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Theorem List for Metamath Proof Explorer - 14901-15000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsylow3lem1 14901* Lemma for sylow3 14907, 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 14902* Lemma for sylow3 14907, 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 14903* Lemma for sylow3 14907, 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 14904* Lemma for sylow3 14907, 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 14905* Lemma for sylow3 14907, second part. Reduce the group action of sylow3lem1 14901 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 14906* Lemma for sylow3 14907, second part. Using the lemma sylow2a 14893, 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 14907 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 14908 Extend class notation with subgroup sum.
 class  LSSum
 
Syntaxcpj1 14909 Extend class notation with left projection.
 class  proj 1
 
Definitiondf-lsm 14910* 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 14911* 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 14912* 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 14913* Subspace sum value (for a group or vector space). Extended domain version of lsmval 14922. (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 14914* Subspace sum membership (for a group or vector space). Extended domain version of lsmelval 14923. (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 14915 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 14916 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 14917 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 14918 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 14919 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 14920 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 14921 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 14922* 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 14923* 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 14924 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 14925* Subgroup sum membership analog of lsmelval 14923 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 14926 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 14927 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 14928 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 14929 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 14930 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 14931 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 14932 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 14933 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 14934 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 14935 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 14936 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 14937 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 14938 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 14939 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 14940 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 14941 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 14942 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 14943 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 14944 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 14945 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 14946 Equivalent expressions for "not less than". (chnlei 22025 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 14947 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 14948 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 14949* 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 14950 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 14951 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 14952 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 14953 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 14954 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 14955 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 14956 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 14957 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 14958 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 14959 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 14960 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 14961 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 14962 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 14963 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 )
 
Theoremsubgdisj2 14964 Vectors belonging to disjoint subgroups are uniquely determined by their sum. (Contributed by NM, 12-Jul-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  ->  B  =  D )
 
Theoremsubgdisjb 14965 Vectors belonging to disjoint subgroups are uniquely determined by their sum. Analogous to opth 4217, this theorem shows a way of representing a pair of vectors. (Contributed by NM, 5-Jul-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 )  <->  ( A  =  C  /\  B  =  D ) ) )
 
Theorempj1fval 14966* The left projection function (for a direct product of group subspaces). (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .(+)  =  (
 LSSum `  G )   &    |-  P  =  ( proj 1 `  G )   =>    |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  ( T P U )  =  ( z  e.  ( T  .(+)  U ) 
 |->  ( iota_ x  e.  T E. y  e.  U  z  =  ( x  .+  y ) ) ) )
 
Theorempj1val 14967* The left projection function (for a direct product of group subspaces). (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .(+)  =  (
 LSSum `  G )   &    |-  P  =  ( proj 1 `  G )   =>    |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  X  e.  ( T  .(+)  U ) )  ->  ( ( T P U ) `  X )  =  ( iota_ x  e.  T E. y  e.  U  X  =  ( x  .+  y
 ) ) )
 
Theorempj1eu 14968* Uniqueness of a left projection. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |- 
 .+  =  ( +g  `  G )   &    |-  .(+)  =  ( LSSum `  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  /\  X  e.  ( T  .(+)  U ) )  ->  E! x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) )
 
Theorempj1f 14969 The left projection function maps a direct subspace sum onto the left factor. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |- 
 .+  =  ( +g  `  G )   &    |-  .(+)  =  ( LSSum `  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 ) )   &    |-  P  =  ( proj 1 `
  G )   =>    |-  ( ph  ->  ( T P U ) : ( T  .(+)  U ) --> T )
 
Theorempj2f 14970 The right projection function maps a direct subspace sum onto the right factor. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |- 
 .+  =  ( +g  `  G )   &    |-  .(+)  =  ( LSSum `  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 ) )   &    |-  P  =  ( proj 1 `
  G )   =>    |-  ( ph  ->  ( U P T ) : ( T  .(+)  U ) --> U )
 
Theorempj1id 14971 Any element of a direct subspace sum can be decomposed into projections onto the left and right factors. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |- 
 .+  =  ( +g  `  G )   &    |-  .(+)  =  ( LSSum `  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 ) )   &    |-  P  =  ( proj 1 `
  G )   =>    |-  ( ( ph  /\  X  e.  ( T 
 .(+)  U ) )  ->  X  =  ( (
 ( T P U ) `  X )  .+  ( ( U P T ) `  X ) ) )
 
