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Theorem List for Metamath Proof Explorer - 36601-36700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcdlemg44b 36601 Eliminate  ( F `  P )  =/=  P,  ( G `  P )  =/=  P from cdlemg44a 36600. (Contributed by NM, 3-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
  F )  =/=  ( R `  G ) )  ->  ( F `
  ( G `  P ) )  =  ( G `  ( F `  P ) ) )
 
Theoremcdlemg44 36602 Part of proof of Lemma G of [Crawley] p. 116, fifth line of third paragraph on p. 117: "and hence fg = gf." (Contributed by NM, 3-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T ) 
 /\  ( R `  F )  =/=  ( R `  G ) ) 
 ->  ( F  o.  G )  =  ( G  o.  F ) )
 
Theoremcdlemg47a 36603 TODO: fix comment. TODO: Use this above in place of  ( F `  P )  =  P antecedents? (Contributed by NM, 5-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T ) 
 /\  F  =  (  _I  |`  B )
 )  ->  ( F  o.  G )  =  ( G  o.  F ) )
 
Theoremcdlemg46 36604* Part of proof of Lemma G of [Crawley] p. 116, seventh line of third paragraph on p. 117: "hf and f have different traces." (Contributed by NM, 5-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) ) 
 ->  ( R `  ( h  o.  F ) )  =/=  ( R `  F ) )
 
Theoremcdlemg47 36605* Part of proof of Lemma G of [Crawley] p. 116, ninth line of third paragraph on p. 117: "we conclude that gf = fg." (Contributed by NM, 5-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  F  e.  T  /\  G  e.  T ) 
 /\  ( h  e.  T  /\  ( R `
  F )  =  ( R `  G ) )  /\  ( F  =/=  (  _I  |`  B ) 
 /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) ) 
 ->  ( F  o.  G )  =  ( G  o.  F ) )
 
Theoremcdlemg48 36606 Elmininate  h from cdlemg47 36605. (Contributed by NM, 5-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  G ) ) ) 
 ->  ( F  o.  G )  =  ( G  o.  F ) )
 
Theoremltrncom 36607 Composition is commutative for translations. Part of proof of Lemma G of [Crawley] p. 116 (Contributed by NM, 5-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  ->  ( F  o.  G )  =  ( G  o.  F ) )
 
Theoremltrnco4 36608 Rearrange a composition of 4 translations, analogous to an4 824. (Contributed by NM, 10-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  E  e.  T  /\  F  e.  T )  ->  ( ( D  o.  E )  o.  ( F  o.  G ) )  =  ( ( D  o.  F )  o.  ( E  o.  G ) ) )
 
Theoremtrljco 36609 Trace joined with trace of composition. (Contributed by NM, 15-Jun-2013.)
 |-  .\/  =  ( join `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  ->  ( ( R `  F )  .\/  ( R `
  ( F  o.  G ) ) )  =  ( ( R `
  F )  .\/  ( R `  G ) ) )
 
Theoremtrljco2 36610 Trace joined with trace of composition. (Contributed by NM, 16-Jun-2013.)
 |-  .\/  =  ( join `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  ->  ( ( R `  F )  .\/  ( R `
  ( F  o.  G ) ) )  =  ( ( R `
  G )  .\/  ( R `  ( F  o.  G ) ) ) )
 
Syntaxctgrp 36611 Extend class notation with translation group.
 class  TGrp
 
Definitiondf-tgrp 36612* Define the class of all translation groups.  k is normally a member of  HL. Each base set is the set of all lattice translations with respect to a hyperplane  w, and the operation is function composition. Similar to definition of G in [Crawley] p. 116, third paragraph (which defines this for geomodular lattices). (Contributed by NM, 5-Jun-2013.)
 |-  TGrp  =  ( 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 ) )
 >. } ) )
 
