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Theorem List for Metamath Proof Explorer - 17001-17100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremidresperm 17001 The identity function restricted to a set is a permutation of this set. (Contributed by AV, 17-Mar-2019.)
 |-  G  =  ( SymGrp `  A )   =>    |-  ( A  e.  V  ->  (  _I  |`  A )  e.  ( Base `  G ) )
 
Theoremidressubgsymg 17002 The singleton containing only the identity function restricted to a set is a subgroup of the symmetric group of this set. (Contributed by AV, 17-Mar-2019.)
 |-  G  =  ( SymGrp `  A )   =>    |-  ( A  e.  V  ->  { (  _I  |`  A ) }  e.  (SubGrp `  G ) )
 
Theoremidrespermg 17003 The structure with the singleton containing only the identity function restricted to a set as base set and the function composition as group operation (constructed by (structure) restricting the symmetric group to that singleton) is a permutation group (group consisting of permutations). (Contributed by AV, 17-Mar-2019.)
 |-  G  =  ( SymGrp `  A )   &    |-  E  =  ( Gs 
 { (  _I  |`  A ) } )   =>    |-  ( A  e.  V  ->  ( E  e.  Grp  /\  ( Base `  E )  C_  ( Base `  G )
 ) )
 
10.2.8.2  Cayley's theorem
 
Theoremcayleylem1 17004* Lemma for cayley 17006. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  H  =  ( SymGrp `  X )   &    |-  S  =  (
 Base `  H )   &    |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g  .+  a ) ) )   =>    |-  ( G  e.  Grp  ->  F  e.  ( G  GrpHom  H ) )
 
Theoremcayleylem2 17005* Lemma for cayley 17006. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  H  =  ( SymGrp `  X )   &    |-  S  =  (
 Base `  H )   &    |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g  .+  a ) ) )   =>    |-  ( G  e.  Grp  ->  F : X -1-1-> S )
 
Theoremcayley 17006* Cayley's Theorem (constructive version): given group  G,  F is an isomorphism between  G and the subgroup  S of the symmetric group  H on the underlying set  X of  G. See also Theorem 3.15 in [Rotman] p. 42. (Contributed by Paul Chapman, 3-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  H  =  (
 SymGrp `  X )   &    |-  .+  =  ( +g  `  G )   &    |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g  .+  a ) ) )   &    |-  S  =  ran  F   =>    |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  H )  /\  F  e.  ( G  GrpHom  ( Hs  S ) )  /\  F : X -1-1-onto-> S ) )
 
Theoremcayleyth 17007* Cayley's Theorem (existence version): every group  G is isomorphic to a subgroup of the symmetric group on the underlying set of  G. (For any group  G there exists an isomorphism  f between  G and a subgroup  h of the symmetric group on the underlying set of  G.) See also Theorem 3.15 in [Rotman] p. 42. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  H  =  (
 SymGrp `  X )   =>    |-  ( G  e.  Grp 
 ->  E. s  e.  (SubGrp `  H ) E. f  e.  ( G  GrpHom  ( Hs  s ) ) f : X -1-1-onto-> s )
 
10.2.8.3  Permutations fixing one element
 
Theoremsymgfix2 17008* If a permutation does not move a certain element of a set to a second element, there is a third element which is moved to the second element. (Contributed by AV, 2-Jan-2019.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   =>    |-  ( L  e.  N  ->  ( Q  e.  ( P  \  { q  e.  P  |  ( q `
  K )  =  L } )  ->  E. k  e.  ( N  \  { K }
 ) ( Q `  k )  =  L ) )
 
Theoremsymgextf 17009* The extension of a permutation, fixing the additional element, is a function. (Contributed by AV, 6-Jan-2019.)
 |-  S  =  ( Base `  ( SymGrp `  ( N  \  { K } )
 ) )   &    |-  E  =  ( x  e.  N  |->  if ( x  =  K ,  K ,  ( Z `
  x ) ) )   =>    |-  ( ( K  e.  N  /\  Z  e.  S )  ->  E : N --> N )
 
Theoremsymgextfv 17010* The function value of the extension of a permutation, fixing the additional element, for elements in the original domain. (Contributed by AV, 6-Jan-2019.)
 |-  S  =  ( Base `  ( SymGrp `  ( N  \  { K } )
 ) )   &    |-  E  =  ( x  e.  N  |->  if ( x  =  K ,  K ,  ( Z `
  x ) ) )   =>    |-  ( ( K  e.  N  /\  Z  e.  S )  ->  ( X  e.  ( N  \  { K } )  ->  ( E `
  X )  =  ( Z `  X ) ) )
 
Theoremsymgextfve 17011* The function value of the extension of a permutation, fixing the additional element, for the additional element. (Contributed by AV, 6-Jan-2019.)
 |-  S  =  ( Base `  ( SymGrp `  ( N  \  { K } )
 ) )   &    |-  E  =  ( x  e.  N  |->  if ( x  =  K ,  K ,  ( Z `
  x ) ) )   =>    |-  ( K  e.  N  ->  ( X  =  K  ->  ( E `  X )  =  K )
 )
 
Theoremsymgextf1lem 17012* Lemma for symgextf1 17013. (Contributed by AV, 6-Jan-2019.)
 |-  S  =  ( Base `  ( SymGrp `  ( N  \  { K } )
 ) )   &    |-  E  =  ( x  e.  N  |->  if ( x  =  K ,  K ,  ( Z `
  x ) ) )   =>    |-  ( ( K  e.  N  /\  Z  e.  S )  ->  ( ( X  e.  ( N  \  { K } )  /\  Y  e.  { K } )  ->  ( E `
  X )  =/=  ( E `  Y ) ) )
 
