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Theorem List for Metamath Proof Explorer - 15201-15300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdprddisj2 15201 The function  S is a family of subgroups. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  C  C_  I
 )   &    |-  ( ph  ->  D  C_  I )   &    |-  ( ph  ->  ( C  i^i  D )  =  (/) )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ph  ->  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd 
 ( S  |`  D ) ) )  =  {  .0.  } )
 
Theoremdprd2dlem2 15202* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  Rel  A )   &    |-  ( ph  ->  S : A --> (SubGrp `  G )
 )   &    |-  ( ph  ->  dom  A  C_  I )   &    |-  ( ( ph  /\  i  e.  I ) 
 ->  G dom DProd  ( j  e.  ( A " {
 i } )  |->  ( i S j ) ) )   &    |-  ( ph  ->  G dom DProd  ( i  e.  I  |->  ( G DProd  (
 j  e.  ( A
 " { i }
 )  |->  ( i S j ) ) ) ) )   &    |-  K  =  (mrCls `  (SubGrp `  G )
 )   =>    |-  ( ( ph  /\  X  e.  A )  ->  ( S `  X )  C_  ( G DProd  ( j  e.  ( A " {
 ( 1st `  X ) } )  |->  ( ( 1st `  X ) S j ) ) ) )
 
Theoremdprd2dlem1 15203* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  Rel  A )   &    |-  ( ph  ->  S : A --> (SubGrp `  G )
 )   &    |-  ( ph  ->  dom  A  C_  I )   &    |-  ( ( ph  /\  i  e.  I ) 
 ->  G dom DProd  ( j  e.  ( A " {
 i } )  |->  ( i S j ) ) )   &    |-  ( ph  ->  G dom DProd  ( i  e.  I  |->  ( G DProd  (
 j  e.  ( A
 " { i }
 )  |->  ( i S j ) ) ) ) )   &    |-  K  =  (mrCls `  (SubGrp `  G )
 )   &    |-  ( ph  ->  C  C_  I )   =>    |-  ( ph  ->  ( K `  U. ( S
 " ( A  |`  C ) ) )  =  ( G DProd  ( i  e.  C  |->  ( G DProd  (
 j  e.  ( A
 " { i }
 )  |->  ( i S j ) ) ) ) ) )
 
Theoremdprd2da 15204* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  Rel  A )   &    |-  ( ph  ->  S : A --> (SubGrp `  G )
 )   &    |-  ( ph  ->  dom  A  C_  I )   &    |-  ( ( ph  /\  i  e.  I ) 
 ->  G dom DProd  ( j  e.  ( A " {
 i } )  |->  ( i S j ) ) )   &    |-  ( ph  ->  G dom DProd  ( i  e.  I  |->  ( G DProd  (
 j  e.  ( A
 " { i }
 )  |->  ( i S j ) ) ) ) )   &    |-  K  =  (mrCls `  (SubGrp `  G )
 )   =>    |-  ( ph  ->  G dom DProd  S )
 
Theoremdprd2db 15205* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( ph  ->  Rel  A )   &    |-  ( ph  ->  S : A --> (SubGrp `  G )
 )   &    |-  ( ph  ->  dom  A  C_  I )   &    |-  ( ( ph  /\  i  e.  I ) 
 ->  G dom DProd  ( j  e.  ( A " {
 i } )  |->  ( i S j ) ) )   &    |-  ( ph  ->  G dom DProd  ( i  e.  I  |->  ( G DProd  (
 j  e.  ( A
 " { i }
 )  |->  ( i S j ) ) ) ) )   &    |-  K  =  (mrCls `  (SubGrp `  G )
 )   =>    |-  ( ph  ->  ( G DProd  S )  =  ( G DProd  ( i  e.  I  |->  ( G DProd  (
 j  e.  ( A
 " { i }
 )  |->  ( i S j ) ) ) ) ) )
 
Theoremdprd2d2 15206* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ( ph  /\  (
 i  e.  I  /\  j  e.  J )
 )  ->  S  e.  (SubGrp `  G ) )   &    |-  ( ( ph  /\  i  e.  I )  ->  G dom DProd  ( j  e.  J  |->  S ) )   &    |-  ( ph  ->  G dom DProd  ( i  e.  I  |->  ( G DProd 
 ( j  e.  J  |->  S ) ) ) )   =>    |-  ( ph  ->  ( G dom DProd  ( i  e.  I ,  j  e.  J  |->  S )  /\  ( G DProd  ( i  e.  I ,  j  e.  J  |->  S ) )  =  ( G DProd  (
 i  e.  I  |->  ( G DProd  ( j  e.  J  |->  S ) ) ) ) ) )
 
Theoremdmdprdsplit2lem 15207 Lemma for dmdprdsplit 15209. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  S : I --> (SubGrp `  G )
 )   &    |-  ( ph  ->  ( C  i^i  D )  =  (/) )   &    |-  ( ph  ->  I  =  ( C  u.  D ) )   &    |-  Z  =  (Cntz `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G dom DProd  ( S  |`  C ) )   &    |-  ( ph  ->  G dom DProd  ( S  |`  D ) )   &    |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  C_  ( Z `  ( G DProd  ( S  |`  D ) ) ) )   &    |-  ( ph  ->  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  =  {  .0.  }
 )   &    |-  K  =  (mrCls `  (SubGrp `  G ) )   =>    |-  ( ( ph  /\  X  e.  C )  ->  (
 ( Y  e.  I  ->  ( X  =/=  Y  ->  ( S `  X )  C_  ( Z `  ( S `  Y ) ) ) )  /\  ( ( S `  X )  i^i  ( K `
  U. ( S "
 ( I  \  { X } ) ) ) )  C_  {  .0.  } ) )
 