Theoremtgrpfset 36613* The translation group maps for a lattice  K. (Contributed by NM, 5-Jun-2013.)
 |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  V  ->  (
 TGrp `  K )  =  ( w  e.  H  |->  {
 <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  w ) >. , 
 <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K ) `  w ) ,  g  e.  ( (
 LTrn `  K ) `  w )  |->  ( f  o.  g ) )
 >. } ) )
 
Theoremtgrpset 36614* The translation group for a fiducial co-atom  W. (Contributed by NM, 5-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  G  =  ( ( TGrp `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  G  =  { <. ( Base `  ndx ) ,  T >. , 
 <. ( +g  `  ndx ) ,  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g ) ) >. } )
 
Theoremtgrpbase 36615 The base set of the translation group is the set of all translations (for a fiducial co-atom  W). (Contributed by NM, 5-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  G  =  ( ( TGrp `  K ) `  W )   &    |-  C  =  ( Base `  G )   =>    |-  (
 ( K  e.  V  /\  W  e.  H ) 
 ->  C  =  T )
 
Theoremtgrpopr 36616* The group operation of the translation group is function composition. (Contributed by NM, 5-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  G  =  ( ( TGrp `  K ) `  W )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( K  e.  V  /\  W  e.  H ) 
 ->  .+  =  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g ) ) )
 
Theoremtgrpov 36617 The group operation value of the translation group is the composition of translations. (Contributed by NM, 5-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  G  =  ( ( TGrp `  K ) `  W )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( K  e.  V  /\  W  e.  H  /\  ( X  e.  T  /\  Y  e.  T ) )  ->  ( X  .+  Y )  =  ( X  o.  Y ) )
 
Theoremtgrpgrplem 36618 Lemma for tgrpgrp 36619. (Contributed by NM, 6-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  G  =  ( ( TGrp `  K ) `  W )   &    |-  .+  =  ( +g  `  G )   &    |-  B  =  ( Base `  K )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  G  e.  Grp )
 
Theoremtgrpgrp 36619 The translation group is a group. (Contributed by NM, 6-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  G  =  ( ( TGrp `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  G  e.  Grp )
 
Theoremtgrpabl 36620 The translation group is an Abelian group. Lemma G of [Crawley] p. 116. (Contributed by NM, 6-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  G  =  ( ( TGrp `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  G  e.  Abel )
 
Syntaxctendo 36621 Extend class notation with translation group endomorphisms.
 class  TEndo
 
Syntaxcedring 36622 Extend class notation with division ring on trace-preserving endomorphisms.
 class  EDRing
 
Syntaxcedring-rN 36623 Extend class notation with division ring on trace-preserving endomorphisms, with multiplication reversed. TODO: remove  EDRingR theorems if not used.
 class  EDRingR
 
Definitiondf-tendo 36624* Define trace-preserving endomorphisms on the set of translations. (Contributed by NM, 8-Jun-2013.)
 |-  TEndo  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  { f  |  ( f : ( ( LTrn `  k ) `  w ) --> ( (
 LTrn `  k ) `  w )  /\  A. x  e.  ( ( LTrn `  k
 ) `  w ) A. y  e.  (
 ( LTrn `  k ) `  w ) ( f `
  ( x  o.  y ) )  =  ( ( f `  x )  o.  (
 f `  y )
 )  /\  A. x  e.  ( ( LTrn `  k
 ) `  w )
 ( ( ( trL `  k ) `  w ) `  ( f `  x ) ) ( le `  k ) ( ( ( trL `  k ) `  w ) `  x ) ) } ) )
 
Definitiondf-edring-rN 36625* Define division ring on trace-preserving endomorphisms. Definition of E of [Crawley] p. 117, 4th line from bottom. (Contributed by NM, 8-Jun-2013.)
 |-  EDRingR  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  {
 <. ( Base `  ndx ) ,  ( ( TEndo `  k
 ) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( s  e.  ( ( TEndo `  k
 ) `  w ) ,  t  e.  (
 ( TEndo `  k ) `  w )  |->  ( f  e.  ( ( LTrn `  k ) `  w )  |->  ( ( s `
  f )  o.  ( t `  f
 ) ) ) )
 >. ,  <. ( .r `  ndx ) ,  ( s  e.  ( ( TEndo `  k ) `  w ) ,  t  e.  ( ( TEndo `  k
 ) `  w )  |->  ( t  o.  s
 ) ) >. } )
 )
 