Theorempj1eq 14972 Any element of a direct subspace sum can be decomposed uniquely into projections onto the left and right factors. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |- 
 .+  =  ( +g  `  G )   &    |-  .(+)  =  ( LSSum `  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 ) )   &    |-  P  =  ( proj 1 `
  G )   &    |-  ( ph  ->  X  e.  ( T  .(+)  U ) )   &    |-  ( ph  ->  B  e.  T )   &    |-  ( ph  ->  C  e.  U )   =>    |-  ( ph  ->  ( X  =  ( B 
 .+  C )  <->  ( ( ( T P U ) `
  X )  =  B  /\  ( ( U P T ) `
  X )  =  C ) ) )
 
Theorempj1lid 14973 The left projection function is the identity on the left subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |- 
 .+  =  ( +g  `  G )   &    |-  .(+)  =  ( LSSum `  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 ) )   &    |-  P  =  ( proj 1 `
  G )   =>    |-  ( ( ph  /\  X  e.  T ) 
 ->  ( ( T P U ) `  X )  =  X )
 
Theorempj1rid 14974 The left projection function is the zero operator on the right subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |- 
 .+  =  ( +g  `  G )   &    |-  .(+)  =  ( LSSum `  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 ) )   &    |-  P  =  ( proj 1 `
  G )   =>    |-  ( ( ph  /\  X  e.  U ) 
 ->  ( ( T P U ) `  X )  =  .0.  )
 
Theorempj1ghm 14975 The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |- 
 .+  =  ( +g  `  G )   &    |-  .(+)  =  ( LSSum `  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 ) )   &    |-  P  =  ( proj 1 `
  G )   =>    |-  ( ph  ->  ( T P U )  e.  ( ( Gs  ( T  .(+)  U )
 )  GrpHom  G ) )
 
Theorempj1ghm2 14976 The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |- 
 .+  =  ( +g  `  G )   &    |-  .(+)  =  ( LSSum `  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 ) )   &    |-  P  =  ( proj 1 `
  G )   =>    |-  ( ph  ->  ( T P U )  e.  ( ( Gs  ( T  .(+)  U )
 )  GrpHom  ( Gs  T ) ) )
 
Theoremlsmhash 14977 The order of the direct product of groups. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  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  ->  T  e.  Fin )   &    |-  ( ph  ->  U  e.  Fin )   =>    |-  ( ph  ->  ( # `
  ( T  .(+)  U ) )  =  ( ( # `  T )  x.  ( # `  U ) ) )
 
10.2.11  Free groups
 
Syntaxcefg 14978 Extend class notation with the free group equivalence relation.
 class ~FG
 
Syntaxcfrgp 14979 Extend class notation with the free group construction.
 class freeGrp
 
Syntaxcvrgp 14980 Extend class notation with free group injection.
 class varFGrp
 
Definitiondf-efg 14981* Define the free group equivalence relation, which is the smallest equivalence relation  ~~ such that for any words 
A ,  B and formal symbol  x with inverse  inv g x,  A B  ~~  A x ( inv g
x ) B. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |- ~FG  =  ( i  e.  _V  |->  |^|
 { r  |  ( r  Er Word  ( i  X.  2o )  /\  A. x  e. Word  ( i  X.  2o ) A. n  e.  ( 0 ... ( # `
  x ) )
 A. y  e.  i  A. z  e.  2o  x r ( x splice  <. n ,  n ,  <" <. y ,  z >.
 <. y ,  ( 1o  \  z ) >. "> >.
 ) ) } )
 
Definitiondf-frgp 14982 Define the free group on a set  I of generators, defined as the quotient of the free monoid on  I  X.  2o (representing the generator elements and their formal inverses) by the free group equivalence relation df-efg 14981. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |- freeGrp  =  ( i  e.  _V  |->  ( (freeMnd `  ( i  X.  2o ) )  /.s  ( ~FG  `  i
 ) ) )
 
Definitiondf-vrgp 14983* Define the canonical injection from the generating set  I into the base set of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |- varFGrp  =  ( i  e.  _V  |->  ( j  e.  i  |->  [ <" <. j ,  (/) >. "> ] ( ~FG  `  i
 ) ) )
 
Theoremefgmval 14984* Value of the formal inverse operation for the generating set of a free group. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z ) >. )   =>    |-  ( ( A  e.  I  /\  B  e.  2o )  ->  ( A M B )  =  <. A ,  ( 1o  \  B ) >. )
 
Theoremefgmf 14985* The formal inverse operation is an endofunction on the generating set. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z ) >. )   =>    |-  M : ( I  X.  2o ) --> ( I  X.  2o )
 
Theoremefgmnvl 14986* The inversion function on the generators is an involution. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z ) >. )   =>    |-  ( A  e.  ( I  X.  2o )  ->  ( M `  ( M `
  A ) )  =  A )
 
Theoremefgrcl 14987 Lemma for efgval 14989. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   =>    |-  ( A  e.  W  ->  ( I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
 
