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Theorem List for Metamath Proof Explorer - 16601-16700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempmtrffv 16601 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 16602* 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 16603 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 16604 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 16605 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 16606 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 16607 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 16608 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 16609* 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 16610* 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 16611 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 16612* 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 16613 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 16614 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 16615 Lemma 1 for pmtr3ncom 16617. (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 16616 Lemma 2 for pmtr3ncom 16617. (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 16617* 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 16618 Lemma 1 for pmtrdifel 16622. (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 16619 Lemma 2 for pmtrdifel 16622. (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 16620* Lemma 3 for pmtrdifel 16622. (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 16621 Lemma 4 for pmtrdifel 16622. (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 16622* 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 16623* Lemma 1 for pmtrdifwrdel 16627. (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 16624* Lemma 2 for pmtrdifwrdel 16627. (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 16625* Lemma 3 for pmtrdifwrdel 16627. (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 16626* Lemma 1 for pmtrdifwrdel2 16628. (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 16627* 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 16628* 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 16629* 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 16630 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 16631 Syntax for the sign of a permutation.
 class pmSgn
 
Syntaxcevpm 16632 Syntax for even permutations.
 class pmEven
 
Definitiondf-psgn 16633* 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 16634 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 16635* Lemma for psgnuni 16641. 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 16636* Lemma for psgnuni 16641. 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 16637* Lemma for psgnuni 16641. 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 16638* Lemma for psgnuni 16641. 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 16639 Lemma for psgnuni 16641. 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 16640 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 16641 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 16642* 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 16643* 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 16644 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 16645 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 16646* 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 16647 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 16648* 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 16649* 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 16650* 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 16651 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 16652 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 16653 The permutation sign function for an empty set at an empty set is 1. (Contributed by AV, 27-Feb-2019.)
 |-  ( (pmSgn `  (/) ) `  (/) )  =  1
 
Theoremsygbasnfpfi 16654 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 16655* 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 16656* 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 16657 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 16658 The group sum of transpositions of a finite set is a permutation, see also psgneldm2i 16647. (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 )
 
Theorempsgnfitr 16659* A permutation of a finite set is generated by transpositions. (Contributed by AV, 13-Jan-2019.)
 |-  G  =  ( SymGrp `  N )   &    |-  B  =  (
 Base `  G )   &    |-  T  =  ran  (pmTrsp `  N )   =>    |-  ( N  e.  Fin  ->  ( Q  e.  B  <->  E. w  e. Word  T Q  =  ( G  gsumg  w ) ) )
 
Theorempsgnfieu 16660* A permutation of a finite set has exactly one parity. (Contributed by AV, 13-Jan-2019.)
 |-  G  =  ( SymGrp `  N )   &    |-  B  =  (
 Base `  G )   &    |-  T  =  ran  (pmTrsp `  N )   =>    |-  ( ( N  e.  Fin  /\  Q  e.  B ) 
 ->  E! s E. w  e. Word  T ( Q  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `
  w ) ) ) )
 
Theorempmtrsn 16661 The value of the transposition generator function for a singleton is empty, i.e. there is no transposition for a singleton. This also holds for  A  e/  _V, i.e. for the empty set  { A }  =  (/) resulting in  (pmTrsp `  (/) )  =  (/). (Contributed by AV, 6-Aug-2019.)
 |-  (pmTrsp `  { A }
 )  =  (/)
 
Theorempsgnsn 16662 The permutation sign function for a singleton. (Contributed by AV, 6-Aug-2019.)
 |-  D  =  { A }   &    |-  G  =  ( SymGrp `  D )   &    |-  B  =  (
 Base `  G )   &    |-  N  =  (pmSgn `  D )   =>    |-  (
 ( A  e.  V  /\  X  e.  B ) 
 ->  ( N `  X )  =  1 )
 
Theorempsgnprfval 16663* The permutation sign function for a pair. (Contributed by AV, 10-Dec-2018.)
 |-  D  =  { 1 ,  2 }   &    |-  G  =  ( SymGrp `  D )   &    |-  B  =  ( Base `  G )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( X  e.  B  ->  ( N `  X )  =  ( iota s E. w  e. Word  T ( X  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `
  w ) ) ) ) )
 
Theorempsgnprfval1 16664 The permutation sign of the identity for a pair. (Contributed by AV, 11-Dec-2018.)
 |-  D  =  { 1 ,  2 }   &    |-  G  =  ( SymGrp `  D )   &    |-  B  =  ( Base `  G )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( N `  { <. 1 ,  1 >. ,  <. 2 ,  2 >. } )  =  1
 