Definitiondf-edring 36626* Define division ring on trace-preserving endomorphisms. The multiplication operation is reversed composition, per the definition of E of [Crawley] p. 117, 4th line from bottom. (Contributed by NM, 8-Jun-2013.)
 |-  EDRing  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  { <. (
 Base `  ndx ) ,  ( ( TEndo `  k
 ) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( s  e.  ( ( TEndo `  k
 ) `  w ) ,  t  e.  (
 ( TEndo `  k ) `  w )  |->  ( f  e.  ( ( LTrn `  k ) `  w )  |->  ( ( s `
  f )  o.  ( t `  f
 ) ) ) )
 >. ,  <. ( .r `  ndx ) ,  ( s  e.  ( ( TEndo `  k ) `  w ) ,  t  e.  ( ( TEndo `  k
 ) `  w )  |->  ( s  o.  t
 ) ) >. } )
 )
 
Theoremtendofset 36627* The set of all trace-preserving endomorphisms on the set of translations for a lattice  K. (Contributed by NM, 8-Jun-2013.)
 |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  V  ->  ( TEndo `  K )  =  ( w  e.  H  |->  { s  |  ( s : ( ( LTrn `  K ) `  w )
 --> ( ( LTrn `  K ) `  w )  /\  A. f  e.  ( (
 LTrn `  K ) `  w ) A. g  e.  ( ( LTrn `  K ) `  w ) ( s `  ( f  o.  g ) )  =  ( ( s `
  f )  o.  ( s `  g
 ) )  /\  A. f  e.  ( ( LTrn `  K ) `  w ) ( ( ( trL `  K ) `  w ) `  ( s `  f
 ) )  .<_  ( ( ( trL `  K ) `  w ) `  f ) ) }
 ) )
 
Theoremtendoset 36628* The set of trace-preserving endomorphisms on the set of translations for a fiducial co-atom  W. (Contributed by NM, 8-Jun-2013.)
 |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  E  =  { s  |  ( s : T --> T  /\  A. f  e.  T  A. g  e.  T  (
 s `  ( f  o.  g ) )  =  ( ( s `  f )  o.  (
 s `  g )
 )  /\  A. f  e.  T  ( R `  ( s `  f
 ) )  .<_  ( R `
  f ) ) } )
 
Theoremistendo 36629* The predicate "is a trace-preserving endomorphism". Similar to definition of trace-preserving endomorphism in [Crawley] p. 117, penultimate line. (Contributed by NM, 8-Jun-2013.)
 |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( S  e.  E  <->  ( S : T
 --> T  /\  A. f  e.  T  A. g  e.  T  ( S `  ( f  o.  g
 ) )  =  ( ( S `  f
 )  o.  ( S `
  g ) ) 
 /\  A. f  e.  T  ( R `  ( S `
  f ) ) 
 .<_  ( R `  f
 ) ) ) )
 
Theoremtendotp 36630 Trace-preserving property of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
 |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  S  e.  E  /\  F  e.  T )  ->  ( R `  ( S `  F ) ) 
 .<_  ( R `  F ) )
 
Theoremistendod 36631* Deduce the predicate "is a trace-preserving endomorphism". (Contributed by NM, 9-Jun-2013.)
 |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  V  /\  W  e.  H ) )   &    |-  ( ph  ->  S : T --> T )   &    |-  ( ( ph  /\  f  e.  T  /\  g  e.  T )  ->  ( S `  ( f  o.  g ) )  =  ( ( S `  f )  o.  ( S `  g ) ) )   &    |-  ( ( ph  /\  f  e.  T ) 
 ->  ( R `  ( S `  f ) ) 
 .<_  ( R `  f
 ) )   =>    |-  ( ph  ->  S  e.  E )
 
