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Theorem List for Metamath Proof Explorer - 31901-32000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcdlemk19ylem 31901* Lemma for cdlemk19y 31903. (Contributed by NM, 30-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  (
 ( P  .\/  ( R `  f ) ) 
 ./\  ( ( N `
  P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )   &    |-  C  =  ( e  e.  T  |->  (
 iota_ j  e.  T ( j `  P )  =  ( ( P  .\/  ( R `  e ) )  ./\  ( ( ( S `
  b ) `  P )  .\/  ( R `
  ( e  o.  `' b ) ) ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F ) ) ) )  ->  [_ F  /  g ]_ Y  =  ( N `  P ) )
 
Theoremcdlemk11tb 31902* Part of proof of Lemma K of [Crawley] p. 118. Lemma for Eq. 5, p. 119.  G,  I stand for g, h. cdlemk11ta 31900 with hypotheses removed. TODO: Can this be proved directly with no quantification? (Contributed by NM, 21-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) ) 
 /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
 b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G ) ) 
 /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  I ) ) ) )  ->  [_ G  /  g ]_ Y  .<_  (
 [_ I  /  g ]_ Y  .\/  ( R `
  ( I  o.  `' G ) ) ) )
 
Theoremcdlemk19y 31903* cdlemk19 31840 with simpler hypotheses. TODO: Clean all this up. (Contributed by NM, 30-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) ) 
 /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
 b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F ) ) ) ) 
 ->  [_ F  /  g ]_ Y  =  ( N `  P ) )
 
Theoremcdlemkid3N 31904* Lemma for cdlemkid 31907. (Contributed by NM, 25-Jul-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B )
 ) )  ->  [_ G  /  g ]_ X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G ) )  ->  ( z `  P )  =  P )
 ) )
 
Theoremcdlemkid4 31905* Lemma for cdlemkid 31907. (Contributed by NM, 25-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B )
 ) )  ->  [_ G  /  g ]_ X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G ) )  ->  z  =  (  _I  |`  B ) ) ) )
 
Theoremcdlemkid5 31906* Lemma for cdlemkid 31907. (Contributed by NM, 25-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B )
 ) )  ->  [_ G  /  g ]_ X  e.  T )
 
Theoremcdlemkid 31907* The value of the tau function (in Lemma K of [Crawley] p. 118) on the identity relation. (Contributed by NM, 25-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B )
 ) )  ->  [_ G  /  g ]_ X  =  (  _I  |`  B )
 )
 
Theoremcdlemk35s 31908* Substitution version of cdlemk35 31883. (Contributed by NM, 22-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) ) )  ->  [_ G  /  g ]_ X  e.  T )
 
Theoremcdlemk35s-id 31909* Substitution version of cdlemk35 31883. (Contributed by NM, 26-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  N  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) ) )  ->  [_ G  /  g ]_ X  e.  T )
 
Theoremcdlemk39s 31910* Substitution version of cdlemk39 31887. TODO: Can any commonality with cdlemk35s 31908 be exploited? (Contributed by NM, 23-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) ) )  ->  ( R `  [_ G  /  g ]_ X ) 
 .<_  ( R `  G ) )
 
Theoremcdlemk39s-id 31911* Substitution version of cdlemk39 31887 with non-identity requirement on  G removed. TODO: Can any commonality with cdlemk35s 31908 be exploited? (Contributed by NM, 26-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  N  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) ) )  ->  ( R `  [_ G  /  g ]_ X ) 
 .<_  ( R `  G ) )
 
Theoremcdlemk42 31912* Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 20-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) ) 
 /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
 b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G ) ) ) )  ->  ( [_ G  /  g ]_ X `  P )  =  [_ G  /  g ]_ Y )
 
Theoremcdlemk19xlem 31913* Lemma for cdlemk19x 31914. (Contributed by NM, 30-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  (
 b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F ) ) ) ) 
 ->  ( [_ F  /  g ]_ X `  P )  =  ( N `  P ) )
 
