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Theorem List for Metamath Proof Explorer - 30201-30300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcdlemg10c 30201 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 30202 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 30203 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 30204 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 30205 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 30206 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 30207 The triples  <. P ,  ( F `  P ) ,  ( F `  ( G `  P ) ) >. and  <. Q , 
( F `  Q
) ,  ( F `
 ( G `  Q ) ) >. are axially perspective by dalaw 29448. 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 30208 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 30209 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 30210 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 30211 TODO: FIX COMMENT TODO: Combine with cdlemg12f 30210. (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 30212 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 30213 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 30214 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 30215 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 30216 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 30217 Eliminate the  ( F `  P )  =/=  P condition from cdlemg13 30214. 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 30218 Eliminate the  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) condition from cdlemg13 30214. 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 30219 Part of proof of Lemma G of [Crawley] p. 116; 2nd line p. 117, which says that (our) cdlemg10 30203 "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 29448, in order to make this inference. This final step eliminates the  ( R `  F )  =/=  ( R `  G ) condition from cdlemg12 30212. 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 30220 This version of cdlemg16 30219 uses cdlemg15a 30217 instead of cdlemg15 30218, in case cdlemg15 30218 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 30221 Eliminate  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) condition from cdlemg16 30219. TODO: would it help to also eliminate  P  =/=  Q here or later? (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  /\  P  =/=  Q ) 
 /\  ( -.  ( R `  F )  .<_  ( P  .\/  Q )  /\  -.  ( R `  G )  .<_  ( P 
 .\/  Q ) ) ) 
 ->  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  W )  =  ( ( Q  .\/  ( F `  ( G `
  Q ) ) )  ./\  W )
 )
 
Theoremcdlemg16zz 30222 Eliminate  P  =/=  Q from cdlemg16z 30221. TODO: Use this only if needed. (Contributed by NM, 26-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 )  .<_  ( P 
 .\/  Q )  /\  -.  ( R `  G ) 
 .<_  ( P  .\/  Q ) ) )  ->  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  W )  =  ( ( Q  .\/  ( F `  ( G `
  Q ) ) )  ./\  W )
 )
 
Theoremcdlemg17a 30223 TODO: FIX COMMENT (Contributed by NM, 8-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 ) ) 
 /\  ( G  e.  T  /\  ( R `  G )  .<_  ( P 
 .\/  Q ) ) ) 
 ->  ( G `  P )  .<_  ( P  .\/  Q ) )
 
Theoremcdlemg17b 30224* Part of proof of Lemma G in [Crawley] p. 117, 4th line. Whenever (in their terminology) p  \/ q/0 (i.e. the sublattice from 0 to p  \/ q) contains precisely three atoms and g is not the identity, g(p) = q. See also comments under cdleme0nex 29852. (Contributed by NM, 8-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 ) ) 
 /\  ( G  e.  T  /\  P  =/=  Q )  /\  ( ( G `
  P )  =/= 
 P  /\  ( R `  G )  .<_  ( P 
 .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( G `  P )  =  Q )
 
Theoremcdlemg17dN 30225* TODO: fix comment. (Contributed by NM, 9-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  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  ( ( R `  G )  .<_  ( P 
 .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  /\  ( G `  P )  =/=  P ) ) 
 ->  ( R `  G )  =  ( ( P  .\/  Q )  ./\  W ) )
 
Theoremcdlemg17dALTN 30226 Same as cdlemg17dN 30225 with fewer antecedents but longer proof TODO: fix comment. (Contributed by NM, 9-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  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( ( R `
  G )  .<_  ( P  .\/  Q )  /\  ( G `  P )  =/=  P ) ) 
 ->  ( R `  G )  =  ( ( P  .\/  Q )  ./\  W ) )
 
Theoremcdlemg17e 30227* TODO: fix comment. (Contributed by NM, 8-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 ) 
 /\  ( ( G `
  P )  =/= 
 P  /\  ( R `  G )  .<_  ( P 
 .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( ( F `  P )  .\/  ( F `  Q ) )  =  ( ( F `  P ) 
 .\/  ( R `  G ) ) )
 