Theoremsymgextf1 17013* The extension of a permutation, fixing the additional element, is a 1-1 function. (Contributed by AV, 6-Jan-2019.)
 |-  S  =  ( Base `  ( SymGrp `  ( N  \  { K } )
 ) )   &    |-  E  =  ( x  e.  N  |->  if ( x  =  K ,  K ,  ( Z `
  x ) ) )   =>    |-  ( ( K  e.  N  /\  Z  e.  S )  ->  E : N -1-1-> N )
 
Theoremsymgextfo 17014* The extension of a permutation, fixing the additional element, is an onto function. (Contributed by AV, 7-Jan-2019.)
 |-  S  =  ( Base `  ( SymGrp `  ( N  \  { K } )
 ) )   &    |-  E  =  ( x  e.  N  |->  if ( x  =  K ,  K ,  ( Z `
  x ) ) )   =>    |-  ( ( K  e.  N  /\  Z  e.  S )  ->  E : N -onto-> N )
 
Theoremsymgextf1o 17015* The extension of a permutation, fixing the additional element, is a bijection. (Contributed by AV, 7-Jan-2019.)
 |-  S  =  ( Base `  ( SymGrp `  ( N  \  { K } )
 ) )   &    |-  E  =  ( x  e.  N  |->  if ( x  =  K ,  K ,  ( Z `
  x ) ) )   =>    |-  ( ( K  e.  N  /\  Z  e.  S )  ->  E : N -1-1-onto-> N )
 
Theoremsymgextsymg 17016* The extension of a permutation is an element of the extended symmetric group. (Contributed by AV, 9-Mar-2019.)
 |-  S  =  ( Base `  ( SymGrp `  ( N  \  { K } )
 ) )   &    |-  E  =  ( x  e.  N  |->  if ( x  =  K ,  K ,  ( Z `
  x ) ) )   =>    |-  ( ( N  e.  V  /\  K  e.  N  /\  Z  e.  S ) 
 ->  E  e.  ( Base `  ( SymGrp `  N )
 ) )
 
Theoremsymgextres 17017* The restriction of the extension of a permutation, fixing the additional element, to the original domain. (Contributed by AV, 6-Jan-2019.)
 |-  S  =  ( Base `  ( SymGrp `  ( N  \  { K } )
 ) )   &    |-  E  =  ( x  e.  N  |->  if ( x  =  K ,  K ,  ( Z `
  x ) ) )   =>    |-  ( ( K  e.  N  /\  Z  e.  S )  ->  ( E  |`  ( N 
 \  { K }
 ) )  =  Z )
 
Theoremgsumccatsymgsn 17018 Homomorphic property of composites of permutations with a singleton. (Contributed by AV, 20-Jan-2019.)
 |-  G  =  ( SymGrp `  A )   &    |-  B  =  (
 Base `  G )   =>    |-  ( ( A  e.  V  /\  W  e. Word  B  /\  Z  e.  B )  ->  ( G 
 gsumg  ( W ++  <" Z "> ) )  =  ( ( G  gsumg  W )  o.  Z ) )
 
Theoremgsmsymgrfixlem1 17019* Lemma 1 for gsmsymgrfix 17020. (Contributed by AV, 20-Jan-2019.)
 |-  S  =  ( SymGrp `  N )   &    |-  B  =  (
 Base `  S )   =>    |-  ( ( ( W  e. Word  B  /\  P  e.  B )  /\  ( N  e.  Fin  /\  K  e.  N ) 
 /\  ( A. i  e.  ( 0..^ ( # `  W ) ) ( ( W `  i
 ) `  K )  =  K  ->  ( ( S  gsumg 
 W ) `  K )  =  K )
 )  ->  ( A. i  e.  ( 0..^ ( ( # `  W )  +  1 )
 ) ( ( ( W ++  <" P "> ) `  i ) `
  K )  =  K  ->  ( ( S  gsumg  ( W ++  <" P "> ) ) `  K )  =  K ) )
 
Theoremgsmsymgrfix 17020* The composition of permutations fixing one element also fixes this element. (Contributed by AV, 20-Jan-2019.)
 |-  S  =  ( SymGrp `  N )   &    |-  B  =  (
 Base `  S )   =>    |-  ( ( N  e.  Fin  /\  K  e.  N  /\  W  e. Word  B )  ->  ( A. i  e.  ( 0..^ ( # `  W ) ) ( ( W `  i
 ) `  K )  =  K  ->  ( ( S  gsumg 
 W ) `  K )  =  K )
 )
 
Theoremfvcosymgeq 17021* The values of two compositions of permutations are equal if the values of the composed permutations are pairwise equal. (Contributed by AV, 26-Jan-2019.)
 |-  S  =  ( SymGrp `  N )   &    |-  B  =  (
 Base `  S )   &    |-  Z  =  ( SymGrp `  M )   &    |-  P  =  ( Base `  Z )   &    |-  I  =  ( N  i^i  M )   =>    |-  ( ( G  e.  B  /\  K  e.  P )  ->  ( ( X  e.  I  /\  ( G `  X )  =  ( K `  X )  /\  A. n  e.  I  ( F `  n )  =  ( H `  n ) ) 
 ->  ( ( F  o.  G ) `  X )  =  ( ( H  o.  K ) `  X ) ) )
 