Theoremcdlemk19x 31914* cdlemk19 31840 with simpler hypotheses. TODO: Clean all this up. (Contributed by NM, 30-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) 
 ->  ( [_ F  /  g ]_ X `  P )  =  ( N `  P ) )
 
Theoremcdlemk42yN 31915* Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 20-Jul-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) ) 
 /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
 b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G ) ) ) )  ->  ( [_ G  /  g ]_ X `  P )  =  ( ( P 
 .\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `
  ( G  o.  `' b ) ) ) ) )
 
Theoremcdlemk11tc 31916* Part of proof of Lemma K of [Crawley] p. 118. Lemma for Eq. 5, p. 119.  G,  I stand for g, h. TODO: fix comment. (Contributed by NM, 21-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) ) 
 /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
 b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G ) ) 
 /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  I ) ) ) )  ->  ( [_ G  /  g ]_ X `  P ) 
 .<_  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `
  ( I  o.  `' G ) ) ) )
 
Theoremcdlemk11t 31917* Part of proof of Lemma K of [Crawley] p. 118. Eq. 5, line 36, p. 119.  G,  I stand for g, h.  X represents tau. (Contributed by NM, 21-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) ) 
 /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  ( [_ G  /  g ]_ X `  P ) 
 .<_  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `
  ( I  o.  `' G ) ) ) )
 
Theoremcdlemk45 31918* Part of proof of Lemma K of [Crawley] p. 118. Line 37, p. 119.  G,  I stand for g, h.  X represents tau. They do not explicitly mention the requirement  ( G  o.  I
)  =/=  (  _I  |  `  B ). (Contributed by NM, 22-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) ) 
 /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) 
 /\  ( G  o.  I )  =/=  (  _I  |`  B ) ) )  ->  ( [_ ( G  o.  I
 )  /  g ]_ X `  P )  .<_  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  G ) ) )
 
Theoremcdlemk46 31919* Part of proof of Lemma K of [Crawley] p. 118. Line 38 (last line), p. 119.  G,  I stand for g, h.  X represents tau. (Contributed by NM, 22-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) ) 
 /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) 
 /\  ( G  o.  I )  =/=  (  _I  |`  B ) ) )  ->  ( [_ ( G  o.  I
 )  /  g ]_ X `  P )  .<_  ( ( [_ G  /  g ]_ X `  P )  .\/  ( R `  I ) ) )
 
Theoremcdlemk47 31920* Part of proof of Lemma K of [Crawley] p. 118. Line 2, p. 120.  G,  I stand for g, h.  X represents tau. (Contributed by NM, 22-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) ) 
 /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) 
 /\  ( R `  G )  =/=  ( R `  I ) ) )  ->  ( [_ ( G  o.  I
 )  /  g ]_ X `  P )  =  ( ( ( [_ G  /  g ]_ X `  P )  .\/  ( R `  I ) ) 
 ./\  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  G ) ) ) )
 
Theoremcdlemk48 31921* Part of proof of Lemma K of [Crawley] p. 118. Line 4, p. 120.  G,  I stand for g, h.  X represents tau. (Contributed by NM, 22-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) ) 
 /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  (
 ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) `
  P )  .<_  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  [_ G  /  g ]_ X ) ) )
 
Theoremcdlemk49 31922* Part of proof of Lemma K of [Crawley] p. 118. Line 5, p. 120.  G,  I stand for g, h.  X represents tau. (Contributed by NM, 23-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) ) 
 /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  (
 ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) `
  P )  .<_  ( ( [_ G  /  g ]_ X `  P )  .\/  ( R `  [_ I  /  g ]_ X ) ) )
 
