P. 139 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13801-13900   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-srg 13801*	Define class of all semirings. A semiring is a set equipped with two everywhere-defined internal operations, whose first one is an additive commutative monoid structure and the second one is a multiplicative monoid structure, and where multiplication is (left- and right-) distributive over addition. Like with rings, the additive identity is an absorbing element of the multiplicative law, but in the case of semirings, this has to be part of the definition, as it cannot be deduced from distributivity alone. Definition of [Golan] p. 1. Note that our semirings are unital. Such semirings are sometimes called "rigs", being "rings without negatives". (Contributed by Thierry Arnoux, 21-Mar-2018.)
|- SRing = { f e. CMnd | ( (mulGrp ` f ) e. Mnd /\ [. ( Base ` f ) / r ]. [. ( +g ` f ) / p ]. [. ( .r ` f ) / t ]. [. ( 0g ` f ) / n ]. 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 ) ) ) /\ ( ( n t x ) = n /\ ( x t n ) = n ) ) ) }
Theorem	issrg 13802*	The predicate "is a semiring". (Contributed by Thierry Arnoux, 21-Mar-2018.)
|- B = ( Base ` R )   &    |- G = (mulGrp ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. SRing <-> ( R e. CMnd /\ 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 ) ) ) /\ ( ( .0. .x. x ) = .0. /\ ( x .x. .0. ) = .0. ) ) ) )
Theorem	srgcmn 13803	A semiring is a commutative monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
|- ( R e. SRing -> R e. CMnd )
Theorem	srgmnd 13804	A semiring is a monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
|- ( R e. SRing -> R e. Mnd )
Theorem	srgmgp 13805	A semiring is a monoid under multiplication. (Contributed by Thierry Arnoux, 21-Mar-2018.)
|- G = (mulGrp ` R )   =>    |- ( R e. SRing -> G e. Mnd )
Theorem	srgdilem 13806	Lemma for srgdi 13811 and srgdir 13812. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. SRing /\ ( 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 ) ) ) )
Theorem	srgcl 13807	Closure of the multiplication operation of a semiring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. SRing /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
Theorem	srgass 13808	Associative law for the multiplication operation of a semiring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. SRing /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) )
Theorem	srgideu 13809*	The unity element of a semiring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. SRing -> E! u e. B A. x e. B ( ( u .x. x ) = x /\ ( x .x. u ) = x ) )
Theorem	srgfcl 13810	Functionality of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by AV, 24-Aug-2021.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. SRing /\ .x. Fn ( B X. B ) ) -> .x. : ( B X. B ) --> B )
Theorem	srgdi 13811	Distributive law for the multiplication operation of a semiring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Thierry Arnoux, 1-Apr-2018.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. SRing /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) )
Theorem	srgdir 13812	Distributive law for the multiplication operation of a semiring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Thierry Arnoux, 1-Apr-2018.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. SRing /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .x. Z ) = ( ( X .x. Z ) .+ ( Y .x. Z ) ) )
Theorem	srgidcl 13813	The unity element of a semiring belongs to the base set of the semiring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
|- B = ( Base ` R )   &    |- .1. = ( 1r ` R )   =>    |- ( R e. SRing -> .1. e. B )
Theorem	srg0cl 13814	The zero element of a semiring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
|- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. SRing -> .0. e. B )
Theorem	srgidmlem 13815	Lemma for srglidm 13816 and srgridm 13817. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) (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 /\ ( X .x. .1. ) = X ) )
Theorem	srglidm 13816	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 )
Theorem	srgridm 13817	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 )
Theorem	issrgid 13818*	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 ) )
Theorem	srgacl 13819	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 )
Theorem	srgcom 13820	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 ) )
Theorem	srgrz 13821	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. )
Theorem	srglz 13822	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. )
Theorem	srgisid 13823*	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. )
Theorem	srg1zr 13824	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 >. } ) ) )
Theorem	srgen1zr 13825	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 >. } ) ) )
Theorem	srgmulgass 13826	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 ) ) )
Theorem	srgpcomp 13827	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 ) ) )
Theorem	srgpcompp 13828	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 ) ) )
Theorem	srgpcomppsc 13829	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 ) ) ) )
Theorem	srglmhm 13830*	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 ) )
Theorem	srgrmhm 13831*	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 ) )
Theorem	srg1expzeq1 13832	The exponentiation (by a nonnegative integer) of the multiplicative identity of a semiring, analogous to mulgnn0z 13560. (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
Syntax	crg 13833	Extend class notation with class of all (unital) rings.
class Ring
Syntax	ccrg 13834	Extend class notation with class of all (unital) commutative rings.
class CRing
Definition	df-ring 13835*	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 13868), 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 ) ) ) ) }
Definition	df-cring 13836	Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.)
|- CRing = { f e. Ring | (mulGrp ` f ) e. CMnd }
Theorem	isring 13837*	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 ) ) ) ) )
Theorem	ringgrp 13838	A ring is a group. (Contributed by NM, 15-Sep-2011.)
|- ( R e. Ring -> R e. Grp )
Theorem	ringmgp 13839	A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.)
