HomeHome Metamath Proof Explorer
Theorem List (p. 148 of 311)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21328)
  Hilbert Space Explorer  Hilbert Space Explorer
(21329-22851)
  Users' Mathboxes  Users' Mathboxes
(22852-31058)
 

Theorem List for Metamath Proof Explorer - 14701-14800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremodval2 14701* 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 14702 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 14703 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 14704 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 14705 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 14706* 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 14707 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 14708 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 14709 The order of the inverse of a group element. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  O  =  ( od
 `  G )   &    |-  I  =  ( inv g `  G )   &    |-  X  =  (
 Base `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( O `
  ( I `  A ) )  =  ( O `  A ) )
 
Theoremodf1 14710* The multiples of an element with infinite order form an infinite cyclic subgroup of  G. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  F  =  ( x  e.  ZZ  |->  ( x  .x.  A ) )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X ) 
 ->  ( ( O `  A )  =  0  <->  F : ZZ -1-1-> X ) )
 
Theoremodinf 14711* The multiples of an element with infinite order form an infinite cyclic subgroup of  G. (Contributed by Mario Carneiro, 14-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  F  =  ( x  e.  ZZ  |->  ( x  .x.  A ) )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  ->  -.  ran  F  e.  Fin )
 
Theoremdfod2 14712* An alternative definition of the order of a group element is as the cardinality of the cyclic subgroup generated by the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  F  =  ( x  e.  ZZ  |->  ( x  .x.  A ) )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X ) 
 ->  ( O `  A )  =  if ( ran  F  e.  Fin ,  ( # `  ran  F ) ,  0 )
 )
 
Theoremodcl2 14713 The order of an element of a finite group is finite. (Contributed by Mario Carneiro, 14-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  A  e.  X )  ->  ( O `  A )  e.  NN )
 
Theoremoddvds2 14714 The order of an element of a finite group divides the order (cardinality) of the group. Corollary of Lagrange's theorem for the order of a subgroup. (Contributed by Mario Carneiro, 14-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  A  e.  X )  ->  ( O `  A )  ||  ( # `  X ) )
 
Theoremsubmod 14715 The order of an element is the same in a subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  H  =  ( Gs  Y )   &    |-  O  =  ( od `  G )   &    |-  P  =  ( od `  H )   =>    |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  ( O `  A )  =  ( P `  A ) )
 
Theoremsubgod 14716 The order of an element is the same in a subgroup. (Contributed by Mario Carneiro, 14-Jan-2015.) (Proof shortened by Stefan O'Rear, 12-Sep-2015.)
 |-  H  =  ( Gs  Y )   &    |-  O  =  ( od `  G )   &    |-  P  =  ( od `  H )   =>    |-  ( ( Y  e.  (SubGrp `  G )  /\  A  e.  Y )  ->  ( O `  A )  =  ( P `  A ) )
 
Theoremodsubdvds 14717 The order of an element of a subgroup divides the order of the subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  O  =  ( od
 `  G )   =>    |-  ( ( S  e.  (SubGrp `  G )  /\  S  e.  Fin  /\  A  e.  S ) 
 ->  ( O `  A )  ||  ( # `  S ) )
 
Theoremodf1o1 14718* An element with zero order has infinitely many multiples. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  O  =  ( od `  G )   &    |-  K  =  (mrCls `  (SubGrp `  G ) )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0
 )  ->  ( x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ -1-1-onto-> ( K `  { A }
 ) )
 
Theoremodf1o2 14719* An element with nonzero order has as many multiples as its order. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  O  =  ( od `  G )   &    |-  K  =  (mrCls `  (SubGrp `  G ) )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  ->  ( x  e.  ( 0..^ ( O `
  A ) ) 
 |->  ( x  .x.  A ) ) : ( 0..^ ( O `  A ) ) -1-1-onto-> ( K `
  { A }
 ) )
 
Theoremodhash 14720 An element of zero order generates an infinite subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  K  =  (mrCls `  (SubGrp `  G ) )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0
 )  ->  ( # `  ( K `  { A }
 ) )  =  +oo )
 
Theoremodhash2 14721 If an element has nonzero order, it generates a subgroup with size equal to the order. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  K  =  (mrCls `  (SubGrp `  G ) )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  ->  ( # `  ( K `  { A }
 ) )  =  ( O `  A ) )
 
