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Theorem List for Metamath Proof Explorer - 15301-15400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremablfac1eulem 15301* Lemma for ablfac1eu 15302. (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 15302* The factorization of ablfac1b 15299 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 15303* Lemma for pgpfac1 15309. (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 15304* Lemma for pgpfac1 15309. (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 15305* Lemma for pgpfac1 15309. (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 15306* Lemma for pgpfac1 15309. (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 15307* Lemma for pgpfac1 15309. (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 15308* Lemma for pgpfac1 15309 (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 15309* 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 15310* Lemma for pgpfac 15313. (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 15311* Lemma for pgpfac 15313. (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 15312* Lemma for pgpfac 15313. (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 15313* Full factorization of a finite abelian p-group, by iterating pgpfac1 15309. 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 15314* Lemma for ablfac 15317. (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 15315* Lemma for ablfac 15317. (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 15316* Lemma for ablfac 15317. (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 15317* 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 15318* Choose generators for each cyclic group in ablfac 15317. (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 15319 Multiplicative group.
 class mulGrp
 
Definitiondf-mgp 15320 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 15443 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 15341) or "the multiplicative identity" in terms of the identity of a monoid (df-1r 8682). (Contributed by Mario Carneiro, 21-Dec-2014.)
 |- mulGrp  =  ( w  e.  _V  |->  ( w sSet  <. ( +g  ` 
 ndx ) ,  ( .r `  w ) >. ) )
 
Theoremfnmgp 15321 The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |- mulGrp  Fn  _V
 
Theoremmgpval 15322 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 15323 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 15324 Lemma for mgpbas 15325. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  N  =/=  2   =>    |-  ( E `  R )  =  ( E `  M )
 
Theoremmgpbas 15325 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 15326 The multiplication monoid has the same (if any) scalars as the original ring. Mostly to simplify pwsmgp 15395. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  S  =  (Scalar `  R )   =>    |-  S  =  (Scalar `  M )
 
Theoremmgptset 15327 Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   =>    |-  (TopSet `  R )  =  (TopSet `  M )
 
Theoremmgptopn 15328 Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  J  =  ( TopOpen `  R )   =>    |-  J  =  ( TopOpen `  M )
 
Theoremmgpds 15329 Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  B  =  ( dist `  R )   =>    |-  B  =  ( dist `  M )
 
Theoremmgpress 15330 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 15331 Extend class notation with class of all (unital) rings.
 class  Ring
 
Syntaxccrg 15332 Extend class notation with class of all (unital) commutative rings.
 class  CRing
 
Syntaxcur 15333 Extend class notation with ring unit.
 class  1r
 
Definitiondf-rng 15334* 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 15363), 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 15335 Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |- 
 CRing  =  { f  e.  Ring  |  (mulGrp `  f
 )  e. CMnd }
 
Definitiondf-ur 15336 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 15320). See also dfur2 15338, 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 15337 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 15338* 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 15339* 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 15340 A ring is a group. (Contributed by NM, 15-Sep-2011.)
 |-  ( R  e.  Ring  ->  R  e.  Grp )
 
Theoremrngmgp 15341 A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  G  =  (mulGrp `  R )   =>    |-  ( R  e.  Ring  ->  G  e.  Mnd )
 
Theoremiscrng 15342 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 15343 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 15344 A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( R  e.  Ring  ->  R  e.  Mnd )
 
Theoremcrngrng 15345 A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( R  e.  CRing  ->  R  e.  Ring )
 
Theoremmgpf 15346 Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (mulGrp  |`  Ring ) : Ring --> Mnd
 
Theoremrngi 15347 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 15348 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 15349 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 15350* 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 15351 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 15352* 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 15353 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 15354 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 15355 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 15356 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 15357 Lemma for rnglidm 15358 and rngridm 15359. (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 15358 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 15359 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 15360* 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 15361 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 15362 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 15363 Commutativity of the additive group of a ring. (See also lmodcom 15665.) (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 15364 A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.)
 |-  ( R  e.  Ring  ->  R  e.  Abel )
 
Theoremrngcmn 15365 A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( R  e.  Ring  ->  R  e. CMnd )
 
Theoremrngpropd 15366* 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 15367* 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 15368 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 15369* 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 15370* 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 15371 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 15372 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 15373 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 15374 Negation in a ring is the same as left multiplication by -1. (rngonegmn1l 25979 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 15375 Negation in a ring is the same as right multiplication by -1. (rngonegmn1r 25980 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 15376 Negation of a product in a ring. (mulneg1 9211 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 15377 Negation of a product in a ring. (mulneg2 9212 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 15378 Double negation of a product in a ring. (mul2neg 9214 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 15379 Ring multiplication distributes over subtraction. (subdi 9208 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 15380 Ring multiplication distributes over subtraction. (subdir 9209 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 15381 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 15382* 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 15383* 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 15384* 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 15385* 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 15386* 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 15387 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 15388 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 15389 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 15390 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 15391 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 15392 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 15393 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 15394 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 )
 
Theorempwsmgp 15395 The multiplicative group of the power structure resembles the power of the multiplicative group. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  M  =  (mulGrp `  R )   &    |-  Z  =  ( M  ^s  I )   &    |-  N  =  (mulGrp `  Y )   &    |-  B  =  (
 Base `  N )   &    |-  C  =  ( Base `  Z )   &    |-  .+  =  ( +g  `  N )   &    |-  .+b  =  ( +g  `  Z )   =>    |-  (
 ( R  e.  V  /\  I  e.  W )  ->  ( B  =  C  /\  .+  =  .+b  )
 )
 
Theoremimasrng 15396* The image structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |- 
 .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ( ph  /\  ( a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `
  a )  =  ( F `  p )  /\  ( F `  b )  =  ( F `  q ) ) 
 ->  ( F `  (
 a  .+  b )
 )  =  ( F `
  ( p  .+  q ) ) ) )   &    |-  ( ( ph  /\  ( a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `
  a )  =  ( F `  p )  /\  ( F `  b )  =  ( F `  q ) ) 
 ->  ( F `  (
 a  .x.  b )
 )  =  ( F `
  ( p  .x.  q ) ) ) )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  ( U  e.  Ring  /\  ( F `  .1.  )  =  ( 1r `  U ) ) )
 
Theoremdivsrng2 15397* The quotient structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  ( ph  ->  U  =  ( R  /.s  .~  ) )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |- 
 .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  .~  Er  V )   &    |-  ( ph  ->  ( ( a  .~  p  /\  b  .~  q ) 
 ->  ( a  .+  b
 )  .~  ( p  .+  q ) ) )   &    |-  ( ph  ->  ( (
 a  .~  p  /\  b  .~  q )  ->  ( a  .x.  b ) 
 .~  ( p  .x.  q ) ) )   &    |-  ( ph  ->  R  e.  Ring
 )   =>    |-  ( ph  ->  ( U  e.  Ring  /\  [  .1.  ]  .~  =  ( 1r `  U ) ) )
 
10.4.3  Opposite ring
 
Syntaxcoppr 15398 The opposite ring operation.
 class oppr
 
Definitiondf-oppr 15399 Define an opposite ring, which is the same as the original ring but with multiplication written the other way around. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |- oppr  =  ( f  e.  _V  |->  ( f sSet  <. ( .r
 `  ndx ) , tpos  ( .r `  f ) >. ) )
 
Theoremopprval 15400 Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  O  =  (oppr `  R )   =>    |-  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
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