Theoremdmdprdsplit2 15208 The direct product splits into the direct product of any partition of the index set. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( ph  ->  S : I --> (SubGrp `  G )
 )   &    |-  ( ph  ->  ( C  i^i  D )  =  (/) )   &    |-  ( ph  ->  I  =  ( C  u.  D ) )   &    |-  Z  =  (Cntz `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G dom DProd  ( S  |`  C ) )   &    |-  ( ph  ->  G dom DProd  ( S  |`  D ) )   &    |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  C_  ( Z `  ( G DProd  ( S  |`  D ) ) ) )   &    |-  ( ph  ->  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  =  {  .0.  }
 )   =>    |-  ( ph  ->  G dom DProd  S )
 
Theoremdmdprdsplit 15209 The direct product splits into the direct product of any partition of the index set. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( ph  ->  S : I --> (SubGrp `  G )
 )   &    |-  ( ph  ->  ( C  i^i  D )  =  (/) )   &    |-  ( ph  ->  I  =  ( C  u.  D ) )   &    |-  Z  =  (Cntz `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ph  ->  ( G dom DProd  S  <->  ( ( G dom DProd  ( S  |`  C ) 
 /\  G dom DProd  ( S  |`  D ) )  /\  ( G DProd  ( S  |`  C ) )  C_  ( Z `  ( G DProd  ( S  |`  D ) ) ) 
 /\  ( ( G DProd 
 ( S  |`  C ) )  i^i  ( G DProd 
 ( S  |`  D ) ) )  =  {  .0.  } ) ) )
 
Theoremdprdsplit 15210 The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( ph  ->  S : I --> (SubGrp `  G )
 )   &    |-  ( ph  ->  ( C  i^i  D )  =  (/) )   &    |-  ( ph  ->  I  =  ( C  u.  D ) )   &    |-  .(+)  =  (
 LSSum `  G )   &    |-  ( ph  ->  G dom DProd  S )   =>    |-  ( ph  ->  ( G DProd  S )  =  ( ( G DProd  ( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
 ) ) )
 
Theoremdmdprdpr 15211 A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  Z  =  (Cntz `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  T  e.  (SubGrp `  G ) )   =>    |-  ( ph  ->  ( G dom DProd  `' ( { S }  +c  { T } )  <->  ( S  C_  ( Z `  T )  /\  ( S  i^i  T )  =  {  .0.  } )
 ) )
 
Theoremdprdpr 15212 A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  Z  =  (Cntz `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  T  e.  (SubGrp `  G ) )   &    |-  .(+) 
 =  ( LSSum `  G )   &    |-  ( ph  ->  S  C_  ( Z `  T ) )   &    |-  ( ph  ->  ( S  i^i  T )  =  {  .0.  }
 )   =>    |-  ( ph  ->  ( G DProd  `' ( { S }  +c  { T } )
 )  =  ( S 
 .(+)  T ) )
 
Theoremdpjlem 15213 Lemma for theorems about direct product projection. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( G DProd  ( S  |`  { X } ) )  =  ( S `  X ) )
 
Theoremdpjcntz 15214 The two subgroups that appear in dpjval 15218 commute. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  X  e.  I )   &    |-  Z  =  (Cntz `  G )   =>    |-  ( ph  ->  ( S `  X )  C_  ( Z `  ( G DProd 
 ( S  |`  ( I 
 \  { X }
 ) ) ) ) )
 
Theoremdpjdisj 15215 The two subgroups that appear in dpjval 15218 are disjoint. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  X  e.  I )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ph  ->  ( ( S `  X )  i^i  ( G DProd  ( S  |`  ( I  \  { X } ) ) ) )  =  {  .0.  } )
 
Theoremdpjlsm 15216 The two subgroups that appear in dpjval 15218 add to the full direct product. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  X  e.  I )   &    |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ph  ->  ( G DProd  S )  =  ( ( S `  X )  .(+)  ( G DProd  ( S  |`  ( I  \  { X } ) ) ) ) )
 
Theoremdpjfval 15217* Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  P  =  ( GdProj S )   &    |-  Q  =  (
 proj 1 `  G )   =>    |-  ( ph  ->  P  =  ( i  e.  I  |->  ( ( S `  i ) Q ( G DProd  ( S  |`  ( I 
 \  { i }
 ) ) ) ) ) )
 
Theoremdpjval 15218 Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  P  =  ( GdProj S )   &    |-  Q  =  (
 proj 1 `  G )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( P `  X )  =  ( ( S `  X ) Q ( G DProd  ( S  |`  ( I 
 \  { X }
 ) ) ) ) )
 
Theoremdpjf 15219 The  X-th index projection is a function from the direct product to the  X-th factor. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  P  =  ( GdProj S )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( P `  X ) : ( G DProd  S )
 --> ( S `  X ) )
 
