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Theorem List for Metamath Proof Explorer - 17001-17100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsymgextf 17001* 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 17002* 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 17003* 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 17004* Lemma for symgextf1 17005. (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 17005* 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 17006* 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 17007* 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 17008* 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 17009* 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 17010 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 17011* Lemma 1 for gsmsymgrfix 17012. (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 17012* 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 17013* 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 17014* Lemma 1 for gsmsymgreq 17016. (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 17015* Lemma 2 for gsmsymgreq 17016. (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 17016* 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 17017* 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 17018* 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 17019* 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 17020* 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 17021* 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 17022* Lemma 1 for symgfixfo 17023. (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 17023* 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 17024* 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 17026.

 
Syntaxcpmtr 17025 Syntax for the transposition generator function.
 class pmTrsp
 
Definitiondf-pmtr 17026* 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 17027 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 17028 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 17029 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 17030 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 17031 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 17032 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 17033 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 17034* 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 17035* 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 17036 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 17037 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 17038 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 17039 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 17040 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 17041 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 17042 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 17043 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 17044* 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 17045 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 17046 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 17047 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 17048 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 17049 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 17050 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 17051* 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 17052* 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 17053 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 17054* 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 17055 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 17056 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 17057 Lemma 1 for pmtr3ncom 17059. (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 17058 Lemma 2 for pmtr3ncom 17059. (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 17059* 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 17060 Lemma 1 for pmtrdifel 17064. (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 17061 Lemma 2 for pmtrdifel 17064. (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 17062* Lemma 3 for pmtrdifel 17064. (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 17063 Lemma 4 for pmtrdifel 17064. (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 17064* 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 17065* Lemma 1 for pmtrdifwrdel 17069. (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 17066* Lemma 2 for pmtrdifwrdel 17069. (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 17067* Lemma 3 for pmtrdifwrdel 17069. (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 17068* Lemma 1 for pmtrdifwrdel2 17070. (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 17069* 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 17070* 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 17071* 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 17072 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 17073 Syntax for the sign of a permutation.
 class pmSgn
 
Syntaxcevpm 17074 Syntax for even permutations.
 class pmEven
 
Definitiondf-psgn 17075* 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 17076 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 17077* Lemma for psgnuni 17083. 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 17078* Lemma for psgnuni 17083. 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 17079* Lemma for psgnuni 17083. 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 17080* Lemma for psgnuni 17083. 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 17081 Lemma for psgnuni 17083. 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 17082 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 17083 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 17084* 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 17085* 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 17086 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 17087 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 17088* 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 17089 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 17090* 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 17091* 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 17092* 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 ) ) ) )
 
Theorempsgnvalii 17093 Any representation of a permutation is length matching the permutation sign. (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 )  ->  ( N `  ( G  gsumg 
 W ) )  =  ( -u 1 ^ ( # `
  W ) ) )
 
Theorempsgnpmtr 17094 All transpositions are odd. (Contributed by Stefan O'Rear, 29-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( P  e.  T  ->  ( N `  P )  =  -u 1 )
 
Theorempsgn0fv0 17095 The permutation sign function for an empty set at an empty set is 1. (Contributed by AV, 27-Feb-2019.)
 |-  ( (pmSgn `  (/) ) `  (/) )  =  1
 
Theoremsygbasnfpfi 17096 The class of non-fixed points of a permutation of a finite set is finite. (Contributed by AV, 13-Jan-2019.)
 |-  G  =  ( SymGrp `  D )   &    |-  B  =  (
 Base `  G )   =>    |-  ( ( D  e.  Fin  /\  P  e.  B )  ->  dom  ( P  \  _I  )  e. 
 Fin )
 
Theorempsgnfvalfi 17097* Function definition of the permutation sign function for permutations of finite sets. (Contributed by AV, 13-Jan-2019.)
 |-  G  =  ( SymGrp `  D )   &    |-  B  =  (
 Base `  G )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( D  e.  Fin  ->  N  =  ( x  e.  B  |->  ( iota s E. w  e. Word  T ( x  =  ( G 
 gsumg  w )  /\  s  =  ( -u 1 ^ ( # `
  w ) ) ) ) ) )
 
Theorempsgnvalfi 17098* Value of the permutation sign function for permutations of finite sets. (Contributed by AV, 13-Jan-2019.)
 |-  G  =  ( SymGrp `  D )   &    |-  B  =  (
 Base `  G )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( ( D  e.  Fin  /\  P  e.  B ) 
 ->  ( N `  P )  =  ( iota s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `
  w ) ) ) ) )
 
Theorempsgnran 17099 The range of the permutation sign function for finite permutations. (Contributed by AV, 1-Jan-2019.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  S  =  (pmSgn `  N )   =>    |-  ( ( N  e.  Fin  /\  Q  e.  P ) 
 ->  ( S `  Q )  e.  { 1 ,  -u 1 } )
 
Theoremgsmtrcl 17100 The group sum of transpositions of a finite set is a permutation, see also psgneldm2i 17089. (Contributed by AV, 19-Jan-2019.)
 |-  S  =  ( SymGrp `  N )   &    |-  B  =  (
 Base `  S )   &    |-  T  =  ran  (pmTrsp `  N )   =>    |-  ( ( N  e.  Fin  /\  W  e. Word  T )  ->  ( S  gsumg 
 W )  e.  B )
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