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Theorem List for Metamath Proof Explorer - 14901-15000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgexdvds3 14901 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 14902 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 14903* 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 14904 Reverse closure for the first argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  ( P pGrp  G  ->  P  e.  Prime )
 
Theorempgpgrp 14905 Reverse closure for the second argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  ( P pGrp  G  ->  G  e.  Grp )
 
Theorempgpfi1 14906 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 14907 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 14908 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 14909* Lemma for sylow1 14914. 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 14910* Lemma for sylow1 14914. 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 14911* Lemma for sylow1 14914. 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 14912* Lemma for sylow1 14914. 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 14913* Lemma for sylow1 14914. 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 14914* 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 14915* 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 14916* The converse to pgpfi1 14906. 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 14917 Alternate version of pgpfi 14916. (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 14918 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 14919* 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 14920 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 14921 A Sylow  P-subgroup is a subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( H  e.  ( P pSyl  G )  ->  H  e.  (SubGrp `  G )
 )
 
Theoremslwispgp 14922 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 14923 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 14924 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 14925* 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 14926 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 14927 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 14928* Lemma for sylow2a 14930. 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 14929* Lemma for sylow2a 14930. 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 14930* 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 14931* Lemma for sylow2b 14934. 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 14932* Lemma for sylow2b 14934. 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 14933* Sylow's second theorem. Putting together the results of sylow2a 14930 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 14934* 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 14935 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 14936 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 14937* Sylow's second theorem. See also sylow2b 14934 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 14936). (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 14938* Lemma for sylow3 14944, 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 14939* Lemma for sylow3 14944, 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 14940* Lemma for sylow3 14944, 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 14941* Lemma for sylow3 14944, 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 14942* Lemma for sylow3 14944, second part. Reduce the group action of sylow3lem1 14938 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 14943* Lemma for sylow3 14944, second part. Using the lemma sylow2a 14930, 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 14944 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 14945 Extend class notation with subgroup sum.
 class  LSSum
 
Syntaxcpj1 14946 Extend class notation with left projection.
 class  proj 1
 
Definitiondf-lsm 14947* 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 14948* 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 14949* 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 14950* Subspace sum value (for a group or vector space). Extended domain version of lsmval 14959. (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 14951* Subspace sum membership (for a group or vector space). Extended domain version of lsmelval 14960. (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 14952 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 14953 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 14954 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 14955 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 14956 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 14957 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 14958 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 14959* 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 14960* 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 14961 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 14962* Subgroup sum membership analog of lsmelval 14960 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 14963 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 14964 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 14965 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 14966 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 14967 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 14968 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 14969 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 14970 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 14971 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 14972 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 14973 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 14974 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 14975 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 )
 
Theoremlsmss1b 14976 Subgroup sum with a subset. (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  ( T 
 C_  U  <->  ( T  .(+)  U )  =  U ) )
 
Theoremlsmss2 14977 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 )  /\  U  C_  T )  ->  ( T  .(+)  U )  =  T )
 
Theoremlsmss2b 14978 Subgroup sum with a subset. (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  ( U 
 C_  T  <->  ( T  .(+)  U )  =  T ) )
 
Theoremlsmass 14979 Subgroup sum is associative. (Contributed by NM, 2-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( R  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  ( ( R  .(+)  T )  .(+)  U )  =  ( R  .(+)  ( T 
 .(+)  U ) ) )
 
Theoremlsm01 14980 Subgroup sum with the zero subgroup. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |- 
 .0.  =  ( 0g `  G )   &    |-  .(+)  =  ( LSSum `  G )   =>    |-  ( X  e.  (SubGrp `  G )  ->  ( X  .(+)  {  .0.  }
 )  =  X )
 
Theoremlsm02 14981 Subgroup sum with the zero subgroup. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |- 
 .0.  =  ( 0g `  G )   &    |-  .(+)  =  ( LSSum `  G )   =>    |-  ( X  e.  (SubGrp `  G )  ->  ( {  .0.  }  .(+)  X )  =  X )
 
Theoremsubglsm 14982 The subgroup sum evaluated within a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  H  =  ( Gs  S )   &    |-  .(+)  =  ( LSSum `  G )   &    |-  A  =  (
 LSSum `  H )   =>    |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  ( T  .(+)  U )  =  ( T A U ) )
 
Theoremlssnle 14983 Equivalent expressions for "not less than". (chnlei 22064 analog.) (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   =>    |-  ( ph  ->  ( -.  U  C_  T  <->  T  C.  ( T 
 .(+)  U ) ) )
 
Theoremlsmmod 14984 The modular law holds for subgroup sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
 )  /\  S  C_  U )  ->  ( S  .(+)  ( T  i^i  U ) )  =  ( ( S  .(+)  T )  i^i  U ) )
 
Theoremlsmmod2 14985 Modular law dual for subgroup sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 8-Jan-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
 )  /\  U  C_  S )  ->  ( S  i^i  ( T  .(+)  U ) )  =  ( ( S  i^i  T ) 
 .(+)  U ) )
 
Theoremlsmpropd 14986* If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 29-Jun-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ph  ->  K  e.  _V )   &    |-  ( ph  ->  L  e.  _V )   =>    |-  ( ph  ->  (
 LSSum `  K )  =  ( LSSum `  L )
 )
 
