q ) )
 
Theoremcdleme1b 29694 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma showing  F is a lattice element.  F represents their f(r). (Contributed by NM, 6-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
 ) )   &    |-  B  =  (
 Base `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
 )  ->  F  e.  B )
 
Theoremcdleme1 29695 Part of proof of Lemma E in [Crawley] p. 113.  F represents their f(r). Here we show r  \/ f(r) = r  \/ u (7th through 5th lines from bottom on p. 113). (Contributed by NM, 4-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
 ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  F )  =  ( R  .\/  U ) )
 
Theoremcdleme2 29696 Part of proof of Lemma E in [Crawley] p. 113. .  F represents f(r).  W is the fiducial co-atom (hyperplane) w. Here we show that (r  \/ f(r))  /\ w = u in their notation (4th line from bottom on p. 113). (Contributed by NM, 5-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
 ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  (
 ( R  .\/  F )  ./\  W )  =  U )
 
Theoremcdleme3b 29697 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 29704 and cdleme3 29705. (Contributed by NM, 6-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
 ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  P  =/=  Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  F  =/=  R )
 
Theoremcdleme3c 29698 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 29704 and cdleme3 29705. (Contributed by NM, 6-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
 ) )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  P  =/=  Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  F  =/=  .0.  )
 
Theoremcdleme3d 29699 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 29704 and cdleme3 29705. (Contributed by NM, 6-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
 ) )   &    |-  V  =  ( ( P  .\/  R )  ./\  W )   =>    |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  V ) )
 
Theoremcdleme3e 29700 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 29704 and cdleme3 29705. (Contributed by NM, 6-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
 ) )   &    |-  V  =  ( ( P  .\/  R )  ./\  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  ( P  .\/  Q )
 ) ) )  ->  V  e.  A )
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