Theoremodhash3 14722 An element which generates a finite subgroup has order the size of that subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  K  =  (mrCls `  (SubGrp `  G ) )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( K `  { A } )  e. 
 Fin )  ->  ( O `  A )  =  ( # `  ( K `  { A }
 ) ) )
 
Theoremodngen 14723* A cyclic subgroup of size  ( O `  A
) has  ( phi `  ( O `  A ) ) generators. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  K  =  (mrCls `  (SubGrp `  G ) )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  ->  ( # `  { x  e.  ( K `  { A } )  |  ( O `  x )  =  ( O `  A ) } )  =  ( phi `  ( O `  A ) ) )
 
Theoremgexval 14724* Value of the exponent of a group. (Contributed by Mario Carneiro, 23-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  E  =  (gEx `  G )   &    |-  I  =  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0.  }   =>    |-  ( G  e.  V  ->  E  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
 
Theoremgexlem1 14725* The group element order is either zero or a nonzero multiplier that annihilates the element. (Contributed by Mario Carneiro, 23-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  E  =  (gEx `  G )   &    |-  I  =  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0.  }   =>    |-  ( G  e.  V  ->  ( ( E  =  0  /\  I  =  (/) )  \/  E  e.  I
 ) )
 
Theoremgexcl 14726 The exponent of a group is a nonnegative integer. (Contributed by Mario Carneiro, 23-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   =>    |-  ( G  e.  V  ->  E  e.  NN0 )
 
Theoremgexid 14727 Any element to the power of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( A  e.  X  ->  ( E  .x.  A )  =  .0.  )
 
Theoremgexlem2 14728* Any positive annihilator of all the group elements is an upper bound on the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  V  /\  N  e.  NN  /\ 
 A. x  e.  X  ( N  .x.  x )  =  .0.  )  ->  E  e.  ( 1 ... N ) )
 
Theoremgexdvdsi 14729 Any group element is annihilated by any multiple of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  ->  ( N  .x.  A )  =  .0.  )
 
Theoremgexdvds 14730* The only  N that annihilate all the elements of the group are the multiples of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  N  e.  ZZ )  ->  ( E  ||  N  <->  A. x  e.  X  ( N  .x.  x )  =  .0.  ) )
 
Theoremgexdvds2 14731* An integer divides the group exponent iff it divides all the group orders. In other words, the group exponent is the LCM of the orders of all the elements. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( ( G  e.  Grp  /\  N  e.  ZZ )  ->  ( E  ||  N  <->  A. x  e.  X  ( O `  x ) 
 ||  N ) )
 
Theoremgexod 14732 Any group element is annihilated by any multiple of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X ) 
 ->  ( O `  A )  ||  E )
 
Theoremgexcl3 14733* If the order of every group element is bounded by  N, the group has finite exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( ( G  e.  Grp  /\  A. x  e.  X  ( O `  x )  e.  ( 1 ...
 N ) )  ->  E  e.  NN )
 
Theoremgexnnod 14734 Every group element has finite order if the exponent is finite. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( ( G  e.  Grp  /\  E  e.  NN  /\  A  e.  X )  ->  ( O `  A )  e.  NN )
 
Theoremgexcl2 14735 The exponent of a finite group is finite. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  Fin )  ->  E  e.  NN )
 
Theoremgexdvds3 14736 The exponent of a finite group divides the order (cardinality) of the group. Corollary of Lagrange's theorem for the order of a subgroup. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  Fin )  ->  E  ||  ( # `  X ) )
 
Theoremgex1 14737 A group or monoid has exponent 1 iff it is trivial. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   =>    |-  ( G  e.  Mnd  ->  ( E  =  1  <->  X 
 ~~  1o ) )
 
Theoremispgp 14738* A group is a  P-group if every element has some power of  P as its order. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( P pGrp  G  <->  ( P  e.  Prime  /\  G  e.  Grp  /\ 
 A. x  e.  X  E. n  e.  NN0  ( O `  x )  =  ( P ^ n ) ) )
 
Theorempgpprm 14739 Reverse closure for the first argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  ( P pGrp  G  ->  P  e.  Prime )
 
Theorempgpgrp 14740 Reverse closure for the second argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  ( P pGrp  G  ->  G  e.  Grp )
 
