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Theorem List for Metamath Proof Explorer - 31301-31400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcdlemftr2 31301* Special case of cdlemf 31298 showing existence of non-identity translation with trace different from any 2 given lattice elements. (Contributed by NM, 25-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. f  e.  T  ( f  =/=  (  _I  |`  B ) 
 /\  ( R `  f )  =/=  X  /\  ( R `  f )  =/=  Y ) )
 
Theoremcdlemftr1 31302* Part of proof of Lemma G of [Crawley] p. 116, sixth line of third paragraph on p. 117: there is "a translation h, different from the identity, such that tr h  =/= tr f." (Contributed by NM, 25-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. f  e.  T  ( f  =/=  (  _I  |`  B ) 
 /\  ( R `  f )  =/=  X ) )
 
Theoremcdlemftr0 31303* Special case of cdlemf 31298 showing existence of a non-identity translation. (Contributed by NM, 1-Aug-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. f  e.  T  f  =/=  (  _I  |`  B ) )
 
Theoremtrlord 31304* The ordering of two Hilbert lattice elements (under the fiducial hyperplane  W) is determined by the translations whose traces are under them. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) ) 
 ->  ( X  .<_  Y  <->  A. f  e.  T  ( ( R `  f )  .<_  X  ->  ( R `  f ) 
 .<_  Y ) ) )
 
Theoremcdlemg1a 31305* Shorter expression for  G. TODO: fix comment. TODO: shorten using cdleme 31295 or vice-versa? Also, if not shortened with cdleme 31295, then it can be moved up to save repeating hypotheses. (Contributed by NM, 15-Apr-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  G  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( x  ./\ 
 W ) )  =  x )  ->  z  =  ( if ( s 
 .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  E ) ) , 
 [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) ) ,  x ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  G  =  ( iota_ f  e.  T ( f `  P )  =  Q )
 )
 
Theoremcdlemg1b2 31306* This theorem can be used to shorten 
G  = hypothesis. TODO: Fix comment. (Contributed by NM, 18-Apr-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  G  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( x  ./\ 
 W ) )  =  x )  ->  z  =  ( if ( s 
 .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  E ) ) , 
 [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) ) ,  x ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  F  =  (
 iota_ f  e.  T ( f `  P )  =  Q )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 ->  F  =  G )
 
Theoremcdlemg1idlemN 31307* Lemma for cdlemg1idN 31312. (Contributed by NM, 18-Apr-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  G  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( x  ./\ 
 W ) )  =  x )  ->  z  =  ( if ( s 
 .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  E ) ) , 
 [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) ) ,  x ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  F  =  (
 iota_ f  e.  T ( f `  P )  =  Q )   =>    |-  (
 ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  /\  P  =  Q )  ->  ( F `  X )  =  X )
 
Theoremcdlemg1fvawlemN 31308* Lemma for ltrniotafvawN 31313. (Contributed by NM, 18-Apr-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  G  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( x  ./\ 
 W ) )  =  x )  ->  z  =  ( if ( s 
 .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  E ) ) , 
 [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) ) ,  x ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  F  =  (
 iota_ f  e.  T ( f `  P )  =  Q )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 ->  ( ( F `  R )  e.  A  /\  -.  ( F `  R )  .<_  W ) )
 
Theoremcdlemg1ltrnlem 31309* Lemma for ltrniotacl 31314. (Contributed by NM, 18-Apr-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  G  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( x  ./\ 
 W ) )  =  x )  ->  z  =  ( if ( s 
 .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  E ) ) , 
 [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) ) ,  x ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  F  =  (
 iota_ f  e.  T ( f `  P )  =  Q )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 ->  F  e.  T )
 
Theoremcdlemg1finvtrlemN 31310* Lemma for ltrniotacnvN 31315. (Contributed by NM, 18-Apr-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  G  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( x  ./\ 
 W ) )  =  x )  ->  z  =  ( if ( s 
 .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  E ) ) , 
 [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) ) ,  x ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  F  =  (
 iota_ f  e.  T ( f `  P )  =  Q )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 ->  `' F  e.  T )
 
Theoremcdlemg1bOLDN 31311* This theorem can be used to shorten 
F  = hypothesis that have the form of the conclusion. TODO: fix comment. (Contributed by NM, 16-Apr-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  F  =  (
 iota_ f  e.  T ( f `  P )  =  Q )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( x  ./\ 
 W ) )  =  x )  ->  z  =  ( if ( s 
 .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  E ) ) , 
 [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) ) ,  x ) ) )
 
Theoremcdlemg1idN 31312* Version of cdleme31id 31129 with simpler hypotheses. TODO: Fix comment. (Contributed by NM, 18-Apr-2013.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  F  =  ( iota_ f  e.  T ( f `  P )  =  Q )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  B  =  ( Base `  K )   =>    |-  (
 ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  /\  P  =  Q )  ->  ( F `  X )  =  X )
 
TheoremltrniotafvawN 31313* Version of cdleme46fvaw 31236 with simpler hypotheses. TODO: Fix comment. (Contributed by NM, 18-Apr-2013.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  F  =  ( iota_ f  e.  T ( f `  P )  =  Q )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 ->  ( ( F `  R )  e.  A  /\  -.  ( F `  R )  .<_  W ) )
 