Theoremgsmsymgreqlem1 17022* Lemma 1 for gsmsymgreq 17024. (Contributed by AV, 26-Jan-2019.)
 |-  S  =  ( SymGrp `  N )   &    |-  B  =  (
 Base `  S )   &    |-  Z  =  ( SymGrp `  M )   &    |-  P  =  ( Base `  Z )   &    |-  I  =  ( N  i^i  M )   =>    |-  ( ( ( N  e.  Fin  /\  M  e.  Fin  /\  J  e.  I ) 
 /\  ( ( X  e. Word  B  /\  C  e.  B )  /\  ( Y  e. Word  P  /\  R  e.  P )  /\  ( # `  X )  =  ( # `  Y ) ) )  ->  ( ( A. n  e.  I  ( ( S  gsumg  X ) `
  n )  =  ( ( Z  gsumg  Y ) `
  n )  /\  ( C `  J )  =  ( R `  J ) )  ->  ( ( S  gsumg  ( X ++ 
 <" C "> ) ) `  J )  =  ( ( Z  gsumg  ( Y ++  <" R "> ) ) `  J ) ) )
 
Theoremgsmsymgreqlem2 17023* Lemma 2 for gsmsymgreq 17024. (Contributed by AV, 26-Jan-2019.)
 |-  S  =  ( SymGrp `  N )   &    |-  B  =  (
 Base `  S )   &    |-  Z  =  ( SymGrp `  M )   &    |-  P  =  ( Base `  Z )   &    |-  I  =  ( N  i^i  M )   =>    |-  ( ( ( N  e.  Fin  /\  M  e.  Fin )  /\  ( ( X  e. Word  B  /\  C  e.  B )  /\  ( Y  e. Word  P  /\  R  e.  P ) 
 /\  ( # `  X )  =  ( # `  Y ) ) )  ->  ( ( A. i  e.  ( 0..^ ( # `  X ) ) A. n  e.  I  (
 ( X `  i
 ) `  n )  =  ( ( Y `  i ) `  n )  ->  A. n  e.  I  ( ( S  gsumg  X ) `
  n )  =  ( ( Z  gsumg  Y ) `
  n ) ) 
 ->  ( A. i  e.  ( 0..^ ( # `  ( X ++  <" C "> ) ) )
 A. n  e.  I  ( ( ( X ++ 
 <" C "> ) `  i ) `  n )  =  (
 ( ( Y ++  <" R "> ) `  i ) `  n )  ->  A. n  e.  I  ( ( S  gsumg  ( X ++ 
 <" C "> ) ) `  n )  =  ( ( Z  gsumg  ( Y ++  <" R "> ) ) `  n ) ) ) )
 
Theoremgsmsymgreq 17024* Two combination of permutations moves an element of the intersection of the base sets of the permutations to the same element if each pair of corresponding permutations moves such an element to the same element. (Contributed by AV, 20-Jan-2019.)
 |-  S  =  ( SymGrp `  N )   &    |-  B  =  (
 Base `  S )   &    |-  Z  =  ( SymGrp `  M )   &    |-  P  =  ( Base `  Z )   &    |-  I  =  ( N  i^i  M )   =>    |-  ( ( ( N  e.  Fin  /\  M  e.  Fin )  /\  ( W  e. Word  B  /\  U  e. Word  P 
 /\  ( # `  W )  =  ( # `  U ) ) )  ->  ( A. i  e.  (
 0..^ ( # `  W ) ) A. n  e.  I  ( ( W `  i ) `  n )  =  (
 ( U `  i
 ) `  n )  ->  A. n  e.  I  ( ( S  gsumg  W ) `
  n )  =  ( ( Z  gsumg  U ) `
  n ) ) )
 
Theoremsymgfixelq 17025* A permutation of a set fixing an element of the set. (Contributed by AV, 4-Jan-2019.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  Q  =  { q  e.  P  |  ( q `
  K )  =  K }   =>    |-  ( F  e.  V  ->  ( F  e.  Q  <->  ( F : N -1-1-onto-> N  /\  ( F `
  K )  =  K ) ) )
 
Theoremsymgfixels 17026* The restriction of a permutation to a set with one element removed is an element of the restricted symmetric group if the restriction is a 1-1 onto function. (Contributed by AV, 4-Jan-2019.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  Q  =  { q  e.  P  |  ( q `
  K )  =  K }   &    |-  S  =  (
 Base `  ( SymGrp `  ( N  \  { K }
 ) ) )   &    |-  D  =  ( N  \  { K } )   =>    |-  ( F  e.  V  ->  ( ( F  |`  D )  e.  S  <->  ( F  |`  D ) : D -1-1-onto-> D ) )
 
Theoremsymgfixelsi 17027* The restriction of a permutation fixing an element to the set with this element removed is an element of the restricted symmetric group. (Contributed by AV, 4-Jan-2019.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  Q  =  { q  e.  P  |  ( q `
  K )  =  K }   &    |-  S  =  (
 Base `  ( SymGrp `  ( N  \  { K }
 ) ) )   &    |-  D  =  ( N  \  { K } )   =>    |-  ( ( K  e.  N  /\  F  e.  Q )  ->  ( F  |`  D )  e.  S )
 
Theoremsymgfixf 17028* The mapping of a permutation of a set fixing an element to a permutation of the set without the fixed element is a function. (Contributed by AV, 4-Jan-2019.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  Q  =  { q  e.  P  |  ( q `
  K )  =  K }   &    |-  S  =  (
 Base `  ( SymGrp `  ( N  \  { K }
 ) ) )   &    |-  H  =  ( q  e.  Q  |->  ( q  |`  ( N 
 \  { K }
 ) ) )   =>    |-  ( K  e.  N  ->  H : Q --> S )
 