Theorempsgnprfval2 16665 The permutation sign of the transposition for a pair. (Contributed by AV, 10-Dec-2018.)
 |-  D  =  { 1 ,  2 }   &    |-  G  =  ( SymGrp `  D )   &    |-  B  =  ( Base `  G )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( N `  { <. 1 ,  2 >. ,  <. 2 ,  1 >. } )  =  -u 1
 
10.2.9  p-Groups and Sylow groups; Sylow's theorems
 
Syntaxcod 16666 Extend class notation to include the order function on the elements of a group.
 class  od
 
Syntaxcgex 16667 Extend class notation to include the order function on the elements of a group.
 class gEx
 
Syntaxcpgp 16668 Extend class notation to include the class of all p-groups.
 class pGrp
 
Syntaxcslw 16669 Extend class notation to include the class of all Sylow p-subgroups of a group.
 class pSyl
 
Definitiondf-od 16670* Define the order of an element in a group. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 4-Sep-2015.)
 |- 
 od  =  ( g  e.  _V  |->  ( x  e.  ( Base `  g
 )  |->  [_ { n  e. 
 NN  |  ( n (.g `  g ) x )  =  ( 0g
 `  g ) }  /  i ]_ if (
 i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) ) )
 
Definitiondf-gex 16671* Define the exponent of a group. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 4-Sep-2015.)
 |- gEx 
 =  ( g  e. 
 _V  |->  [_ { n  e. 
 NN  |  A. x  e.  ( Base `  g )
 ( n (.g `  g
 ) x )  =  ( 0g `  g
 ) }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )
 
Definitiondf-pgp 16672* Define the set of p-groups, which are groups such that every element has a power of  p as its order. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |- pGrp  =  { <. p ,  g >.  |  ( ( p  e.  Prime  /\  g  e. 
 Grp )  /\  A. x  e.  ( Base `  g ) E. n  e.  NN0  ( ( od
 `  g ) `  x )  =  ( p ^ n ) ) }
 
Definitiondf-slw 16673* Define the set of Sylow p-subgroups of a group  g. A Sylow p-subgroup is a p-group that is not a subgroup of any other p-groups in  g. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |- pSyl  =  ( p  e.  Prime ,  g  e.  Grp  |->  { h  e.  (SubGrp `  g )  |  A. k  e.  (SubGrp `  g ) ( ( h  C_  k  /\  p pGrp  ( gs  k ) )  <->  h  =  k
 ) } )
 
Theoremodfval 16674* Value of the order function. (Contributed by Mario Carneiro, 13-Jul-2014.)
 |-  X  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  O  =  ( od `  G )   =>    |-  O  =  ( x  e.  X  |->  [_ { y  e.  NN  |  ( y 
 .x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )
 
Theoremodval 16675* Second substitution for the group order definition. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 5-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  O  =  ( od `  G )   &    |-  I  =  {
 y  e.  NN  |  ( y  .x.  A )  =  .0.  }   =>    |-  ( A  e.  X  ->  ( O `  A )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
 
Theoremodlem1 16676* The group element order is either zero or a nonzero multiplier that annihilates the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  O  =  ( od `  G )   &    |-  I  =  {
 y  e.  NN  |  ( y  .x.  A )  =  .0.  }   =>    |-  ( A  e.  X  ->  ( ( ( O `  A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I ) )
 
Theoremodcl 16677 The order of a group element is always a nonnegative integer. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
 
Theoremodf 16678 Functionality of the group element order. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   =>    |-  O : X --> NN0
 
Theoremodid 16679 Any element to the power of its order is the identity. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( A  e.  X  ->  ( ( O `  A )  .x.  A )  =  .0.  )
 
Theoremodlem2 16680 Any positive annihilator of a group element is an upper bound on the (positive) order of the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( A  e.  X  /\  N  e.  NN  /\  ( N  .x.  A )  =  .0.  )  ->  ( O `  A )  e.  ( 1 ... N ) )
 
Theoremodmodnn0 16681 Reduce the argument of a group multiple by modding out the order of the element. (Contributed by Mario Carneiro, 23-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  e.  NN )  ->  ( ( N 
 mod  ( O `  A ) )  .x.  A )  =  ( N 
 .x.  A ) )
 
Theoremmndodconglem 16682 Lemma for mndodcong 16683. (Contributed by Mario Carneiro, 23-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  ( O `  A )  e.  NN )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  M  <  ( O `  A ) )   &    |-  ( ph  ->  N  <  ( O `  A ) )   &    |-  ( ph  ->  ( M  .x.  A )  =  ( N  .x.  A )
 )   =>    |-  ( ( ph  /\  M  <_  N )  ->  M  =  N )
 