Theoremltrniotacl 31314* Version of cdleme50ltrn 31292 with simpler hypotheses. TODO: Fix comment. (Contributed by NM, 17-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  F  =  ( iota_ f  e.  T ( f `  P )  =  Q )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 ->  F  e.  T )
 
TheoremltrniotacnvN 31315* Version of cdleme51finvtrN 31293 with simpler hypotheses. TODO: Fix comment. (Contributed by NM, 18-Apr-2013.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  F  =  ( iota_ f  e.  T ( f `  P )  =  Q )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 ->  `' F  e.  T )
 
Theoremltrniotaval 31316* Value of the unique translation specified by a value. (Contributed by NM, 21-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  F  =  ( iota_ f  e.  T ( f `  P )  =  Q )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 ->  ( F `  P )  =  Q )
 
Theoremltrniotacnvval 31317* Converse value of the unique translation specified by a value. (Contributed by NM, 21-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  F  =  ( iota_ f  e.  T ( f `  P )  =  Q )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 ->  ( `' F `  Q )  =  P )
 
TheoremltrniotaidvalN 31318* Value of the unique translation specified by identity value. (Contributed by NM, 25-Aug-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  F  =  ( iota_ f  e.  T ( f `  P )  =  P )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  =  (  _I  |`  B ) )
 
TheoremltrniotavalbN 31319* Value of the unique translation specified by a value. (Contributed by NM, 10-Mar-2014.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  F  e.  T )  ->  ( ( F `
  P )  =  Q  <->  F  =  ( iota_
 f  e.  T ( f `  P )  =  Q ) ) )
 
Theoremcdlemeiota 31320* A translation is uniquely determined by one of its values. (Contributed by NM, 18-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  F  e.  T )  ->  F  =  (
 iota_ f  e.  T ( f `  P )  =  ( F `  P ) ) )
 
Theoremcdlemg1ci2 31321* Any function of the form of the function constructed for cdleme 31295 is a translation. TODO: Fix comment. (Contributed by NM, 18-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  F  =  (
 iota_ f  e.  T ( f `  P )  =  Q )
 )  ->  F  e.  T )
 
Theoremcdlemg1cN 31322* Any translation belongs to the set of functions constructed for cdleme 31295. TODO: Fix comment. (Contributed by NM, 18-Apr-2013.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F `  P )  =  Q )  ->  ( F  e.  T 
 <->  F  =  ( iota_ f  e.  T ( f `
  P )  =  Q ) ) )
 
Theoremcdlemg1cex 31323* Any translation is one of our  F s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf 31298? (Contributed by NM, 17-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( F  e.  T  <->  E. p  e.  A  E. q  e.  A  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  ( iota_ f  e.  T ( f `  p )  =  q )
 ) ) )
 
Theoremcdlemg2cN 31324* Any translation belongs to the set of functions constructed for cdleme 31295. TODO: Fix comment. (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  G  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( x  ./\ 
 W ) )  =  x )  ->  z  =  ( if ( s 
 .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  E ) ) , 
 [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) ) ,  x ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `  P )  =  Q )  ->  ( F  e.  T  <->  F  =  G ) )
 
Theoremcdlemg2dN 31325* This theorem can be used to shorten 
G  = hypothesis. TODO: Fix comment. (Contributed by NM, 21-Apr-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  G  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( x  ./\ 
 W ) )  =  x )  ->  z  =  ( if ( s 
 .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  E ) ) , 
 [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) ) ,  x ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  ( F `  P )  =  Q ) ) 
 ->  F  =  G )
 
Theoremcdlemg2cex 31326* Any translation is one of our  F s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf 31298? (Contributed by NM, 22-Apr-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( p  .\/  q )  ./\  W )   &    |-  D  =  ( (
 t  .\/  U )  ./\  ( q  .\/  (
 ( p  .\/  t
 )  ./\  W ) ) )   &    |-  E  =  ( ( p  .\/  q
 )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  G  =  ( x  e.  B  |->  if ( ( p  =/=  q  /\  -.  x  .<_  W ) ,  ( iota_
 z  e.  B A. s  e.  A  (
 ( -.  s  .<_  W 
 /\  ( s  .\/  ( x  ./\  W ) )  =  x ) 
 ->  z  =  ( if ( s  .<_  ( p 
 .\/  q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( p  .\/  q
 ) )  ->  y  =  E ) ) , 
 [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) ) ,  x ) )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( F  e.  T 
 <-> 
 E. p  e.  A  E. q  e.  A  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G ) ) )
 
Theoremcdlemg2ce 31327* Utility theorem to eliminate p,q when converting theorems with explicit f. TODO: fix comment. (Contributed by NM, 22-Apr-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( p  .\/  q )  ./\  W )   &    |-  D  =  ( (
 t  .\/  U )  ./\  ( q  .\/  (
 ( p  .\/  t
 )  ./\  W ) ) )   &    |-  E  =  ( ( p  .\/  q
 )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  G  =  ( x  e.  B  |->  if ( ( p  =/=  q  /\  -.  x  .<_  W ) ,  ( iota_
 z  e.  B A. s  e.  A  (
 ( -.  s  .<_  W 
 /\  ( s  .\/  ( x  ./\  W ) )  =  x ) 
 ->  z  =  ( if ( s  .<_  ( p 
 .\/  q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( p  .\/  q
 ) )  ->  y  =  E ) ) , 
 [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) ) ,  x ) )   &    |-  ( F  =  G  ->  ( ps  <->  ch ) )   &    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  -.  p  .<_  W )  /\  ( q  e.  A  /\  -.  q  .<_  W ) )  /\  ph )  ->  ch )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ph )  ->  ps )
 