Theoremdpjidcl 15220* The key property of projections: the sum of all the projections of  A is  A. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  P  =  ( GdProj S )   &    |-  ( ph  ->  A  e.  ( G DProd  S ) )   &    |-  .0.  =  ( 0g `  G )   &    |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
 Fin }   =>    |-  ( ph  ->  (
 ( x  e.  I  |->  ( ( P `  x ) `  A ) )  e.  W  /\  A  =  ( G 
 gsumg  ( x  e.  I  |->  ( ( P `  x ) `  A ) ) ) ) )
 
Theoremdpjeq 15221* Decompose a group sum into projections. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  P  =  ( GdProj S )   &    |-  ( ph  ->  A  e.  ( G DProd  S ) )   &    |-  .0.  =  ( 0g `  G )   &    |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
 Fin }   &    |-  ( ph  ->  ( x  e.  I  |->  C )  e.  W )   =>    |-  ( ph  ->  ( A  =  ( G  gsumg  ( x  e.  I  |->  C ) )  <->  A. x  e.  I  ( ( P `  x ) `  A )  =  C )
 )
 
Theoremdpjid 15222* The key property of projections: the sum of all the projections of  A is  A. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  P  =  ( GdProj S )   &    |-  ( ph  ->  A  e.  ( G DProd  S ) )   =>    |-  ( ph  ->  A  =  ( G  gsumg  ( x  e.  I  |->  ( ( P `  x ) `  A ) ) ) )
 
Theoremdpjlid 15223 The  X-th index projection acts as the identity on elements of the  X-th factor. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  P  =  ( GdProj S )   &    |-  ( ph  ->  X  e.  I )   &    |-  ( ph  ->  A  e.  ( S `  X ) )   =>    |-  ( ph  ->  ( ( P `  X ) `  A )  =  A )
 
Theoremdpjrid 15224 The  Y-th index projection annihilates elements of other factors. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  P  =  ( GdProj S )   &    |-  ( ph  ->  X  e.  I )   &    |-  ( ph  ->  A  e.  ( S `  X ) )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  Y  e.  I )   &    |-  ( ph  ->  Y  =/=  X )   =>    |-  ( ph  ->  (
 ( P `  Y ) `  A )  =  .0.  )
 
Theoremdpjghm 15225 The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  P  =  ( GdProj S )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( P `  X )  e.  ( ( Gs  ( G DProd  S ) ) 
 GrpHom  G ) )
 
Theoremdpjghm2 15226 The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  P  =  ( GdProj S )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( P `  X )  e.  ( ( Gs  ( G DProd  S ) ) 
 GrpHom  ( Gs  ( S `  X ) ) ) )
 
10.3.5  The Fundamental Theorem of Abelian Groups
 
Theoremablfacrplem 15227* Lemma for ablfacrp2 15229. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  K  =  { x  e.  B  |  ( O `
  x )  ||  M }   &    |-  L  =  { x  e.  B  |  ( O `  x ) 
 ||  N }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  ( M  gcd  N )  =  1 )   &    |-  ( ph  ->  ( # `  B )  =  ( M  x.  N ) )   =>    |-  ( ph  ->  ( ( # `  K )  gcd  N )  =  1 )
 
Theoremablfacrp 15228* A finite abelian group whose order factors into relatively prime integers, itself "factors" into two subgroups  K ,  L that have trivial intersection and whose product is the whole group. Lemma 6.1C.2 of [Shapiro], p. 199. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  K  =  { x  e.  B  |  ( O `
  x )  ||  M }   &    |-  L  =  { x  e.  B  |  ( O `  x ) 
 ||  N }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  ( M  gcd  N )  =  1 )   &    |-  ( ph  ->  ( # `  B )  =  ( M  x.  N ) )   &    |-  .0.  =  ( 0g `  G )   &    |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ph  ->  (
 ( K  i^i  L )  =  {  .0.  } 
 /\  ( K  .(+)  L )  =  B ) )
 
Theoremablfacrp2 15229* The factors  K ,  L of ablfacrp 15228 have the expected orders (which allows for repeated application to decompose  G into subgroups of prime-power order). Lemma 6.1C.2 of [Shapiro], p. 199. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  K  =  { x  e.  B  |  ( O `
  x )  ||  M }   &    |-  L  =  { x  e.  B  |  ( O `  x ) 
 ||  N }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  ( M  gcd  N )  =  1 )   &    |-  ( ph  ->  ( # `  B )  =  ( M  x.  N ) )   =>    |-  ( ph  ->  ( ( # `  K )  =  M  /\  ( # `  L )  =  N ) )
 
Theoremablfac1lem 15230* Lemma for ablfac1b 15232. Satisfy the assumptions of ablfacrp. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  ( p ^
 ( p  pCnt  ( # `
  B ) ) ) } )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  A 
 C_  Prime )   &    |-  M  =  ( P ^ ( P 
 pCnt  ( # `  B ) ) )   &    |-  N  =  ( ( # `  B )  /  M )   =>    |-  ( ( ph  /\  P  e.  A ) 
 ->  ( ( M  e.  NN  /\  N  e.  NN )  /\  ( M  gcd  N )  =  1  /\  ( # `  B )  =  ( M  x.  N ) ) )
 
Theoremablfac1a 15231* The factors of ablfac1b 15232 are of prime power order. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  ( p ^
 ( p  pCnt  ( # `
  B ) ) ) } )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  A 
 C_  Prime )   =>    |-  ( ( ph  /\  P  e.  A )  ->  ( # `
  ( S `  P ) )  =  ( P ^ ( P  pCnt  ( # `  B ) ) ) )
 