Theoremsymgfixf1 17029* The mapping of a permutation of a set fixing an element to a permutation of the set without the fixed element is a 1-1 function. (Contributed by AV, 4-Jan-2019.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  Q  =  { q  e.  P  |  ( q `
  K )  =  K }   &    |-  S  =  (
 Base `  ( SymGrp `  ( N  \  { K }
 ) ) )   &    |-  H  =  ( q  e.  Q  |->  ( q  |`  ( N 
 \  { K }
 ) ) )   =>    |-  ( K  e.  N  ->  H : Q -1-1-> S )
 
Theoremsymgfixfolem1 17030* Lemma 1 for symgfixfo 17031. (Contributed by AV, 7-Jan-2019.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  Q  =  { q  e.  P  |  ( q `
  K )  =  K }   &    |-  S  =  (
 Base `  ( SymGrp `  ( N  \  { K }
 ) ) )   &    |-  H  =  ( q  e.  Q  |->  ( q  |`  ( N 
 \  { K }
 ) ) )   &    |-  E  =  ( x  e.  N  |->  if ( x  =  K ,  K ,  ( Z `
  x ) ) )   =>    |-  ( ( N  e.  V  /\  K  e.  N  /\  Z  e.  S ) 
 ->  E  e.  Q )
 
Theoremsymgfixfo 17031* The mapping of a permutation of a set fixing an element to a permutation of the set without the fixed element is an onto function. (Contributed by AV, 7-Jan-2019.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  Q  =  { q  e.  P  |  ( q `
  K )  =  K }   &    |-  S  =  (
 Base `  ( SymGrp `  ( N  \  { K }
 ) ) )   &    |-  H  =  ( q  e.  Q  |->  ( q  |`  ( N 
 \  { K }
 ) ) )   =>    |-  ( ( N  e.  V  /\  K  e.  N )  ->  H : Q -onto-> S )
 
Theoremsymgfixf1o 17032* The mapping of a permutation of a set fixing an element to a permutation of the set without the fixed element is a bijection. (Contributed by AV, 7-Jan-2019.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  Q  =  { q  e.  P  |  ( q `
  K )  =  K }   &    |-  S  =  (
 Base `  ( SymGrp `  ( N  \  { K }
 ) ) )   &    |-  H  =  ( q  e.  Q  |->  ( q  |`  ( N 
 \  { K }
 ) ) )   =>    |-  ( ( N  e.  V  /\  K  e.  N )  ->  H : Q -1-1-onto-> S )
 
10.2.8.4  Transpositions in the symmetric group

Transpositions are special cases of "cycles" as defined in [Rotman] p. 28: "Let i1 , i2 , ... , ir be distinct integers between 1 and n. If α in Sn fixes the other integers and α(i1) = i2, α(i2) = i3, ..., α(ir-1 ) = ir, α(ir) = i1, then α is an r-cycle. We also say that α is a cycle of length r." and in [Rotman] p. 31: "A 2-cycle is also called transposition.".

We (currently) do not have/need a definition for cycles, so transpositions are explicitly defined in df-pmtr 17034.

 
Syntaxcpmtr 17033 Syntax for the transposition generator function.
 class pmTrsp
 
Definitiondf-pmtr 17034* Define a function that generates the transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |- pmTrsp  =  ( d  e.  _V  |->  ( p  e.  { y  e.  ~P d  |  y 
 ~~  2o }  |->  ( z  e.  d  |->  if (
 z  e.  p ,  U. ( p  \  {
 z } ) ,  z ) ) ) )
 
Theoremf1omvdmvd 17035 A permutation of any class moves a point which is moved to a different point which is moved. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  ( ( F : A
 -1-1-onto-> A  /\  X  e.  dom  ( F  \  _I  )
 )  ->  ( F `  X )  e.  ( dom  ( F  \  _I  )  \  { X }
 ) )
 
Theoremf1omvdcnv 17036 A permutation and its inverse move the same points. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  ( F : A -1-1-onto-> A  ->  dom  ( `' F  \  _I  )  =  dom  ( F  \  _I  )
 )
 
Theoremmvdco 17037 Composing two permutations moves at most the union of the points. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |- 
 dom  ( ( F  o.  G )  \  _I  )  C_  ( dom  ( F  \  _I  )  u.  dom  ( G  \  _I  ) )
 
Theoremf1omvdconj 17038 Conjugation of a permutation takes the image of the moved subclass. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  ( ( F : A
 --> A  /\  G : A
 -1-1-onto-> A )  ->  dom  (
 ( ( G  o.  F )  o.  `' G )  \  _I  )  =  ( G " dom  ( F  \  _I  )
 ) )
 
Theoremf1otrspeq 17039 A transposition is characterized by the points it moves. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  ( ( ( F : A -1-1-onto-> A  /\  G : A
 -1-1-onto-> A )  /\  ( dom  ( F  \  _I  )  ~~  2o  /\  dom  ( G  \  _I  )  =  dom  ( F  \  _I  ) ) )  ->  F  =  G )
 
Theoremf1omvdco2 17040 If exactly one of two permutations is limited to a set of points, then the composition will not be. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( ( F : A
 -1-1-onto-> A  /\  G : A -1-1-onto-> A  /\  ( dom  ( F 
 \  _I  )  C_  X  \/_  dom  ( G  \  _I  )  C_  X ) )  ->  -.  dom  ( ( F  o.  G )  \  _I  )  C_  X )
 
Theoremf1omvdco3 17041 If a point is moved by exactly one of two permutations, then it will be moved by their composite. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( ( F : A
 -1-1-onto-> A  /\  G : A -1-1-onto-> A  /\  ( X  e.  dom  ( F  \  _I  )  \/_  X  e.  dom  ( G  \  _I  ) ) )  ->  X  e.  dom  ( ( F  o.  G )  \  _I  )
 )
 
