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Theorem List for Intuitionistic Logic Explorer - 13301-13400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsrglidm 13301 The unity element of a semiring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011.) (Revised by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e. SRing  /\  X  e.  B ) 
 ->  (  .1.  .x.  X )  =  X )
 
Theoremsrgridm 13302 The unity element of a semiring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.) (Revised by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e. SRing  /\  X  e.  B ) 
 ->  ( X  .x.  .1.  )  =  X )
 
Theoremissrgid 13303* Properties showing that an element 
I is the unity element of a semiring. (Contributed by NM, 7-Aug-2013.) (Revised by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e. SRing  ->  ( ( I  e.  B  /\  A. x  e.  B  ( ( I  .x.  x )  =  x  /\  ( x  .x.  I
 )  =  x ) )  <->  .1.  =  I ) )
 
Theoremsrgacl 13304 Closure of the addition operation of a semiring. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   =>    |-  ( ( R  e. SRing  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .+  Y )  e.  B )
 
Theoremsrgcom 13305 Commutativity of the additive group of a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   =>    |-  ( ( R  e. SRing  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .+  Y )  =  ( Y  .+  X ) )
 
Theoremsrgrz 13306 The zero of a semiring is a right-absorbing element. (Contributed by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. SRing  /\  X  e.  B ) 
 ->  ( X  .x.  .0.  )  =  .0.  )
 
Theoremsrglz 13307 The zero of a semiring is a left-absorbing element. (Contributed by AV, 23-Aug-2019.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. SRing  /\  X  e.  B ) 
 ->  (  .0.  .x.  X )  =  .0.  )
 
Theoremsrgisid 13308* In a semiring, the only left-absorbing element is the additive identity. Remark in [Golan] p. 1. (Contributed by Thierry Arnoux, 1-May-2018.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  ( ph  ->  R  e. SRing )   &    |-  ( ph  ->  Z  e.  B )   &    |-  (
 ( ph  /\  x  e.  B )  ->  ( Z  .x.  x )  =  Z )   =>    |-  ( ph  ->  Z  =  .0.  )
 
Theoremsrg1zr 13309 The only semiring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .*  =  ( .r `  R )   =>    |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B )  /\  .* 
 Fn  ( B  X.  B ) )  /\  Z  e.  B )  ->  ( B  =  { Z }  <->  (  .+  =  { <.
 <. Z ,  Z >. ,  Z >. }  /\  .*  =  { <. <. Z ,  Z >. ,  Z >. } )
 ) )
 
Theoremsrgen1zr 13310 The only semiring with one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 14-Feb-2010.) (Revised by AV, 25-Jan-2020.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .*  =  ( .r `  R )   &    |-  Z  =  ( 0g
 `  R )   =>    |-  ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B )  /\  .* 
 Fn  ( B  X.  B ) )  ->  ( B  ~~  1o  <->  (  .+  =  { <.
 <. Z ,  Z >. ,  Z >. }  /\  .*  =  { <. <. Z ,  Z >. ,  Z >. } )
 ) )
 
Theoremsrgmulgass 13311 An associative property between group multiple and ring multiplication for semirings. (Contributed by AV, 23-Aug-2019.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  (.g `  R )   &    |-  .X.  =  ( .r `  R )   =>    |-  ( ( R  e. SRing  /\  ( N  e.  NN0  /\  X  e.  B  /\  Y  e.  B )
 )  ->  ( ( N  .x.  X )  .X.  Y )  =  ( N 
 .x.  ( X  .X.  Y ) ) )
 
Theoremsrgpcomp 13312 If two elements of a semiring commute, they also commute if one of the elements is raised to a higher power. (Contributed by AV, 23-Aug-2019.)
 |-  S  =  ( Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  G  =  (mulGrp `  R )   &    |-  .^  =  (.g `  G )   &    |-  ( ph  ->  R  e. SRing )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  K  e.  NN0 )   &    |-  ( ph  ->  ( A  .X.  B )  =  ( B  .X.  A ) )   =>    |-  ( ph  ->  (
 ( K  .^  B )  .X.  A )  =  ( A  .X.  ( K  .^  B ) ) )
 
Theoremsrgpcompp 13313 If two elements of a semiring commute, they also commute if the elements are raised to a higher power. (Contributed by AV, 23-Aug-2019.)
 |-  S  =  ( Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  G  =  (mulGrp `  R )   &    |-  .^  =  (.g `  G )   &    |-  ( ph  ->  R  e. SRing )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  K  e.  NN0 )   &    |-  ( ph  ->  ( A  .X.  B )  =  ( B  .X.  A ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( ( N  .^  A )  .X.  ( K 
 .^  B ) ) 
 .X.  A )  =  ( ( ( N  +  1 )  .^  A ) 
 .X.  ( K  .^  B ) ) )
 