Theoremcdlemg17f 30228* TODO: fix comment. (Contributed by NM, 8-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 ) 
 /\  ( ( G `
  P )  =/= 
 P  /\  ( R `  G )  .<_  ( P 
 .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( ( F `  P )  .\/  ( F `  Q ) )  =  ( ( F `  P ) 
 .\/  ( G `  ( F `  P ) ) ) )
 
Theoremcdlemg17g 30229* TODO: fix comment. (Contributed by NM, 9-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 ) 
 /\  ( ( G `
  P )  =/= 
 P  /\  ( R `  G )  .<_  ( P 
 .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( G `  ( F `  P ) )  .<_  ( ( F `  P ) 
 .\/  ( F `  Q ) ) )
 
Theoremcdlemg17h 30230* TODO: fix comment. (Contributed by NM, 10-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 ) ) 
 /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) 
 /\  ( P  =/=  Q 
 /\  S  .<_  ( ( F `  P ) 
 .\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
 .\/  r )  =  ( Q  .\/  r
 ) ) ) ) 
 ->  ( S  =  ( F `  P )  \/  S  =  ( F `  Q ) ) )
 
Theoremcdlemg17i 30231* TODO: fix comment. (Contributed by NM, 10-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 ) 
 /\  ( ( G `
  P )  =/= 
 P  /\  ( R `  G )  .<_  ( P 
 .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( G `  ( F `  P ) )  =  ( F `  Q ) )
 
Theoremcdlemg17ir 30232* TODO: fix comment. (Contributed by NM, 13-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 ) 
 /\  ( ( G `
  P )  =/= 
 P  /\  ( R `  G )  .<_  ( P 
 .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( F `  ( G `  P ) )  =  ( F `  Q ) )
 
Theoremcdlemg17j 30233* TODO: fix comment. (Contributed by NM, 11-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 ) 
 /\  ( ( G `
  P )  =/= 
 P  /\  ( R `  G )  .<_  ( P 
 .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( G `  ( F `  P ) )  =  ( F `  ( G `  P ) ) )
 
Theoremcdlemg17pq 30234* Utility theorem for swapping  P and  Q. TODO: fix comment. (Contributed by NM, 11-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 ) 
 /\  ( ( G `
  P )  =/= 
 P  /\  ( R `  G )  .<_  ( P 
 .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( (
 ( K  e.  HL  /\  W  e.  H ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) 
 /\  ( F  e.  T  /\  G  e.  T  /\  Q  =/=  P ) 
 /\  ( ( G `
  Q )  =/= 
 Q  /\  ( R `  G )  .<_  ( Q 
 .\/  P )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( Q  .\/  r )  =  ( P  .\/  r ) ) ) ) )
 
Theoremcdlemg17bq 30235* cdlemg17b 30224 with  P and  Q swapped. Antecedent  F  e.  ( T `  W ) is redundant for easier use. TODO: should we have redundant antecedent for cdlemg17b 30224 also? (Contributed by NM, 13-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 ) 
 /\  ( ( G `
  P )  =/= 
 P  /\  ( R `  G )  .<_  ( P 
 .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( G `  Q )  =  P )
 
Theoremcdlemg17iqN 30236* cdlemg17i 30231 with  P and  Q swapped. (Contributed by NM, 13-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 )  /\  ( ( R `
  G )  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
 .\/  r )  =  ( Q  .\/  r
 ) )  /\  ( G `  P )  =/= 
 P ) )  ->  ( G `  ( F `
  Q ) )  =  ( F `  P ) )
 
Theoremcdlemg17irq 30237* cdlemg17ir 30232 with  P and  Q swapped. (Contributed by NM, 13-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 ) 
 /\  ( ( G `
  P )  =/= 
 P  /\  ( R `  G )  .<_  ( P 
 .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( F `  ( G `  Q ) )  =  ( F `  P ) )
 