Theoremcdlemg2jlemOLDN 31328* Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT . f preserves join: f(r  \/ s) = f(r)  \/ s, p. 115 10th line from bottom. TODO: Combine with cdlemg2jOLDN 31333? (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( p  .\/  q )  ./\  W )   &    |-  D  =  ( (
 t  .\/  U )  ./\  ( q  .\/  (
 ( p  .\/  t
 )  ./\  W ) ) )   &    |-  E  =  ( ( p  .\/  q
 )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  G  =  ( x  e.  B  |->  if ( ( p  =/=  q  /\  -.  x  .<_  W ) ,  ( iota_
 z  e.  B A. s  e.  A  (
 ( -.  s  .<_  W 
 /\  ( s  .\/  ( x  ./\  W ) )  =  x ) 
 ->  z  =  ( if ( s  .<_  ( p 
 .\/  q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( p  .\/  q
 ) )  ->  y  =  E ) ) , 
 [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) ) ,  x ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  ->  ( F `  ( P  .\/  Q ) )  =  ( ( F `  P )  .\/  ( F `  Q ) ) )
 
Theoremcdlemg2fvlem 31329* Lemma for cdlemg2fv 31334. (Contributed by NM, 23-Apr-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( p  .\/  q )  ./\  W )   &    |-  D  =  ( (
 t  .\/  U )  ./\  ( q  .\/  (
 ( p  .\/  t
 )  ./\  W ) ) )   &    |-  E  =  ( ( p  .\/  q
 )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  G  =  ( x  e.  B  |->  if ( ( p  =/=  q  /\  -.  x  .<_  W ) ,  ( iota_
 z  e.  B A. s  e.  A  (
 ( -.  s  .<_  W 
 /\  ( s  .\/  ( x  ./\  W ) )  =  x ) 
 ->  z  =  ( if ( s  .<_  ( p 
 .\/  q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( p  .\/  q
 ) )  ->  y  =  E ) ) , 
 [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) ) ,  x ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  ( P  .\/  ( X 
 ./\  W ) )  =  X ) )  ->  ( F `  X )  =  ( ( F `
  P )  .\/  ( X  ./\  W ) ) )
 
Theoremcdlemg2klem 31330* cdleme42keg 31221 with simpler hypotheses. TODO: FIX COMMENT (Contributed by NM, 22-Apr-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( p  .\/  q )  ./\  W )   &    |-  D  =  ( (
 t  .\/  U )  ./\  ( q  .\/  (
 ( p  .\/  t
 )  ./\  W ) ) )   &    |-  E  =  ( ( p  .\/  q
 )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  G  =  ( x  e.  B  |->  if ( ( p  =/=  q  /\  -.  x  .<_  W ) ,  ( iota_
 z  e.  B A. s  e.  A  (
 ( -.  s  .<_  W 
 /\  ( s  .\/  ( x  ./\  W ) )  =  x ) 
 ->  z  =  ( if ( s  .<_  ( p 
 .\/  q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( p  .\/  q
 ) )  ->  y  =  E ) ) , 
 [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) ) ,  x ) )   &    |-  V  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  F  e.  T )  ->  ( ( F `
  P )  .\/  ( F `  Q ) )  =  ( ( F `  P ) 
 .\/  V ) )
 
Theoremcdlemg2idN 31331 Version of cdleme31id 31129 with simpler hypotheses. TODO: Fix comment. (Contributed by NM, 21-Apr-2013.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  B  =  ( Base `  K )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  Q  /\  X  e.  B )  /\  P  =  Q )  ->  ( F `  X )  =  X )
 
Theoremcdlemg3a 31332 Part of proof of Lemma G in [Crawley] p. 116, line 19. Show p  \/ q = p  \/ u. TODO: reformat cdleme0cp 30949 to match this, then replace with cdleme0cp 30949. (Contributed by NM, 19-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  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  ( P 
 .\/  Q )  =  ( P  .\/  U )
 )
 
Theoremcdlemg2jOLDN 31333 TODO: Replace this with ltrnj 30867. (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  F  e.  T )  ->  ( F `  ( P  .\/  Q ) )  =  ( ( F `  P ) 
 .\/  ( F `  Q ) ) )
 
Theoremcdlemg2fv 31334 Value of a translation in terms of an associated atom. cdleme48fvg 31235 with simpler hypotheses. TODO: Use ltrnj 30867 to vastly simplify. (Contributed by NM, 23-Apr-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  B  =  ( Base `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( F `
  X )  =  ( ( F `  P )  .\/  ( X 
 ./\  W ) ) )
 