Theorempmtrfval 17042* The function generating transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   =>    |-  ( D  e.  V  ->  T  =  ( p  e.  { y  e. 
 ~P D  |  y 
 ~~  2o }  |->  ( z  e.  D  |->  if (
 z  e.  p ,  U. ( p  \  {
 z } ) ,  z ) ) ) )
 
Theorempmtrval 17043* A generated transposition, expressed in a symmetric form. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   =>    |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  =  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z }
 ) ,  z ) ) )
 
Theorempmtrfv 17044 General value of mapping a point under a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   =>    |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D )  ->  ( ( T `  P ) `
  Z )  =  if ( Z  e.  P ,  U. ( P 
 \  { Z }
 ) ,  Z ) )
 
Theorempmtrprfv 17045 In a transposition of two given points, each maps to the other. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   =>    |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  X  =/=  Y ) )  ->  ( ( T `  { X ,  Y } ) `  X )  =  Y )
 
Theorempmtrprfv3 17046 In a transposition of two given points, all other points are mapped to themselves. (Contributed by AV, 17-Mar-2019.)
 |-  T  =  (pmTrsp `  D )   =>    |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D ) 
 /\  ( X  =/=  Y 
 /\  X  =/=  Z  /\  Y  =/=  Z ) )  ->  ( ( T `  { X ,  Y } ) `  Z )  =  Z )
 
Theorempmtrf 17047 Functionality of a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   =>    |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P ) : D --> D )
 
Theorempmtrmvd 17048 A transposition moves precisely the transposed points. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   =>    |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  dom  ( ( T `
  P )  \  _I  )  =  P )
 
Theorempmtrrn 17049 Transposing two points gives a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  (
 ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  e.  R )
 
Theorempmtrfrn 17050 A transposition (as a kind of function) is the function transposing the two points it moves. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   &    |-  P  =  dom  ( F  \  _I  )   =>    |-  ( F  e.  R  ->  ( ( D  e.  _V 
 /\  P  C_  D  /\  P  ~~  2o )  /\  F  =  ( T `
  P ) ) )
 
Theorempmtrffv 17051 Mapping of a point under a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   &    |-  P  =  dom  ( F  \  _I  )   =>    |-  ( ( F  e.  R  /\  Z  e.  D )  ->  ( F `  Z )  =  if ( Z  e.  P ,  U. ( P  \  { Z } ) ,  Z ) )
 
Theorempmtrrn2 17052* For any transposition there are two points it is transposing. (Contributed by SO, 15-Jul-2018.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  ( F  e.  R  ->  E. x  e.  D  E. y  e.  D  ( x  =/=  y  /\  F  =  ( T `  { x ,  y } ) ) )
 
Theorempmtrfinv 17053 A transposition function is an involution. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  ( F  e.  R  ->  ( F  o.  F )  =  (  _I  |`  D ) )
 
Theorempmtrfmvdn0 17054 A transposition moves at least one point. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  ( F  e.  R  ->  dom  ( F  \  _I  )  =/=  (/) )
 
Theorempmtrff1o 17055 A transposition function is a permutation. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  ( F  e.  R  ->  F : D -1-1-onto-> D )
 
Theorempmtrfcnv 17056 A transposition function is its own inverse. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  ( F  e.  R  ->  `' F  =  F )
 
Theorempmtrfb 17057 An intrinsic characterization of the transposition permutations. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  ( F  e.  R  <->  ( D  e.  _V 
 /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )
 )
 
Theorempmtrfconj 17058 Any conjugate of a transposition is a transposition. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  (
 ( F  e.  R  /\  G : D -1-1-onto-> D )  ->  (
 ( G  o.  F )  o.  `' G )  e.  R )
 
Theoremsymgsssg 17059* The symmetric group has subgroups restricting the set of non-fixed points. (Contributed by Stefan O'Rear, 24-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  B  =  (
 Base `  G )   =>    |-  ( D  e.  V  ->  { x  e.  B  |  dom  ( x  \  _I  )  C_  X }  e.  (SubGrp `  G ) )
 
Theoremsymgfisg 17060* The symmetric group has a subgroup of permutations that move finitely many points. (Contributed by Stefan O'Rear, 24-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  B  =  (
 Base `  G )   =>    |-  ( D  e.  V  ->  { x  e.  B  |  dom  ( x  \  _I  )  e. 
 Fin }  e.  (SubGrp `  G ) )
 
Theoremsymgtrf 17061 Transpositions are elements of the symmetric group. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  T  =  ran  (pmTrsp `  D )   &    |-  G  =  (
 SymGrp `  D )   &    |-  B  =  ( Base `  G )   =>    |-  T  C_  B
 
Theoremsymggen 17062* The span of the transpositions is the subgroup that moves finitely many points. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  T  =  ran  (pmTrsp `  D )   &    |-  G  =  (
 SymGrp `  D )   &    |-  B  =  ( Base `  G )   &    |-  K  =  (mrCls `  (SubMnd `  G ) )   =>    |-  ( D  e.  V  ->  ( K `  T )  =  { x  e.  B  |  dom  ( x  \  _I  )  e. 
 Fin } )
 
Theoremsymggen2 17063 A finite permutation group is generated by the transpositions, see also Theorem 3.4 in [Rotman] p. 31. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  T  =  ran  (pmTrsp `  D )   &    |-  G  =  (
 SymGrp `  D )   &    |-  B  =  ( Base `  G )   &    |-  K  =  (mrCls `  (SubMnd `  G ) )   =>    |-  ( D  e.  Fin  ->  ( K `  T )  =  B )
 