Theoremtendof 36632 Functionality of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  S  e.  E ) 
 ->  S : T --> T )
 
Theoremtendoeq1 36633* Condition determining equality of two trace-preserving endomorphisms. (Contributed by NM, 11-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) 
 /\  A. f  e.  T  ( U `  f )  =  ( V `  f ) )  ->  U  =  V )
 
Theoremtendovalco 36634 Value of composition of translations in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H  /\  S  e.  E )  /\  ( F  e.  T  /\  G  e.  T ) )  ->  ( S `  ( F  o.  G ) )  =  (
 ( S `  F )  o.  ( S `  G ) ) )
 
Theoremtendocoval 36635 Value of composition of endomorphisms in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-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 ) ) )
 
Theoremtendocl 36636 Closure of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  S  e.  E  /\  F  e.  T )  ->  ( S `  F )  e.  T )
 
Theoremtendoco2 36637 Distribution of compositions in preparation for endomorphism sum definition. (Contributed by NM, 10-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) 
 /\  ( F  e.  T  /\  G  e.  T ) )  ->  ( ( U `  ( F  o.  G ) )  o.  ( V `  ( F  o.  G ) ) )  =  ( ( ( U `
  F )  o.  ( V `  F ) )  o.  (
 ( U `  G )  o.  ( V `  G ) ) ) )
 
Theoremtendoidcl 36638 The identity is a trace-preserving endomorphism. (Contributed by NM, 30-Jul-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  E )
 
Theoremtendo1mul 36639 Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E ) 
 ->  ( (  _I  |`  T )  o.  U )  =  U )
 
Theoremtendo1mulr 36640 Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E ) 
 ->  ( U  o.  (  _I  |`  T ) )  =  U )
 
Theoremtendococl 36641 The composition of two trace-preserving endomorphisms (multiplication in the endormorphism ring) is a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E  /\  T  e.  E )  ->  ( S  o.  T )  e.  E )
 
Theoremtendoid 36642 The identity value of a trace-preserving endomorphism. (Contributed by NM, 21-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E ) 
 ->  ( S `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
 
Theoremtendoeq2 36643* Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 36693, we show that we only need to consider a single non-identity translation. (Contributed by NM, 21-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) 
 /\  A. f  e.  T  ( f  =/=  (  _I  |`  B )  ->  ( U `  f )  =  ( V `  f ) ) ) 
 ->  U  =  V )
 
Theoremtendoplcbv 36644* Define sum operation for trace-perserving endomorphisms. Change bound variables to isolate them later. (Contributed by NM, 11-Jun-2013.)
 |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   =>    |-  P  =  ( u  e.  E ,  v  e.  E  |->  ( g  e.  T  |->  ( ( u `  g )  o.  ( v `  g ) ) ) )
 
Theoremtendopl 36645* Value of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)
 |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( U  e.  E  /\  V  e.  E )  ->  ( U P V )  =  ( g  e.  T  |->  ( ( U `  g )  o.  ( V `  g ) ) ) )
 
Theoremtendopl2 36646* Value of result of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)
 |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  ( ( U P V ) `
  F )  =  ( ( U `  F )  o.  ( V `  F ) ) )
 
Theoremtendoplcl2 36647* Value of result of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) 
 /\  F  e.  T )  ->  ( ( U P V ) `  F )  e.  T )
 
Theoremtendoplco2 36648* Value of result of endomorphism sum operation on a translation composition. (Contributed by NM, 10-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) 
 /\  ( F  e.  T  /\  G  e.  T ) )  ->  ( ( U P V ) `
  ( F  o.  G ) )  =  ( ( ( U P V ) `  F )  o.  (
 ( U P V ) `  G ) ) )
 