Theoremcntzrecd 14987 Commute the "subgroups commute" predicate. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  T  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  U  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  T 
 C_  ( Z `  U ) )   =>    |-  ( ph  ->  U 
 C_  ( Z `  T ) )
 
Theoremlsmcntz 14988 The "subgroups commute" predicate applied to a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  Z  =  (Cntz `  G )   =>    |-  ( ph  ->  (
 ( S  .(+)  T ) 
 C_  ( Z `  U )  <->  ( S  C_  ( Z `  U ) 
 /\  T  C_  ( Z `  U ) ) ) )
 
Theoremlsmcntzr 14989 The "subgroups commute" predicate applied to a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  Z  =  (Cntz `  G )   =>    |-  ( ph  ->  ( S  C_  ( Z `  ( T  .(+)  U ) )  <->  ( S  C_  ( Z `  T ) 
 /\  S  C_  ( Z `  U ) ) ) )
 
Theoremlsmdisj 14990 Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |- 
 .0.  =  ( 0g `  G )   &    |-  ( ph  ->  ( ( S  .(+)  T )  i^i  U )  =  {  .0.  } )   =>    |-  ( ph  ->  ( ( S  i^i  U )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  }
 ) )
 
Theoremlsmdisj2 14991 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |- 
 .0.  =  ( 0g `  G )   &    |-  ( ph  ->  ( ( S  .(+)  T )  i^i  U )  =  {  .0.  } )   &    |-  ( ph  ->  ( S  i^i  T )  =  {  .0.  } )   =>    |-  ( ph  ->  ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  } )
 
Theoremlsmdisj3 14992 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |- 
 .0.  =  ( 0g `  G )   &    |-  ( ph  ->  ( ( S  .(+)  T )  i^i  U )  =  {  .0.  } )   &    |-  ( ph  ->  ( S  i^i  T )  =  {  .0.  } )   &    |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  S 
 C_  ( Z `  T ) )   =>    |-  ( ph  ->  ( S  i^i  ( T 
 .(+)  U ) )  =  {  .0.  } )
 
Theoremlsmdisjr 14993 Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |- 
 .0.  =  ( 0g `  G )   &    |-  ( ph  ->  ( S  i^i  ( T 
 .(+)  U ) )  =  {  .0.  } )   =>    |-  ( ph  ->  ( ( S  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
 ) )
 
Theoremlsmdisj2r 14994 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |- 
 .0.  =  ( 0g `  G )   &    |-  ( ph  ->  ( S  i^i  ( T 
 .(+)  U ) )  =  {  .0.  } )   &    |-  ( ph  ->  ( T  i^i  U )  =  {  .0.  } )   =>    |-  ( ph  ->  (
 ( S  .(+)  U )  i^i  T )  =  {  .0.  } )
 
Theoremlsmdisj3r 14995 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |- 
 .0.  =  ( 0g `  G )   &    |-  ( ph  ->  ( S  i^i  ( T 
 .(+)  U ) )  =  {  .0.  } )   &    |-  ( ph  ->  ( T  i^i  U )  =  {  .0.  } )   &    |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  T 
 C_  ( Z `  U ) )   =>    |-  ( ph  ->  ( ( S  .(+)  T )  i^i  U )  =  {  .0.  } )
 
Theoremlsmdisj2a 14996 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |- 
 .0.  =  ( 0g `  G )   =>    |-  ( ph  ->  (
 ( ( ( S 
 .(+)  T )  i^i  U )  =  {  .0.  } 
 /\  ( S  i^i  T )  =  {  .0.  } )  <->  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } )
 ) )
 
Theoremlsmdisj2b 14997 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |- 
 .0.  =  ( 0g `  G )   =>    |-  ( ph  ->  (
 ( ( ( S 
 .(+)  U )  i^i  T )  =  {  .0.  } 
 /\  ( S  i^i  U )  =  {  .0.  } )  <->  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } )
 ) )
 
Theoremlsmdisj3a 14998 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |- 
 .0.  =  ( 0g `  G )   &    |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  S 
 C_  ( Z `  T ) )   =>    |-  ( ph  ->  ( ( ( ( S 
 .(+)  T )  i^i  U )  =  {  .0.  } 
 /\  ( S  i^i  T )  =  {  .0.  } )  <->  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } )
 ) )
 
Theoremlsmdisj3b 14999 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |- 
 .0.  =  ( 0g `  G )   &    |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  T 
 C_  ( Z `  U ) )   =>    |-  ( ph  ->  ( ( ( ( S 
 .(+)  T )  i^i  U )  =  {  .0.  } 
 /\  ( S  i^i  T )  =  {  .0.  } )  <->  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } )
 ) )
 
Theoremsubgdisj1 15000 Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. (Contributed by NM, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |- 
 .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  ( T  i^i  U )  =  {  .0.  } )   &    |-  ( ph  ->  T  C_  ( Z `  U ) )   &    |-  ( ph  ->  A  e.  T )   &    |-  ( ph  ->  C  e.  T )   &    |-  ( ph  ->  B  e.  U )   &    |-  ( ph  ->  D  e.  U )   &    |-  ( ph  ->  ( A  .+  B )  =  ( C  .+  D ) )   =>    |-  ( ph  ->  A  =  C )
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