Theorempgpfi1 14741 A finite group with order a power of a prime  P is a 
P-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  X  =  ( Base `  G )   =>    |-  ( ( G  e.  Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  ->  (
 ( # `  X )  =  ( P ^ N )  ->  P pGrp  G ) )
 
Theorempgp0 14742 The identity subgroup is a  P-group for every prime  P. (Contributed by Mario Carneiro, 19-Jan-2015.)
 |- 
 .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  P  e.  Prime )  ->  P pGrp  ( Gs  {  .0.  } ) )
 
Theoremsubgpgp 14743 A subgroup of a p-group is a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  ( ( P pGrp  G  /\  S  e.  (SubGrp `  G ) )  ->  P pGrp  ( Gs  S ) )
 
Theoremsylow1lem1 14744* Lemma for sylow1 14749. The p-adic valuation of the size of  S is equal to the number of excess powers of  P in  ( # `  X
)  /  ( P ^ N ). (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  ( P ^ N ) 
 ||  ( # `  X ) )   &    |-  .+  =  ( +g  `  G )   &    |-  S  =  { s  e.  ~P X  |  ( # `  s
 )  =  ( P ^ N ) }   =>    |-  ( ph  ->  ( ( # `  S )  e.  NN  /\  ( P  pCnt  ( # `
  S ) )  =  ( ( P 
 pCnt  ( # `  X ) )  -  N ) ) )
 
Theoremsylow1lem2 14745* Lemma for sylow1 14749. The function  .(+) is a group action on  S. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  ( P ^ N ) 
 ||  ( # `  X ) )   &    |-  .+  =  ( +g  `  G )   &    |-  S  =  { s  e.  ~P X  |  ( # `  s
 )  =  ( P ^ N ) }   &    |-  .(+)  =  ( x  e.  X ,  y  e.  S  |->  ran  (  z  e.  y  |->  ( x  .+  z ) ) )   =>    |-  ( ph  ->  .(+)  e.  ( G  GrpAct  S ) )
 
Theoremsylow1lem3 14746* Lemma for sylow1 14749. One of the orbits of the group action has p-adic valuation less than the prime count of the set  S. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  ( P ^ N ) 
 ||  ( # `  X ) )   &    |-  .+  =  ( +g  `  G )   &    |-  S  =  { s  e.  ~P X  |  ( # `  s
 )  =  ( P ^ N ) }   &    |-  .(+)  =  ( x  e.  X ,  y  e.  S  |->  ran  (  z  e.  y  |->  ( x  .+  z ) ) )   &    |-  .~  =  { <. x ,  y >.  |  ( { x ,  y }  C_  S  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }   =>    |-  ( ph  ->  E. w  e.  S  ( P  pCnt  ( # `  [ w ]  .~  ) )  <_  ( ( P  pCnt  ( # `  X ) )  -  N ) )
 
Theoremsylow1lem4 14747* Lemma for sylow1 14749. The stabilizer subgroup of any element of  S is at most  P ^ N in size. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  ( P ^ N ) 
 ||  ( # `  X ) )   &    |-  .+  =  ( +g  `  G )   &    |-  S  =  { s  e.  ~P X  |  ( # `  s
 )  =  ( P ^ N ) }   &    |-  .(+)  =  ( x  e.  X ,  y  e.  S  |->  ran  (  z  e.  y  |->  ( x  .+  z ) ) )   &    |-  .~  =  { <. x ,  y >.  |  ( { x ,  y }  C_  S  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }   &    |-  ( ph  ->  B  e.  S )   &    |-  H  =  { u  e.  X  |  ( u 
 .(+)  B )  =  B }   =>    |-  ( ph  ->  ( # `
  H )  <_  ( P ^ N ) )
 
Theoremsylow1lem5 14748* Lemma for sylow1 14749. Using Lagrange's theorem and the orbit-stabilizer theorem, show that there is a subgroup with size exactly  P ^ N. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  ( P ^ N ) 
 ||  ( # `  X ) )   &    |-  .+  =  ( +g  `  G )   &    |-  S  =  { s  e.  ~P X  |  ( # `  s
 )  =  ( P ^ N ) }   &    |-  .(+)  =  ( x  e.  X ,  y  e.  S  |->  ran  (  z  e.  y  |->  ( x  .+  z ) ) )   &    |-  .~  =  { <. x ,  y >.  |  ( { x ,  y }  C_  S  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }   &    |-  ( ph  ->  B  e.  S )   &    |-  H  =  { u  e.  X  |  ( u 
 .(+)  B )  =  B }   &    |-  ( ph  ->  ( P  pCnt  ( # `  [ B ]  .~  ) )  <_  ( ( P  pCnt  ( # `  X ) )  -  N ) )   =>    |-  ( ph  ->  E. h  e.  (SubGrp `  G )
 ( # `  h )  =  ( P ^ N ) )
 