Theoremtendopltp 36649* Trace-preserving property of endomorphism sum operation  P, based on theorem trlco 36596. Part of remark in [Crawley] p. 118, 2nd line, "it is clear from the second part of G (our trlco 36596) that Delta is a subring of E." (In our development, we will bypass their E and go directly to their Delta, whose base set is our  ( TEndo `  K
) `  W.) (Contributed by NM, 9-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T ) 
 ->  ( R `  (
 ( U P V ) `  F ) ) 
 .<_  ( R `  F ) )
 
Theoremtendoplcl 36650* Endomorphism sum is a trace-preserving endomorphism. (Contributed by NM, 10-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  ->  ( U P V )  e.  E )
 
Theoremtendoplcom 36651* The endomorphism sum operation is commutative. (Contributed by NM, 11-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  ->  ( U P V )  =  ( V P U ) )
 
Theoremtendoplass 36652* The endomorphism sum operation is associative. (Contributed by NM, 11-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  E  /\  U  e.  E  /\  V  e.  E )
 )  ->  ( ( S P U ) P V )  =  ( S P ( U P V ) ) )
 
Theoremtendodi1 36653* Endomorphism composition distributes over sum. (Contributed by NM, 13-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  E  /\  U  e.  E  /\  V  e.  E )
 )  ->  ( S  o.  ( U P V ) )  =  (
 ( S  o.  U ) P ( S  o.  V ) ) )
 
Theoremtendodi2 36654* Endomorphism composition distributes over sum. (Contributed by NM, 13-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  E  /\  U  e.  E  /\  V  e.  E )
 )  ->  ( ( S P U )  o.  V )  =  ( ( S  o.  V ) P ( U  o.  V ) ) )
 
Theoremtendo0cbv 36655* Define additive identity for trace-perserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 11-Jun-2013.)
 |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  O  =  ( g  e.  T  |->  (  _I  |`  B ) )
 
Theoremtendo02 36656* Value of additive identity endomorphism. (Contributed by NM, 11-Jun-2013.)
 |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  B  =  ( Base `  K )   =>    |-  ( F  e.  T  ->  ( O `  F )  =  (  _I  |`  B ) )
 
Theoremtendo0co2 36657* The additive identity trace-perserving endormorphism preserves composition of translations. TODO: why isn't this a special case of tendospdi1 36890? (Contributed by NM, 11-Jun-2013.)
 |-  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 )  /\  F  e.  T  /\  G  e.  T )  ->  ( O `
  ( F  o.  G ) )  =  ( ( O `  F )  o.  ( O `  G ) ) )
 
Theoremtendo0tp 36658* Trace-preserving property of endomorphism additive identity. (Contributed by NM, 11-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |- 
 .<_  =  ( le `  K )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  ->  ( R `  ( O `  F ) )  .<_  ( R `  F ) )
 
Theoremtendo0cl 36659* The additive identity is a trace-perserving endormorphism. (Contributed by NM, 12-Jun-2013.)
 |-  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 )  ->  O  e.  E )
 
Theoremtendo0pl 36660* Property of the additive identity endormorphism. (Contributed by NM, 12-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `
  f )  o.  ( t `  f
 ) ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  ->  ( O P S )  =  S )
 
Theoremtendo0plr 36661* Property of the additive identity endormorphism. (Contributed by NM, 21-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `
  f )  o.  ( t `  f
 ) ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  ->  ( S P O )  =  S )
 
Theoremtendoicbv 36662* Define inverse function for trace-perserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 12-Jun-2013.)
 |-  I  =  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f
 ) ) )   =>    |-  I  =  ( u  e.  E  |->  ( g  e.  T  |->  `' ( u `  g
 ) ) )
 
Theoremtendoi 36663* Value of inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
 |-  I  =  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f
 ) ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( S  e.  E  ->  ( I `  S )  =  (
 g  e.  T  |->  `' ( S `  g
 ) ) )
 
Theoremtendoi2 36664* Value of additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
 |-  I  =  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f
 ) ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( S  e.  E  /\  F  e.  T )  ->  (
 ( I `  S ) `  F )  =  `' ( S `  F ) )
 