Theoremefglem 14988* Lemma for efgval 14989. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   =>    |- 
 E. r ( r  Er  W  /\  A. x  e.  W  A. n  e.  ( 0 ... ( # `
  x ) )
 A. y  e.  I  A. z  e.  2o  x r ( x splice  <. n ,  n ,  <" <. y ,  z >.
 <. y ,  ( 1o  \  z ) >. "> >.
 ) )
 
Theoremefgval 14989* Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   =>    |- 
 .~  =  |^| { r  |  ( r  Er  W  /\  A. x  e.  W  A. n  e.  ( 0
 ... ( # `  x ) ) A. y  e.  I  A. z  e. 
 2o  x r ( x splice  <. n ,  n ,  <" <. y ,  z >. <. y ,  ( 1o  \  z ) >. "> >. ) ) }
 
Theoremefger 14990 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   =>    |- 
 .~  Er  W
 
Theoremefgi 14991 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   =>    |-  ( ( ( A  e.  W  /\  N  e.  ( 0 ... ( # `
  A ) ) )  /\  ( J  e.  I  /\  K  e.  2o ) )  ->  A  .~  ( A splice  <. N ,  N ,  <" <. J ,  K >. <. J ,  ( 1o  \  K )
 >. "> >. ) )
 
Theoremefgi0 14992 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   =>    |-  ( ( A  e.  W  /\  N  e.  (
 0 ... ( # `  A ) )  /\  J  e.  I )  ->  A  .~  ( A splice  <. N ,  N ,  <" <. J ,  (/) >. <. J ,  1o >. "> >. ) )
 
Theoremefgi1 14993 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   =>    |-  ( ( A  e.  W  /\  N  e.  (
 0 ... ( # `  A ) )  /\  J  e.  I )  ->  A  .~  ( A splice  <. N ,  N ,  <" <. J ,  1o >. <. J ,  (/)
 >. "> >. ) )
 
Theoremefgtf 14994* Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   &    |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z
 ) >. )   &    |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0
 ... ( # `  v
 ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. ) ) )   =>    |-  ( X  e.  W  ->  ( ( T `  X )  =  (
 a  e.  ( 0
 ... ( # `  X ) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a ,  <" b
 ( M `  b
 ) "> >. ) ) 
 /\  ( T `  X ) : ( ( 0 ... ( # `
  X ) )  X.  ( I  X.  2o ) ) --> W ) )
 
Theoremefgtval 14995* Value of the extension function, which maps a word (a representation of the group element as a sequence of elements and their inverses) to its direct extensions, defined as the original representation with an element and its inverse inserted somewhere in the string. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   &    |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z
 ) >. )   &    |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0
 ... ( # `  v
 ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. ) ) )   =>    |-  ( ( X  e.  W  /\  N  e.  (
 0 ... ( # `  X ) )  /\  A  e.  ( I  X.  2o )
 )  ->  ( N ( T `  X ) A )  =  ( X splice  <. N ,  N ,  <" A ( M `  A ) "> >. ) )
 
Theoremefgval2 14996* Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   &    |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z
 ) >. )   &    |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0
 ... ( # `  v
 ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. ) ) )   =>    |- 
 .~  =  |^| { r  |  ( r  Er  W  /\  A. x  e.  W  ran  (  T `  x )  C_  [ x ]
 r ) }
 
Theoremefgi2 14997* Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   &    |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z
 ) >. )   &    |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0
 ... ( # `  v
 ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. ) ) )   =>    |-  ( ( A  e.  W  /\  B  e.  ran  (  T `  A ) )  ->  A  .~  B )
 
Theoremefgtlen 14998* Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   &    |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z
 ) >. )   &    |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0
 ... ( # `  v
 ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. ) ) )   =>    |-  ( ( X  e.  W  /\  A  e.  ran  (  T `  X ) )  ->  ( # `  A )  =  ( ( # `
  X )  +  2 ) )
 
Theoremefginvrel2 14999* The inverse of the reverse of a word composed with the word relates to the identity. (This provides an explicit expression for the representation of the group inverse, given a representative of the free group equivalence class.) (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   &    |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z
 ) >. )   &    |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0
 ... ( # `  v
 ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. ) ) )   =>    |-  ( A  e.  W  ->  ( A concat  ( M  o.  (reverse `  A )
 ) )  .~  (/) )
 
Theoremefginvrel1 15000* The inverse of the reverse of a word composed with the word relates to the identity. (This provides an explicit expression for the representation of the group inverse, given a representative of the free group equivalence class.) (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   &    |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z
 ) >. )   &    |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0
 ... ( # `  v
 ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. ) ) )   =>    |-  ( A  e.  W  ->  ( ( M  o.  (reverse `  A ) ) concat  A )  .~  (/) )
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