Theoremcdlemg17jq 30238* cdlemg17j 30233 with  P and  Q swapped. (Contributed by NM, 13-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 ) 
 /\  ( ( G `
  P )  =/= 
 P  /\  ( R `  G )  .<_  ( P 
 .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( G `  ( F `  Q ) )  =  ( F `  ( G `  Q ) ) )
 
Theoremcdlemg17 30239* Part of Lemma G of [Crawley] p. 117, lines 7 and 8. We show an argument whose value at  G equals itself. TODO: fix comment. (Contributed by NM, 12-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 ) 
 /\  ( ( G `
  P )  =/= 
 P  /\  ( R `  G )  .<_  ( P 
 .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( G `  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q ) ) ) ) )  =  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q ) ) ) ) )
 
Theoremcdlemg18a 30240 Show two lines are different. TODO: fix comment. (Contributed by NM, 14-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  /\  Q  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  ( ( F `  Q )  .\/  ( F `
  P ) )  =/=  ( P  .\/  Q ) ) )  ->  ( P  .\/  ( F `
  Q ) )  =/=  ( Q  .\/  ( F `  P ) ) )
 
Theoremcdlemg18b 30241 Lemma for cdlemg18c 30242. TODO: fix comment. (Contributed by NM, 15-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  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 )  /\  ( P  =/=  Q 
 /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q )  .\/  ( F `
  P ) )  =/=  ( P  .\/  Q ) ) )  ->  -.  P  .<_  ( U  .\/  ( F `  Q ) ) )
 
Theoremcdlemg18c 30242 Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  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 )  /\  ( P  =/=  Q 
 /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q )  .\/  ( F `
  P ) )  =/=  ( P  .\/  Q ) ) )  ->  ( ( P  .\/  ( F `  Q ) )  ./\  ( Q  .\/  ( F `  P ) ) )  e.  A )
 
Theoremcdlemg18d 30243* Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-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  /\  ( G `
  P )  =/= 
 P )  /\  (
 ( R `  G )  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
  P ) ) 
 .\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -. 
 E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q ) ) ) )  e.  A )
 
Theoremcdlemg18 30244* Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-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  /\  ( G `
  P )  =/= 
 P )  /\  (
 ( R `  G )  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
  P ) ) 
 .\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -. 
 E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q ) ) ) )  .<_  W )
 
Theoremcdlemg19a 30245* Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-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  /\  ( G `
  P )  =/= 
 P )  /\  (
 ( R `  G )  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
  P ) ) 
 .\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -. 
 E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q ) ) ) )  =  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  W )
 )
 
Theoremcdlemg19 30246* Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-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  /\  ( G `
  P )  =/= 
 P )  /\  (
 ( R `  G )  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
  P ) ) 
 .\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -. 
 E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W )  =  ( ( Q 
 .\/  ( F `  ( G `  Q ) ) )  ./\  W ) )
 
Theoremcdlemg20 30247* Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 23-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 )  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
 .\/  r )  =  ( Q  .\/  r
 ) ) ) ) 
 ->  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  W )  =  ( ( Q  .\/  ( F `  ( G `
  Q ) ) )  ./\  W )
 )
 
Theoremcdlemg21 30248* Version of cdlemg19 with  ( R `  F
)  .<_  ( P  .\/  Q ) instead of  ( R `  G )  .<_  ( P 
.\/  Q ) as a condition. (Contributed by NM, 23-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 `
  P )  =/= 
 P )  /\  (
 ( R `  F )  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
  P ) ) 
 .\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -. 
 E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W )  =  ( ( Q 
 .\/  ( F `  ( G `  Q ) ) )  ./\  W ) )
 
Theoremcdlemg22 30249* cdlemg21 30248 with  ( F `  P )  =/=  P condition removed. (Contributed by NM, 23-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 )  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
 .\/  r )  =  ( Q  .\/  r
 ) ) ) ) 
 ->  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  W )  =  ( ( Q  .\/  ( F `  ( G `
  Q ) ) )  ./\  W )
 )
 
