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Theorem List for Metamath Proof Explorer - 15601-15700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremablfac 15601* 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 15602* Choose generators for each cyclic group in ablfac 15601. (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 15603 Multiplicative group.
 class mulGrp
 
Definitiondf-mgp 15604 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 15727 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 15625) or "the multiplicative identity" in terms of the identity of a monoid (df-1r 8896). (Contributed by Mario Carneiro, 21-Dec-2014.)
 |- mulGrp  =  ( w  e.  _V  |->  ( w sSet  <. ( +g  ` 
 ndx ) ,  ( .r `  w ) >. ) )
 
Theoremfnmgp 15605 The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |- mulGrp  Fn  _V
 
Theoremmgpval 15606 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 15607 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 15608 Lemma for mgpbas 15609. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  N  =/=  2   =>    |-  ( E `  R )  =  ( E `  M )
 
Theoremmgpbas 15609 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 15610 The multiplication monoid has the same (if any) scalars as the original ring. Mostly to simplify pwsmgp 15679. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  S  =  (Scalar `  R )   =>    |-  S  =  (Scalar `  M )
 
Theoremmgptset 15611 Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   =>    |-  (TopSet `  R )  =  (TopSet `  M )
 
Theoremmgptopn 15612 Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  J  =  ( TopOpen `  R )   =>    |-  J  =  ( TopOpen `  M )
 
Theoremmgpds 15613 Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  B  =  ( dist `  R )   =>    |-  B  =  ( dist `  M )
 
Theoremmgpress 15614 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 15615 Extend class notation with class of all (unital) rings.
 class  Ring
 
Syntaxccrg 15616 Extend class notation with class of all (unital) commutative rings.
 class  CRing
 
Syntaxcur 15617 Extend class notation with ring unit.
 class  1r
 
Definitiondf-rng 15618* 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 15647), 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 15619 Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |- 
 CRing  =  { f  e.  Ring  |  (mulGrp `  f
 )  e. CMnd }
 
Definitiondf-ur 15620 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 15604). See also dfur2 15622, 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 15621 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 15622* 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 15623* 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 15624 A ring is a group. (Contributed by NM, 15-Sep-2011.)
 |-  ( R  e.  Ring  ->  R  e.  Grp )
 
Theoremrngmgp 15625 A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  G  =  (mulGrp `  R )   =>    |-  ( R  e.  Ring  ->  G  e.  Mnd )
 
Theoremiscrng 15626 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 15627 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 15628 A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( R  e.  Ring  ->  R  e.  Mnd )
 
Theoremcrngrng 15629 A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( R  e.  CRing  ->  R  e.  Ring )
 
Theoremmgpf 15630 Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (mulGrp  |`  Ring ) : Ring --> Mnd
 
Theoremrngi 15631 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 15632 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 15633 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 15634* 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 15635 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 15636* 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 15637 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 15638 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 15639 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 15640 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 15641 Lemma for rnglidm 15642 and rngridm 15643. (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 15642 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 15643 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 15644* 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 15645 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 15646 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 15647 Commutativity of the additive group of a ring. (See also lmodcom 15945.) (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 15648 A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.)
 |-  ( R  e.  Ring  ->  R  e.  Abel )
 
Theoremrngcmn 15649 A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( R  e.  Ring  ->  R  e. CMnd )
 
Theoremrngpropd 15650* 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 15651* 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 15652 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 15653* 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 15654* 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 15655 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 15656 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 15657 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 15658 Negation in a ring is the same as left multiplication by -1. (rngonegmn1l 26455 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 15659 Negation in a ring is the same as right multiplication by -1. (rngonegmn1r 26456 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 15660 Negation of a product in a ring. (mulneg1 9426 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 15661 Negation of a product in a ring. (mulneg2 9427 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 15662 Double negation of a product in a ring. (mul2neg 9429 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 15663 Ring multiplication distributes over subtraction. (subdi 9423 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 15664 Ring multiplication distributes over subtraction. (subdir 9424 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 15665 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 15666* 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 15667* 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 15668* 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 15669* 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 15670* 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 15671 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 15672 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 15673 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 15674 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 15675 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 15676 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 15677 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 15678 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 15679 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 15680* 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 15681* 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 15682 The opposite ring operation.
 class oppr
 
Definitiondf-oppr 15683 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 15684 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.  >. )
 
Theoremopprmulfval 15685 Value of the multiplication operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  O  =  (oppr `  R )   &    |-  .xb  =  ( .r `  O )   =>    |-  .xb  = tpos  .x.
 
Theoremopprmul 15686 Value of the multiplication operation of an opposite ring. Hypotheses eliminated by a suggestion of Stefan O'Rear, 30-Aug-2015. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  O  =  (oppr `  R )   &    |-  .xb  =  ( .r `  O )   =>    |-  ( X  .xb  Y )  =  ( Y 
 .x.  X )
 
Theoremcrngoppr 15687 In a commutative ring, the opposite ring is equivalent to the original ring (for theorems like unitpropd 15757). (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  O  =  (oppr `  R )   &    |-  .xb  =  ( .r `  O )   =>    |-  ( ( R  e.  CRing  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  Y )  =  ( X 
 .xb  Y ) )
 
Theoremopprlem 15688 Lemma for opprbas 15689 and oppradd 15690. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  O  =  (oppr `  R )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  N  <  3   =>    |-  ( E `  R )  =  ( E `  O )
 
Theoremopprbas 15689 Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  O  =  (oppr `  R )   &    |-  B  =  ( Base `  R )   =>    |-  B  =  ( Base `  O )
 
Theoremoppradd 15690 Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  O  =  (oppr `  R )   &    |- 
 .+  =  ( +g  `  R )   =>    |- 
 .+  =  ( +g  `  O )
 
Theoremopprrng 15691 An opposite ring is a ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e.  Ring  ->  O  e.  Ring )
 
Theoremopprrngb 15692 Bidirectional form of opprrng 15691. (Contributed by Mario Carneiro, 6-Dec-2014.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e.  Ring  <->  O  e.  Ring )
 
Theoremoppr0 15693 Additive identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  O  =  (oppr `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |- 
 .0.  =  ( 0g `  O )
 
Theoremoppr1 15694 Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  O  =  (oppr `  R )   &    |- 
 .1.  =  ( 1r `  R )   =>    |- 
 .1.  =  ( 1r `  O )
 
Theoremopprneg 15695 The negative function in an opposite ring. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  O  =  (oppr `  R )   &    |-  N  =  ( inv
 g `  R )   =>    |-  N  =  ( inv g `  O )
 
Theoremopprsubg 15696 Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
 |-  O  =  (oppr `  R )   =>    |-  (SubGrp `  R )  =  (SubGrp `  O )
 
Theoremmulgass3 15697 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 ) )  ->  ( X  .X.  ( N 
 .x.  Y ) )  =  ( N  .x.  ( X  .X.  Y ) ) )
 
10.4.4  Divisibility
 
Syntaxcdsr 15698 Ring divides relation.
 class  ||r
 
Syntaxcui 15699 Ring unit.
 class Unit
 
Syntaxcir 15700 Ring irreducibles.
 class Irred
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