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Theorem List for Metamath Proof Explorer - 15301-15400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremablfacrp 15301* 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 15302* The factors  K ,  L of ablfacrp 15301 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 15303* Lemma for ablfac1b 15305. 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 15304* The factors of ablfac1b 15305 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 15305* 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 15306* The factors of ablfac1b 15305 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 15307* Lemma for ablfac1eu 15308. (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 15308* The factorization of ablfac1b 15305 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 15309* Lemma for pgpfac1 15315. (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 15310* Lemma for pgpfac1 15315. (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 15311* Lemma for pgpfac1 15315. (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 15312* Lemma for pgpfac1 15315. (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 15313* Lemma for pgpfac1 15315. (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 15314* Lemma for pgpfac1 15315 (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 15315* 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 15316* Lemma for pgpfac 15319. (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 15317* Lemma for pgpfac 15319. (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 15318* Lemma for pgpfac 15319. (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 15319* Full factorization of a finite abelian p-group, by iterating pgpfac1 15315. 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 15320* Lemma for ablfac 15323. (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 15321* Lemma for ablfac 15323. (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 15322* Lemma for ablfac 15323. (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 15323* 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 15324* Choose generators for each cyclic group in ablfac 15323. (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 15325 Multiplicative group.
 class mulGrp
 
Definitiondf-mgp 15326 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 15449 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 15347) or "the multiplicative identity" in terms of the identity of a monoid (df-1r 8687). (Contributed by Mario Carneiro, 21-Dec-2014.)
 |- mulGrp  =  ( w  e.  _V  |->  ( w sSet  <. ( +g  ` 
 ndx ) ,  ( .r `  w ) >. ) )
 
Theoremfnmgp 15327 The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |- mulGrp  Fn  _V
 
Theoremmgpval 15328 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 15329 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 15330 Lemma for mgpbas 15331. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  N  =/=  2   =>    |-  ( E `  R )  =  ( E `  M )
 
Theoremmgpbas 15331 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 15332 The multiplication monoid has the same (if any) scalars as the original ring. Mostly to simplify pwsmgp 15401. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  S  =  (Scalar `  R )   =>    |-  S  =  (Scalar `  M )
 
Theoremmgptset 15333 Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   =>    |-  (TopSet `  R )  =  (TopSet `  M )
 
Theoremmgptopn 15334 Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  J  =  ( TopOpen `  R )   =>    |-  J  =  ( TopOpen `  M )
 
Theoremmgpds 15335 Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  B  =  ( dist `  R )   =>    |-  B  =  ( dist `  M )
 
Theoremmgpress 15336 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 15337 Extend class notation with class of all (unital) rings.
 class  Ring
 
Syntaxccrg 15338 Extend class notation with class of all (unital) commutative rings.
 class  CRing
 
Syntaxcur 15339 Extend class notation with ring unit.
 class  1r
 
Definitiondf-rng 15340* 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 15369), 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 15341 Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |- 
 CRing  =  { f  e.  Ring  |  (mulGrp `  f
 )  e. CMnd }
 
Definitiondf-ur 15342 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 15326). See also dfur2 15344, 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 15343 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 15344* 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 15345* 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 15346 A ring is a group. (Contributed by NM, 15-Sep-2011.)
 |-  ( R  e.  Ring  ->  R  e.  Grp )
 
Theoremrngmgp 15347 A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  G  =  (mulGrp `  R )   =>    |-  ( R  e.  Ring  ->  G  e.  Mnd )
 
Theoremiscrng 15348 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 15349 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 15350 A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( R  e.  Ring  ->  R  e.  Mnd )
 
Theoremcrngrng 15351 A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( R  e.  CRing  ->  R  e.  Ring )
 
Theoremmgpf 15352 Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (mulGrp  |`  Ring ) : Ring --> Mnd
 
Theoremrngi 15353 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 15354 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 15355 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 15356* 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 15357 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 15358* 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 15359 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 15360 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 15361 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 15362 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 15363 Lemma for rnglidm 15364 and rngridm 15365. (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 15364 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 15365 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 15366* 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 15367 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 15368 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 15369 Commutativity of the additive group of a ring. (See also lmodcom 15671.) (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 15370 A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.)
 |-  ( R  e.  Ring  ->  R  e.  Abel )
 
Theoremrngcmn 15371 A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( R  e.  Ring  ->  R  e. CMnd )
 
Theoremrngpropd 15372* 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 15373* 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 ) )
 