Theoremsylow1 14749* Sylow's first theorem. If  P ^ N is a prime power that divides the cardinality of  G, then  G has a supgroup with size  P ^ N. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  ( P ^ N ) 
 ||  ( # `  X ) )   =>    |-  ( ph  ->  E. g  e.  (SubGrp `  G )
 ( # `  g )  =  ( P ^ N ) )
 
Theoremodcau 14750* Cauchy's theorem for the order of an element in a group. A finite group whose order divides a prime 
P contains an element of order  P. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X ) )  ->  E. g  e.  X  ( O `  g )  =  P )
 
Theorempgpfi 14751* The converse to pgpfi1 14741. A finite group is a  P-group iff it has size some power of  P. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  X  =  ( Base `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  Fin )  ->  ( P pGrp  G  <->  ( P  e.  Prime  /\  E. n  e. 
 NN0  ( # `  X )  =  ( P ^ n ) ) ) )
 
Theorempgpfi2 14752 Alternate version of pgpfi 14751. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  X  =  ( Base `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  Fin )  ->  ( P pGrp  G  <->  ( P  e.  Prime  /\  ( # `  X )  =  ( P ^ ( P  pCnt  ( # `  X ) ) ) ) ) )
 
Theorempgphash 14753 The order of a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  X  =  ( Base `  G )   =>    |-  ( ( P pGrp  G  /\  X  e.  Fin )  ->  ( # `  X )  =  ( P ^ ( P  pCnt  ( # `  X ) ) ) )
 
Theoremisslw 14754* The property of being a Sylow subgroup. A Sylow  P-subgroup is a  P-group which has no proper supersets that are also  P-groups. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( H  e.  ( P pSyl  G )  <->  ( P  e.  Prime  /\  H  e.  (SubGrp `  G )  /\  A. k  e.  (SubGrp `  G ) ( ( H 
 C_  k  /\  P pGrp  ( Gs  k ) )  <->  H  =  k
 ) ) )
 
Theoremslwprm 14755 Reverse closure for the first argument of a Sylow  P-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.) (Revised by Mario Carneiro, 2-May-2015.)
 |-  ( H  e.  ( P pSyl  G )  ->  P  e.  Prime )
 
Theoremslwsubg 14756 A Sylow  P-subgroup is a subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( H  e.  ( P pSyl  G )  ->  H  e.  (SubGrp `  G )
 )
 
Theoremslwispgp 14757 Defining property of a Sylow  P-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  S  =  ( Gs  K )   =>    |-  ( ( H  e.  ( P pSyl  G )  /\  K  e.  (SubGrp `  G ) )  ->  ( ( H  C_  K  /\  P pGrp  S )  <->  H  =  K ) )
 
Theoremslwpss 14758 A proper superset of a Sylow subgroup is not a  P-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  S  =  ( Gs  K )   =>    |-  ( ( H  e.  ( P pSyl  G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  -.  P pGrp  S )
 
Theoremslwpgp 14759 A Sylow  P-subgroup is a  P-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  S  =  ( Gs  H )   =>    |-  ( H  e.  ( P pSyl  G )  ->  P pGrp  S )
 
Theorempgpssslw 14760* Every  P-subgroup is contained in a Sylow  P-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  S  =  ( Gs  H )   &    |-  F  =  ( x  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  |->  ( # `  x ) )   =>    |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  ->  E. k  e.  ( P pSyl  G ) H  C_  k )
 
Theoremslwn0 14761 Every finite group contains a Sylow 
P-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  X  =  ( Base `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  ->  ( P pSyl  G )  =/=  (/) )
 
Theoremsubgslw 14762 A Sylow subgroup that is contained in a larger subgroup is also Sylow with respect to the subgroup. (The converse may not be true, though.) (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  H  =  ( Gs  S )   =>    |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S )  ->  K  e.  ( P pSyl  H ) )
 