Theoremcdlemg2fv2 31335 Value of a translation in terms of an associated atom. TODO: FIX COMMENT TODO: Is this useful elsewhere e.g. around cdlemeg46fjv 31258 that use more complex proofs? TODO: Use ltrnj 30867 to vastly simplify. (Contributed by NM, 23-Apr-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  ./\  =  ( meet `  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 ) )  /\  F  e.  T )  ->  ( F `  ( R  .\/  U ) )  =  ( ( F `  R )  .\/  U ) )
 
Theoremcdlemg2k 31336 cdleme42keg 31221 with simpler hypotheses. TODO: FIX COMMENT Todo: derive from cdlemg3a 31332, cdlemg2fv2 31335, cdlemg2jOLDN 31333, ltrnel 30874? (Contributed by NM, 22-Apr-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  ->  ( ( F `  P ) 
 .\/  ( F `  Q ) )  =  ( ( F `  P )  .\/  U ) )
 
Theoremcdlemg2kq 31337 cdlemg2k 31336 with  P and  Q swapped. TODO: FIX COMMENT (Contributed by NM, 15-May-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  ->  ( ( F `  P ) 
 .\/  ( F `  Q ) )  =  ( ( F `  Q )  .\/  U ) )
 
Theoremcdlemg2l 31338 TODO: FIX COMMENT (Contributed by NM, 23-Apr-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T ) )  ->  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `  Q ) ) )  =  ( ( F `  ( G `  P ) )  .\/  U )
 )
 
Theoremcdlemg2m 31339 TODO: FIX COMMENT T (Contributed by NM, 25-Apr-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  ->  ( ( ( F `  P )  .\/  ( F `  Q ) )  ./\  W )  =  U )
 
Theoremcdlemg5 31340* TODO: Is there a simpler more direct proof, that could be placed earlier e.g. near lhpexle 30740? TODO: The  .\/ hypothesis is unused. FIX COMMENT (Contributed by NM, 26-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  E. q  e.  A  ( P  =/=  q  /\  -.  q  .<_  W ) )
 
Theoremcdlemb3 31341* Given two atoms not under the fiducial co-atom  W, there is a third. Lemma B in [Crawley] p. 112. TODO: Is there a simpler more direct proof, that could be placed earlier e.g. near lhpexle 30740? Then replace cdlemb2 30776 with it. This is a more general version of cdlemb2 30776 without  P  =/=  Q condition. (Contributed by NM, 27-Apr-2013.)
 |-  .<_  =  ( 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 ) ) 
 ->  E. r  e.  A  ( -.  r  .<_  W  /\  -.  r  .<_  ( P  .\/  Q ) ) )
 
Theoremcdlemg7fvbwN 31342 Properties of a translation of an element not under  W. TODO: Fix comment. Can this be simplified? Perhaps derived from cdleme48bw 31237? Done with a *ltrn* theorem? (Contributed by NM, 28-Apr-2013.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  B  =  ( Base `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T )  ->  ( ( F `  X )  e.  B  /\  -.  ( F `  X ) 
 .<_  W ) )
 
Theoremcdlemg4a 31343 TODO: FIX COMMENT If fg(p) = p, then tr f = tr g. (Contributed by NM, 23-Apr-2013.)
 |-  .<_  =  ( 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  /\  G  e.  T )  /\  ( F `  ( G `  P ) )  =  P ) 
 ->  ( R `  F )  =  ( R `  G ) )
 
Theoremcdlemg4b1 31344 TODO: FIX COMMENT (Contributed by NM, 24-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  .\/  =  ( join `  K )   &    |-  V  =  ( R `  G )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  ( P  .\/  V )  =  ( P  .\/  ( G `  P ) ) )
 
Theoremcdlemg4b2 31345 TODO: FIX COMMENT (Contributed by NM, 24-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  .\/  =  ( join `  K )   &    |-  V  =  ( R `  G )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  ( ( G `  P )  .\/  V )  =  ( P  .\/  ( G `  P ) ) )
 
Theoremcdlemg4b12 31346 TODO: FIX COMMENT (Contributed by NM, 24-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  .\/  =  ( join `  K )   &    |-  V  =  ( R `  G )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  ( ( G `  P )  .\/  V )  =  ( P  .\/  V ) )
 
Theoremcdlemg4c 31347 TODO: FIX COMMENT (Contributed by NM, 24-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  .\/  =  ( join `  K )   &    |-  V  =  ( R `  G )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  G  e.  T )  /\  -.  Q  .<_  ( P  .\/  V ) )  ->  -.  ( G `  Q )  .<_  ( P  .\/  V )
 )
 
Theoremcdlemg4d 31348 TODO: FIX COMMENT (Contributed by NM, 25-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  .\/  =  ( join `  K )   &    |-  V  =  ( R `  G )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T )  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V )  /\  ( F `
  ( G `  P ) )  =  P ) )  ->  -.  ( G `  Q )  .<_  ( ( G `
  P )  .\/  ( F `  ( G `
  P ) ) ) )
 