Theoremtendoicl 36665* Closure of the additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f
 ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E ) 
 ->  ( I `  S )  e.  E )
 
Theoremtendoipl 36666* Property of the additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f
 ) ) )   &    |-  B  =  ( Base `  K )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  ->  (
 ( I `  S ) P S )  =  O )
 
Theoremtendoipl2 36667* Property of the additive inverse endomorphism. (Contributed by NM, 29-Sep-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f
 ) ) )   &    |-  B  =  ( Base `  K )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  ->  ( S P ( I `  S ) )  =  O )
 
Theoremerngfset 36668* The division rings on trace-preserving endomorphisms for a lattice  K. (Contributed by NM, 8-Jun-2013.)
 |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  V  ->  (
 EDRing `  K )  =  ( w  e.  H  |->  {
 <. ( Base `  ndx ) ,  ( ( TEndo `  K ) `  w ) >. , 
 <. ( +g  `  ndx ) ,  ( s  e.  ( ( TEndo `  K ) `  w ) ,  t  e.  ( (
 TEndo `  K ) `  w )  |->  ( f  e.  ( ( LTrn `  K ) `  w )  |->  ( ( s `
  f )  o.  ( t `  f
 ) ) ) )
 >. ,  <. ( .r `  ndx ) ,  ( s  e.  ( ( TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w )  |->  ( s  o.  t ) ) >. } ) )
 
Theoremerngset 36669* The division ring on trace-preserving endomorphisms for a fiducial co-atom  W. (Contributed by NM, 5-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  D  =  { <. ( Base `  ndx ) ,  E >. , 
 <. ( +g  `  ndx ) ,  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `
  f )  o.  ( t `  f
 ) ) ) )
 >. ,  <. ( .r `  ndx ) ,  ( s  e.  E ,  t  e.  E  |->  ( s  o.  t ) ) >. } )
 
Theoremerngbase 36670 The base set of the division ring on trace-preserving endomorphisms is the set of all trace-preserving endomorphisms (for a fiducial co-atom  W). TODO: the .t hypothesis isn't used. (Also look at others.) (Contributed by NM, 9-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  C  =  ( Base `  D )   =>    |-  (
 ( K  e.  V  /\  W  e.  H ) 
 ->  C  =  E )
 
Theoremerngfplus 36671* Ring addition operation. (Contributed by NM, 9-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  .+  =  ( +g  `  D )   =>    |-  (
 ( K  e.  V  /\  W  e.  H ) 
 ->  .+  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `
  f )  o.  ( t `  f
 ) ) ) ) )
 
Theoremerngplus 36672* Ring addition operation. (Contributed by NM, 10-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  .+  =  ( +g  `  D )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) )  ->  ( U 
 .+  V )  =  ( f  e.  T  |->  ( ( U `  f )  o.  ( V `  f ) ) ) )
 
Theoremerngplus2 36673 Ring addition operation. (Contributed by NM, 10-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  .+  =  ( +g  `  D )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  F  e.  T ) )  ->  ( ( U  .+  V ) `  F )  =  (
 ( U `  F )  o.  ( V `  F ) ) )
 
Theoremerngfmul 36674* Ring multiplication operation. (Contributed by NM, 9-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  .x.  =  ( .r `  D )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  .x.  =  (
 s  e.  E ,  t  e.  E  |->  ( s  o.  t ) ) )
 
Theoremerngmul 36675 Ring addition operation. (Contributed by NM, 10-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  .x.  =  ( .r `  D )   =>    |-  ( ( ( K  e.  X  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )
 )  ->  ( U  .x.  V )  =  ( U  o.  V ) )
 