Theoremablfac1b 15232* Any abelian group is the direct product of factors of prime power order (with the exact order further matching the prime factorization of the group order). (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  ( p ^
 ( p  pCnt  ( # `
  B ) ) ) } )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  A 
 C_  Prime )   =>    |-  ( ph  ->  G dom DProd  S )
 
Theoremablfac1c 15233* The factors of ablfac1b 15232 cover the entire group. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  ( p ^
 ( p  pCnt  ( # `
  B ) ) ) } )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  A 
 C_  Prime )   &    |-  D  =  { w  e.  Prime  |  w  ||  ( # `  B ) }   &    |-  ( ph  ->  D 
 C_  A )   =>    |-  ( ph  ->  ( G DProd  S )  =  B )
 
Theoremablfac1eulem 15234* Lemma for ablfac1eu 15235. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  ( p ^
 ( p  pCnt  ( # `
  B ) ) ) } )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  A 
 C_  Prime )   &    |-  D  =  { w  e.  Prime  |  w  ||  ( # `  B ) }   &    |-  ( ph  ->  D 
 C_  A )   &    |-  ( ph  ->  ( G dom DProd  T 
 /\  ( G DProd  T )  =  B )
 )   &    |-  ( ph  ->  dom  T  =  A )   &    |-  ( ( ph  /\  q  e.  A ) 
 ->  C  e.  NN0 )   &    |-  (
 ( ph  /\  q  e.  A )  ->  ( # `
  ( T `  q ) )  =  ( q ^ C ) )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  A  e.  Fin )   =>    |-  ( ph  ->  -.  P  ||  ( # `  ( G DProd  ( T  |`  ( A 
 \  { P }
 ) ) ) ) )
 
Theoremablfac1eu 15235* The factorization of ablfac1b 15232 is unique, in that any other factorization into prime power factors (even if the exponents are different) must be equal to 
S. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  ( p ^
 ( p  pCnt  ( # `
  B ) ) ) } )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  A 
 C_  Prime )   &    |-  D  =  { w  e.  Prime  |  w  ||  ( # `  B ) }   &    |-  ( ph  ->  D 
 C_  A )   &    |-  ( ph  ->  ( G dom DProd  T 
 /\  ( G DProd  T )  =  B )
 )   &    |-  ( ph  ->  dom  T  =  A )   &    |-  ( ( ph  /\  q  e.  A ) 
 ->  C  e.  NN0 )   &    |-  (
 ( ph  /\  q  e.  A )  ->  ( # `
  ( T `  q ) )  =  ( q ^ C ) )   =>    |-  ( ph  ->  T  =  S )
 
Theorempgpfac1lem1 15236* Lemma for pgpfac1 15242. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  K  =  (mrCls `  (SubGrp `  G ) )   &    |-  S  =  ( K ` 
 { A } )   &    |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  E  =  (gEx `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .(+) 
 =  ( LSSum `  G )   &    |-  ( ph  ->  P pGrp  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  ( O `  A )  =  E )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  W  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  ( S  i^i  W )  =  {  .0.  }
 )   &    |-  ( ph  ->  ( S  .(+)  W )  C_  U )   &    |-  ( ph  ->  A. w  e.  (SubGrp `  G ) ( ( w 
 C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W )  C.  w ) )   =>    |-  ( ( ph  /\  C  e.  ( U 
 \  ( S  .(+)  W ) ) )  ->  ( ( S  .(+)  W )  .(+)  ( K ` 
 { C } )
 )  =  U )
 
Theorempgpfac1lem2 15237* Lemma for pgpfac1 15242. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  K  =  (mrCls `  (SubGrp `  G ) )   &    |-  S  =  ( K ` 
 { A } )   &    |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  E  =  (gEx `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .(+) 
 =  ( LSSum `  G )   &    |-  ( ph  ->  P pGrp  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  ( O `  A )  =  E )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  W  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  ( S  i^i  W )  =  {  .0.  }
 )   &    |-  ( ph  ->  ( S  .(+)  W )  C_  U )   &    |-  ( ph  ->  A. w  e.  (SubGrp `  G ) ( ( w 
 C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W )  C.  w ) )   &    |-  ( ph  ->  C  e.  ( U  \  ( S  .(+)  W ) ) )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ph  ->  ( P  .x.  C )  e.  ( S  .(+)  W ) )
 
Theorempgpfac1lem3a 15238* Lemma for pgpfac1 15242. (Contributed by Mario Carneiro, 4-Jun-2016.)
 |-  K  =  (mrCls `  (SubGrp `  G ) )   &    |-  S  =  ( K ` 
 { A } )   &    |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  E  =  (gEx `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .(+) 
 =  ( LSSum `  G )   &    |-  ( ph  ->  P pGrp  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  ( O `  A )  =  E )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  W  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  ( S  i^i  W )  =  {  .0.  }
 )   &    |-  ( ph  ->  ( S  .(+)  W )  C_  U )   &    |-  ( ph  ->  A. w  e.  (SubGrp `  G ) ( ( w 
 C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W )  C.  w ) )   &    |-  ( ph  ->  C  e.  ( U  \  ( S  .(+)  W ) ) )   &    |-  .x.  =  (.g `  G )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  ( ( P 
 .x.  C ) ( +g  `  G ) ( M 
 .x.  A ) )  e.  W )   =>    |-  ( ph  ->  ( P  ||  E  /\  P  ||  M ) )
 