Theoremsylow2alem1 14763* Lemma for sylow2a 14765. An equivalence class of fixed points is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  .(+) 
 e.  ( G  GrpAct  Y ) )   &    |-  ( ph  ->  P pGrp 
 G )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  Y  e.  Fin )   &    |-  Z  =  { u  e.  Y  |  A. h  e.  X  ( h  .(+)  u )  =  u }   &    |-  .~  =  { <. x ,  y >.  |  ( { x ,  y }  C_  Y  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }   =>    |-  (
 ( ph  /\  A  e.  Z )  ->  [ A ]  .~  =  { A } )
 
Theoremsylow2alem2 14764* Lemma for sylow2a 14765. All the orbits which are not for fixed points have size  |  G  |  /  |  G x  | (where  G x is the stabilizer subgroup) and thus are powers of  P. And since they are all nontrivial (because any orbit which is a singleton is a fixed point), they all divide  P, and so does the sum of all of them. (Contributed by Mario Carneiro, 17-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  .(+) 
 e.  ( G  GrpAct  Y ) )   &    |-  ( ph  ->  P pGrp 
 G )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  Y  e.  Fin )   &    |-  Z  =  { u  e.  Y  |  A. h  e.  X  ( h  .(+)  u )  =  u }   &    |-  .~  =  { <. x ,  y >.  |  ( { x ,  y }  C_  Y  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }   =>    |-  ( ph  ->  P  ||  sum_ z  e.  ( ( Y /.  .~  )  \  ~P Z ) ( # `  z
 ) )
 
Theoremsylow2a 14765* A named lemma of Sylow's second and third theorems. If  G is a finite  P-group that acts on the finite set  Y, then the set  Z of all points of  Y fixed by every element of  G has cardinality equivalent to the cardinality of  Y, 
mod  P. (Contributed by Mario Carneiro, 17-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  .(+) 
 e.  ( G  GrpAct  Y ) )   &    |-  ( ph  ->  P pGrp 
 G )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  Y  e.  Fin )   &    |-  Z  =  { u  e.  Y  |  A. h  e.  X  ( h  .(+)  u )  =  u }   &    |-  .~  =  { <. x ,  y >.  |  ( { x ,  y }  C_  Y  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }   =>    |-  ( ph  ->  P  ||  (
 ( # `  Y )  -  ( # `  Z ) ) )
 
Theoremsylow2blem1 14766* Lemma for sylow2b 14769. Evaluate the group action on a left coset. (Contributed by Mario Carneiro, 17-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  H  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  K  e.  (SubGrp `  G ) )   &    |-  .+  =  ( +g  `  G )   &    |-  .~  =  ( G ~QG  K )   &    |-  .x.  =  ( x  e.  H ,  y  e.  ( X /.  .~  )  |->  ran  (  z  e.  y  |->  ( x  .+  z ) ) )   =>    |-  ( ( ph  /\  B  e.  H  /\  C  e.  X )  ->  ( B 
 .x.  [ C ]  .~  )  =  [ ( B  .+  C ) ]  .~  )
 
Theoremsylow2blem2 14767* Lemma for sylow2b 14769. Left multiplication in a subgroup  H is a group action on the set of all left cosets of  K. (Contributed by Mario Carneiro, 17-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  H  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  K  e.  (SubGrp `  G ) )   &    |-  .+  =  ( +g  `  G )   &    |-  .~  =  ( G ~QG  K )   &    |-  .x.  =  ( x  e.  H ,  y  e.  ( X /.  .~  )  |->  ran  (  z  e.  y  |->  ( x  .+  z ) ) )   =>    |-  ( ph  ->  .x.  e.  ( ( Gs  H ) 
 GrpAct  ( X /.  .~  ) ) )
 
Theoremsylow2blem3 14768* Sylow's second theorem. Putting together the results of sylow2a 14765 and the orbit-stabilizer theorem to show that  P does not divide the set of all fixed points under the group action, we get that there is a fixed point of the group action, so that there is some  g  e.  X with  h g K  =  g K for all  h  e.  H. This implies that  inv g ( g ) h g  e.  K, so  h is in the conjugated subgroup  g K inv g ( g ). (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  H  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  K  e.  (SubGrp `  G ) )   &    |-  .+  =  ( +g  `  G )   &    |-  .~  =  ( G ~QG  K )   &    |-  .x.  =  ( x  e.  H ,  y  e.  ( X /.  .~  )  |->  ran  (  z  e.  y  |->  ( x  .+  z ) ) )   &    |-  ( ph  ->  P pGrp 
 ( Gs  H ) )   &    |-  ( ph  ->  ( # `  K )  =  ( P ^ ( P  pCnt  ( # `  X ) ) ) )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ph  ->  E. g  e.  X  H  C_ 
 ran  (  x  e.  K  |->  ( ( g 
 .+  x )  .-  g ) ) )
 