Theoremrngprop 15374 If two structures have the same ring components (properties), one is a ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.)
 |-  ( Base `  K )  =  ( Base `  L )   &    |-  ( +g  `  K )  =  ( +g  `  L )   &    |-  ( .r `  K )  =  ( .r `  L )   =>    |-  ( K  e.  Ring  <->  L  e.  Ring )
 
Theoremisrngd 15375* Properties that determine a ring. (Contributed by NM, 2-Aug-2013.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  R )
 )   &    |-  ( ph  ->  .x.  =  ( .r `  R ) )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  (
 ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .x.  y )  e.  B )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  ( ( x  .x.  y )  .x.  z )  =  ( x  .x.  ( y  .x.  z ) ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B )
 )  ->  ( x  .x.  ( y  .+  z
 ) )  =  ( ( x  .x.  y
 )  .+  ( x  .x.  z ) ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B )
 )  ->  ( ( x  .+  y )  .x.  z )  =  (
 ( x  .x.  z
 )  .+  ( y  .x.  z ) ) )   &    |-  ( ph  ->  .1.  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  (  .1.  .x.  x )  =  x )   &    |-  (
 ( ph  /\  x  e.  B )  ->  ( x  .x.  .1.  )  =  x )   =>    |-  ( ph  ->  R  e.  Ring )
 
Theoremiscrngd 15376* Properties that determine a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  R )
 )   &    |-  ( ph  ->  .x.  =  ( .r `  R ) )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  (
 ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .x.  y )  e.  B )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  ( ( x  .x.  y )  .x.  z )  =  ( x  .x.  ( y  .x.  z ) ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B )
 )  ->  ( x  .x.  ( y  .+  z
 ) )  =  ( ( x  .x.  y
 )  .+  ( x  .x.  z ) ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B )
 )  ->  ( ( x  .+  y )  .x.  z )  =  (
 ( x  .x.  z
 )  .+  ( y  .x.  z ) ) )   &    |-  ( ph  ->  .1.  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  (  .1.  .x.  x )  =  x )   &    |-  (
 ( ph  /\  x  e.  B )  ->  ( x  .x.  .1.  )  =  x )   &    |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .x.  y
 )  =  ( y 
 .x.  x ) )   =>    |-  ( ph  ->  R  e.  CRing
 )
 
Theoremrnglz 15377 The zero of a unital ring is a left absorbing element. (Contributed by FL, 31-Aug-2009.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  (  .0.  .x.  X )  =  .0.  )
 
Theoremrngrz 15378 The zero of a unital ring is a right absorbing element. (Contributed by FL, 31-Aug-2009.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  ( X  .x.  .0.  )  =  .0.  )
 
Theoremrng1eq0 15379 If one and zero are equal, then any two elements of a ring are equal. Alternatively, every ring has one distinct from zero except the zero ring containing the single element  { 0 }. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  B  =  ( Base `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (  .1.  =  .0. 
 ->  X  =  Y ) )
 
Theoremrngnegl 15380 Negation in a ring is the same as left multiplication by -1. (rngonegmn1l 26580 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  N  =  ( inv
 g `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  (
 ( N `  .1.  )  .x.  X )  =  ( N `  X ) )
 
Theoremrngnegr 15381 Negation in a ring is the same as right multiplication by -1. (rngonegmn1r 26581 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  N  =  ( inv
 g `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( X  .x.  ( N `  .1.  ) )  =  ( N `  X ) )
 
Theoremrngmneg1 15382 Negation of a product in a ring. (mulneg1 9216 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  N  =  ( inv g `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( N `  X )  .x.  Y )  =  ( N `  ( X  .x.  Y ) ) )
 
Theoremrngmneg2 15383 Negation of a product in a ring. (mulneg2 9217 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  N  =  ( inv g `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .x.  ( N `  Y ) )  =  ( N `  ( X  .x.  Y ) ) )
 
Theoremrngm2neg 15384 Double negation of a product in a ring. (mul2neg 9219 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  N  =  ( inv g `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( N `  X )  .x.  ( N `  Y ) )  =  ( X  .x.  Y ) )
 
Theoremrngsubdi 15385 Ring multiplication distributes over subtraction. (subdi 9213 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .-  =  ( -g `  R )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  ( X  .x.  ( Y  .-  Z ) )  =  ( ( X  .x.  Y )  .-  ( X  .x.  Z ) ) )
 
Theoremrngsubdir 15386 Ring multiplication distributes over subtraction. (subdir 9214 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .-  =  ( -g `  R )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 ( X  .-  Y )  .x.  Z )  =  ( ( X  .x.  Z )  .-  ( Y  .x.  Z ) ) )
 