Theoremcdlemg24 30250* Combine cdlemg16z 30221 and cdlemg22 30249. TODO: Fix comment. (Contributed by NM, 24-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 )  /\  -. 
 E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W )  =  ( ( Q 
 .\/  ( F `  ( G `  Q ) ) )  ./\  W ) )
 
Theoremcdlemg37 30251* Use cdlemg8 30193 to eliminate the  =/=  ( P  .\/  Q
) condition of cdlemg24 30250. (Contributed by NM, 31-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  /\  -. 
 E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W )  =  ( ( Q 
 .\/  ( F `  ( G `  Q ) ) )  ./\  W ) )
 
Theoremcdlemg25zz 30252 cdlemg16zz 30222 restated for easier studying. TODO: Discard this after everything is figured out. (Contributed by NM, 26-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 )  /\  ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T )  /\  ( G  e.  T  /\  -.  ( R `  F )  .<_  ( P 
 .\/  z )  /\  -.  ( R `  G )  .<_  ( P  .\/  z ) ) ) 
 ->  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  W )  =  ( ( z  .\/  ( F `  ( G `
  z ) ) )  ./\  W )
 )
 
Theoremcdlemg26zz 30253 cdlemg16zz 30222 restated for easier studying. TODO: Discard this after everything is figured out. (Contributed by NM, 26-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 )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T )  /\  ( G  e.  T  /\  -.  ( R `  F )  .<_  ( Q 
 .\/  z )  /\  -.  ( R `  G )  .<_  ( Q  .\/  z ) ) ) 
 ->  ( ( Q  .\/  ( F `  ( G `
  Q ) ) )  ./\  W )  =  ( ( z  .\/  ( F `  ( G `
  z ) ) )  ./\  W )
 )
 
Theoremcdlemg27a 30254 For use with case when  ( P  .\/  v
)  ./\  ( Q  .\/  ( R `  F
) ) or  ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) is zero, letting us establish  -.  z  .<_  W  /\  z  .<_  ( P 
.\/  v ) via 4atex 29638. TODO: Fix comment. (Contributed by NM, 28-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 )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T ) 
 /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
 .\/  v )  /\  ( F `  P )  =/=  P ) ) 
 ->  -.  ( R `  F )  .<_  ( P 
 .\/  z ) )
 
Theoremcdlemg28a 30255 Part of proof of Lemma G of [Crawley] p. 116. First equality of the equation of line 14 on p. 117. (Contributed by NM, 29-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 )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( v  =/=  ( R `
  F )  /\  v  =/=  ( R `  G ) )  /\  z  .<_  ( P  .\/  v )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/= 
 P ) ) ) 
 ->  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  W )  =  ( ( z  .\/  ( F `  ( G `
  z ) ) )  ./\  W )
 )
 
Theoremcdlemg31b0N 30256 TODO: Fix comment. (Contributed by NM, 30-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 )   &    |-  N  =  ( ( P  .\/  v )  ./\  ( Q 
 .\/  ( R `  F ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `
  P )  =/= 
 P ) )  ->  ( N  e.  A  \/  N  =  ( 0. `  K ) ) )
 
Theoremcdlemg31b0a 30257 TODO: Fix comment. (Contributed by NM, 30-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  N  =  ( ( P  .\/  v )  ./\  ( Q 
 .\/  ( R `  F ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F ) ) )  ->  ( N  e.  A  \/  N  =  ( 0. `  K ) ) )
 
Theoremcdlemg27b 30258 TODO: Fix comment. (Contributed by NM, 28-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  N  =  ( ( P  .\/  v )  ./\  ( Q 
 .\/  ( R `  F ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 z  e.  A  /\  ( v  e.  A  /\  v  .<_  W ) 
 /\  ( F  e.  T  /\  z  =/=  N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P ) ) 
 ->  -.  ( R `  F )  .<_  ( Q 
 .\/  z ) )
 