Theoremerngfset-rN 36676* The division rings on trace-preserving endomorphisms for a lattice  K. (Contributed by NM, 8-Jun-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  V  ->  (
 EDRingR `  K )  =  ( w  e.  H  |->  { <. ( Base `  ndx ) ,  ( ( TEndo `  K ) `  w ) >. ,  <. (
 +g  `  ndx ) ,  ( s  e.  (
 ( TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w )  |->  ( f  e.  ( (
 LTrn `  K ) `  w )  |->  ( ( s `  f )  o.  ( t `  f ) ) ) ) >. ,  <. ( .r
 `  ndx ) ,  (
 s  e.  ( (
 TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w )  |->  ( t  o.  s ) ) >. } ) )
 
Theoremerngset-rN 36677* The division ring on trace-preserving endomorphisms for a fiducial co-atom  W. (Contributed by NM, 5-Jun-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRingR `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  D  =  { <. ( Base `  ndx ) ,  E >. ,  <. ( +g  ` 
 ndx ) ,  (
 s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  ( t `  f ) ) ) ) >. ,  <. ( .r
 `  ndx ) ,  (
 s  e.  E ,  t  e.  E  |->  ( t  o.  s ) )
 >. } )
 
Theoremerngbase-rN 36678 The base set of the division ring on trace-preserving endomorphisms is the set of all trace-preserving endomorphisms (for a fiducial co-atom  W). (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRingR `  K ) `  W )   &    |-  C  =  ( Base `  D )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  C  =  E )
 
Theoremerngfplus-rN 36679* Ring addition operation. (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRingR `  K ) `  W )   &    |- 
 .+  =  ( +g  `  D )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  .+  =  (
 s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  ( t `  f ) ) ) ) )
 
Theoremerngplus-rN 36680* Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRingR `  K ) `  W )   &    |- 
 .+  =  ( +g  `  D )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )
 )  ->  ( U  .+  V )  =  ( f  e.  T  |->  ( ( U `  f
 )  o.  ( V `
  f ) ) ) )
 
Theoremerngplus2-rN 36681 Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRingR `  K ) `  W )   &    |- 
 .+  =  ( +g  `  D )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  F  e.  T )
 )  ->  ( ( U  .+  V ) `  F )  =  (
 ( U `  F )  o.  ( V `  F ) ) )
 
Theoremerngfmul-rN 36682* Ring multiplication operation. (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRingR `  K ) `  W )   &    |- 
 .x.  =  ( .r `  D )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  .x.  =  (
 s  e.  E ,  t  e.  E  |->  ( t  o.  s ) ) )
 
Theoremerngmul-rN 36683 Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRingR `  K ) `  W )   &    |- 
 .x.  =  ( .r `  D )   =>    |-  ( ( ( K  e.  X  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )
 )  ->  ( U  .x.  V )  =  ( V  o.  U ) )
 
Theoremcdlemh1 36684 Part of proof of Lemma H of [Crawley] p. 118. (Contributed by NM, 17-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( ( P  .\/  ( R `  G ) )  ./\  ( Q  .\/  ( R `  ( G  o.  `' F ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  F  e.  T  /\  G  e.  T ) 
 /\  ( P  e.  A  /\  Q  e.  A )  /\  ( Q  .<_  ( P  .\/  ( R `  F ) )  /\  ( R `  F )  =/=  ( R `  G ) ) ) 
 ->  ( S  .\/  ( R `  ( G  o.  `' F ) ) )  =  ( Q  .\/  ( R `  ( G  o.  `' F ) ) ) )
 
Theoremcdlemh2 36685 Part of proof of Lemma H of [Crawley] p. 118. (Contributed by NM, 16-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( ( P  .\/  ( R `  G ) )  ./\  ( Q  .\/  ( R `  ( G  o.  `' F ) ) ) )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  ( S  ./\  W )  =  .0.  )
 
Theoremcdlemh 36686 Lemma H of [Crawley] p. 118. (Contributed by NM, 17-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( ( P  .\/  ( R `  G ) )  ./\  ( Q  .\/  ( R `  ( G  o.  `' F ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  F  e.  T  /\  G  e.  T ) 
 /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  Q  .<_  ( P 
 .\/  ( R `  F ) ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  ( S  e.  A  /\  -.  S  .<_  W ) )
 