Theoremcdlemg4e 31349 TODO: FIX COMMENT (Contributed by NM, 25-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  .\/  =  ( join `  K )   &    |-  V  =  ( R `  G )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T )  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V )  /\  ( F `
  ( G `  P ) )  =  P ) )  ->  ( F `  ( G `
  Q ) )  =  ( ( ( G `  Q ) 
 .\/  ( R `  F ) )  ./\  ( ( F `  ( G `  P ) )  .\/  ( (
 ( G `  P )  .\/  ( G `  Q ) )  ./\  W ) ) ) )
 
Theoremcdlemg4f 31350 TODO: FIX COMMENT (Contributed by NM, 25-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  .\/  =  ( join `  K )   &    |-  V  =  ( R `  G )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T )  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V )  /\  ( F `
  ( G `  P ) )  =  P ) )  ->  ( F `  ( G `
  Q ) )  =  ( ( Q 
 .\/  V )  ./\  ( P  .\/  ( ( P 
 .\/  Q )  ./\  W ) ) ) )
 
Theoremcdlemg4g 31351 TODO: FIX COMMENT (Contributed by NM, 25-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  .\/  =  ( join `  K )   &    |-  V  =  ( R `  G )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T )  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V )  /\  ( F `
  ( G `  P ) )  =  P ) )  ->  ( F `  ( G `
  Q ) )  =  ( ( Q 
 .\/  V )  ./\  ( P  .\/  Q ) ) )
 
Theoremcdlemg4 31352 TODO: FIX COMMENT (Contributed by NM, 25-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  .\/  =  ( join `  K )   &    |-  V  =  ( R `  G )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T )  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V )  /\  ( F `
  ( G `  P ) )  =  P ) )  ->  ( F `  ( G `
  Q ) )  =  Q )
 
Theoremcdlemg6a 31353* TODO: FIX COMMENT TODO: replace with cdlemg4 31352. (Contributed by NM, 27-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  .\/  =  ( join `  K )   &    |-  V  =  ( R `  G )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( r  e.  A  /\  -.  r  .<_  W )  /\  F  e.  T )  /\  ( G  e.  T  /\  -.  r  .<_  ( P  .\/  V )  /\  ( F `
  ( G `  P ) )  =  P ) )  ->  ( F `  ( G `
  r ) )  =  r )
 
Theoremcdlemg6b 31354* TODO: FIX COMMENT TODO: replace with cdlemg4 31352. (Contributed by NM, 27-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  .\/  =  ( join `  K )   &    |-  V  =  ( R `  G )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( r  e.  A  /\  -.  r  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T )  /\  ( G  e.  T  /\  -.  Q  .<_  ( r  .\/  V )  /\  ( F `
  ( G `  r ) )  =  r ) )  ->  ( F `  ( G `
  Q ) )  =  Q )
 
Theoremcdlemg6c 31355* TODO: FIX COMMENT (Contributed by NM, 27-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  .\/  =  ( join `  K )   &    |-  V  =  ( R `  G )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T )  /\  ( G  e.  T  /\  Q  .<_  ( P  .\/  V )  /\  ( F `  ( G `  P ) )  =  P ) )  ->  ( (
 ( r  e.  A  /\  -.  r  .<_  W ) 
 /\  -.  r  .<_  ( P  .\/  V )
 )  ->  ( F `  ( G `  Q ) )  =  Q ) )
 
Theoremcdlemg6d 31356* TODO: FIX COMMENT (Contributed by NM, 27-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  .\/  =  ( join `  K )   &    |-  V  =  ( R `  G )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T )  /\  ( G  e.  T  /\  Q  .<_  ( P  .\/  V )  /\  ( F `  ( G `  P ) )  =  P ) )  ->  ( (
 ( r  e.  A  /\  -.  r  .<_  W ) 
 /\  -.  r  .<_  ( P  .\/  ( G `  P ) ) ) 
 ->  ( F `  ( G `  Q ) )  =  Q ) )
 
Theoremcdlemg6e 31357 TODO: FIX COMMENT (Contributed by NM, 27-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  .\/  =  ( join `  K )   &    |-  V  =  ( R `  G )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T )  /\  ( G  e.  T  /\  Q  .<_  ( P  .\/  V )  /\  ( F `  ( G `  P ) )  =  P ) )  ->  ( F `  ( G `  Q ) )  =  Q )
 
Theoremcdlemg6 31358 TODO: FIX COMMENT (Contributed by NM, 27-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) ) 
 ->  ( F `  ( G `  Q ) )  =  Q )
 
Theoremcdlemg7fvN 31359 Value of a translation composition in terms of an associated atom. (Contributed by NM, 28-Apr-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) ) 
 ->  ( F `  ( G `  X ) )  =  ( ( F `
  ( G `  P ) )  .\/  ( X  ./\  W ) ) )
 
Theoremcdlemg7aN 31360 TODO: FIX COMMENT (Contributed by NM, 28-Apr-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) ) 
 ->  ( F `  ( G `  X ) )  =  X )
 
Theoremcdlemg7N 31361 TODO: FIX COMMENT (Contributed by NM, 28-Apr-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `
  ( G `  P ) )  =  P ) )  ->  ( F `  ( G `
  X ) )  =  X )
 
Theoremcdlemg8a 31362 TODO: FIX COMMENT (Contributed by NM, 29-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) ) 
 ->  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  W )  =  ( ( Q  .\/  ( F `  ( G `
  Q ) ) )  ./\  W )
 )
 