Theoremcdlemk50 31923* Part of proof of Lemma K of [Crawley] p. 118. Line 6, p. 120.  G,  I stand for g, h.  X represents tau. TODO: Combine into cdlemk52 31925? (Contributed by NM, 23-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) ) 
 /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  (
 ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) `
  P )  .<_  ( ( ( [_ G  /  g ]_ X `  P )  .\/  ( R `
  [_ I  /  g ]_ X ) )  ./\  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `
  [_ G  /  g ]_ X ) ) ) )
 
Theoremcdlemk51 31924* Part of proof of Lemma K of [Crawley] p. 118. Line 6, p. 120.  G,  I stand for g, h.  X represents tau. TODO: Combine into cdlemk52 31925? (Contributed by NM, 23-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) ) 
 /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  (
 ( ( [_ G  /  g ]_ X `  P )  .\/  ( R `
  [_ I  /  g ]_ X ) )  ./\  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `
  [_ G  /  g ]_ X ) ) ) 
 .<_  ( ( ( [_ G  /  g ]_ X `  P )  .\/  ( R `  I ) ) 
 ./\  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  G ) ) ) )
 
Theoremcdlemk52 31925* Part of proof of Lemma K of [Crawley] p. 118. Line 6, p. 120.  G,  I stand for g, h.  X represents tau. (Contributed by NM, 23-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) ) 
 /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) 
 /\  ( R `  G )  =/=  ( R `  I ) ) )  ->  ( ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) `
  P )  =  ( [_ ( G  o.  I )  /  g ]_ X `  P ) )
 
Theoremcdlemk53a 31926* Lemma for cdlemk53 31928. (Contributed by NM, 26-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) ) 
 /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) 
 /\  ( R `  G )  =/=  ( R `  I ) ) )  ->  [_ ( G  o.  I )  /  g ]_ X  =  (
 [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) )
 
Theoremcdlemk53b 31927* Lemma for cdlemk53 31928. (Contributed by NM, 26-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) 
 /\  ( R `  G )  =/=  ( R `  I ) ) )  ->  [_ ( G  o.  I )  /  g ]_ X  =  (
 [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) )
 
Theoremcdlemk53 31928* Part of proof of Lemma K of [Crawley] p. 118. Line 7, p. 120.  G,  I stand for g, h.  X represents tau. (Contributed by NM, 26-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G )  =/=  ( R `  I
 ) ) )  ->  [_ ( G  o.  I
 )  /  g ]_ X  =  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) )
 
Theoremcdlemk54 31929* Part of proof of Lemma K of [Crawley] p. 118. Line 10, p. 120.  G,  I stand for g, h.  X represents tau. (Contributed by NM, 26-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( I  e.  T  /\  ( R `  G )  =  ( R `  I ) )  /\  j  e.  T  /\  ( j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G )  /\  ( R `
  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  ( [_ ( G  o.  I )  /  g ]_ X  o.  [_ j  /  g ]_ X )  =  ( ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X )  o.  [_ j  /  g ]_ X ) )
 
Theoremcdlemk55a 31930* Lemma for cdlemk55 31932. (Contributed by NM, 26-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( I  e.  T  /\  ( R `  G )  =  ( R `  I ) )  /\  j  e.  T  /\  ( j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G )  /\  ( R `
  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  [_ ( G  o.  I
 )  /  g ]_ X  =  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) )
 
Theoremcdlemk55b 31931* Lemma for cdlemk55 31932. (Contributed by NM, 26-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G )  =  ( R `  I
 ) ) )  ->  [_ ( G  o.  I
 )  /  g ]_ X  =  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) )
 
Theoremcdlemk55 31932* Part of proof of Lemma K of [Crawley] p. 118. Line 11, p. 120.  G,  I stand for g, h.  X represents tau. (Contributed by NM, 26-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  [_ ( G  o.  I )  /  g ]_ X  =  (
 [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) )
 
