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Theorem List for Metamath Proof Explorer - 14601-14700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremorbstaval 14601* Value of the function at a given equivalence class element. (Contributed by Mario Carneiro, 15-Jan-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
 |-  X  =  ( Base `  G )   &    |-  H  =  { u  e.  X  |  ( u  .(+)  A )  =  A }   &    |-  .~  =  ( G ~QG  H )   &    |-  F  =  ran  (  k  e.  X  |->  <. [ k ]  .~  ,  ( k  .(+)  A )
 >. )   =>    |-  ( ( (  .(+)  e.  ( G  GrpAct  Y ) 
 /\  A  e.  Y )  /\  B  e.  X )  ->  ( F `  [ B ]  .~  )  =  ( B  .(+)  A ) )
 
Theoremorbsta 14602* The Orbit-Stabilizer theorem. The mapping  F is a bijection from the cosets of the stabilizer subgroup of  A to the orbit of  A. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  H  =  { u  e.  X  |  ( u  .(+)  A )  =  A }   &    |-  .~  =  ( G ~QG  H )   &    |-  F  =  ran  (  k  e.  X  |->  <. [ k ]  .~  ,  ( k  .(+)  A )
 >. )   &    |-  O  =  { <. x ,  y >.  |  ( { x ,  y }  C_  Y  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }   =>    |-  ( (  .(+)  e.  ( G  GrpAct  Y ) 
 /\  A  e.  Y )  ->  F : ( X /.  .~  ) -1-1-onto-> [ A ] O )
 
Theoremorbsta2 14603* Relation between the size of the orbit and the size of the stabilizer of a point in a finite group action. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  H  =  { u  e.  X  |  ( u  .(+)  A )  =  A }   &    |-  .~  =  ( G ~QG  H )   &    |-  O  =  { <. x ,  y >.  |  ( { x ,  y }  C_  Y  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }   =>    |-  ( ( ( 
 .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( # `  X )  =  ( ( # `
  [ A ] O )  x.  ( # `
  H ) ) )
 
10.2.6  Symmetry groups and Cayley's Theorem
 
Syntaxcsymg 14604 Extend class notation to include the class of symmetry groups.
 class  SymGrp
 
Definitiondf-symg 14605* Define the symmetry group on set  x. We represent the group as the set of 1-1-onto functions from  x to itself under function composition, and topologize it as a function space assuming the set is discrete. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  SymGrp  =  ( x  e. 
 _V  |->  [_ { h  |  h : x -1-1-onto-> x }  /  b ]_ { <. ( Base `  ndx ) ,  b >. , 
 <. ( +g  `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( f  o.  g ) ) >. , 
 <. (TopSet `  ndx ) ,  ( Xt_ `  ( x  X.  { ~P x } ) ) >. } )
 
Theoremsymgval 14606* The value of the symmetry group function at  A. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   &    |-  B  =  { x  |  x : A
 -1-1-onto-> A }   &    |-  .+  =  (
 f  e.  B ,  g  e.  B  |->  ( f  o.  g ) )   &    |-  J  =  ( Xt_ `  ( A  X.  { ~P A } ) )   =>    |-  ( A  e.  V  ->  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. } )
 
Theoremsymgbas 14607* The base set of the symmetric group. (Contributed by Mario Carneiro, 12-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   &    |-  B  =  (
 Base `  G )   =>    |-  B  =  { x  |  x : A
 -1-1-onto-> A }
 
Theoremelsymgbas2 14608 Two ways of saying a function is a 1-1-onto mapping of A to itself. (Contributed by Mario Carneiro, 28-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   &    |-  B  =  (
 Base `  G )   =>    |-  ( F  e.  V  ->  ( F  e.  B 
 <->  F : A -1-1-onto-> A ) )
 
Theoremelsymgbas 14609 Two ways of saying a function is a 1-1-onto mapping of A to itself. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   &    |-  B  =  (
 Base `  G )   =>    |-  ( A  e.  V  ->  ( F  e.  B 
 <->  F : A -1-1-onto-> A ) )
 