Theoremsylow2b 14769* Sylow's second theorem. Any  P-group  H is a subgroup of a conjugated  P-group  K of order  P ^ n  ||  ( # `  X
) with  n maximal. This is usually stated under the assumption that  K is a Sylow subgroup, but we use a slightly different definition, whose equivalence to this one requires this theorem. (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  H  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  K  e.  (SubGrp `  G ) )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  P pGrp  ( Gs  H ) )   &    |-  ( ph  ->  ( # `  K )  =  ( P ^ ( P  pCnt  ( # `  X ) ) ) )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ph  ->  E. g  e.  X  H  C_  ran  (  x  e.  K  |->  ( ( g  .+  x ) 
 .-  g ) ) )
 
Theoremslwhash 14770 A sylow subgroup has cardinality equal to the maximum power of  P dividing the group. (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  H  e.  ( P pSyl  G ) )   =>    |-  ( ph  ->  ( # `  H )  =  ( P ^ ( P  pCnt  ( # `  X ) ) ) )
 
Theoremfislw 14771 The sylow subgroups of a finite group are exactly the groups which have cardinality equal to the maximum power of  P dividing the group. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  X  =  ( Base `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  ->  ( H  e.  ( P pSyl  G )  <->  ( H  e.  (SubGrp `  G )  /\  ( # `  H )  =  ( P ^
 ( P  pCnt  ( # `
  X ) ) ) ) ) )
 
Theoremsylow2 14772* Sylow's second theorem. See also sylow2b 14769 for the "hard" part of the proof. Any two Sylow  P-subgroups are conjugate to one another, and hence the same size, namely 
P ^ ( P 
pCnt  |  X  | 
) (see fislw 14771). (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  H  e.  ( P pSyl  G ) )   &    |-  ( ph  ->  K  e.  ( P pSyl  G ) )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ph  ->  E. g  e.  X  H  =  ran  (  x  e.  K  |->  ( ( g  .+  x )  .-  g ) ) )
 
Theoremsylow3lem1 14773* Lemma for sylow3 14779, first part. (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  .(+) 
 =  ( x  e.  X ,  y  e.  ( P pSyl  G ) 
 |->  ran  (  z  e.  y  |->  ( ( x 
 .+  z )  .-  x ) ) )   =>    |-  ( ph  ->  .(+)  e.  ( G  GrpAct  ( P pSyl  G ) ) )
 
Theoremsylow3lem2 14774* Lemma for sylow3 14779, first part. The stabilizer of a given Sylow subgroup  K in the group action  .(+) acting on all of  G is the normalizer NG(K). (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  .(+) 
 =  ( x  e.  X ,  y  e.  ( P pSyl  G ) 
 |->  ran  (  z  e.  y  |->  ( ( x 
 .+  z )  .-  x ) ) )   &    |-  ( ph  ->  K  e.  ( P pSyl  G ) )   &    |-  H  =  { u  e.  X  |  ( u 
 .(+)  K )  =  K }   &    |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x 
 .+  y )  e.  K  <->  ( y  .+  x )  e.  K ) }   =>    |-  ( ph  ->  H  =  N )
 
Theoremsylow3lem3 14775* Lemma for sylow3 14779, first part. The number of Sylow subgroups is the same as the index (number of cosets) of the normalizer of the Sylow subgroup  K. (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  .(+) 
 =  ( x  e.  X ,  y  e.  ( P pSyl  G ) 
 |->  ran  (  z  e.  y  |->  ( ( x 
 .+  z )  .-  x ) ) )   &    |-  ( ph  ->  K  e.  ( P pSyl  G ) )   &    |-  H  =  { u  e.  X  |  ( u 
 .(+)  K )  =  K }   &    |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x 
 .+  y )  e.  K  <->  ( y  .+  x )  e.  K ) }   =>    |-  ( ph  ->  ( # `
  ( P pSyl  G ) )  =  ( # `
  ( X /. ( G ~QG  N ) ) ) )
 