TheoremcdlemkyyN 31933* Part of proof of Lemma K of [Crawley] p. 118. TODO: clean up  ( b Y G ) stuff. (Contributed by NM, 21-Jul-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `
  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )   &    |-  V  =  ( d  e.  T ,  e  e.  T  |->  ( iota_ j  e.  T ( j `
  P )  =  ( ( P  .\/  ( R `  e ) )  ./\  ( (
 ( S `  d
 ) `  P )  .\/  ( R `  (
 e  o.  `' d
 ) ) ) ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) 
 /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  (
 b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `
  b )  =/=  ( R `  G ) ) ) ) 
 ->  ( [_ G  /  g ]_ X `  P )  =  ( (
 b V G ) `
  P ) )
 
Theoremcdlemk43N 31934* Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 31-Jul-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   &    |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  N  e.  T  /\  F  =/=  N ) 
 /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  (
 b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `
  b )  =/=  ( R `  G ) ) ) ) 
 ->  ( ( U `  G ) `  P )  =  [_ G  /  g ]_ Y )
 
Theoremcdlemk35u 31935* Substitution version of cdlemk35 31883. (Contributed by NM, 31-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   &    |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( U `  G )  e.  T )
 
Theoremcdlemk55u1 31936* Lemma for cdlemk55u 31937. (Contributed by NM, 31-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   &    |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( ( ( R `
  F )  =  ( R `  N )  /\  F  =/=  N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( U `  ( G  o.  I ) )  =  ( ( U `  G )  o.  ( U `  I ) ) )
 
Theoremcdlemk55u 31937* Part of proof of Lemma K of [Crawley] p. 118. Line 11, p. 120.  G,  I stand for g, h.  X represents tau. (Contributed by NM, 31-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   &    |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( ( R `  F )  =  ( R `  N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( U `  ( G  o.  I
 ) )  =  ( ( U `  G )  o.  ( U `  I ) ) )
 
Theoremcdlemk39u1 31938* Lemma for cdlemk39u 31939. (Contributed by NM, 31-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   &    |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( ( R `  F )  =  ( R `  N )  /\  F  =/=  N  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) 
 ->  ( R `  ( U `  G ) ) 
 .<_  ( R `  G ) )
 
Theoremcdlemk39u 31939* Part of proof of Lemma K of [Crawley] p. 118. Line 31, p. 119. Trace-preserving property of the value of tau, represented by  ( U `  G ). (Contributed by NM, 31-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   &    |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  ( U `  G ) )  .<_  ( R `
  G ) )
 
Theoremcdlemk19u1 31940* cdlemk19 31840 with simpler hypotheses. TODO: Clean all this up. (Contributed by NM, 31-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   &    |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( U `  F ) `  P )  =  ( N `  P ) )
 
Theoremcdlemk19u 31941* Part of Lemma K of [Crawley] p. 118. Line 12, p. 120, "f (exponent) tau = k". We represent f, k, tau with  F,  N,  U. (Contributed by NM, 31-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   &    |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X )
 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( U `  F )  =  N )
 
Theoremcdlemk56 31942* Part of Lemma K of [Crawley] p. 118. Line 11, p. 120, "tau is in Delta" i.e.  U is a trace-preserving endormorphism. (Contributed by NM, 31-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   &    |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X )
 )   &    |-  E  =  ( (
 TEndo `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) 
 ->  U  e.  E )
 
Theoremcdlemk19w 31943* Use a fixed element to eliminate  P in cdlemk19u 31941. (Contributed by NM, 1-Aug-2013.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  P  =  (  ._|_  `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) ) 
 ./\  ( Z  .\/  ( R `  ( g  o.  `' b ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   &    |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N ) )  ->  ( U `  F )  =  N )
 
Theoremcdlemk56w 31944* Use a fixed element to eliminate  P in cdlemk56 31942. (Contributed by NM, 1-Aug-2013.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  P  =  (  ._|_  `  W )   &    |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( b  o.  `' F ) ) ) )   &    |-  Y  =  ( ( P  .\/  ( R `  g ) ) 
 ./\  ( Z  .\/  ( R `  ( g  o.  `' b ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  P )  =  Y )
 )   &    |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X )
 )   &    |-  E  =  ( (
 TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N ) )  ->  ( U  e.  E  /\  ( U `  F )  =  N )
 )
 