Theoremcdlemg8b 31363 TODO: FIX COMMENT (Contributed by NM, 29-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T )  /\  ( G  e.  T  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `  Q ) ) )  =  ( P  .\/  Q )  /\  ( F `  ( G `  P ) )  =/=  P ) )  ->  ( P  .\/  ( F `  ( G `  P ) ) )  =  ( P 
 .\/  Q ) )
 
Theoremcdlemg8c 31364 TODO: FIX COMMENT (Contributed by NM, 29-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T )  /\  ( G  e.  T  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `  Q ) ) )  =  ( P  .\/  Q )  /\  ( F `  ( G `  P ) )  =/=  P ) )  ->  ( Q  .\/  ( F `  ( G `  Q ) ) )  =  ( P 
 .\/  Q ) )
 
Theoremcdlemg8d 31365 TODO: FIX COMMENT (Contributed by NM, 29-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T )  /\  ( G  e.  T  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `  Q ) ) )  =  ( P  .\/  Q )  /\  ( F `  ( G `  P ) )  =/=  P ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W )  =  ( ( Q 
 .\/  ( F `  ( G `  Q ) ) )  ./\  W ) )
 
Theoremcdlemg8 31366 TODO: FIX COMMENT (Contributed by NM, 29-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T )  /\  ( G  e.  T  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `  Q ) ) )  =  ( P  .\/  Q ) ) )  ->  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  W )  =  ( ( Q  .\/  ( F `  ( G `
  Q ) ) )  ./\  W )
 )
 
Theoremcdlemg9a 31367 TODO: FIX COMMENT (Contributed by NM, 1-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  F  e.  T )  /\  ( G  e.  T  /\  P  =/=  Q  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q ) ) )  ->  ( ( P  .\/  U )  ./\  ( ( F `  ( G `  P ) )  .\/  U ) )  .<_  ( ( G `  P ) 
 .\/  U ) )
 
Theoremcdlemg9b 31368 The triples  <. P ,  ( F `  ( G `
 P ) ) ,  ( F `  P ) >. and  <. Q , 
( F `  ( G `  Q )
) ,  ( F `
 Q ) >. are centrally perspective. TODO: FIX COMMENT (Contributed by NM, 1-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T )  /\  ( G  e.  T  /\  P  =/=  Q  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q ) ) )  ->  ( ( P  .\/  Q )  ./\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
  Q ) ) ) )  .<_  ( ( G `  P ) 
 .\/  ( G `  Q ) ) )
 
Theoremcdlemg9 31369 The triples  <. P ,  ( F `  ( G `
 P ) ) ,  ( F `  P ) >. and  <. Q , 
( F `  ( G `  Q )
) ,  ( F `
 Q ) >. are axially perspective by dalaw 30621. Part of Lemma G of [Crawley] p. 116, last 2 lines. TODO: FIX COMMENT (Contributed by NM, 1-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T )  /\  ( G  e.  T  /\  P  =/=  Q  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q ) ) )  ->  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q ) ) ) )  .<_  ( ( ( ( F `  ( G `  P ) )  .\/  ( G `  P ) )  ./\  ( ( F `  ( G `  Q ) )  .\/  ( G `  Q ) ) ) 
 .\/  ( ( ( G `  P ) 
 .\/  P )  ./\  (
 ( G `  Q )  .\/  Q ) ) ) )
 
Theoremcdlemg10b 31370 TODO: FIX COMMENT TODO: Can this be moved up as a stand-alone theorem in ltrn* area? (Contributed by NM, 4-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  F  e.  T )  ->  ( ( ( F `  P ) 
 .\/  ( F `  Q ) )  ./\  W )  =  ( ( P  .\/  Q )  ./\ 
 W ) )
 
Theoremcdlemg10bALTN 31371 TODO: FIX COMMENT TODO: Can this be moved up as a stand-alone theorem in ltrn* area? TODO: Compare this proof to cdlemg2m 31339 and pick best, if moved to ltrn* area. (Contributed by NM, 4-May-2013.) (New usage is discouraged.)
 |-  .<_  =  ( 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 `  P )  .\/  ( F `
  Q ) ) 
 ./\  W )  =  ( ( P  .\/  Q )  ./\  W ) )
 
Theoremcdlemg11a 31372 TODO: FIX COMMENT (Contributed by NM, 4-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q ) ) )  ->  ( F `  ( G `
  P ) )  =/=  P )
 
Theoremcdlemg11aq 31373 TODO: FIX COMMENT TODO: can proof using this be restructured to use cdlemg11a 31372? (Contributed by NM, 4-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q ) ) )  ->  ( F `  ( G `
  Q ) )  =/=  Q )
 
Theoremcdlemg10c 31374 TODO: FIX COMMENT TODO: Can this be moved up as a stand-alone theorem in trl* area? (Contributed by NM, 4-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 )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T ) )  ->  ( ( R `  F ) 
 .<_  ( ( G `  P )  .\/  ( G `
  Q ) )  <-> 
 ( R `  F )  .<_  ( P  .\/  Q ) ) )
 