Theoremmulgass2 15387 An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  (.g `  R )   &    |-  .X.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  ( ( N  .x.  X )  .X.  Y )  =  ( N  .x.  ( X  .X.  Y ) ) )
 
Theoremrnglghm 15388* Left-multiplication in a ring by a fixed element of the ring is a group homomorphism. (It is not usually a ring homomorphism.) (Contributed by Mario Carneiro, 4-May-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( x  e.  B  |->  ( X 
 .x.  x ) )  e.  ( R  GrpHom  R ) )
 
Theoremrngrghm 15389* Right-multiplication in a ring by a fixed element of the ring is a group homomorphism. (It is not usually a ring homomorphism.) (Contributed by Mario Carneiro, 4-May-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( x  e.  B  |->  ( x 
 .x.  X ) )  e.  ( R  GrpHom  R ) )
 
Theoremgsummulc1 15390* A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  .+  =  ( +g  `  R )   &    |- 
 .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ( ph  /\  k  e.  A )  ->  X  e.  B )   &    |-  ( ph  ->  ( `' ( k  e.  A  |->  X ) " ( _V  \  {  .0.  }
 ) )  e.  Fin )   =>    |-  ( ph  ->  ( R  gsumg  ( k  e.  A  |->  ( X  .x.  Y ) ) )  =  ( ( R  gsumg  ( k  e.  A  |->  X ) )  .x.  Y ) )
 
Theoremgsummulc2 15391* A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  .+  =  ( +g  `  R )   &    |- 
 .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ( ph  /\  k  e.  A )  ->  X  e.  B )   &    |-  ( ph  ->  ( `' ( k  e.  A  |->  X ) " ( _V  \  {  .0.  }
 ) )  e.  Fin )   =>    |-  ( ph  ->  ( R  gsumg  ( k  e.  A  |->  ( Y  .x.  X ) ) )  =  ( Y  .x.  ( R  gsumg  (
 k  e.  A  |->  X ) ) ) )
 
Theoremgsumdixp 15392* Distribute a binary product of sums to a sum of binary products in a ring. (Contributed by Mario Carneiro, 8-Mar-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  J  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ( ph  /\  x  e.  I )  ->  X  e.  B )   &    |-  ( ( ph  /\  y  e.  J ) 
 ->  Y  e.  B )   &    |-  ( ph  ->  ( `' ( x  e.  I  |->  X ) " ( _V  \  {  .0.  }
 ) )  e.  Fin )   &    |-  ( ph  ->  ( `' ( y  e.  J  |->  Y ) " ( _V  \  {  .0.  }
 ) )  e.  Fin )   =>    |-  ( ph  ->  (
 ( R  gsumg  ( x  e.  I  |->  X ) )  .x.  ( R  gsumg  ( y  e.  J  |->  Y ) ) )  =  ( R  gsumg  ( x  e.  I ,  y  e.  J  |->  ( X  .x.  Y ) ) ) )
 
Theoremprdsmgp 15393 The multiplicative monoid of a product is the product of the multiplicative monoids of the factors. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  M  =  (mulGrp `  Y )   &    |-  Z  =  ( S X_s (mulGrp  o.  R )
 )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  R  Fn  I
 )   =>    |-  ( ph  ->  (
 ( Base `  M )  =  ( Base `  Z )  /\  ( +g  `  M )  =  ( +g  `  Z ) ) )
 
Theoremprdsmulrcl 15394 A structure product of rings has closed binary operation. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  .x.  =  ( .r `  Y )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R : I --> Ring )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   =>    |-  ( ph  ->  ( F  .x.  G )  e.  B )
 
Theoremprdsrngd 15395 A product of rings is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I --> Ring )   =>    |-  ( ph  ->  Y  e.  Ring )
 
Theoremprdscrngd 15396 A product of commutative rings is a commutative ring. Since the resulting ring will have zero divisors in all nontrivial cases, this cannot be strengthened much further. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I --> CRing )   =>    |-  ( ph  ->  Y  e.  CRing )
 
Theoremprds1 15397 Value of the ring unit in a structure family product. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I --> Ring )   =>    |-  ( ph  ->  ( 1r  o.  R )  =  ( 1r `  Y ) )
 
Theorempwsrng 15398 A structure power of a ring is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  Y  =  ( R 
 ^s  I )   =>    |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  Y  e.  Ring )
 
Theorempws1 15399 Value of the ring unit in a structure power. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ( I  X.  {  .1.  } )  =  ( 1r `  Y ) )
 
Theorempwscrng 15400 A structure power of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  Y  =  ( R 
 ^s  I )   =>    |-  ( ( R  e.  CRing  /\  I  e.  V )  ->  Y  e.  CRing )
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