Theoremsrgpcomppsc 13314 If two elements of a semiring commute, they also commute if the elements are raised to a higher power and a scalar multiplication is involved. (Contributed by AV, 23-Aug-2019.)
 |-  S  =  ( Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  G  =  (mulGrp `  R )   &    |-  .^  =  (.g `  G )   &    |-  ( ph  ->  R  e. SRing )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  K  e.  NN0 )   &    |-  ( ph  ->  ( A  .X.  B )  =  ( B  .X.  A ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  .x.  =  (.g `  R )   &    |-  ( ph  ->  C  e.  NN0 )   =>    |-  ( ph  ->  (
 ( C  .x.  (
 ( N  .^  A )  .X.  ( K  .^  B ) ) ) 
 .X.  A )  =  ( C  .x.  ( (
 ( N  +  1 )  .^  A )  .X.  ( K  .^  B ) ) ) )
 
Theoremsrglmhm 13315* Left-multiplication in a semiring by a fixed element of the ring is a monoid homomorphism. (Contributed by AV, 23-Aug-2019.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e. SRing  /\  X  e.  B )  ->  ( x  e.  B  |->  ( X  .x.  x ) )  e.  ( R MndHom  R )
 )
 
Theoremsrgrmhm 13316* Right-multiplication in a semiring by a fixed element of the ring is a monoid homomorphism. (Contributed by AV, 23-Aug-2019.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e. SRing  /\  X  e.  B )  ->  ( x  e.  B  |->  ( x  .x.  X ) )  e.  ( R MndHom  R ) )
 
Theoremsrg1expzeq1 13317 The exponentiation (by a nonnegative integer) of the multiplicative identity of a semiring, analogous to mulgnn0z 13062. (Contributed by AV, 25-Nov-2019.)
 |-  G  =  (mulGrp `  R )   &    |- 
 .x.  =  (.g `  G )   &    |- 
 .1.  =  ( 1r `  R )   =>    |-  ( ( R  e. SRing  /\  N  e.  NN0 )  ->  ( N  .x.  .1.  )  =  .1.  )
 
7.3.5  Definition and basic properties of unital rings
 
Syntaxcrg 13318 Extend class notation with class of all (unital) rings.
 class  Ring
 
Syntaxccrg 13319 Extend class notation with class of all (unital) commutative rings.
 class  CRing
 
Definitiondf-ring 13320* 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. Definition 1 in [BourbakiAlg1] p. 92 or definition of a ring with identity in part Preliminaries of [Roman] p. 19. So that the additive structure must be abelian (see ringcom 13353), 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 13321 Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |- 
 CRing  =  { f  e.  Ring  |  (mulGrp `  f
 )  e. CMnd }
 
Theoremisring 13322* 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 ) ) ) ) )
 
Theoremringgrp 13323 A ring is a group. (Contributed by NM, 15-Sep-2011.)
 |-  ( R  e.  Ring  ->  R  e.  Grp )
 
Theoremringmgp 13324 A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  G  =  (mulGrp `  R )   =>    |-  ( R  e.  Ring  ->  G  e.  Mnd )
 
Theoremiscrng 13325 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 13326 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  G  =  (mulGrp `  R )   =>    |-  ( R  e.  CRing  ->  G  e. CMnd )
 
Theoremringgrpd 13327 A ring is a group. (Contributed by SN, 16-May-2024.)
 |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  R  e.  Grp )
 
Theoremringmnd 13328 A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( R  e.  Ring  ->  R  e.  Mnd )
 
Theoremringmgm 13329 A ring is a magma. (Contributed by AV, 31-Jan-2020.)
 |-  ( R  e.  Ring  ->  R  e. Mgm )
 
Theoremcrngring 13330 A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( R  e.  CRing  ->  R  e.  Ring )
 
Theoremcrngringd 13331 A commutative ring is a ring. (Contributed by SN, 16-May-2024.)
 |-  ( ph  ->  R  e.  CRing )   =>    |-  ( ph  ->  R  e.  Ring )
 
Theoremcrnggrpd 13332 A commutative ring is a group. (Contributed by SN, 16-May-2024.)
 |-  ( ph  ->  R  e.  CRing )   =>    |-  ( ph  ->  R  e.  Grp )
 
Theoremmgpf 13333 Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (mulGrp  |`  Ring ) : Ring --> Mnd
 
Theoremringdilem 13334 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 ) ) ) )
 
Theoremringcl 13335 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 13336 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 13337* 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 )
 ) )
 