Theoremsymgtrinv 17064 To invert a permutation represented as a sequence of transpositions, reverse the sequence. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  T  =  ran  (pmTrsp `  D )   &    |-  G  =  (
 SymGrp `  D )   &    |-  I  =  ( invg `  G )   =>    |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I `  ( G  gsumg 
 W ) )  =  ( G  gsumg  (reverse `  W )
 ) )
 
Theorempmtr3ncomlem1 17065 Lemma 1 for pmtr3ncom 17067. (Contributed by AV, 17-Mar-2018.)
 |-  T  =  (pmTrsp `  D )   &    |-  F  =  ( T `
  { X ,  Y } )   &    |-  G  =  ( T `  { Y ,  Z } )   =>    |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z ) ) 
 ->  ( ( G  o.  F ) `  X )  =/=  ( ( F  o.  G ) `  X ) )
 
Theorempmtr3ncomlem2 17066 Lemma 2 for pmtr3ncom 17067. (Contributed by AV, 17-Mar-2018.)
 |-  T  =  (pmTrsp `  D )   &    |-  F  =  ( T `
  { X ,  Y } )   &    |-  G  =  ( T `  { Y ,  Z } )   =>    |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z ) ) 
 ->  ( G  o.  F )  =/=  ( F  o.  G ) )
 
Theorempmtr3ncom 17067* Transpositions over sets with at least 3 elements are not commutative, see also the remark in [Rotman] p. 28. (Contributed by AV, 21-Mar-2019.)
 |-  T  =  (pmTrsp `  D )   =>    |-  ( ( D  e.  V  /\  3  <_  ( # `
  D ) ) 
 ->  E. f  e.  ran  T E. g  e.  ran  T ( g  o.  f
 )  =/=  ( f  o.  g ) )
 
Theorempmtrdifellem1 17068 Lemma 1 for pmtrdifel 17072. (Contributed by AV, 15-Jan-2019.)
 |-  T  =  ran  (pmTrsp `  ( N  \  { K } ) )   &    |-  R  =  ran  (pmTrsp `  N )   &    |-  S  =  ( (pmTrsp `  N ) `  dom  ( Q  \  _I  )
 )   =>    |-  ( Q  e.  T  ->  S  e.  R )
 
Theorempmtrdifellem2 17069 Lemma 2 for pmtrdifel 17072. (Contributed by AV, 15-Jan-2019.)
 |-  T  =  ran  (pmTrsp `  ( N  \  { K } ) )   &    |-  R  =  ran  (pmTrsp `  N )   &    |-  S  =  ( (pmTrsp `  N ) `  dom  ( Q  \  _I  )
 )   =>    |-  ( Q  e.  T  ->  dom  ( S  \  _I  )  =  dom  ( Q  \  _I  )
 )
 
Theorempmtrdifellem3 17070* Lemma 3 for pmtrdifel 17072. (Contributed by AV, 15-Jan-2019.)
 |-  T  =  ran  (pmTrsp `  ( N  \  { K } ) )   &    |-  R  =  ran  (pmTrsp `  N )   &    |-  S  =  ( (pmTrsp `  N ) `  dom  ( Q  \  _I  )
 )   =>    |-  ( Q  e.  T  ->  A. x  e.  ( N  \  { K }
 ) ( Q `  x )  =  ( S `  x ) )
 
Theorempmtrdifellem4 17071 Lemma 4 for pmtrdifel 17072. (Contributed by AV, 28-Jan-2019.)
 |-  T  =  ran  (pmTrsp `  ( N  \  { K } ) )   &    |-  R  =  ran  (pmTrsp `  N )   &    |-  S  =  ( (pmTrsp `  N ) `  dom  ( Q  \  _I  )
 )   =>    |-  ( ( Q  e.  T  /\  K  e.  N )  ->  ( S `  K )  =  K )
 
Theorempmtrdifel 17072* A transposition of elements of a set without a special element corresponds to a transposition of elements of the set. (Contributed by AV, 15-Jan-2019.)
 |-  T  =  ran  (pmTrsp `  ( N  \  { K } ) )   &    |-  R  =  ran  (pmTrsp `  N )   =>    |- 
 A. t  e.  T  E. r  e.  R  A. x  e.  ( N 
 \  { K }
 ) ( t `  x )  =  (
 r `  x )
 
Theorempmtrdifwrdellem1 17073* Lemma 1 for pmtrdifwrdel 17077. (Contributed by AV, 15-Jan-2019.)
 |-  T  =  ran  (pmTrsp `  ( N  \  { K } ) )   &    |-  R  =  ran  (pmTrsp `  N )   &    |-  U  =  ( x  e.  ( 0..^ ( # `  W ) ) 
 |->  ( (pmTrsp `  N ) `  dom  ( ( W `  x ) 
 \  _I  ) ) )   =>    |-  ( W  e. Word  T  ->  U  e. Word  R )
 
Theorempmtrdifwrdellem2 17074* Lemma 2 for pmtrdifwrdel 17077. (Contributed by AV, 15-Jan-2019.)
 |-  T  =  ran  (pmTrsp `  ( N  \  { K } ) )   &    |-  R  =  ran  (pmTrsp `  N )   &    |-  U  =  ( x  e.  ( 0..^ ( # `  W ) ) 
 |->  ( (pmTrsp `  N ) `  dom  ( ( W `  x ) 
 \  _I  ) ) )   =>    |-  ( W  e. Word  T  ->  ( # `  W )  =  ( # `  U ) )
 