Theoremsymghash 14610 The symmetric group on  n objects has cardinality  n !. (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   &    |-  B  =  (
 Base `  G )   =>    |-  ( A  e.  Fin 
 ->  ( # `  B )  =  ( ! `  ( # `  A ) ) )
 
Theoremsymgplusg 14611* The value of the symmetry group function at  A. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   &    |-  B  =  (
 Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  .+  =  ( f  e.  B ,  g  e.  B  |->  ( f  o.  g
 ) )
 
Theoremsymgov 14612 The value of the group operation of the symmetry group on  A. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   &    |-  B  =  (
 Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .+  Y )  =  ( X  o.  Y ) )
 
Theoremsymgcl 14613 The group operation of the symmetry group on  A is closed, i.e. a magma. (Contributed by Mario Carneiro, 12-Jan-2015.) (Revised by Mario Carneiro, 28-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   &    |-  B  =  (
 Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .+  Y )  e.  B )
 
Theoremsymgtset 14614 The topology of the symmetry group on  A. This component is defined on a larger set than the true base - the product topology is defined on the set of all functions, not just bijections - but the definition of  TopOpen ensures that it is trimmed down before it gets use. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  G  =  ( SymGrp `  A )   =>    |-  ( A  e.  V  ->  ( Xt_ `  ( A  X.  { ~P A } ) )  =  (TopSet `  G )
 )
 
Theoremsymggrp 14615 The symmetry group on  A is a group. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   =>    |-  ( A  e.  V  ->  G  e.  Grp )
 
Theoremsymgid 14616 The value of the identity element of the symmetry group on  A (Contributed by Paul Chapman, 25-Jul-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   =>    |-  ( A  e.  V  ->  (  _I  |`  A )  =  ( 0g `  G ) )
 
Theoremsymginv 14617 The group inverse in the symmetric group corresponds to the functional inverse. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.)
 |-  G  =  ( SymGrp `  A )   &    |-  B  =  (
 Base `  G )   &    |-  N  =  ( inv g `  G )   =>    |-  ( F  e.  B  ->  ( N `  F )  =  `' F )
 
Theoremgalactghm 14618* The currying of a group action is a group homomorphism between the group  G and the symetry group  ( SymGrp `  Y
). (Contributed by FL, 17-May-2010.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  H  =  (
 SymGrp `  Y )   &    |-  F  =  ( x  e.  X  |->  ( y  e.  Y  |->  ( x  .(+)  y ) ) )   =>    |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  F  e.  ( G  GrpHom  H ) )
 
Theoremlactghmga 14619* The converse of galactghm 14618. The uncurrying of a homomorphism into  ( SymGrp `  Y
) is a group action. Thus group actions and group homomorphisms into a symmetric group are essentially equivalent notions. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  H  =  (
 SymGrp `  Y )   &    |-  .(+)  =  ( x  e.  X ,  y  e.  Y  |->  ( ( F `  x ) `
  y ) )   =>    |-  ( F  e.  ( G  GrpHom  H )  ->  .(+) 
 e.  ( G  GrpAct  Y ) )
 
Theoremsymgtopn 14620 The topology of the symmetry group on  A. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  G  =  ( SymGrp `  X )   &    |-  B  =  (
 Base `  G )   =>    |-  ( X  e.  V  ->  ( ( Xt_ `  ( X  X.  { ~P X } ) )t  B )  =  ( TopOpen `  G ) )
 
Theoremsymgga 14621* The symmetric group induces a group action on its base set. (Contributed by Mario Carneiro, 24-Jan-2015.)
 |-  G  =  ( SymGrp `  X )   &    |-  B  =  (
 Base `  G )   &    |-  F  =  ( f  e.  B ,  x  e.  X  |->  ( f `  x ) )   =>    |-  ( X  e.  V  ->  F  e.  ( G 
 GrpAct  X ) )
 
Theoremcayleylem1 14622* Lemma for cayley 14624. (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 14623* Lemma for cayley 14624. (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 14624* Cayley's Theorem (constructive version): given group  G,  F is an isomorphism between  G and the subgroup  S of the symmetry group  H on the underlying set  X of  G. (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 14625* Cayley's Theorem (existence version): every group  G is isomorphic to a subgroup of the symmetry group on the underlying set of  G. (For any group  G there exists an isomorphism  f between  G and a subgroup  h of the symmetry group on the underlying set of  G.) (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 )
 