Theoremringass 13338 Associative law for multiplication in 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 ) ) )
 
Theoremringideu 13339* The unity 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 ) )
 
Theoremringdi 13340 Distributive law for the multiplication operation of a ring (left-distributivity). (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 ) ) )
 
Theoremringdir 13341 Distributive law for the multiplication operation of a ring (right-distributivity). (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 ) ) )
 
Theoremringidcl 13342 The unity 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 )
 
Theoremring0cl 13343 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 )
 
Theoremringidmlem 13344 Lemma for ringlidm 13345 and ringridm 13346. (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 ) )
 
Theoremringlidm 13345 The unity 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 )
 
Theoremringridm 13346 The unity 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 )
 
Theoremisringid 13347* 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 ) )
 
Theoremringid 13348* The multiplication operation of a unital ring has (one or more) identity elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 24-Aug-2021.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  E. u  e.  B  ( ( u 
 .x.  X )  =  X  /\  ( X  .x.  u )  =  X )
 )
 
Theoremringadd2 13349* A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 24-Aug-2021.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  E. x  e.  B  ( X  .+  X )  =  ( ( x 
 .+  x )  .x.  X ) )
 
Theoremringo2times 13350 A ring element plus itself is two times the element. "Two" in an arbitrary unital ring is the sum of the unity element with itself. (Contributed by AV, 24-Aug-2021.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  A  e.  B ) 
 ->  ( A  .+  A )  =  ( (  .1.  .+  .1.  )  .x.  A ) )
 
Theoremringidss 13351 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 ) )
 
Theoremringacl 13352 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 )
 
Theoremringcom 13353 Commutativity of the additive group of a ring. (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 ) )
 
Theoremringabl 13354 A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.)
 |-  ( R  e.  Ring  ->  R  e.  Abel )
 
Theoremringcmn 13355 A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( R  e.  Ring  ->  R  e. CMnd )
 
Theoremringabld 13356 A ring is an Abelian group. (Contributed by SN, 1-Jun-2024.)
 |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  R  e.  Abel )
 
Theoremringcmnd 13357 A ring is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
 |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  R  e. CMnd )
 
Theoremringrng 13358 A unital ring is a non-unital ring. (Contributed by AV, 6-Jan-2020.)
 |-  ( R  e.  Ring  ->  R  e. Rng )
 
Theoremringssrng 13359 The unital rings are non-unital rings. (Contributed by AV, 20-Mar-2020.)
 |- 
 Ring  C_ Rng
 
Theoremringpropd 13360* 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 13361* 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 ) )
 
Theoremringprop 13362 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 )
 
Theoremisringd 13363* 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 13364* 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
 )
 
Theoremringlz 13365 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.  )
 
Theoremringrz 13366 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.  )
 
Theoremringsrg 13367 Any ring is also a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)
 |-  ( R  e.  Ring  ->  R  e. SRing )
 
Theoremring1eq0 13368 If one and zero are equal, then any two elements of a ring are equal. Alternately, 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 ) )
 
Theoremringinvnz1ne0 13369* In a unital ring, a left invertible element is different from zero iff  .1.  =/=  .0.. (Contributed by FL, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |- 
 .0.  =  ( 0g `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  E. a  e.  B  ( a  .x.  X )  =  .1.  )   =>    |-  ( ph  ->  ( X  =/=  .0.  <->  .1.  =/=  .0.  )
 )
 
Theoremringinvnzdiv 13370* In a unital ring, a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.) (Revised by Jeff Madsen, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |- 
 .0.  =  ( 0g `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  E. a  e.  B  ( a  .x.  X )  =  .1.  )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( X  .x.  Y )  =  .0.  <->  Y  =  .0.  ) )
 
Theoremringnegl 13371 Negation in a ring is the same as left multiplication by -1. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  N  =  ( invg `  R )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  (
 ( N `  .1.  )  .x.  X )  =  ( N `  X ) )
 
Theoremringnegr 13372 Negation in a ring is the same as right multiplication by -1. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  N  =  ( invg `  R )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( X  .x.  ( N `  .1.  ) )  =  ( N `  X ) )
 
Theoremringmneg1 13373 Negation of a product in a ring. (mulneg1 8372 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  N  =  ( invg `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( N `  X )  .x.  Y )  =  ( N `  ( X  .x.  Y ) ) )
 
Theoremringmneg2 13374 Negation of a product in a ring. (mulneg2 8373 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  N  =  ( invg `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .x.  ( N `  Y ) )  =  ( N `  ( X  .x.  Y ) ) )
 
Theoremringm2neg 13375 Double negation of a product in a ring. (mul2neg 8375 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  N  =  ( invg `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( N `  X )  .x.  ( N `  Y ) )  =  ( X  .x.  Y ) )
 