Theorempmtrdifwrdellem3 17075* Lemma 3 for pmtrdifwrdel 17077. (Contributed by AV, 15-Jan-2019.)
 |-  T  =  ran  (pmTrsp `  ( N  \  { K } ) )   &    |-  R  =  ran  (pmTrsp `  N )   &    |-  U  =  ( x  e.  ( 0..^ ( # `  W ) ) 
 |->  ( (pmTrsp `  N ) `  dom  ( ( W `  x ) 
 \  _I  ) ) )   =>    |-  ( W  e. Word  T  ->  A. i  e.  (
 0..^ ( # `  W ) ) A. n  e.  ( N  \  { K } ) ( ( W `  i ) `
  n )  =  ( ( U `  i ) `  n ) )
 
Theorempmtrdifwrdel2lem1 17076* Lemma 1 for pmtrdifwrdel2 17078. (Contributed by AV, 31-Jan-2019.)
 |-  T  =  ran  (pmTrsp `  ( N  \  { K } ) )   &    |-  R  =  ran  (pmTrsp `  N )   &    |-  U  =  ( x  e.  ( 0..^ ( # `  W ) ) 
 |->  ( (pmTrsp `  N ) `  dom  ( ( W `  x ) 
 \  _I  ) ) )   =>    |-  ( ( W  e. Word  T 
 /\  K  e.  N )  ->  A. i  e.  (
 0..^ ( # `  W ) ) ( ( U `  i ) `
  K )  =  K )
 
Theorempmtrdifwrdel 17077* A sequence of transpositions of elements of a set without a special element corresponds to a sequence of transpositions of elements of the set. (Contributed by AV, 15-Jan-2019.)
 |-  T  =  ran  (pmTrsp `  ( N  \  { K } ) )   &    |-  R  =  ran  (pmTrsp `  N )   =>    |- 
 A. w  e. Word  T E. u  e. Word  R ( ( # `  w )  =  ( # `  u )  /\  A. i  e.  ( 0..^ ( # `  w ) ) A. x  e.  ( N  \  { K } )
 ( ( w `  i ) `  x )  =  ( ( u `  i ) `  x ) )
 
Theorempmtrdifwrdel2 17078* A sequence of transpositions of elements of a set without a special element corresponds to a sequence of transpositions of elements of the set not moving the special element. (Contributed by AV, 31-Jan-2019.)
 |-  T  =  ran  (pmTrsp `  ( N  \  { K } ) )   &    |-  R  =  ran  (pmTrsp `  N )   =>    |-  ( K  e.  N  ->  A. w  e. Word  T E. u  e. Word  R ( ( # `  w )  =  ( # `  u )  /\  A. i  e.  ( 0..^ ( # `  w ) ) ( ( ( u `  i ) `  K )  =  K  /\  A. x  e.  ( N 
 \  { K }
 ) ( ( w `
  i ) `  x )  =  (
 ( u `  i
 ) `  x )
 ) ) )
 
Theorempmtrprfval 17079* The transpositions on a pair. (Contributed by AV, 9-Dec-2018.)
 |-  (pmTrsp `  { 1 ,  2 } )  =  ( p  e.  { { 1 ,  2 } }  |->  ( z  e.  { 1 ,  2 }  |->  if (
 z  =  1 ,  2 ,  1 ) ) )
 
Theorempmtrprfvalrn 17080 The range of the transpositions on a pair is actually a singleton: the transposition of the two elements of the pair. (Contributed by AV, 9-Dec-2018.)
 |- 
 ran  (pmTrsp `  { 1 ,  2 } )  =  { { <. 1 ,  2 >. ,  <. 2 ,  1 >. } }
 
10.2.8.5  The sign of a permutation
 
Syntaxcpsgn 17081 Syntax for the sign of a permutation.
 class pmSgn
 
Syntaxcevpm 17082 Syntax for even permutations.
 class pmEven
 
Definitiondf-psgn 17083* Define a function which takes the value  1 for even permutations and  -u 1 for odd. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |- pmSgn  =  ( d  e.  _V  |->  ( x  e.  { p  e.  ( Base `  ( SymGrp `  d
 ) )  |  dom  ( p  \  _I  )  e.  Fin }  |->  ( iota
 s E. w  e. Word  ran  (pmTrsp `  d )
 ( x  =  ( ( SymGrp `  d )  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `
  w ) ) ) ) ) )
 
Definitiondf-evpm 17084 Define the set of even permutations on a given set. (Contributed by Stefan O'Rear, 9-Jul-2018.)
 |- pmEven  =  ( d  e.  _V  |->  ( `' (pmSgn `  d ) " { 1 } )
 )
 
Theorempsgnunilem1 17085* Lemma for psgnuni 17091. Given two consequtive transpositions in a representation of a permutation, either they are equal and therefore equivalent to the identity, or they are not and it is possible to commute them such that a chosen point in the left transposition is preserved in the right. By repeating this process, a point can be removed from a representation of the identity. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  ran  (pmTrsp `  D )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  P  e.  T )   &    |-  ( ph  ->  Q  e.  T )   &    |-  ( ph  ->  A  e.  dom  ( P  \  _I  ) )   =>    |-  ( ph  ->  ( ( P  o.  Q )  =  (  _I  |`  D )  \/  E. r  e.  T  E. s  e.  T  ( ( P  o.  Q )  =  ( r  o.  s
 )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e.  dom  ( r  \  _I  ) ) ) )
 