10.2.7  Centralizers and centers
 
Syntaxccntz 14626 Syntax for the centralizer of a set in a monoid.
 class Cntz
 
Syntaxccntr 14627 Syntax for the centralizer of a monoid.
 class Cntr
 
Definitiondf-cntz 14628* Define the centralizer of a subset of a magma, which is the set of elements each of which commutes with each element of the given subset. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |- Cntz  =  ( m  e.  _V  |->  ( s  e.  ~P ( Base `  m )  |->  { x  e.  ( Base `  m )  | 
 A. y  e.  s  ( x ( +g  `  m ) y )  =  ( y ( +g  `  m ) x ) } ) )
 
Definitiondf-cntr 14629 Define the center of a magma, which is the elements that commute with all others. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |- Cntr  =  ( m  e.  _V  |->  ( (Cntz `  m ) `  ( Base `  m )
 ) )
 
Theoremcntrval 14630 Substitute definition of the center. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( Z `  B )  =  (Cntr `  M )
 
Theoremcntzfval 14631* First level substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( M  e.  V  ->  Z  =  ( s  e. 
 ~P B  |->  { x  e.  B  |  A. y  e.  s  ( x  .+  y )  =  ( y  .+  x ) } ) )
 
Theoremcntzval 14632* Definition substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( S  C_  B  ->  ( Z `  S )  =  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) } )
 
Theoremelcntz 14633* Elementhood in the centralizer. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( S  C_  B  ->  ( A  e.  ( Z `  S )  <->  ( A  e.  B  /\  A. y  e.  S  ( A  .+  y )  =  (
 y  .+  A )
 ) ) )
 
Theoremcntzel 14634* Membership in a centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  (
 ( S  C_  B  /\  X  e.  B ) 
 ->  ( X  e.  ( Z `  S )  <->  A. y  e.  S  ( X  .+  y )  =  ( y  .+  X ) ) )
 
Theoremcntzsnval 14635* Special substitution for the centralizer of a singleton. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( Y  e.  B  ->  ( Z `  { Y } )  =  { x  e.  B  |  ( x  .+  Y )  =  ( Y  .+  x ) } )
 
Theoremelcntzsn 14636 Value of the centralizer of a singleton. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( Y  e.  B  ->  ( X  e.  ( Z `
  { Y }
 ) 
 <->  ( X  e.  B  /\  ( X  .+  Y )  =  ( Y  .+  X ) ) ) )
 
Theoremsscntz 14637* A centralizer expression for two sets elementwise commuting. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  (
 ( S  C_  B  /\  T  C_  B )  ->  ( S  C_  ( Z `  T )  <->  A. x  e.  S  A. y  e.  T  ( x  .+  y )  =  ( y  .+  x ) ) )
 
Theoremcntzrcl 14638 Reverse closure for elements of the centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( X  e.  ( Z `  S )  ->  ( M  e.  _V  /\  S  C_  B )
 )
 
Theoremcntzssv 14639 The centralizer is unconditionally a subset. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( Z `  S )  C_  B
 
Theoremcntzi 14640 Membership in a centralizer (inference). (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |- 
 .+  =  ( +g  `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( ( X  e.  ( Z `  S ) 
 /\  Y  e.  S )  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
 
Theoremcntri 14641 Defining property of the center of a group. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  Z  =  (Cntr `  M )   =>    |-  (
 ( X  e.  Z  /\  Y  e.  B ) 
 ->  ( X  .+  Y )  =  ( Y  .+  X ) )
 
Theoremresscntz 14642 Centralizer in a substructure. (Contributed by Mario Carneiro, 3-Oct-2015.)
 |-  H  =  ( Gs  A )   &    |-  Z  =  (Cntz `  G )   &    |-  Y  =  (Cntz `  H )   =>    |-  ( ( A  e.  V  /\  S  C_  A )  ->  ( Y `  S )  =  (
 ( Z `  S )  i^i  A ) )
 