Theorempgpfac1lem3 15239* Lemma for pgpfac1 15242. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  K  =  (mrCls `  (SubGrp `  G ) )   &    |-  S  =  ( K ` 
 { A } )   &    |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  E  =  (gEx `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .(+) 
 =  ( LSSum `  G )   &    |-  ( ph  ->  P pGrp  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  ( O `  A )  =  E )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  W  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  ( S  i^i  W )  =  {  .0.  }
 )   &    |-  ( ph  ->  ( S  .(+)  W )  C_  U )   &    |-  ( ph  ->  A. w  e.  (SubGrp `  G ) ( ( w 
 C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W )  C.  w ) )   &    |-  ( ph  ->  C  e.  ( U  \  ( S  .(+)  W ) ) )   &    |-  .x.  =  (.g `  G )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  ( ( P 
 .x.  C ) ( +g  `  G ) ( M 
 .x.  A ) )  e.  W )   &    |-  D  =  ( C ( +g  `  G ) ( ( M 
 /  P )  .x.  A ) )   =>    |-  ( ph  ->  E. t  e.  (SubGrp `  G )
 ( ( S  i^i  t )  =  {  .0.  }  /\  ( S 
 .(+)  t )  =  U ) )
 
Theorempgpfac1lem4 15240* Lemma for pgpfac1 15242. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  K  =  (mrCls `  (SubGrp `  G ) )   &    |-  S  =  ( K ` 
 { A } )   &    |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  E  =  (gEx `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .(+) 
 =  ( LSSum `  G )   &    |-  ( ph  ->  P pGrp  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  ( O `  A )  =  E )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  W  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  ( S  i^i  W )  =  {  .0.  }
 )   &    |-  ( ph  ->  ( S  .(+)  W )  C_  U )   &    |-  ( ph  ->  A. w  e.  (SubGrp `  G ) ( ( w 
 C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W )  C.  w ) )   &    |-  ( ph  ->  C  e.  ( U  \  ( S  .(+)  W ) ) )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ph  ->  E. t  e.  (SubGrp `  G )
 ( ( S  i^i  t )  =  {  .0.  }  /\  ( S 
 .(+)  t )  =  U ) )
 
Theorempgpfac1lem5 15241* Lemma for pgpfac1 15242 (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  K  =  (mrCls `  (SubGrp `  G ) )   &    |-  S  =  ( K ` 
 { A } )   &    |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  E  =  (gEx `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .(+) 
 =  ( LSSum `  G )   &    |-  ( ph  ->  P pGrp  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  ( O `  A )  =  E )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  A. s  e.  (SubGrp `  G ) ( ( s 
 C.  U  /\  A  e.  s )  ->  E. t  e.  (SubGrp `  G )
 ( ( S  i^i  t )  =  {  .0.  }  /\  ( S 
 .(+)  t )  =  s ) ) )   =>    |-  ( ph  ->  E. t  e.  (SubGrp `  G ) ( ( S  i^i  t )  =  {  .0.  }  /\  ( S  .(+)  t )  =  U ) )
 
Theorempgpfac1 15242* Factorization of a finite abelian p-group. There is a direct product decomposition of any abelian group of prime-power order where one of the factors is cyclic and generated by an element of maximal order. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  K  =  (mrCls `  (SubGrp `  G ) )   &    |-  S  =  ( K ` 
 { A } )   &    |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  E  =  (gEx `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .(+) 
 =  ( LSSum `  G )   &    |-  ( ph  ->  P pGrp  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  ( O `  A )  =  E )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  E. t  e.  (SubGrp `  G )
 ( ( S  i^i  t )  =  {  .0.  }  /\  ( S 
 .(+)  t )  =  B ) )
 
Theorempgpfaclem1 15243* Lemma for pgpfac 15246. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  {
 r  e.  (SubGrp `  G )  |  ( Gs  r
 )  e.  (CycGrp  i^i  ran pGrp  ) }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  P pGrp 
 G )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  U  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  A. t  e.  (SubGrp `  G ) ( t 
 C.  U  ->  E. s  e. Word  C ( G dom DProd  s 
 /\  ( G DProd  s
 )  =  t ) ) )   &    |-  H  =  ( Gs  U )   &    |-  K  =  (mrCls `  (SubGrp `  H )
 )   &    |-  O  =  ( od
 `  H )   &    |-  E  =  (gEx `  H )   &    |-  .0.  =  ( 0g `  H )   &    |-  .(+)  =  ( LSSum `  H )   &    |-  ( ph  ->  E  =/=  1 )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  ( O `  X )  =  E )   &    |-  ( ph  ->  W  e.  (SubGrp `  H )
 )   &    |-  ( ph  ->  (
 ( K `  { X } )  i^i  W )  =  {  .0.  }
 )   &    |-  ( ph  ->  (
 ( K `  { X } )  .(+)  W )  =  U )   &    |-  ( ph  ->  S  e. Word  C )   &    |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  ( G DProd  S )  =  W )   &    |-  T  =  ( S concat  <" ( K `
  { X }
 ) "> )   =>    |-  ( ph  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
 
