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Theorem List for Metamath Proof Explorer - 15201-15300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremodinf 15201* 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 15202* 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 15203 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 15204 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 15205 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 15206 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 15207 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 15208* 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 15209* 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 15210 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 15211 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 15212 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 15213* 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 15214* 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 15215* 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 15216 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 15217 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 15218* 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 15219 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 15220* 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 15221* 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 15222 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 15223* 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 15224 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 15225 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 15226 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 15227 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 15228* 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 15229 Reverse closure for the first argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  ( P pGrp  G  ->  P  e.  Prime )
 
Theorempgpgrp 15230 Reverse closure for the second argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  ( P pGrp  G  ->  G  e.  Grp )
 
Theorempgpfi1 15231 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 15232 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 15233 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 15234* Lemma for sylow1 15239. 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 15235* Lemma for sylow1 15239. 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 15236* Lemma for sylow1 15239. 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 15237* Lemma for sylow1 15239. 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 15238* Lemma for sylow1 15239. 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 15239* 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 15240* 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 15241* The converse to pgpfi1 15231. 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 15242 Alternate version of pgpfi 15241. (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 15243 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 15244* 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 15245 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 15246 A Sylow  P-subgroup is a subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( H  e.  ( P pSyl  G )  ->  H  e.  (SubGrp `  G )
 )
 
Theoremslwispgp 15247 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 15248 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 15249 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 15250* 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 15251 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 15252 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 15253* Lemma for sylow2a 15255. 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 15254* Lemma for sylow2a 15255. 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 15255* 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 15256* Lemma for sylow2b 15259. 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 15257* Lemma for sylow2b 15259. 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 15258* Sylow's second theorem. Putting together the results of sylow2a 15255 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 15259* 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 15260 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 15261 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 15262* Sylow's second theorem. See also sylow2b 15259 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 15261). (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 15263* Lemma for sylow3 15269, 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 15264* Lemma for sylow3 15269, 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 15265* Lemma for sylow3 15269, 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 15266* Lemma for sylow3 15269, 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 15267* Lemma for sylow3 15269, second part. Reduce the group action of sylow3lem1 15263 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 15268* Lemma for sylow3 15269, second part. Using the lemma sylow2a 15255, 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 15269 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 15270 Extend class notation with subgroup sum.
 class  LSSum
 
Syntaxcpj1 15271 Extend class notation with left projection.
 class  proj 1
 
Definitiondf-lsm 15272* 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 15273* 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 15274* 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 15275* Subspace sum value (for a group or vector space). Extended domain version of lsmval 15284. (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 15276* Subspace sum membership (for a group or vector space). Extended domain version of lsmelval 15285. (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 15277 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 15278 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 15279 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 15280 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 15281 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 15282 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 15283 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 15284* 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 15285* 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 15286 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 15287* Subgroup sum membership analog of lsmelval 15285 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 15288 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 15289 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 15290 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 ) )
 
Theoremlsmcom2 15291 Subgroup sum commutes. (Contributed by Mario Carneiro, 22-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  Z  =  (Cntz `  G )   =>    |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `  U ) ) 
 ->  ( T  .(+)  U )  =  ( U  .(+)  T ) )
 
Theoremlsmub1 15292 Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  T  C_  ( T  .(+)  U ) )
 
Theoremlsmub2 15293 Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  U  C_  ( T  .(+)  U ) )
 
Theoremlsmunss 15294 Union of subgroups is a subset of subgroup sum. (Contributed by NM, 6-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  ( T  u.  U )  C_  ( T  .(+)  U ) )
 
Theoremlsmless1 15295 Subset implies subgroup sum subset. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  S  C_  T )  ->  ( S  .(+)  U )  C_  ( T  .(+)  U ) )
 
Theoremlsmless2 15296 Subset implies subgroup sum subset. (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  ( S  .(+)  T )  C_  ( S  .(+)  U ) )
 
Theoremlsmless12 15297 Subset implies subgroup sum subset. (Contributed by NM, 14-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  ( R  C_  S  /\  T  C_  U ) ) 
 ->  ( R  .(+)  T ) 
 C_  ( S  .(+)  U ) )
 
Theoremlsmidm 15298 Subgroup sum is idempotent. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( U  e.  (SubGrp `  G )  ->  ( U  .(+)  U )  =  U )
 
Theoremlsmlub 15299 The least upper bound property of subgroup sum. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  ( ( S  C_  U  /\  T  C_  U ) 
 <->  ( S  .(+)  T ) 
 C_  U ) )
 
Theoremlsmss1 15300 Subgroup sum with a subset. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  ( T  .(+)  U )  =  U )
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