Theoremcdlemi1 36687 Part of proof of Lemma I of [Crawley] p. 118. (Contributed by NM, 18-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  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  /\  G  e.  T ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  (
 ( U `  G ) `  P )  .<_  ( P  .\/  ( R `  G ) ) )
 
Theoremcdlemi2 36688 Part of proof of Lemma I of [Crawley] p. 118. (Contributed by NM, 18-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  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  /\  F  e.  T  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( U `  G ) `  P )  .<_  ( ( ( U `  F ) `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) )
 
Theoremcdlemi 36689 Lemma I of [Crawley] p. 118. (Contributed by NM, 19-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  S  =  ( ( P  .\/  ( R `  G ) )  ./\  ( (
 ( U `  F ) `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  F  e.  T  /\  G  e.  T ) 
 /\  ( U  e.  E  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F  =/=  (  _I  |`  B ) 
 /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =/=  ( R `  G ) ) ) 
 ->  ( ( U `  G ) `  P )  =  S )
 
Theoremcdlemj1 36690 Part of proof of Lemma J of [Crawley] p. 118. (Contributed by NM, 19-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T ) )  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `  F )  =/=  ( R `  g )  /\  ( R `
  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) ) 
 ->  ( ( U `  h ) `  p )  =  ( ( V `  h ) `  p ) )
 
Theoremcdlemj2 36691 Part of proof of Lemma J of [Crawley] p. 118. Eliminate  p. (Contributed by NM, 20-Jun-2013.)
 |-  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  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T ) )  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `  F )  =/=  ( R `  g )  /\  ( R `
  g )  =/=  ( R `  h ) ) )  ->  ( U `  h )  =  ( V `  h ) )
 
Theoremcdlemj3 36692 Part of proof of Lemma J of [Crawley] p. 118. Eliminate  g. (Contributed by NM, 20-Jun-2013.)
 |-  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  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T ) )  /\  h  =/=  (  _I  |`  B ) )  ->  ( U `  h )  =  ( V `  h ) )
 
Theoremtendocan 36693 Cancellation law: if the values of two trace-preserving endormorphisms are equal, so are the endormorphisms. Lemma J of [Crawley] p. 118. (Contributed by NM, 21-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) ) 
 ->  U  =  V )
 
Theoremtendoid0 36694* A trace-preserving endomorphism is the additive identity iff at least one of its values (at a non-identity translation) is the identity translation. (Contributed by NM, 1-Aug-2013.)
 |-  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 )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  (
 ( U `  F )  =  (  _I  |`  B )  <->  U  =  O ) )
 
Theoremtendo0mul 36695* Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 1-Aug-2013.)
 |-  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 )  /\  U  e.  E )  ->  ( O  o.  U )  =  O )
 
Theoremtendo0mulr 36696* Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 13-Feb-2014.)
 |-  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 )  /\  U  e.  E )  ->  ( U  o.  O )  =  O )
 
Theoremtendo1ne0 36697* The identity (unity) is not equal to the zero trace-preserving endomorphism. (Contributed by NM, 8-Aug-2013.)
 |-  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 )  ->  (  _I  |`  T )  =/=  O )
 
Theoremtendoconid 36698* The composition (product) of trace-preserving endormorphisms is nonzero when each argument is nonzero. (Contributed by NM, 8-Aug-2013.)
 |-  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 )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O ) )  ->  ( U  o.  V )  =/=  O )
 
Theoremtendotr 36699* The trace of the value of a non-zero trace-preserving endomorphism equals the trace of the argument. (Contributed by NM, 11-Aug-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  ->  ( R `  ( U `  F ) )  =  ( R `  F ) )
 
Theoremcdlemk1 36700 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 22-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T ) 
 /\  ( ( R `
  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( P  .\/  ( N `
  P ) )  =  ( ( F `
  P )  .\/  ( R `  F ) ) )
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