Theoremcdlemg31a 30259 TODO: fix comment. (Contributed by NM, 29-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  N  =  ( ( P  .\/  v )  ./\  ( Q 
 .\/  ( R `  F ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( v  e.  A  /\  F  e.  T ) )  ->  N  .<_  ( P  .\/  v )
 )
 
Theoremcdlemg31b 30260 TODO: fix comment. (Contributed by NM, 29-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  N  =  ( ( P  .\/  v )  ./\  ( Q 
 .\/  ( R `  F ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( v  e.  A  /\  F  e.  T ) )  ->  N  .<_  ( Q  .\/  ( R `  F ) ) )
 
Theoremcdlemg31c 30261 Show that when  N is an atom, it is not under  W. TODO: Is there a shorter direct proof? Todo: should we eliminate  ( F `  P )  =/=  P here? (Contributed by NM, 29-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  N  =  ( ( P  .\/  v )  ./\  ( Q 
 .\/  ( R `  F ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( v  e.  A  /\  v  .<_  W ) 
 /\  F  e.  T )  /\  ( v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P  /\  N  e.  A )
 )  ->  -.  N  .<_  W )
 
Theoremcdlemg31d 30262 Eliminate  ( F `  P )  =/=  P from cdlemg31c 30261. TODO: Prove directly. Todo: do we need to eliminate  ( F `  P )  =/=  P? It might be better to do this all at once at the end. See also cdlemg29 30267 vs. cdlemg28 30266. (Contributed by NM, 29-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  N  =  ( ( P  .\/  v )  ./\  ( Q 
 .\/  ( R `  F ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )  /\  N  e.  A )
 )  ->  -.  N  .<_  W )
 
Theoremcdlemg33b0 30263* TODO: Fix comment. (Contributed by NM, 30-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  N  =  ( ( P  .\/  v )  ./\  ( Q 
 .\/  ( R `  F ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( v  e.  A  /\  v  .<_  W ) 
 /\  N  e.  A  /\  F  e.  T ) 
 /\  ( P  =/=  Q 
 /\  v  =/=  ( R `  F )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/=  N  /\  z  .<_  ( P  .\/  v
 ) ) ) )
 
Theoremcdlemg33c0 30264* TODO: Fix comment. (Contributed by NM, 30-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  N  =  ( ( P  .\/  v )  ./\  ( Q 
 .\/  ( R `  F ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( v  e.  A  /\  v  .<_  W ) 
 /\  F  e.  T )  /\  ( P  =/=  Q 
 /\  v  =/=  ( R `  F )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  z  .<_  ( P  .\/  v )
 ) )
 
Theoremcdlemg28b 30265* Part of proof of Lemma G of [Crawley] p. 116. Second equality of the equation of line 14 on p. 117. Note that  -.  z  .<_  W is redundant here (but simplifies cdlemg28 30266.) (Contributed by NM, 29-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  N  =  ( ( P  .\/  v )  ./\  ( Q 
 .\/  ( R `  F ) ) )   &    |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `
  G ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( v  e.  A  /\  v  .<_  W ) 
 /\  ( z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )
 )  /\  ( (
 z  =/=  N  /\  z  =/=  O  /\  z  .<_  ( P  .\/  v
 ) )  /\  (
 v  =/=  ( R `  F )  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/= 
 P ) ) ) 
 ->  ( ( Q  .\/  ( F `  ( G `
  Q ) ) )  ./\  W )  =  ( ( z  .\/  ( F `  ( G `
  z ) ) )  ./\  W )
 )
 