Theorempsgnunilem5 17086* Lemma for psgnuni 17091. It is impossible to shift a transposition off the end because if the active transposition is at the right end, it is the only transposition moving  A in contradiction to this being a representation of the identity. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  W  e. Word  T )   &    |-  ( ph  ->  ( G  gsumg 
 W )  =  (  _I  |`  D )
 )   &    |-  ( ph  ->  ( # `
  W )  =  L )   &    |-  ( ph  ->  I  e.  ( 0..^ L ) )   &    |-  ( ph  ->  A  e.  dom  ( ( W `  I )  \  _I  ) )   &    |-  ( ph  ->  A. k  e.  ( 0..^ I )  -.  A  e.  dom  ( ( W `
  k )  \  _I  ) )   =>    |-  ( ph  ->  ( I  +  1 )  e.  ( 0..^ L ) )
 
Theorempsgnunilem2 17087* Lemma for psgnuni 17091. Induction step for moving a transposition as far to the right as possible. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  W  e. Word  T )   &    |-  ( ph  ->  ( G  gsumg 
 W )  =  (  _I  |`  D )
 )   &    |-  ( ph  ->  ( # `
  W )  =  L )   &    |-  ( ph  ->  I  e.  ( 0..^ L ) )   &    |-  ( ph  ->  A  e.  dom  ( ( W `  I )  \  _I  ) )   &    |-  ( ph  ->  A. k  e.  ( 0..^ I )  -.  A  e.  dom  ( ( W `
  k )  \  _I  ) )   &    |-  ( ph  ->  -. 
 E. x  e. Word  T ( ( # `  x )  =  ( L  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
 ) )   =>    |-  ( ph  ->  E. w  e. Word  T ( ( ( G  gsumg  w )  =  (  _I  |`  D )  /\  ( # `  w )  =  L )  /\  ( ( I  +  1 )  e.  (
 0..^ L )  /\  A  e.  dom  ( ( w `  ( I  +  1 ) ) 
 \  _I  )  /\  A. j  e.  ( 0..^ ( I  +  1 ) )  -.  A  e.  dom  ( ( w `
  j )  \  _I  ) ) ) )
 
Theorempsgnunilem3 17088* Lemma for psgnuni 17091. Any nonempty representation of the identity can be incrementally transformed into a representation two shorter. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  W  e. Word  T )   &    |-  ( ph  ->  ( # `  W )  =  L )   &    |-  ( ph  ->  ( # `  W )  e. 
 NN )   &    |-  ( ph  ->  ( G  gsumg 
 W )  =  (  _I  |`  D )
 )   &    |-  ( ph  ->  -.  E. x  e. Word  T ( ( # `  x )  =  ( L  -  2
 )  /\  ( G  gsumg  x )  =  (  _I  |`  D ) ) )   =>    |-  -.  ph
 
Theorempsgnunilem4 17089 Lemma for psgnuni 17091. An odd-length representation of the identity is impossible, as it could be repeatedly shortened to a length of 1, but a length 1 permutation must be a transposition. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  W  e. Word  T )   &    |-  ( ph  ->  ( G  gsumg 
 W )  =  (  _I  |`  D )
 )   =>    |-  ( ph  ->  ( -u 1 ^ ( # `  W ) )  =  1 )
 
Theoremm1expaddsub 17090 Addition and subtraction of parities are the same. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( -u 1 ^ ( X  -  Y ) )  =  ( -u 1 ^ ( X  +  Y )
 ) )
 
Theorempsgnuni 17091 If the same permutation can be written in more than one way as a product of transpositions, the parity of those products must agree; otherwise the product of one with the inverse of the other would be an odd representation of the identity. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  W  e. Word  T )   &    |-  ( ph  ->  X  e. Word  T )   &    |-  ( ph  ->  ( G  gsumg  W )  =  ( G  gsumg  X ) )   =>    |-  ( ph  ->  ( -u 1 ^ ( # `  W ) )  =  ( -u 1 ^ ( # `
  X ) ) )
 
Theorempsgnfval 17092* Function definition of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  B  =  (
 Base `  G )   &    |-  F  =  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  N  =  ( x  e.  F  |->  ( iota
 s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `
  w ) ) ) ) )
 
Theorempsgnfn 17093* Functionality and domain of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  B  =  (
 Base `  G )   &    |-  F  =  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }   &    |-  N  =  (pmSgn `  D )   =>    |-  N  Fn  F
 
Theorempsgndmsubg 17094 The finitary permutations are a subgroup. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( D  e.  V  ->  dom  N  e.  (SubGrp `  G ) )
 
Theorempsgneldm 17095 Property of being a finitary permutation. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  N  =  (pmSgn `  D )   &    |-  B  =  (
 Base `  G )   =>    |-  ( P  e.  dom 
 N 
 <->  ( P  e.  B  /\  dom  ( P  \  _I  )  e.  Fin ) )
 
Theorempsgneldm2 17096* The finitary permutations are the span of the transpositions. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( D  e.  V  ->  ( P  e.  dom  N  <->  E. w  e. Word  T P  =  ( G  gsumg  w ) ) )
 
Theorempsgneldm2i 17097 A sequence of transpositions describes a finitary permutation. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  (
 ( D  e.  V  /\  W  e. Word  T )  ->  ( G  gsumg 
 W )  e.  dom  N )
 
Theorempsgneu 17098* A finitary permutation has exactly one parity. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( P  e.  dom  N  ->  E! s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `
  w ) ) ) )
 
Theorempsgnval 17099* Value of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( P  e.  dom  N  ->  ( N `  P )  =  ( iota s E. w  e. Word  T ( P  =  ( G 
 gsumg  w )  /\  s  =  ( -u 1 ^ ( # `
  w ) ) ) ) )
 
Theorempsgnvali 17100* A finitary permutation has at least one representation for its parity. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( P  e.  dom  N  ->  E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  ( N `  P )  =  ( -u 1 ^ ( # `  w ) ) ) )
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