Theorempgpfaclem2 15244* Lemma for pgpfac 15246. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  {
 r  e.  (SubGrp `  G )  |  ( Gs  r
 )  e.  (CycGrp  i^i  ran pGrp  ) }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  P pGrp 
 G )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  U  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  A. t  e.  (SubGrp `  G ) ( t 
 C.  U  ->  E. s  e. Word  C ( G dom DProd  s 
 /\  ( G DProd  s
 )  =  t ) ) )   &    |-  H  =  ( Gs  U )   &    |-  K  =  (mrCls `  (SubGrp `  H )
 )   &    |-  O  =  ( od
 `  H )   &    |-  E  =  (gEx `  H )   &    |-  .0.  =  ( 0g `  H )   &    |-  .(+)  =  ( LSSum `  H )   &    |-  ( ph  ->  E  =/=  1 )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  ( O `  X )  =  E )   &    |-  ( ph  ->  W  e.  (SubGrp `  H )
 )   &    |-  ( ph  ->  (
 ( K `  { X } )  i^i  W )  =  {  .0.  }
 )   &    |-  ( ph  ->  (
 ( K `  { X } )  .(+)  W )  =  U )   =>    |-  ( ph  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd 
 s )  =  U ) )
 
Theorempgpfaclem3 15245* Lemma for pgpfac 15246. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  {
 r  e.  (SubGrp `  G )  |  ( Gs  r
 )  e.  (CycGrp  i^i  ran pGrp  ) }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  P pGrp 
 G )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  U  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  A. t  e.  (SubGrp `  G ) ( t 
 C.  U  ->  E. s  e. Word  C ( G dom DProd  s 
 /\  ( G DProd  s
 )  =  t ) ) )   =>    |-  ( ph  ->  E. s  e. Word  C ( G dom DProd  s 
 /\  ( G DProd  s
 )  =  U ) )
 
Theorempgpfac 15246* Full factorization of a finite abelian p-group, by iterating pgpfac1 15242. There is a direct product decomposition of any abelian group of prime-power order into cyclic subgroups. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  {
 r  e.  (SubGrp `  G )  |  ( Gs  r
 )  e.  (CycGrp  i^i  ran pGrp  ) }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  P pGrp 
 G )   &    |-  ( ph  ->  B  e.  Fin )   =>    |-  ( ph  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd 
 s )  =  B ) )
 
Theoremablfaclem1 15247* Lemma for ablfac 15250. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  {
 r  e.  (SubGrp `  G )  |  ( Gs  r
 )  e.  (CycGrp  i^i  ran pGrp  ) }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  O  =  ( od `  G )   &    |-  A  =  { w  e.  Prime  |  w  ||  ( # `  B ) }   &    |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x ) 
 ||  ( p ^
 ( p  pCnt  ( # `
  B ) ) ) } )   &    |-  W  =  ( g  e.  (SubGrp `  G )  |->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd 
 s )  =  g ) } )   =>    |-  ( U  e.  (SubGrp `  G )  ->  ( W `  U )  =  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd 
 s )  =  U ) } )
 
Theoremablfaclem2 15248* Lemma for ablfac 15250. (Contributed by Mario Carneiro, 27-Apr-2016.) (Proof shortened by Mario Carneiro, 3-May-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  {
 r  e.  (SubGrp `  G )  |  ( Gs  r
 )  e.  (CycGrp  i^i  ran pGrp  ) }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  O  =  ( od `  G )   &    |-  A  =  { w  e.  Prime  |  w  ||  ( # `  B ) }   &    |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x ) 
 ||  ( p ^
 ( p  pCnt  ( # `
  B ) ) ) } )   &    |-  W  =  ( g  e.  (SubGrp `  G )  |->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd 
 s )  =  g ) } )   &    |-  ( ph  ->  F : A -->Word  C )   &    |-  ( ph  ->  A. y  e.  A  ( F `  y )  e.  ( W `  ( S `  y ) ) )   &    |-  L  =  U_ y  e.  A  ( { y }  X.  dom  (  F `  y
 ) )   &    |-  ( ph  ->  H : ( 0..^ ( # `  L ) ) -1-1-onto-> L )   =>    |-  ( ph  ->  ( W `  B )  =/=  (/) )
 
Theoremablfaclem3 15249* Lemma for ablfac 15250. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  {
 r  e.  (SubGrp `  G )  |  ( Gs  r
 )  e.  (CycGrp  i^i  ran pGrp  ) }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  O  =  ( od `  G )   &    |-  A  =  { w  e.  Prime  |  w  ||  ( # `  B ) }   &    |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x ) 
 ||  ( p ^
 ( p  pCnt  ( # `
  B ) ) ) } )   &    |-  W  =  ( g  e.  (SubGrp `  G )  |->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd 
 s )  =  g ) } )   =>    |-  ( ph  ->  ( W `  B )  =/=  (/) )
 
Theoremablfac 15250* The Fundamental Theorem of (finite) Abelian Groups. Any finite abelian group is a direct product of cyclic p-groups. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  {
 r  e.  (SubGrp `  G )  |  ( Gs  r
 )  e.  (CycGrp  i^i  ran pGrp  ) }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   =>    |-  ( ph  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd 
 s )  =  B ) )
 
Theoremablfac2 15251* Choose generators for each cyclic group in ablfac 15250. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  {
 r  e.  (SubGrp `  G )  |  ( Gs  r
 )  e.  (CycGrp  i^i  ran pGrp  ) }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  .x.  =  (.g `  G )   &    |-  S  =  ( k  e.  dom  w  |->  ran  (  n  e.  ZZ  |->  ( n  .x.  ( w `
  k ) ) ) )   =>    |-  ( ph  ->  E. w  e. Word  B ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd  S )  =  B ) )
 