Theoremcdlemg10a 31375 TODO: FIX COMMENT (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 ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q ) 
 /\  ( ( ( F `  ( G `
  P ) ) 
 .\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  ( R `  F )  .<_  ( P  .\/  Q )  /\  -.  ( R `  G )  .<_  ( P  .\/  Q )
 ) )  ->  (
 ( P  .\/  ( F `  ( G `  P ) ) ) 
 ./\  ( Q  .\/  ( F `  ( G `
  Q ) ) ) )  .<_  ( ( R `  F ) 
 .\/  ( R `  G ) ) )
 
Theoremcdlemg10 31376 TODO: FIX COMMENT (Contributed by NM, 4-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 ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q ) 
 /\  ( ( ( F `  ( G `
  P ) ) 
 .\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  ( R `  F )  .<_  ( P  .\/  Q )  /\  -.  ( R `  G )  .<_  ( P  .\/  Q )
 ) )  ->  (
 ( P  .\/  ( F `  ( G `  P ) ) ) 
 ./\  ( Q  .\/  ( F `  ( G `
  Q ) ) ) )  .<_  W )
 
Theoremcdlemg11b 31377 TODO: FIX COMMENT (Contributed by NM, 5-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 )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( G  e.  T  /\  P  =/=  Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q )
 ) )  ->  ( P  .\/  Q )  =/=  ( ( G `  P )  .\/  ( G `
  Q ) ) )
 
Theoremcdlemg12a 31378 TODO: FIX COMMENT. (Contributed by NM, 5-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  F  e.  T )  /\  ( G  e.  T  /\  P  =/=  Q  /\  ( P  .\/  U )  =/=  ( ( G `
  P )  .\/  U ) ) )  ->  ( ( P  .\/  U )  ./\  ( ( G `  P )  .\/  U ) )  .<_  ( ( F `  ( G `
  P ) ) 
 .\/  U ) )
 
Theoremcdlemg12b 31379 The triples  <. P ,  ( F `  P ) ,  ( F `  ( G `  P ) ) >. and  <. Q , 
( F `  Q
) ,  ( F `
 ( G `  Q ) ) >. are centrally perspective. TODO: FIX COMMENT (Contributed by NM, 5-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 )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T )  /\  ( G  e.  T  /\  P  =/=  Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q ) ) )  ->  ( ( P  .\/  Q )  ./\  ( ( G `  P )  .\/  ( G `  Q ) ) )  .<_  ( ( F `  ( G `
  P ) ) 
 .\/  ( F `  ( G `  Q ) ) ) )
 
Theoremcdlemg12c 31380 The triples  <. P ,  ( F `  P ) ,  ( F `  ( G `  P ) ) >. and  <. Q , 
( F `  Q
) ,  ( F `
 ( G `  Q ) ) >. are axially perspective by dalaw 30621. TODO: FIX COMMENT (Contributed by NM, 5-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 )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T )  /\  ( G  e.  T  /\  P  =/=  Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q ) ) )  ->  ( ( P  .\/  ( G `  P ) )  ./\  ( Q  .\/  ( G `  Q ) ) )  .<_  ( ( ( ( G `
  P )  .\/  ( F `  ( G `
  P ) ) )  ./\  ( ( G `  Q )  .\/  ( F `  ( G `
  Q ) ) ) )  .\/  (
 ( ( F `  ( G `  P ) )  .\/  P )  ./\  ( ( F `  ( G `  Q ) )  .\/  Q )
 ) ) )
 
Theoremcdlemg12d 31381 TODO: FIX COMMENT (Contributed by NM, 5-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 ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q 
 /\  -.  ( R `  F )  .<_  ( P 
 .\/  Q )  /\  -.  ( R `  G ) 
 .<_  ( P  .\/  Q ) ) )  ->  ( R `  G ) 
 .<_  ( ( R `  F )  .\/  ( ( ( F `  ( G `  P ) ) 
 .\/  P )  ./\  (
 ( F `  ( G `  Q ) ) 
 .\/  Q ) ) ) )
 
Theoremcdlemg12e 31382 TODO: FIX COMMENT (Contributed by NM, 6-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q )  /\  ( -.  ( R `  F )  .<_  ( P 
 .\/  Q )  /\  -.  ( R `  G ) 
 .<_  ( P  .\/  Q )  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  ( (
 ( F `  ( G `  P ) ) 
 .\/  P )  ./\  (
 ( F `  ( G `  Q ) ) 
 .\/  Q ) )  =/= 
 .0.  )
 
Theoremcdlemg12f 31383 TODO: FIX COMMENT (Contributed by NM, 6-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 ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q ) 
 /\  ( ( -.  ( R `  F )  .<_  ( P  .\/  Q )  /\  -.  ( R `  G )  .<_  ( P  .\/  Q )
 )  /\  ( R `  F )  =/=  ( R `  G )  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q ) ) )  ->  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q ) ) ) )  .<_  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W ) )
 
Theoremcdlemg12g 31384 TODO: FIX COMMENT TODO: Combine with cdlemg12f 31383. (Contributed by NM, 6-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 ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q ) 
 /\  ( ( -.  ( R `  F )  .<_  ( P  .\/  Q )  /\  -.  ( R `  G )  .<_  ( P  .\/  Q )
 )  /\  ( R `  F )  =/=  ( R `  G )  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q ) ) )  ->  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q ) ) ) )  =  ( ( P  .\/  ( F `  ( G `  P ) ) ) 
 ./\  W ) )
 