Theoremmndodcong 16683 If two multipliers are congruent relative to the base point's order, the corresponding multiples are the same. (Contributed by Mario Carneiro, 23-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( ( G  e.  Mnd  /\  A  e.  X )  /\  ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( O `
  A )  e. 
 NN )  ->  (
 ( O `  A )  ||  ( M  -  N )  <->  ( M  .x.  A )  =  ( N 
 .x.  A ) ) )
 
Theoremmndodcongi 16684 If two multipliers are congruent relative to the base point's order, the corresponding multiples are the same. For monoids, the reverse implication is false for elements with infinite order. For example, the powers of  2 mod  10 are 1,2,4,8,6,2,4,8,6,... so that the identity 1 never repeats, which is infinite order by our definition, yet other numbers like 6 appear many times in the sequence. (Contributed by Mario Carneiro, 23-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  ->  ( ( O `  A )  ||  ( M  -  N )  ->  ( M  .x.  A )  =  ( N  .x.  A ) ) )
 
Theoremoddvdsnn0 16685 The only multiples of  A that are equal to the identity are the multiples of the order of  A. (Contributed by Mario Carneiro, 23-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  ->  ( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  )
 )
 
Theoremodnncl 16686 If a nonzero multiple of an element is zero, the element has positive order. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A )  =  .0.  )
 )  ->  ( O `  A )  e.  NN )
 
Theoremodmod 16687 Reduce the argument of a group multiple by modding out the order of the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 6-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A )  e.  NN )  ->  ( ( N 
 mod  ( O `  A ) )  .x.  A )  =  ( N 
 .x.  A ) )
 
Theoremoddvds 16688 The only multiples of  A that are equal to the identity are the multiples of the order of  A. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  )
 )
 
Theoremoddvdsi 16689 Any group element is annihilated by any multiple of its order. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A ) 
 ||  N )  ->  ( N  .x.  A )  =  .0.  )
 
Theoremodcong 16690 If two multipliers are congruent relative to the base point's order, the corresponding multiples are the same. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( M  e.  ZZ  /\  N  e.  ZZ )
 )  ->  ( ( O `  A )  ||  ( M  -  N ) 
 <->  ( M  .x.  A )  =  ( N  .x.  A ) ) )
 
Theoremodeq 16691* The oddvds 16688 property uniquely defines the group order. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  ->  ( N  =  ( O `  A )  <->  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  )
 ) )
 
Theoremodval2 16692* A non-conditional definition of the group order. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X ) 
 ->  ( O `  A )  =  ( iota_ x  e. 
 NN0  A. y  e.  NN0  ( x  ||  y  <->  ( y  .x.  A )  =  .0.  )
 ) )
 
Theoremodmulgid 16693 A relationship between the order of a multiple and the order of the basepoint. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   =>    |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( O `  ( N  .x.  A ) )  ||  K  <->  ( O `  A )  ||  ( K  x.  N ) ) )
 
Theoremodmulg2 16694 The order of a multiple divides the order of the base point. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( O `  ( N  .x.  A ) )  ||  ( O `  A ) )
 
Theoremodmulg 16695 Relationship between the order of an element and that of a multiple. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( O `  A )  =  (
 ( N  gcd  ( O `  A ) )  x.  ( O `  ( N  .x.  A ) ) ) )
 
Theoremodmulgeq 16696 A multiple of a point of finite order only has the same order if the multiplier is relatively prime. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   =>    |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A )  e.  NN )  ->  ( ( O `  ( N  .x.  A ) )  =  ( O `
  A )  <->  ( N  gcd  ( O `  A ) )  =  1 ) )
 
Theoremodbezout 16697* If  N is coprime to the order of  A, there is a modular inverse  x to cancel multiplication by  N. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   =>    |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  gcd  ( O `  A ) )  =  1 )  ->  E. x  e.  ZZ  ( x  .x.  ( N 
 .x.  A ) )  =  A )
 
Theoremod1 16698 The order of the group identity is one. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
 |-  O  =  ( od
 `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( G  e.  Grp  ->  ( O `  .0.  )  =  1 )
 
Theoremodeq1 16699 The group identity is the unique element of a group with order one. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
 |-  O  =  ( od
 `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  X  =  ( Base `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X ) 
 ->  ( ( O `  A )  =  1  <->  A  =  .0.  ) )
 
Theoremodinv 16700 The order of the inverse of a group element. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  O  =  ( od
 `  G )   &    |-  I  =  ( invg `  G )   &    |-  X  =  (
 Base `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( O `
  ( I `  A ) )  =  ( O `  A ) )
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