10.4  Rings
 
10.4.1  Multiplicative Group
 
Syntaxcmgp 15252 Multiplicative group.
 class mulGrp
 
Definitiondf-mgp 15253 Define a structure that puts the multiplication operation of a ring in the addition slot. Note that this will not actually be a group for the average ring, or even for a field, but it will be a monoid, and unitgrp 15376 shows that we get a group if we restrict to the elements that have inverses. This allows us to formalize such notions as "the multiplication operation of a ring is a monoid" (rngmgp 15274) or "the multiplicative identity" in terms of the identity of a monoid (df-1r 8620). (Contributed by Mario Carneiro, 21-Dec-2014.)
 |- mulGrp  =  ( w  e.  _V  |->  ( w sSet  <. ( +g  ` 
 ndx ) ,  ( .r `  w ) >. ) )
 
Theoremfnmgp 15254 The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |- mulGrp  Fn  _V
 
Theoremmgpval 15255 Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.)
 |-  M  =  (mulGrp `  R )   &    |- 
 .x.  =  ( .r `  R )   =>    |-  M  =  ( R sSet  <. ( +g  `  ndx ) ,  .x.  >. )
 
Theoremmgpplusg 15256 Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.)
 |-  M  =  (mulGrp `  R )   &    |- 
 .x.  =  ( .r `  R )   =>    |- 
 .x.  =  ( +g  `  M )
 
Theoremmgplem 15257 Lemma for mgpbas 15258. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  N  =/=  2   =>    |-  ( E `  R )  =  ( E `  M )
 
Theoremmgpbas 15258 Base set of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  B  =  ( Base `  R )   =>    |-  B  =  ( Base `  M )
 
Theoremmgpsca 15259 The multiplication monoid has the same (if any) scalars as the original ring. Mostly to simplify pwsmgp 15328. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  S  =  (Scalar `  R )   =>    |-  S  =  (Scalar `  M )
 
Theoremmgptset 15260 Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   =>    |-  (TopSet `  R )  =  (TopSet `  M )
 
Theoremmgptopn 15261 Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  J  =  ( TopOpen `  R )   =>    |-  J  =  ( TopOpen `  M )
 
Theoremmgpds 15262 Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  B  =  ( dist `  R )   =>    |-  B  =  ( dist `  M )
 
Theoremmgpress 15263 Subgroup commutes with the multiplication group operator. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  S  =  ( Rs  A )   &    |-  M  =  (mulGrp `  R )   =>    |-  ( ( R  e.  V  /\  A  e.  W )  ->  ( Ms  A )  =  (mulGrp `  S ) )
 
10.4.2  Definition and basic properties
 
Syntaxcrg 15264 Extend class notation with class of all (unital) rings.
 class  Ring
 
Syntaxccrg 15265 Extend class notation with class of all (unital) commutative rings.
 class  CRing
 
Syntaxcur 15266 Extend class notation with ring unit.
 class  1r
 
Definitiondf-ring 15267* Define class of all (unital) rings. A unital ring is a set equipped with two everywhere-defined internal operations, whose first one is an additive group structure and the second one is a multiplicative monoid structure, and where the addition is left- and right-distributive for the multiplication. So that the additive structure must be abelian (see rngcom 15296), care must be taken that in the case of a non-unital ring, the commutativity of addition must be postulated and cannot be proved from the other conditions. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |- 
 Ring  =  { f  e.  Grp  |  ( (mulGrp `  f )  e.  Mnd  /\  [. ( Base `  f )  /  r ]. [. ( +g  `  f )  /  p ]. [. ( .r
 `  f )  /  t ]. A. x  e.  r  A. y  e.  r  A. z  e.  r  ( ( x t ( y p z ) )  =  ( ( x t y ) p ( x t z ) )  /\  ( ( x p y ) t z )  =  ( ( x t z ) p ( y t z ) ) ) ) }
 
Definitiondf-cring 15268 Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |- 
 CRing  =  { f  e.  Ring  |  (mulGrp `  f
 )  e. CMnd }
 
Definitiondf-ur 15269 Define the multiplicative neutral element of a ring. This definition works by extracting the  0g element, i.e. the neutral element in a group or monoid, and transfering it to the multiplicative monoid via the mulGrp function (df-mgp 15253). See also dfur2 15271, which derives the "traditional" definition as the unique element of a ring which is left- and right-neutral under multiplication. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |- 
 1r  =  ( 0g 
 o. mulGrp )
 
Theoremrngidval 15270 The value of the unity element of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  G  =  (mulGrp `  R )   &    |- 
 .1.  =  ( 1r `  R )   =>    |- 
 .1.  =  ( 0g `  G )
 
Theoremdfur2 15271* The multiplicative identity is the unique element of the ring that is left- and right-neutral on all elements under multiplication. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |- 
 .1.  =  ( iota e ( e  e.  B  /\  A. x  e.  B  ( ( e 
 .x.  x )  =  x  /\  ( x 
 .x.  e )  =  x ) ) )
 