Theoremsylow3lem4 14776* Lemma for sylow3 14779, first part. The number of Sylow subgroups is a divisor of the size of  G reduced by the size of a Sylow subgroup of  G. (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  .(+) 
 =  ( x  e.  X ,  y  e.  ( P pSyl  G ) 
 |->  ran  (  z  e.  y  |->  ( ( x 
 .+  z )  .-  x ) ) )   &    |-  ( ph  ->  K  e.  ( P pSyl  G ) )   &    |-  H  =  { u  e.  X  |  ( u 
 .(+)  K )  =  K }   &    |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x 
 .+  y )  e.  K  <->  ( y  .+  x )  e.  K ) }   =>    |-  ( ph  ->  ( # `
  ( P pSyl  G ) )  ||  ( ( # `  X )  /  ( P ^ ( P 
 pCnt  ( # `  X ) ) ) ) )
 
Theoremsylow3lem5 14777* Lemma for sylow3 14779, second part. Reduce the group action of sylow3lem1 14773 to a given Sylow subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  K  e.  ( P pSyl  G ) )   &    |-  .(+) 
 =  ( x  e.  K ,  y  e.  ( P pSyl  G ) 
 |->  ran  (  z  e.  y  |->  ( ( x 
 .+  z )  .-  x ) ) )   =>    |-  ( ph  ->  .(+)  e.  (
 ( Gs  K )  GrpAct  ( P pSyl 
 G ) ) )
 
Theoremsylow3lem6 14778* Lemma for sylow3 14779, second part. Using the lemma sylow2a 14765, show that the number of sylow subgroups is equivalent  mod  P to the number of fixed points under the group action. But  K is the unique element of the set of Sylow subgroups that is fixed under the group action, so there is exactly one fixed point and so  ( ( # `  ( P pSyl  G ) )  mod  P )  =  1. (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  K  e.  ( P pSyl  G ) )   &    |-  .(+) 
 =  ( x  e.  K ,  y  e.  ( P pSyl  G ) 
 |->  ran  (  z  e.  y  |->  ( ( x 
 .+  z )  .-  x ) ) )   &    |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x 
 .+  y )  e.  s  <->  ( y  .+  x )  e.  s
 ) }   =>    |-  ( ph  ->  (
 ( # `  ( P pSyl 
 G ) )  mod  P )  =  1 )
 
Theoremsylow3 14779 Sylow's third theorem. The number of Sylow subgroups is a divisor of  |  G  |  /  d, where  d is the common order of a Sylow subgroup, and is equivalent to  1  mod  P. (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  N  =  ( # `  ( P pSyl  G ) )   =>    |-  ( ph  ->  ( N  ||  ( ( # `  X )  /  ( P ^ ( P  pCnt  ( # `  X ) ) ) )  /\  ( N  mod  P )  =  1 ) )
 
10.2.10  Direct products
 
Syntaxclsm 14780 Extend class notation with subgroup sum.
 class  LSSum
 
Syntaxcpj1 14781 Extend class notation with left projection.
 class  proj 1
 
Definitiondf-lsm 14782* Define subgroup sum (inner direct product of subgroups). (Contributed by NM, 28-Jan-2014.)
 |- 
 LSSum  =  ( w  e.  _V  |->  ( t  e. 
 ~P ( Base `  w ) ,  u  e.  ~P ( Base `  w )  |-> 
 ran  (  x  e.  t ,  y  e.  u  |->  ( x (
 +g  `  w )
 y ) ) ) )
 
Definitiondf-pj1 14783* Define the left projection function, which takes two subgroups  t ,  u with trivial intersection and returns a function mapping the elements of the subgroup sum  t  +  u to their projections onto  t. (The other projection function can be obtained by swapping the roles of  t and  u.) (Contributed by Mario Carneiro, 15-Oct-2015.)
 |- 
 proj 1  =  ( w  e.  _V  |->  ( t  e.  ~P ( Base `  w ) ,  u  e.  ~P ( Base `  w )  |->  ( z  e.  ( t ( LSSum `  w ) u ) 
 |->  ( iota_ x  e.  t E. y  e.  u  z  =  ( x ( +g  `  w )
 y ) ) ) ) )
 
Theoremlsmfval 14784* The subgroup sum function (for a group or vector space). (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .(+)  =  (
 LSSum `  G )   =>    |-  ( G  e.  V  ->  .(+)  =  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  (  x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) ) )
 