Theoremcntz2ss 14643 Centralizers reverse the subset relation. (Contributed by Mario Carneiro, 3-Oct-2015.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( ( S  C_  B  /\  T  C_  S )  ->  ( Z `  S )  C_  ( Z `
  T ) )
 
Theoremcntzrec 14644 Reciprocity relationship for centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( ( S  C_  B  /\  T  C_  B )  ->  ( S  C_  ( Z `  T )  <->  T  C_  ( Z `  S ) ) )
 
Theoremcntziinsn 14645* Express any centralizer as an intersection of singleton centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( S  C_  B  ->  ( Z `  S )  =  ( B  i^i  |^|_ x  e.  S  ( Z `  { x } ) ) )
 
Theoremcntzsubm 14646 Centralizers in a monoid are submonoids. (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( ( M  e.  Mnd  /\  S  C_  B )  ->  ( Z `  S )  e.  (SubMnd `  M ) )
 
Theoremcntzsubg 14647 Centralizers in a group are subgroups. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( ( M  e.  Grp  /\  S  C_  B )  ->  ( Z `  S )  e.  (SubGrp `  M ) )
 
Theoremcntzidss 14648 If the elements of  S commute, the elements of a subset 
T also commute. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  Z  =  (Cntz `  G )   =>    |-  ( ( S  C_  ( Z `  S ) 
 /\  T  C_  S )  ->  T  C_  ( Z `  T ) )
 
Theoremcntzmhm 14649 Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  Z  =  (Cntz `  G )   &    |-  Y  =  (Cntz `  H )   =>    |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S ) )  ->  ( F `  A )  e.  ( Y `  ( F " S ) ) )
 
Theoremcntzmhm2 14650 Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  Z  =  (Cntz `  G )   &    |-  Y  =  (Cntz `  H )   =>    |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T ) )  ->  ( F " S ) 
 C_  ( Y `  ( F " T ) ) )
 
Theoremcntrsubgnsg 14651 A central subgroup is normal. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  Z  =  (Cntr `  M )   =>    |-  ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  ->  X  e.  (NrmSGrp `  M ) )
 
Theoremcntrnsg 14652 The center of a group is a normal subgroup. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  Z  =  (Cntr `  M )   =>    |-  ( M  e.  Grp  ->  Z  e.  (NrmSGrp `  M ) )
 
10.2.8  The opposite group
 
Syntaxcoppg 14653 The opposite group operation.
 class oppg
 
Definitiondf-oppg 14654 Define an opposite group, which is the same as the original group but with addition written the other way around. df-oppr 15240 does the same thing for multiplication. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |- oppg  =  ( w  e.  _V  |->  ( w sSet  <. ( +g  ` 
 ndx ) , tpos  ( +g  `  w ) >. ) )
 
Theoremoppgval 14655 Value of the opposite group. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
 |- 
 .+  =  ( +g  `  R )   &    |-  O  =  (oppg `  R )   =>    |-  O  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
 
Theoremoppgplusfval 14656 Value of the addition operation of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
 |- 
 .+  =  ( +g  `  R )   &    |-  O  =  (oppg `  R )   &    |-  .+b  =  ( +g  `  O )   =>    |-  .+b  = tpos  .+
 
Theoremoppgplus 14657 Value of the addition operation of an opposite ring. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
 |- 
 .+  =  ( +g  `  R )   &    |-  O  =  (oppg `  R )   &    |-  .+b  =  ( +g  `  O )   =>    |-  ( X  .+b  Y )  =  ( Y 
 .+  X )
 
Theoremoppglem 14658 Lemma for oppgbas 14659. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  O  =  (oppg `  R )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  N  =/=  2   =>    |-  ( E `  R )  =  ( E `  O )
 
Theoremoppgbas 14659 Base set of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  O  =  (oppg `  R )   &    |-  B  =  ( Base `  R )   =>    |-  B  =  ( Base `  O )
 
Theoremoppgtset 14660 Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  O  =  (oppg `  R )   &    |-  J  =  (TopSet `  R )   =>    |-  J  =  (TopSet `  O )
 