Theoremringsubdi 13376 Ring multiplication distributes over subtraction. (subdi 8362 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 ) ) )
 
Theoremringsubdir 13377 Ring multiplication distributes over subtraction. (subdir 8363 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 13378 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 ) ) )
 
Theoremring1 13379 The (smallest) structure representing a zero ring. (Contributed by AV, 28-Apr-2019.)
 |-  M  =  { <. (
 Base `  ndx ) ,  { Z } >. , 
 <. ( +g  `  ndx ) ,  { <. <. Z ,  Z >. ,  Z >. }
 >. ,  <. ( .r `  ndx ) ,  { <. <. Z ,  Z >. ,  Z >. } >. }   =>    |-  ( Z  e.  V  ->  M  e.  Ring )
 
Theoremringn0 13380 The class of rings is not empty (it is also inhabited, as shown at ring1 13379). (Contributed by AV, 29-Apr-2019.)
 |- 
 Ring  =/=  (/)
 
Theoremringressid 13381 A ring restricted to its base set is a ring. It will usually be the original ring exactly, of course, but to show that needs additional conditions such as those in strressid 12556. (Contributed by Jim Kingdon, 28-Feb-2025.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e.  Ring  ->  ( G ↾s  B )  e.  Ring )
 
Theoremimasring 13382* 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 ) ) )
 
Theoremimasringf1 13383 The image of a ring under an injection is a ring. (Contributed by AV, 27-Feb-2025.)
 |-  U  =  ( F 
 "s 
 R )   &    |-  V  =  (
 Base `  R )   =>    |-  ( ( F : V -1-1-> B  /\  R  e.  Ring )  ->  U  e.  Ring )
 
Theoremqusring2 13384* 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 ) ) )
 
7.3.6  Opposite ring
 
Syntaxcoppr 13385 The opposite ring operation.
 class oppr
 
Definitiondf-oppr 13386 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 ) >. ) )
 
Theoremopprvalg 13387 Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  O  =  (oppr `  R )   =>    |-  ( R  e.  V  ->  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. ) )
 
Theoremopprmulfvalg 13388 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 )   =>    |-  ( R  e.  V  ->  .xb  = tpos  .x.  )
 
Theoremopprmulg 13389 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 )   =>    |-  ( ( R  e.  V  /\  X  e.  W  /\  Y  e.  U )  ->  ( X 
 .xb  Y )  =  ( Y  .x.  X )
 )
 
Theoremcrngoppr 13390 In a commutative ring, the opposite ring is equivalent to the original ring. (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 ) )
 
Theoremopprex 13391 Existence of the opposite ring. If you know that  R is a ring, see opprring 13397. (Contributed by Jim Kingdon, 10-Jan-2025.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e.  V  ->  O  e.  _V )
 
Theoremopprsllem 13392 Lemma for opprbasg 13393 and oppraddg 13394. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.)
 |-  O  =  (oppr `  R )   &    |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   &    |-  ( E `  ndx )  =/=  ( .r `  ndx )   =>    |-  ( R  e.  V  ->  ( E `  R )  =  ( E `  O ) )
 
Theoremopprbasg 13393 Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
 |-  O  =  (oppr `  R )   &    |-  B  =  ( Base `  R )   =>    |-  ( R  e.  V  ->  B  =  ( Base `  O ) )
 
Theoremoppraddg 13394 Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
 |-  O  =  (oppr `  R )   &    |- 
 .+  =  ( +g  `  R )   =>    |-  ( R  e.  V  ->  .+  =  ( +g  `  O ) )
 
Theoremopprrng 13395 An opposite non-unital ring is a non-unital ring. (Contributed by AV, 15-Feb-2025.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e. Rng  ->  O  e. Rng )
 
Theoremopprrngbg 13396 A set is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 13395. (Contributed by AV, 15-Feb-2025.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e.  V  ->  ( R  e. Rng  <->  O  e. Rng ) )
 
Theoremopprring 13397 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 )
 
Theoremopprringbg 13398 Bidirectional form of opprring 13397. (Contributed by Mario Carneiro, 6-Dec-2014.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e.  V  ->  ( R  e.  Ring  <->  O  e.  Ring ) )
 
Theoremoppr0g 13399 Additive identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  O  =  (oppr `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( R  e.  V  ->  .0.  =  ( 0g
 `  O ) )
 
Theoremoppr1g 13400 Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  O  =  (oppr `  R )   &    |- 
 .1.  =  ( 1r `  R )   =>    |-  ( R  e.  V  ->  .1.  =  ( 1r
 `  O ) )
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