Theoremcdlemg12 31385 TODO: FIX COMMENT (Contributed by NM, 6-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 ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q ) 
 /\  ( ( -.  ( R `  F )  .<_  ( P  .\/  Q )  /\  -.  ( R `  G )  .<_  ( P  .\/  Q )
 )  /\  ( R `  F )  =/=  ( R `  G )  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q ) ) )  ->  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  W )  =  ( ( Q  .\/  ( F `  ( G `
  Q ) ) )  ./\  W )
 )
 
Theoremcdlemg13a 31386 TODO: FIX COMMENT (Contributed by NM, 6-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 ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( F `
  P )  =/= 
 P  /\  ( R `  F )  =  ( R `  G ) 
 /\  ( ( F `
  ( G `  P ) )  .\/  ( F `  ( G `
  Q ) ) )  =/=  ( P 
 .\/  Q ) ) ) 
 ->  ( P  .\/  ( F `  ( G `  P ) ) )  =  ( ( G `
  P )  .\/  ( F `  ( G `
  P ) ) ) )
 
Theoremcdlemg13 31387 TODO: FIX COMMENT (Contributed by NM, 6-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 ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( F `
  P )  =/= 
 P  /\  ( R `  F )  =  ( R `  G ) 
 /\  ( ( F `
  ( G `  P ) )  .\/  ( F `  ( G `
  Q ) ) )  =/=  ( P 
 .\/  Q ) ) ) 
 ->  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  W )  =  ( ( Q  .\/  ( F `  ( G `
  Q ) ) )  ./\  W )
 )
 
Theoremcdlemg14f 31388 TODO: FIX COMMENT (Contributed by NM, 6-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 )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  P )  =  P )
 )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W )  =  ( ( Q 
 .\/  ( F `  ( G `  Q ) ) )  ./\  W ) )
 
Theoremcdlemg14g 31389 TODO: FIX COMMENT (Contributed by NM, 22-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 )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T  /\  ( G `  P )  =  P )
 )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W )  =  ( ( Q 
 .\/  ( F `  ( G `  Q ) ) )  ./\  W ) )
 
Theoremcdlemg15a 31390 Eliminate the  ( F `  P )  =/=  P condition from cdlemg13 31387. TODO: FIX COMMENT (Contributed by NM, 6-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 ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( R `
  F )  =  ( R `  G )  /\  ( ( F `
  ( G `  P ) )  .\/  ( F `  ( G `
  Q ) ) )  =/=  ( P 
 .\/  Q ) ) ) 
 ->  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  W )  =  ( ( Q  .\/  ( F `  ( G `
  Q ) ) )  ./\  W )
 )
 
Theoremcdlemg15 31391 Eliminate the  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) condition from cdlemg13 31387. TODO: FIX COMMENT (Contributed by NM, 25-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 ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T )  /\  ( R `  F )  =  ( R `  G ) ) 
 ->  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  W )  =  ( ( Q  .\/  ( F `  ( G `
  Q ) ) )  ./\  W )
 )
 
Theoremcdlemg16 31392 Part of proof of Lemma G of [Crawley] p. 116; 2nd line p. 117, which says that (our) cdlemg10 31376 "implies (2)" (of p. 116). No details are provided by the authors, so there may be a shorter proof; but ours requires the 14 lemmas, one using Desargues' law dalaw 30621, in order to make this inference. This final step eliminates the  ( R `  F )  =/=  ( R `  G ) condition from cdlemg12 31385. TODO: FIX COMMENT TODO: should we also eliminate  P  =/=  Q here (or earlier)? Do it if we don't need to add it in for something else later. (Contributed by NM, 6-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 ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q ) 
 /\  ( -.  ( R `  F )  .<_  ( P  .\/  Q )  /\  -.  ( R `  G )  .<_  ( P 
 .\/  Q )  /\  (
 ( F `  ( G `  P ) ) 
 .\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q ) ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W )  =  ( ( Q 
 .\/  ( F `  ( G `  Q ) ) )  ./\  W ) )
 
Theoremcdlemg16ALTN 31393 This version of cdlemg16 31392 uses cdlemg15a 31390 instead of cdlemg15 31391, in case cdlemg15 31391 ends up not being needed. TODO: FIX COMMENT (Contributed by NM, 6-May-2013.) (New usage is discouraged.)
 |-  .<_  =  ( 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  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  P  =/=  Q )  /\  ( ( ( F `  ( G `
  P ) ) 
 .\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  ( R `  F )  .<_  ( P  .\/  Q )  /\  -.  ( R `  G )  .<_  ( P  .\/  Q )
 ) )  ->  (
 ( P  .\/  ( F `  ( G `  P ) ) ) 
 ./\  W )  =  ( ( Q  .\/  ( F `  ( G `  Q ) ) ) 
 ./\  W ) )
 
Theoremcdlemg16z 31394 Eliminate  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) condition from cdlemg16 31392. TODO: would it help to also eliminate  P  =/=  Q here or later? (Contributed by NM, 25-May-2013.)
 |-  .<_  =  ( le `  K )