Theoremoppgtopn 14661 Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  O  =  (oppg `  R )   &    |-  J  =  ( TopOpen `  R )   =>    |-  J  =  ( TopOpen `  O )
 
Theoremoppgmnd 14662 The opposite of a monoid is a monoid. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
 |-  O  =  (oppg `  R )   =>    |-  ( R  e.  Mnd  ->  O  e.  Mnd )
 
Theoremoppgmndb 14663 Bidirectional form of oppgmnd 14662. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  O  =  (oppg `  R )   =>    |-  ( R  e.  Mnd  <->  O  e.  Mnd )
 
Theoremoppgid 14664 Zero in a monoid is a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
 |-  O  =  (oppg `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |- 
 .0.  =  ( 0g `  O )
 
Theoremoppggrp 14665 The opposite of a group is a group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  O  =  (oppg `  R )   =>    |-  ( R  e.  Grp  ->  O  e.  Grp )
 
Theoremoppggrpb 14666 Bidirectional form of oppggrp 14665. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  O  =  (oppg `  R )   =>    |-  ( R  e.  Grp  <->  O  e.  Grp )
 
Theoremoppginv 14667 Inverses in a group are a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  O  =  (oppg `  R )   &    |-  I  =  ( inv
 g `  R )   =>    |-  ( R  e.  Grp  ->  I  =  ( inv g `  O ) )
 
Theoreminvoppggim 14668 The inverse is an antiautomorphism on any group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  O  =  (oppg `  G )   &    |-  I  =  ( inv
 g `  G )   =>    |-  ( G  e.  Grp  ->  I  e.  ( G GrpIso  O )
 )
 
Theoremoppggic 14669 Every group is (naturally) isomorphic to its opposite. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  O  =  (oppg `  G )   =>    |-  ( G  e.  Grp  ->  G  ~=ph𝑔 
 O )
 
Theoremoppgsubm 14670 Being a submonoid is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  O  =  (oppg `  G )   =>    |-  (SubMnd `  G )  =  (SubMnd `  O )
 
Theoremoppgsubg 14671 Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  O  =  (oppg `  G )   =>    |-  (SubGrp `  G )  =  (SubGrp `  O )
 
Theoremoppgcntz 14672 A centralizer in a group is the same as the centralizer in the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  O  =  (oppg `  G )   &    |-  Z  =  (Cntz `  G )   =>    |-  ( Z `  A )  =  ( (Cntz `  O ) `  A )
 
Theoremoppgcntr 14673 The center of a group is the same as the center of the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  O  =  (oppg `  G )   &    |-  Z  =  (Cntr `  G )   =>    |-  Z  =  (Cntr `  O )
 
Theoremgsumwrev 14674 A sum in an opposite monoid is the regular sum of a reversed word. (Contributed by Stefan O'Rear, 27-Aug-2015.) (Proof shortened by Mario Carneiro, 28-Feb-2016.)
 |-  B  =  ( Base `  M )   &    |-  O  =  (oppg `  M )   =>    |-  ( ( M  e.  Mnd  /\  W  e. Word  B )  ->  ( O  gsumg 
 W )  =  ( M  gsumg  (reverse `  W )
 ) )
 
10.2.9  p-Groups and Sylow groups; Sylow's theorems
 
Syntaxcod 14675 Extend class notation to include the order function on the elements of a group.
 class  od
 
Syntaxcgex 14676 Extend class notation to include the order function on the elements of a group.
 class gEx
 
Syntaxcpgp 14677 Extend class notation to include the class of all p-groups.
 class pGrp
 
Syntaxcslw 14678 Extend class notation to include the class of all Sylow p-subgroups of a group.
 class pSyl
 
Definitiondf-od 14679* 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 14680* 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 14681* 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 14682* 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 14683* 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 14684* 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 14685* 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 14686 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 14687 Functionality of the group element order. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   =>    |-  O : X --> NN0
 
Theoremodid 14688 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 14689 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 14690 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 14691 Lemma for mndodcong 14692. (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 14692 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 14693 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 14694 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 14695 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 14696 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 14697 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 14698 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 14699 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 14700* The oddvds 14697 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.  )
 ) )
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