Theoremcdlemg28 30266* Part of proof of Lemma G of [Crawley] p. 116. Chain the equalities of line 14 on p. 117. TODO: rearrange hypotheses in the order of cdlemg29 30267 (and maybe leading up to this too)? (Contributed by NM, 29-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  N  =  ( ( P  .\/  v )  ./\  ( Q 
 .\/  ( R `  F ) ) )   &    |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `
  G ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( v  e.  A  /\  v  .<_  W ) 
 /\  ( z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )
 )  /\  ( (
 z  =/=  N  /\  z  =/=  O  /\  z  .<_  ( P  .\/  v
 ) )  /\  (
 v  =/=  ( R `  F )  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/= 
 P ) ) ) 
 ->  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  W )  =  ( ( Q  .\/  ( F `  ( G `
  Q ) ) )  ./\  W )
 )
 
Theoremcdlemg29 30267* Eliminate  ( F `  P )  =/=  P and  ( G `  P )  =/=  P from cdlemg28 30266. TODO: would it be better to do this later? (Contributed by NM, 29-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  N  =  ( ( P  .\/  v )  ./\  ( Q 
 .\/  ( R `  F ) ) )   &    |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `
  G ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( v  e.  A  /\  v  .<_  W ) 
 /\  ( z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )
 )  /\  ( (
 z  =/=  N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G ) ) ) ) 
 ->  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  W )  =  ( ( Q  .\/  ( F `  ( G `
  Q ) ) )  ./\  W )
 )
 
Theoremcdlemg33a 30268* TODO: Fix comment. (Contributed by NM, 29-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  N  =  ( ( P  .\/  v )  ./\  ( Q 
 .\/  ( R `  F ) ) )   &    |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `
  G ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( v  e.  A  /\  v  .<_  W ) 
 /\  ( N  e.  A  /\  O  e.  A )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  =/=  Q  /\  N  =/=  O )  /\  v  =/=  ( R `  F )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
 .\/  r )  =  ( Q  .\/  r
 ) ) ) ) 
 ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/=  N  /\  z  =/=  O  /\  z  .<_  ( P  .\/  v
 ) ) ) )
 
Theoremcdlemg33b 30269* TODO: Fix comment. (Contributed by NM, 30-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  N  =  ( ( P  .\/  v )  ./\  ( Q 
 .\/  ( R `  F ) ) )   &    |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `
  G ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( v  e.  A  /\  v  .<_  W ) 
 /\  ( N  e.  A  /\  O  e.  A )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
 .\/  r )  =  ( Q  .\/  r
 ) ) ) ) 
 ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/=  N  /\  z  =/=  O  /\  z  .<_  ( P  .\/  v
 ) ) ) )
 
Theoremcdlemg33c 30270* TODO: Fix comment. (Contributed by NM, 30-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  N  =  ( ( P  .\/  v )  ./\  ( Q 
 .\/  ( R `  F ) ) )   &    |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `
  G ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( v  e.  A  /\  v  .<_  W ) 
 /\  ( N  e.  A  /\  O  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
 .\/  r )  =  ( Q  .\/  r
 ) ) ) ) 
 ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/=  N  /\  z  =/=  O  /\  z  .<_  ( P  .\/  v
 ) ) ) )
 
Theoremcdlemg33d 30271* TODO: Fix comment. (Contributed by NM, 30-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  N  =  ( ( P  .\/  v )  ./\  ( Q 
 .\/  ( R `  F ) ) )   &    |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `
  G ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( v  e.  A  /\  v  .<_  W ) 
 /\  ( N  =  ( 0. `  K ) 
 /\  O  e.  A )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
 .\/  r )  =  ( Q  .\/  r
 ) ) ) ) 
 ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/=  N  /\  z  =/=  O  /\  z  .<_  ( P  .\/  v
 ) ) ) )
 
Theoremcdlemg33e 30272* TODO: Fix comment. (Contributed by NM, 30-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  N  =  ( ( P  .\/  v )  ./\  ( Q 
 .\/  ( R `  F ) ) )   &    |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `
  G ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( v  e.  A  /\  v  .<_  W ) 
 /\  ( N  =  ( 0. `  K ) 
 /\  O  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
 .\/  r )  =  ( Q  .\/  r
 ) ) ) ) 
 ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/=  N  /\  z  =/=  O  /\  z  .<_  ( P  .\/  v
 ) ) ) )
 