Theoremcdlemk 31945* Lemma K of [Crawley] p. 118. Final result, lines 11 and 12 on p. 120: given two translations f and k with the same trace, there exists a trace-preserving endomorphism tau whose value at f is k. We use  F,  N, and  u to represent f, k, and tau. (Contributed by NM, 1-Aug-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T ) 
 /\  ( R `  F )  =  ( R `  N ) ) 
 ->  E. u  e.  E  ( u `  F )  =  N )
 
Theoremtendoex 31946* Generalization of Lemma K of [Crawley] p. 118, cdlemk 31945. TODO: can this be used to shorten uses of cdlemk 31945? (Contributed by NM, 15-Oct-2013.)
 |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T ) 
 /\  ( R `  N )  .<_  ( R `
  F ) ) 
 ->  E. u  e.  E  ( u `  F )  =  N )
 
Theoremcdleml1N 31947 Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  ( f  =/=  (  _I  |`  B )  /\  ( U `  f )  =/=  (  _I  |`  B ) 
 /\  ( V `  f )  =/=  (  _I  |`  B ) ) )  ->  ( R `  ( U `  f
 ) )  =  ( R `  ( V `
  f ) ) )
 
Theoremcdleml2N 31948* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  ( f  =/=  (  _I  |`  B )  /\  ( U `  f )  =/=  (  _I  |`  B ) 
 /\  ( V `  f )  =/=  (  _I  |`  B ) ) )  ->  E. s  e.  E  ( s `  ( U `  f ) )  =  ( V `
  f ) )
 
Theoremcdleml3N 31949* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  .0.  =  ( g  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  ( f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/=  .0.  ) )  ->  E. s  e.  E  ( s  o.  U )  =  V )
 
Theoremcdleml4N 31950* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  .0.  =  ( g  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  ( U  =/=  .0.  /\  V  =/=  .0.  )
 )  ->  E. s  e.  E  ( s  o.  U )  =  V )
 
Theoremcdleml5N 31951* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  .0.  =  ( g  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  ->  E. s  e.  E  ( s  o.  U )  =  V )
 
Theoremcdleml6 31952* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Q  =  ( ( oc `  K ) `  W )   &    |-  Z  =  ( ( Q  .\/  ( R `  b ) )  ./\  ( ( h `  Q )  .\/  ( R `
  ( b  o.  `' ( s `  h ) ) ) ) )   &    |-  Y  =  ( ( Q  .\/  ( R `  g ) ) 
 ./\  ( Z  .\/  ( R `  ( g  o.  `' b ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  ( s `  h ) )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  Q )  =  Y )
 )   &    |-  U  =  ( g  e.  T  |->  if (
 ( s `  h )  =  h ,  g ,  X )
 )   &    |-  E  =  ( (
 TEndo `  K ) `  W )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  ( U  e.  E  /\  ( U `  (
 s `  h )
 )  =  h ) )
 
Theoremcdleml7 31953* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Q  =  ( ( oc `  K ) `  W )   &    |-  Z  =  ( ( Q  .\/  ( R `  b ) )  ./\  ( ( h `  Q )  .\/  ( R `
  ( b  o.  `' ( s `  h ) ) ) ) )   &    |-  Y  =  ( ( Q  .\/  ( R `  g ) ) 
 ./\  ( Z  .\/  ( R `  ( g  o.  `' b ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  ( s `  h ) )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  Q )  =  Y )
 )   &    |-  U  =  ( g  e.  T  |->  if (
 ( s `  h )  =  h ,  g ,  X )
 )   &    |-  E  =  ( (
 TEndo `  K ) `  W )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  ( ( U  o.  s ) `  h )  =  ( (  _I  |`  T ) `  h ) )
 