Theoremlsmvalx 14785* Subspace sum value (for a group or vector space). Extended domain version of lsmval 14794. (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .(+)  =  (
 LSSum `  G )   =>    |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  ( T 
 .(+)  U )  =  ran  (  x  e.  T ,  y  e.  U  |->  ( x  .+  y ) ) )
 
Theoremlsmelvalx 14786* Subspace sum membership (for a group or vector space). Extended domain version of lsmelval 14795. (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .(+)  =  (
 LSSum `  G )   =>    |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  ( X  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. z  e.  U  X  =  ( y  .+  z ) ) )
 
Theoremlsmelvalix 14787 Subspace sum membership (for a group or vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .(+)  =  (
 LSSum `  G )   =>    |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  ( X  e.  T  /\  Y  e.  U )
 )  ->  ( X  .+  Y )  e.  ( T  .(+)  U ) )
 
Theoremoppglsm 14788 The subspace sum operation in the opposite group. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  O  =  (oppg `  G )   &    |-  .(+)  =  ( LSSum `  G )   =>    |-  ( T ( LSSum `  O ) U )  =  ( U  .(+)  T )
 
Theoremlsmssv 14789 Subgroup sum is a subset of the base. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  ->  ( T  .(+)  U ) 
 C_  B )
 
Theoremlsmless1x 14790 Subset implies subgroup sum subset (extended domain version). (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  R  C_  T )  ->  ( R 
 .(+)  U )  C_  ( T  .(+)  U ) )
 
Theoremlsmless2x 14791 Subset implies subgroup sum subset (extended domain version). (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( ( G  e.  V  /\  R  C_  B  /\  U  C_  B )  /\  T  C_  U )  ->  ( R 
 .(+)  T )  C_  ( R  .(+)  U ) )
 
Theoremlsmub1x 14792 Subgroup sum is an upper bound of its arguments. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( T  C_  B  /\  U  e.  (SubMnd `  G ) )  ->  T  C_  ( T  .(+)  U ) )
 
Theoremlsmub2x 14793 Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( T  e.  (SubMnd `  G )  /\  U  C_  B )  ->  U  C_  ( T  .(+)  U ) )
 
Theoremlsmval 14794* Subgroup sum value (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .(+)  =  (
 LSSum `  G )   =>    |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  ( T  .(+)  U )  =  ran  (  x  e.  T ,  y  e.  U  |->  ( x  .+  y ) ) )
 
Theoremlsmelval 14795* Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |- 
 .+  =  ( +g  `  G )   &    |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  ( X  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. z  e.  U  X  =  ( y  .+  z ) ) )
 
Theoremlsmelvali 14796 Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |- 
 .+  =  ( +g  `  G )   &    |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  ( X  e.  T  /\  Y  e.  U ) )  ->  ( X  .+  Y )  e.  ( T  .(+)  U ) )
 
Theoremlsmelvalm 14797* Subgroup sum membership analog of lsmelval 14795 using vector subtraction. TODO: any way to shorten proof? (Contributed by NM, 16-Mar-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .-  =  ( -g `  G )   &    |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  T  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  U  e.  (SubGrp `  G ) )   =>    |-  ( ph  ->  ( X  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. z  e.  U  X  =  ( y  .-  z ) ) )
 
Theoremlsmelvalmi 14798 Membership of vector subtraction in subgroup sum. (Contributed by NM, 27-Apr-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .-  =  ( -g `  G )   &    |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  T  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  U  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  X  e.  T )   &    |-  ( ph  ->  Y  e.  U )   =>    |-  ( ph  ->  ( X  .-  Y )  e.  ( T  .(+)  U ) )
 
Theoremlsmsubm 14799 The sum of two commuting submonoids is a submonoid. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  Z  =  (Cntz `  G )   =>    |-  ( ( T  e.  (SubMnd `  G )  /\  U  e.  (SubMnd `  G )  /\  T  C_  ( Z `  U ) ) 
 ->  ( T  .(+)  U )  e.  (SubMnd `  G ) )
 
Theoremlsmsubg 14800 The sum of two commuting subgroups is a subgroup. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  Z  =  (Cntz `  G )   =>    |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `  U ) ) 
 ->  ( T  .(+)  U )  e.  (SubGrp `  G ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31058
  Copyright terms: Public domain < Previous  Next >