Theoremcdlemg33 30273* Combine cdlemg33b 30269, cdlemg33c 30270, cdlemg33d 30271, cdlemg33e 30272. TODO: Fix comment. (Contributed by NM, 30-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  N  =  ( ( P  .\/  v )  ./\  ( Q 
 .\/  ( R `  F ) ) )   &    |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `
  G ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( v  e.  A  /\  v  .<_  W ) 
 /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q )  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/=  N  /\  z  =/=  O  /\  z  .<_  ( P  .\/  v )
 ) ) )
 
Theoremcdlemg34 30274* Use cdlemg33 to eliminate  z from cdlemg29 30267. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  N  =  ( ( P  .\/  v )  ./\  ( Q 
 .\/  ( R `  F ) ) )   &    |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `
  G ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( v  e.  A  /\  v  .<_  W ) 
 /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q )  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W )  =  ( ( Q 
 .\/  ( F `  ( G `  Q ) ) )  ./\  W ) )
 
Theoremcdlemg35 30275* TODO: Fix comment. TODO: should we have a more general version of hlsupr 28948 to avoid the  =/= conditions? (Contributed by NM, 31-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 )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( F `
  P )  =/= 
 P  /\  ( G `  P )  =/=  P  /\  ( R `  F )  =/=  ( R `  G ) ) ) 
 ->  E. v  e.  A  ( v  .<_  W  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G ) ) ) )
 
Theoremcdlemg36 30276* Use cdlemg35 to eliminate  v from cdlemg34 30274. TODO: Fix comment. (Contributed by NM, 31-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 `  P )  =/=  P  /\  ( G `  P )  =/= 
 P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
 .\/  r )  =  ( Q  .\/  r
 ) ) ) ) 
 ->  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  W )  =  ( ( Q  .\/  ( F `  ( G `
  Q ) ) )  ./\  W )
 )
 
Theoremcdlemg38 30277 Use cdlemg37 30251 to eliminate  E. r  e.  A from cdlemg36 30276. TODO: Fix comment. (Contributed by NM, 31-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 `  P )  =/=  P  /\  ( G `  P )  =/= 
 P )  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  ( ( P  .\/  ( F `  ( G `
  P ) ) )  ./\  W )  =  ( ( Q  .\/  ( F `  ( G `
  Q ) ) )  ./\  W )
 )
 
Theoremcdlemg39 30278 Eliminate  =/= conditions from cdlemg38 30277. TODO: Would this better be done at cdlemg35 30275? TODO: Fix comment. (Contributed by NM, 31-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 ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W )  =  ( ( Q 
 .\/  ( F `  ( G `  Q ) ) )  ./\  W ) )
 
Theoremcdlemg40 30279 Eliminate  P  =/=  Q conditions from cdlemg39 30278. TODO: Fix comment. (Contributed by NM, 31-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  .\/  ( F `  ( G `  P ) ) )  ./\  W )  =  ( ( Q  .\/  ( F `  ( G `  Q ) ) )  ./\  W ) )
 
Theoremcdlemg41 30280 Convert cdlemg40 30279 to function composition. TODO: Fix comment. (Contributed by NM, 31-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  .\/  ( ( F  o.  G ) `  P ) )  ./\  W )  =  ( ( Q  .\/  ( ( F  o.  G ) `  Q ) )  ./\  W ) )
 
Theoremltrnco 30281 The composition of two translations is a translation. Part of proof of Lemma G of [Crawley] p. 116, line 15 on p. 117. (Contributed by NM, 31-May-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  ->  ( F  o.  G )  e.  T )
 
Theoremtrlcocnv 30282 Swap the arguments of the trace of a composition with converse. (Contributed by NM, 1-Jul-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  ->  ( R `  ( F  o.  `' G ) )  =  ( R `
  ( G  o.  `' F ) ) )
 
Theoremtrlcoabs 30283 Absorption into a composition by joining with trace. (Contributed by NM, 22-Jul-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.