Theoremisrng 15272* The predicate "is a (unital) ring." Definition of ring with unit in [Schechter] p. 187. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  R )   &    |-  G  =  (mulGrp `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( R  e.  Ring  <->  ( R  e.  Grp  /\  G  e.  Mnd  /\  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( ( x 
 .x.  ( y  .+  z ) )  =  ( ( x  .x.  y )  .+  ( x 
 .x.  z ) ) 
 /\  ( ( x 
 .+  y )  .x.  z )  =  (
 ( x  .x.  z
 )  .+  ( y  .x.  z ) ) ) ) )
 
Theoremrnggrp 15273 A ring is a group. (Contributed by NM, 15-Sep-2011.)
 |-  ( R  e.  Ring  ->  R  e.  Grp )
 
Theoremrngmgp 15274 A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  G  =  (mulGrp `  R )   =>    |-  ( R  e.  Ring  ->  G  e.  Mnd )
 
Theoremiscrng 15275 A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  G  =  (mulGrp `  R )   =>    |-  ( R  e.  CRing  <->  ( R  e.  Ring  /\  G  e. CMnd ) )
 
Theoremcrngmgp 15276 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  G  =  (mulGrp `  R )   =>    |-  ( R  e.  CRing  ->  G  e. CMnd )
 
Theoremrngmnd 15277 A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( R  e.  Ring  ->  R  e.  Mnd )
 
Theoremcrngrng 15278 A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( R  e.  CRing  ->  R  e.  Ring )
 
Theoremmgpf 15279 Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (mulGrp  |`  Ring ) : Ring --> Mnd
 
Theoremrngi 15280 Properties of a unital ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .x.  ( Y  .+  Z ) )  =  ( ( X  .x.  Y )  .+  ( X 
 .x.  Z ) )  /\  ( ( X  .+  Y )  .x.  Z )  =  ( ( X 
 .x.  Z )  .+  ( Y  .x.  Z ) ) ) )
 
Theoremrngcl 15281 Closure of the multiplication operation of a ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  Y )  e.  B )
 
Theoremcrngcom 15282 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  CRing  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  Y )  =  ( Y 
 .x.  X ) )
 
Theoremiscrng2 15283* A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( R  e.  CRing  <->  ( R  e.  Ring  /\  A. x  e.  B  A. y  e.  B  ( x  .x.  y )  =  (
 y  .x.  x )
 ) )
 
Theoremrngass 15284 Associative law for the multiplication operation of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .x.  Y )  .x.  Z )  =  ( X  .x.  ( Y  .x.  Z ) ) )
 
Theoremrngideu 15285* The unit element of a ring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( R  e.  Ring 
 ->  E! u  e.  B  A. x  e.  B  ( ( u  .x.  x )  =  x  /\  ( x  .x.  u )  =  x ) )
 
Theoremrngdi 15286 Distributive law for the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( X  .x.  ( Y  .+  Z ) )  =  (
 ( X  .x.  Y )  .+  ( X  .x.  Z ) ) )
 
Theoremrngdir 15287 Distributive law for the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .+  Y )  .x.  Z )  =  ( ( X  .x.  Z )  .+  ( Y  .x.  Z ) ) )
 
Theoremrngidcl 15288 The unit element of a ring belongs to the base set of the ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  .1. 
 e.  B )
 
Theoremrng0cl 15289 The zero element of a ring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  .0. 
 e.  B )
 
Theoremrngidmlem 15290 Lemma for rnglidm 15291 and rngridm 15292. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  ( (  .1.  .x.  X )  =  X  /\  ( X  .x.  .1.  )  =  X ) )
 
Theoremrnglidm 15291 The unit element of a ring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  (  .1.  .x.  X )  =  X )
 
Theoremrngridm 15292 The unit element of a ring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  ( X  .x.  .1.  )  =  X )
 
Theoremisrngid 15293* Properties showing that an element 
I is the unity element of a ring. (Contributed by NM, 7-Aug-2013.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  ( ( I  e.  B  /\  A. x  e.  B  ( ( I 
 .x.  x )  =  x  /\  ( x 
 .x.  I )  =  x ) )  <->  .1.  =  I ) )
 
Theoremrngidss 15294 A subset of the multiplicative group has the multiplicative identity as its identity if the identity is in the subset. (Contributed by Mario Carneiro, 27-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  M  =  ( (mulGrp `  R )s  A )   &    |-  B  =  (
 Base `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  .1.  =  ( 0g `  M ) )
 
Theoremrngacl 15295 Closure of the addition operation of a ring. (Contributed by Mario Carneiro, 14-Jan-2014.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  e.  B )
 
Theoremrngcom 15296 Commutativity of the additive group of a ring. (See also lmodcom 15598.) (Contributed by Gérard Lang, 4-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
 
Theoremrngabl 15297 A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.)
 |-  ( R  e.  Ring  ->  R  e.  Abel )
 
Theoremrngcmn 15298 A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( R  e.  Ring  ->  R  e. CMnd )
 
Theoremrngpropd 15299* If two structures have the same group components (properties), one is a ring iff the other one is. (Contributed by Mario Carneiro, 6-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-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  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )   =>    |-  ( ph  ->  ( K  e.  Ring  <->  L  e.  Ring )
 )
 
Theoremcrngpropd 15300* If two structures have the same group components (properties), one is a commutative ring iff the other one is. (Contributed by Mario Carneiro, 8-Feb-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  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )   =>    |-  ( ph  ->  ( K  e.  CRing  <->  L  e.  CRing ) )
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