Theoremcdleml8 31954* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Q  =  ( ( oc `  K ) `  W )   &    |-  Z  =  ( ( Q  .\/  ( R `  b ) )  ./\  ( ( h `  Q )  .\/  ( R `
  ( b  o.  `' ( s `  h ) ) ) ) )   &    |-  Y  =  ( ( Q  .\/  ( R `  g ) ) 
 ./\  ( Z  .\/  ( R `  ( g  o.  `' b ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  ( s `  h ) )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  Q )  =  Y )
 )   &    |-  U  =  ( g  e.  T  |->  if (
 ( s `  h )  =  h ,  g ,  X )
 )   &    |-  E  =  ( (
 TEndo `  K ) `  W )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  ( U  o.  s
 )  =  (  _I  |`  T ) )
 
Theoremcdleml9 31955* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Q  =  ( ( oc `  K ) `  W )   &    |-  Z  =  ( ( Q  .\/  ( R `  b ) )  ./\  ( ( h `  Q )  .\/  ( R `
  ( b  o.  `' ( s `  h ) ) ) ) )   &    |-  Y  =  ( ( Q  .\/  ( R `  g ) ) 
 ./\  ( Z  .\/  ( R `  ( g  o.  `' b ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  ( s `  h ) )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  Q )  =  Y )
 )   &    |-  U  =  ( g  e.  T  |->  if (
 ( s `  h )  =  h ,  g ,  X )
 )   &    |-  E  =  ( (
 TEndo `  K ) `  W )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  U  =/=  .0.  )
 
Theoremdva1dim 31956* Two expressions for the 1-dimensional subspaces of partial vector space A. Remark in [Crawley] p. 120 line 21, but using a non-identity translation (nonzero vector) 
F whose trace is  P rather than  P itself;  F exists by cdlemf 31534. 
E is the division ring base by erngdv 31964, and  s `  F is the scalar product by dvavsca 31988. 
F must be a non-identity translation for the expression to be a 1-dimensional subspace, although the theorem doesn't require it. (Contributed by NM, 14-Oct-2013.)
 |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T ) 
 ->  { g  |  E. s  e.  E  g  =  ( s `  F ) }  =  {
 g  e.  T  |  ( R `  g ) 
 .<_  ( R `  F ) } )
 
Theoremdvhb1dimN 31957* Two expressions for the 1-dimensional subspaces of vector space H, in the isomorphism B case where the 2nd vector component is zero. (Contributed by NM, 23-Feb-2014.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  .0.  =  ( h  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  ->  { g  e.  ( T  X.  E )  |  E. s  e.  E  g  =  <. ( s `  F ) ,  .0.  >. }  =  { g  e.  ( T  X.  E )  |  ( ( R `  ( 1st `  g )
 )  .<_  ( R `  F )  /\  ( 2nd `  g )  =  .0.  ) } )
 
Theoremerng1lem 31958 Value of the endomorphism division ring unit. (Contributed by NM, 12-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  D  e.  Ring )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  ( 1r `  D )  =  (  _I  |`  T ) )
 
Theoremerngdvlem1 31959* Lemma for erngrng 31963. (Contributed by NM, 4-Aug-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  B  =  ( Base `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( a  e.  E ,  b  e.  E  |->  ( f  e.  T  |->  ( ( a `  f )  o.  (
 b `  f )
 ) ) )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( a  e.  E  |->  ( f  e.  T  |->  `' (
 a `  f )
 ) )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Grp )
 
Theoremerngdvlem2N 31960* Lemma for erngrng 31963. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  B  =  ( Base `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( a  e.  E ,  b  e.  E  |->  ( f  e.  T  |->  ( ( a `  f )  o.  (
 b `  f )
 ) ) )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( a  e.  E  |->  ( f  e.  T  |->  `' (
 a `  f )
 ) )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Abel )
 
Theoremerngdvlem3 31961* Lemma for erngrng 31963. (Contributed by NM, 6-Aug-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &