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Theorem List for Metamath Proof Explorer - 30301-30400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcdlemk55 30301* 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 30302* 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 30303* 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 30304* Substitution version of cdlemk35 30252. (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 30305* Lemma for cdlemk55u 30306. (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 30306* 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 30307* Lemma for cdlemk39u 30308. (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 30308* 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 30309* cdlemk19 30209 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 30310* 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 30311* 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 30312* Use a fixed element to eliminate  P in cdlemk19u 30310. (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 30313* Use a fixed element to eliminate  P in cdlemk56 30311. (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 30314* 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 30315* Generalization of Lemma K of [Crawley] p. 118, cdlemk 30314. TODO: can this be used to shorten uses of cdlemk 30314? (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 30316 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 30317* 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 30318* 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 30319* 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 30320* 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 30321* 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 30322* 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 30323* 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 30324* 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 30325* 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 29903. 
E is the division ring base by erngdv 30333, and  s `  F is the scalar product by dvavsca 30357. 
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 30326* 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 30327 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 30328* Lemma for erngrng 30332. (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 30329* Lemma for erngrng 30332. (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 30330* Lemma for erngrng 30332. (Contributed by NM, 6-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 )
 ) )   &    |-  .+  =  (
 a  e.  E ,  b  e.  E  |->  ( a  o.  b ) )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Ring )
 
Theoremerngdvlem4 30331* Lemma for erngdv 30333. (Contributed by NM, 11-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 )
 ) )   &    |-  .+  =  (
 a  e.  E ,  b  e.  E  |->  ( a  o.  b ) )   &    |-  .\/ 
 =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  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 )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  ->  D  e. 
 DivRing )
 
Theoremerngrng 30332 An endomorphism ring is a ring. Todo: fix comment. (Contributed by NM, 4-Aug-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Ring )
 
Theoremerngdv 30333 An endomorphism ring is a division ring. Todo: fix comment. (Contributed by NM, 11-Aug-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e. 
 DivRing )
 
Theoremerng0g 30334* The division ring zero of an endomorphism ring. (Contributed by NM, 5-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |- 
 .0.  =  ( 0g `  D )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  =  O )
 
Theoremerng1r 30335 The division ring unit of an endomorphism ring. (Contributed by NM, 5-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  .1.  =  ( 1r `  D )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .1.  =  (  _I  |`  T ) )
 
Theoremerngdvlem1-rN 30336* Lemma for erngrng 30332. (Contributed by NM, 4-Aug-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing R `  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 ) ) ) )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( a  e.  E  |->  ( f  e.  T  |->  `' (
 a `  f )
 ) )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Grp )
 
Theoremerngdvlem2-rN 30337* Lemma for erngrng 30332. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing R `  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 ) ) ) )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( a  e.  E  |->  ( f  e.  T  |->  `' (
 a `  f )
 ) )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Abel )
 
Theoremerngdvlem3-rN 30338* Lemma for erngrng 30332. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing R `  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 ) ) ) )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( a  e.  E  |->  ( f  e.  T  |->  `' (
 a `  f )
 ) )   &    |-  M  =  ( a  e.  E ,  b  e.  E  |->  ( b  o.  a ) )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Ring )
 
Theoremerngdvlem4-rN 30339* Lemma for erngdv 30333. (Contributed by NM, 11-Aug-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing R `  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 ) ) ) )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( a  e.  E  |->  ( f  e.  T  |->  `' (
 a `  f )
 ) )   &    |-  M  =  ( a  e.  E ,  b  e.  E  |->  ( b  o.  a ) )   &    |-  .\/ 
 =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  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 )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  ->  D  e. 
 DivRing )
 
Theoremerngrng-rN 30340 An endomorphism ring is a ring. Todo: fix comment. (Contributed by NM, 4-Aug-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing R `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Ring )
 
Theoremerngdv-rN 30341 An endomorphism ring is a division ring. Todo: fix comment. (Contributed by NM, 11-Aug-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing R `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )
 
Syntaxcdveca 30342 Extend class notation with constructed vector space A.
 class  DVecA
 
Definitiondf-dveca 30343* Define constructed partial vector space A. (Contributed by NM, 8-Oct-2013.)
 |-  DVecA  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( { <. ( Base `  ndx ) ,  ( ( LTrn `  k
 ) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  k
 ) `  w ) ,  g  e.  (
 ( LTrn `  k ) `  w )  |->  ( f  o.  g ) )
 >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `  k
 ) `  w ) >. }  u.  { <. ( .s `  ndx ) ,  ( s  e.  (
 ( TEndo `  k ) `  w ) ,  f  e.  ( ( LTrn `  k
 ) `  w )  |->  ( s `  f
 ) ) >. } )
 ) )
 
Theoremdvafset 30344* The constructed partial vector space A for a lattice  K. (Contributed by NM, 8-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  V  ->  (
 DVecA `  K )  =  ( w  e.  H  |->  ( { <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  w ) >. ,  <. (
 +g  `  ndx ) ,  ( f  e.  (
 ( LTrn `  K ) `  w ) ,  g  e.  ( ( LTrn `  K ) `  w )  |->  ( f  o.  g ) ) >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `
  K ) `  w ) >. }  u.  {
 <. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K ) `  w ) ,  f  e.  ( (
 LTrn `  K ) `  w )  |->  ( s `
  f ) )
 >. } ) ) )
 
Theoremdvaset 30345* The constructed partial vector space A for a lattice  K. (Contributed by NM, 8-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   =>    |-  ( ( K  e.  X  /\  W  e.  H )  ->  U  =  ( { <. ( Base ` 
 ndx ) ,  T >. ,  <. ( +g  `  ndx ) ,  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g ) ) >. , 
 <. (Scalar `  ndx ) ,  D >. }  u.  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  T  |->  ( s `  f ) ) >. } ) )
 
Theoremdvasca 30346 The ring base set of the constructed partial vector space A are all translation group endomorphisms (for a fiducial co-atom  W). (Contributed by NM, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  F  =  (Scalar `  U )   =>    |-  (
 ( K  e.  X  /\  W  e.  H ) 
 ->  F  =  D )
 
Theoremdvabase 30347 The ring base set of the constructed partial vector space A are all translation group endomorphisms (for a fiducial co-atom  W). (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  C  =  ( Base `  F )   =>    |-  (
 ( K  e.  X  /\  W  e.  H ) 
 ->  C  =  E )
 
Theoremdvafplusg 30348* Ring addition operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  .+  =  ( +g  `  F )   =>    |-  (
 ( K  e.  V  /\  W  e.  H ) 
 ->  .+  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `
  f )  o.  ( t `  f
 ) ) ) ) )
 
Theoremdvaplusg 30349* Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  .+  =  ( +g  `  F )   =>    |-  (
 ( ( K  e.  V  /\  W  e.  H )  /\  ( R  e.  E  /\  S  e.  E ) )  ->  ( R 
 .+  S )  =  ( f  e.  T  |->  ( ( R `  f )  o.  ( S `  f ) ) ) )
 
Theoremdvaplusgv 30350 Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  .+  =  ( +g  `  F )   =>    |-  (
 ( ( K  e.  V  /\  W  e.  H )  /\  ( R  e.  E  /\  S  e.  E  /\  G  e.  T ) )  ->  ( ( R  .+  S ) `  G )  =  (
 ( R `  G )  o.  ( S `  G ) ) )
 
Theoremdvafmulr 30351* Ring multiplication operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  .x.  =  ( .r `  F )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  .x.  =  (
 s  e.  E ,  t  e.  E  |->  ( s  o.  t ) ) )
 
Theoremdvamulr 30352 Ring multiplication operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  .x.  =  ( .r `  F )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( R  e.  E  /\  S  e.  E )
 )  ->  ( R  .x.  S )  =  ( R  o.  S ) )
 
Theoremdvavbase 30353 The vectors (vector base set) of the constructed partial vector space A are all translations (for a fiducial co-atom  W). (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  V  =  ( Base `  U )   =>    |-  (
 ( K  e.  X  /\  W  e.  H ) 
 ->  V  =  T )
 
Theoremdvafvadd 30354* The vector sum operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   =>    |-  (
 ( K  e.  X  /\  W  e.  H ) 
 ->  .+  =  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g ) ) )
 
Theoremdvavadd 30355 Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   =>    |-  (
 ( ( K  e.  V  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T ) )  ->  ( F 
 .+  G )  =  ( F  o.  G ) )
 
Theoremdvafvsca 30356* Ring addition operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  .x.  =  ( .s `  U )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  .x.  =  (
 s  e.  E ,  f  e.  T  |->  ( s `
  f ) ) )
 
Theoremdvavsca 30357 Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  .x.  =  ( .s `  U )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( R  e.  E  /\  F  e.  T )
 )  ->  ( R  .x.  F )  =  ( R `  F ) )
 
Theoremtendospid 30358 Identity property of endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
 |-  ( F  e.  T  ->  ( (  _I  |`  T ) `
  F )  =  F )
 
Theoremtendospcl 30359 Closure of endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  U  e.  E  /\  F  e.  T )  ->  ( U `  F )  e.  T )
 
Theoremtendospass 30360 Associative law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  X  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  F  e.  T )
 )  ->  ( ( U  o.  V ) `  F )  =  ( U `  ( V `  F ) ) )
 
Theoremtendospdi1 30361 Forward distributive law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T )
 )  ->  ( U `  ( F  o.  G ) )  =  (
 ( U `  F )  o.  ( U `  G ) ) )
 
Theoremtendocnv 30362 Converse of a trace-preserving endomorphism value. (Contributed by NM, 7-Apr-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E  /\  F  e.  T )  ->  `' ( S `  F )  =  ( S `  `' F ) )
 
Theoremtendospdi2 30363* Reverse distributive law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
 |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   =>    |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  ( ( U P V ) `
  F )  =  ( ( U `  F )  o.  ( V `  F ) ) )
 
TheoremtendospcanN 30364* Cancellation law for trace-perserving endomorphism values (used as scalar product). (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  E  /\  S  =/=  O )  /\  ( F  e.  T  /\  G  e.  T ) )  ->  ( ( S `  F )  =  ( S `  G ) 
 <->  F  =  G ) )
 
Theoremdvaabl 30365 The constructed partial vector space A for a lattice  K is an abelian group. (Contributed by NM, 11-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  Abel )
 
Theoremdvalveclem 30366 Lemma for dvalvec 30367. (Contributed by NM, 11-Oct-2013.) (Proof shortened by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  (Scalar `  U )   &    |-  B  =  ( Base `  K )   &    |-  .+^  =  (
 +g  `  D )   &    |-  .X.  =  ( .r `  D )   &    |-  .x. 
 =  ( .s `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LVec )
 
Theoremdvalvec 30367 The constructed partial vector space A for a lattice  K is a left vector space. (Contributed by NM, 11-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LVec )
 
Theoremdva0g 30368 The zero vector of partial vector space A. (Contributed by NM, 9-Sep-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  .0.  =  ( 0g `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  =  (  _I  |`  B ) )
 
Syntaxcdia 30369 Extend class notation with partial isomorphism A.
 class  DIsoA
 
Definitiondf-disoa 30370* Define partial isomorphism A. (Contributed by NM, 15-Oct-2013.)
 |-  DIsoA  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( x  e.  { y  e.  ( Base `  k )  |  y ( le `  k
 ) w }  |->  { f  e.  ( (
 LTrn `  k ) `  w )  |  (
 ( ( trL `  k
 ) `  w ) `  f ) ( le `  k ) x }
 ) ) )
 
Theoremdiaffval 30371* The partial isomorphism A for a lattice  K. (Contributed by NM, 15-Oct-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  V  ->  ( DIsoA `  K )  =  ( w  e.  H  |->  ( x  e.  { y  e.  B  |  y  .<_  w }  |->  { f  e.  (
 ( LTrn `  K ) `  w )  |  ( ( ( trL `  K ) `  w ) `  f )  .<_  x }
 ) ) )
 
Theoremdiafval 30372* The partial isomorphism A for a lattice  K. (Contributed by NM, 15-Oct-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  {
 y  e.  B  |  y  .<_  W }  |->  { f  e.  T  |  ( R `  f ) 
 .<_  x } ) )
 
Theoremdiaval 30373* The partial isomorphism A for a lattice  K. Definition of isomorphism map in [Crawley] p. 120 line 24. (Contributed by NM, 15-Oct-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  =  { f  e.  T  |  ( R `
  f )  .<_  X } )
 
Theoremdiaelval 30374 Member of the partial isomorphism A for a lattice  K. (Contributed by NM, 3-Dec-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( F  e.  ( I `  X )  <->  ( F  e.  T  /\  ( R `  F )  .<_  X ) ) )
 
Theoremdiafn 30375* Functionality and domain of the partial isomorphism A. (Contributed by NM, 26-Nov-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { x  e.  B  |  x  .<_  W } )
 
Theoremdiadm 30376* Domain of the partial isomorphism A. (Contributed by NM, 3-Dec-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  { x  e.  B  |  x  .<_  W }
 )
 
Theoremdiaeldm 30377 Member of domain of the partial isomorphism A. (Contributed by NM, 4-Dec-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom 
 I 
 <->  ( X  e.  B  /\  X  .<_  W ) ) )
 
TheoremdiadmclN 30378 A member of domain of the partial isomorphism A is a lattice element. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  X  e.  dom  I
 )  ->  X  e.  B )
 
TheoremdiadmleN 30379 A member of domain of the partial isomorphism A is under the fiducial hyperplane. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  X  e.  dom  I )  ->  X  .<_  W )
 
Theoremdian0 30380 The value of the partial isomorphism A is not empty. (Contributed by NM, 17-Jan-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  =/=  (/) )
 
Theoremdia0eldmN 30381 The lattice zero belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
 |-  .0.  =  ( 0. `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  e.  dom  I )
 
Theoremdia1eldmN 30382 The fiducial hyperplane (largest allowed lattice element) belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W  e.  dom  I )
 
Theoremdiass 30383 The value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  C_  T )
 
Theoremdiael 30384 A member of the value of the partial isomorphism A is a translation i.e. a vector. (Contributed by NM, 17-Jan-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  F  e.  ( I `  X ) )  ->  F  e.  T )
 
Theoremdiatrl 30385 Trace of a member of the partial isomorphism A. (Contributed by NM, 17-Jan-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  F  e.  ( I `  X ) )  ->  ( R `  F ) 
 .<_  X )
 
TheoremdiaelrnN 30386 Any value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  S  e.  ran  I
 )  ->  S  C_  T )
 
Theoremdialss 30387 The value of partial isomorphism A is a subspace of partial vector space A. Part of Lemma M of [Crawley] p. 120 line 26. (Contributed by NM, 17-Jan-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  S  =  (
 LSubSp `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  e.  S )
 
Theoremdiaord 30388 The partial isomorphism A for a lattice  K is order-preserving in the region under co-atom  W. Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 26-Nov-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) ) 
 ->  ( ( I `  X )  C_  ( I `
  Y )  <->  X  .<_  Y ) )
 
Theoremdia11N 30389 The partial isomorphism A for a lattice  K is one-to-one in the region under co-atom  W. Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 25-Nov-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) ) 
 ->  ( ( I `  X )  =  ( I `  Y )  <->  X  =  Y ) )
 
Theoremdiaf11N 30390 The partial isomorphism A for a lattice  K is a one-to-one function. . Part of Lemma M of [Crawley] p. 120 line 27. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : dom  I -1-1-onto-> ran  I )
 
TheoremdiaclN 30391 Closure of partial isomorphism A for a lattice  K. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I
 )  ->  ( I `  X )  e.  ran  I )
 
TheoremdiacnvclN 30392 Closure of partial isomorphism A converse. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I
 )  ->  ( `' I `  X )  e. 
 dom  I )
 
Theoremdia0 30393 The value of the partial isomorphism A at the lattice zero is the singleton of the identity translation i.e. the zero subspace. (Contributed by NM, 26-Nov-2013.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  .0.  )  =  { (  _I  |`  B ) }
 )
 
Theoremdia1N 30394 The value of the partial isomorphism A at the fiducial co-atom is the set of all translations i.e. the entire vector space. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  W )  =  T )
 
Theoremdia1elN 30395 The largest subspace in the range of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  T  e.  ran  I )
 
TheoremdiaglbN 30396* Partial isomorphism A of a lattice glb. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)
 |-  G  =  ( glb `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  dom  I  /\  S  =/=  (/) ) )  ->  ( I `  ( G `
  S ) )  =  |^|_ x  e.  S  ( I `  x ) )
 
TheoremdiameetN 30397 Partial isomorphism A of a lattice meet. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
 |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  dom  I 
 /\  Y  e.  dom  I ) )  ->  ( I `  ( X  ./\  Y ) )  =  ( ( I `  X )  i^i  ( I `  Y ) ) )
 
TheoremdiainN 30398 Inverse partial isomorphism A of an intersection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
 |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  ran  I 
 /\  Y  e.  ran  I ) )  ->  ( X  i^i  Y )  =  ( I `  (
 ( `' I `  X )  ./\  ( `' I `  Y ) ) ) )
 
TheoremdiaintclN 30399 The intersection of partial isomorphism A closed subspaces is a closed subspace. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) ) 
 ->  |^| S  e.  ran  I )
 
TheoremdiasslssN 30400 The partial isomorphism A maps to subspaces of partial vector space A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  ran  I  C_  S )
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