|- G = (mulGrp ` R )   =>    |- ( R e. Ring -> G e. Mnd )
Theorem	iscrng 13840	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 ) )
Theorem	crngmgp 13841	A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
|- G = (mulGrp ` R )   =>    |- ( R e. CRing -> G e. CMnd )
Theorem	ringgrpd 13842	A ring is a group. (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. Ring )   =>    |- ( ph -> R e. Grp )
Theorem	ringmnd 13843	A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.)
|- ( R e. Ring -> R e. Mnd )
Theorem	ringmgm 13844	A ring is a magma. (Contributed by AV, 31-Jan-2020.)
|- ( R e. Ring -> R e. Mgm )
Theorem	crngring 13845	A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
|- ( R e. CRing -> R e. Ring )
Theorem	crngringd 13846	A commutative ring is a ring. (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. CRing )   =>    |- ( ph -> R e. Ring )
Theorem	crnggrpd 13847	A commutative ring is a group. (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. CRing )   =>    |- ( ph -> R e. Grp )
Theorem	mgpf 13848	Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.)
|- (mulGrp |` Ring ) : Ring --> Mnd
Theorem	ringdilem 13849	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 ) ) ) )
Theorem	ringcl 13850	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 )
Theorem	crngcom 13851	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 ) )
Theorem	iscrng2 13852*	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 ) ) )
Theorem	ringass 13853	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 ) ) )
Theorem	ringideu 13854*	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 ) )
Theorem	ringdi 13855	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 ) ) )
Theorem	ringdir 13856	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 ) ) )
Theorem	ringidcl 13857	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 )
Theorem	ring0cl 13858	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 )
Theorem	ringidmlem 13859	Lemma for ringlidm 13860 and ringridm 13861. (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 ) )
Theorem	ringlidm 13860	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 )
Theorem	ringridm 13861	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 )
Theorem	isringid 13862*	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 ) )
Theorem	ringid 13863*	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 ) )
Theorem	ringadd2 13864*	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 ) )
Theorem	ringo2times 13865	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 ) )
Theorem	ringidss 13866	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 ) )
Theorem	ringacl 13867	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 )
Theorem	ringcom 13868	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 ) )
Theorem	ringabl 13869	A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.)
|- ( R e. Ring -> R e. Abel )
Theorem	ringcmn 13870	A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
|- ( R e. Ring -> R e. CMnd )
Theorem	ringabld 13871	A ring is an Abelian group. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> R e. Ring )   =>    |- ( ph -> R e. Abel )
Theorem	ringcmnd 13872	A ring is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> R e. Ring )   =>    |- ( ph -> R e. CMnd )
Theorem	ringrng 13873	A unital ring is a non-unital ring. (Contributed by AV, 6-Jan-2020.)
|- ( R e. Ring -> R e. Rng )
Theorem	ringssrng 13874	The unital rings are non-unital rings. (Contributed by AV, 20-Mar-2020.)
|- Ring C_ Rng
Theorem	ringpropd 13875*	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 ) )
Theorem	crngpropd 13876*	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 ) )
Theorem	ringprop 13877	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 )
Theorem	isringd 13878*	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 )
Theorem	iscrngd 13879*	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 )
Theorem	ringlz 13880	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. )
Theorem	ringrz 13881	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. )
Theorem	ringlzd 13882	The zero of a unital ring is a left-absorbing element. (Contributed by SN, 7-Mar-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( .0. .x. X ) = .0. )
Theorem	ringrzd 13883	The zero of a unital ring is a right-absorbing element. (Contributed by SN, 7-Mar-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( X .x. .0. ) = .0. )
Theorem	ringsrg 13884	Any ring is also a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)
|- ( R e. Ring -> R e. SRing )
Theorem	ring1eq0 13885	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 ) )
Theorem	ringinvnz1ne0 13886*	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. ) )
Theorem	ringinvnzdiv 13887*	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. ) )
Theorem	ringnegl 13888	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 ) )
Theorem	ringnegr 13889	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 ) )
Theorem	ringmneg1 13890	Negation of a product in a ring. (mulneg1 8487 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 ) ) )
Theorem	ringmneg2 13891	Negation of a product in a ring. (mulneg2 8488 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 ) ) )
Theorem	ringm2neg 13892	Double negation of a product in a ring. (mul2neg 8490 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 ) )
Theorem	ringsubdi 13893	Ring multiplication distributes over subtraction. (subdi 8477 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 ) ) )
Theorem	ringsubdir 13894	Ring multiplication distributes over subtraction. (subdir 8478 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 ) ) )
Theorem	mulgass2 13895	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 ) ) )
Theorem	ring1 13896	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 )
Theorem	ringn0 13897	The class of rings is not empty (it is also inhabited, as shown at ring1 13896). (Contributed by AV, 29-Apr-2019.)
|- Ring =/= (/)
Theorem	ringlghm 13898*	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 ) )
Theorem	ringrghm 13899*	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 ) )
Theorem	ringressid 13900	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 12978. (Contributed by Jim Kingdon, 28-Feb-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Ring -> ( G ↾s B ) e. Ring )