P. 136 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13501-13600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rngabl 13501	A non-unital ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)
|- ( R e. Rng -> R e. Abel )
Theorem	rngmgp 13502	A non-unital ring is a semigroup under multiplication. (Contributed by AV, 17-Feb-2020.)
|- G = (mulGrp ` R )   =>    |- ( R e. Rng -> G e. Smgrp )
Theorem	rngmgpf 13503	Restricted functionality of the multiplicative group on non-unital rings (mgpf 13577 analog). (Contributed by AV, 22-Feb-2025.)
|- (mulGrp |` Rng ) :Rng -->Smgrp
Theorem	rnggrp 13504	A non-unital ring is a (additive) group. (Contributed by AV, 16-Feb-2025.)
|- ( R e. Rng -> R e. Grp )
Theorem	rngass 13505	Associative law for the multiplication operation of a non-unital ring. (Contributed by NM, 27-Aug-2011.) (Revised by AV, 13-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) )
Theorem	rngdi 13506	Distributive law for the multiplication operation of a non-unital ring (left-distributivity). (Contributed by AV, 14-Feb-2025.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) )
Theorem	rngdir 13507	Distributive law for the multiplication operation of a non-unital ring (right-distributivity). (Contributed by AV, 17-Apr-2020.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .x. Z ) = ( ( X .x. Z ) .+ ( Y .x. Z ) ) )
Theorem	rngacl 13508	Closure of the addition operation of a non-unital ring. (Contributed by AV, 16-Feb-2025.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   =>    |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )
Theorem	rng0cl 13509	The zero element of a non-unital ring belongs to its base set. (Contributed by AV, 16-Feb-2025.)
|- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. Rng -> .0. e. B )
Theorem	rngcl 13510	Closure of the multiplication operation of a non-unital ring. (Contributed by AV, 17-Apr-2020.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
Theorem	rnglz 13511	The zero of a non-unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringlz 13609. (Revised by AV, 17-Apr-2020.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Rng /\ X e. B ) -> ( .0. .x. X ) = .0. )
Theorem	rngrz 13512	The zero of a non-unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringrz 13610. (Revised by AV, 16-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Rng /\ X e. B ) -> ( X .x. .0. ) = .0. )
Theorem	rngmneg1 13513	Negation of a product in a non-unital ring (mulneg1 8423 analog). In contrast to ringmneg1 13619, the proof does not (and cannot) make use of the existence of a ring unity. (Contributed by AV, 17-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- N = ( invg ` R )   &    |- ( ph -> R e. Rng )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( ( N ` X ) .x. Y ) = ( N ` ( X .x. Y ) ) )
Theorem	rngmneg2 13514	Negation of a product in a non-unital ring (mulneg2 8424 analog). In contrast to ringmneg2 13620, the proof does not (and cannot) make use of the existence of a ring unity. (Contributed by AV, 17-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- N = ( invg ` R )   &    |- ( ph -> R e. Rng )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .x. ( N ` Y ) ) = ( N ` ( X .x. Y ) ) )
Theorem	rngm2neg 13515	Double negation of a product in a non-unital ring (mul2neg 8426 analog). (Contributed by Mario Carneiro, 4-Dec-2014.) Generalization of ringm2neg 13621. (Revised by AV, 17-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- N = ( invg ` R )   &    |- ( ph -> R e. Rng )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( ( N ` X ) .x. ( N ` Y ) ) = ( X .x. Y ) )
Theorem	rngansg 13516	Every additive subgroup of a non-unital ring is normal. (Contributed by AV, 25-Feb-2025.)
|- ( R e. Rng -> (NrmSGrp ` R ) = (SubGrp ` R ) )
Theorem	rngsubdi 13517	Ring multiplication distributes over subtraction. (subdi 8413 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdi 13622. (Revised by AV, 23-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .- = ( -g ` R )   &    |- ( ph -> R e. Rng )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( X .x. ( Y .- Z ) ) = ( ( X .x. Y ) .- ( X .x. Z ) ) )
Theorem	rngsubdir 13518	Ring multiplication distributes over subtraction. (subdir 8414 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdir 13623. (Revised by AV, 23-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .- = ( -g ` R )   &    |- ( ph -> R e. Rng )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( ( X .- Y ) .x. Z ) = ( ( X .x. Z ) .- ( Y .x. Z ) ) )
Theorem	isrngd 13519*	Properties that determine a non-unital ring. (Contributed by AV, 14-Feb-2025.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> .+ = ( +g ` R ) )   &    |- ( ph -> .x. = ( .r ` R ) )   &    |- ( ph -> R e. Abel )   &    |- ( ( 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 -> R e. Rng )
Theorem	rngressid 13520	A non-unital ring restricted to its base set is a non-unital ring. It will usually be the original non-unital ring exactly, of course, but to show that needs additional conditions such as those in strressid 12759. (Contributed by Jim Kingdon, 5-May-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Rng -> ( G ↾s B ) e. Rng )
Theorem	rngpropd 13521*	If two structures have the same base set, and the values of their group (addition) and ring (multiplication) operations are equal for all pairs of elements of the base set, one is a non-unital ring iff the other one is. (Contributed by AV, 15-Feb-2025.)
|- ( 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. Rng <-> L e. Rng ) )
Theorem	imasrng 13522*	The image structure of a non-unital ring is a non-unital ring (imasring 13630 analog). (Contributed by AV, 22-Feb-2025.)
|- ( ph -> U = ( F "s R ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` 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. Rng )   =>    |- ( ph -> U e. Rng )
Theorem	imasrngf1 13523	The image of a non-unital ring under an injection is a non-unital ring. (Contributed by AV, 22-Feb-2025.)
|- U = ( F "s R )   &    |- V = ( Base ` R )   =>    |- ( ( F : V -1-1-> B /\ R e. Rng ) -> U e. Rng )
Theorem	qusrng 13524*	The quotient structure of a non-unital ring is a non-unital ring (qusring2 13632 analog). (Contributed by AV, 23-Feb-2025.)
|- ( ph -> U = ( R /.s .~ ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` 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. Rng )   =>    |- ( ph -> U e. Rng )
7.3.3  Ring unity (multiplicative identity) In Wikipedia "Identity element", see https://en.wikipedia.org/wiki/Identity_element (18-Jan-2025): "... an identity with respect to multiplication is called a multiplicative identity (often denoted as 1). ... The distinction between additive and multiplicative identity is used most often for sets that support both binary operations, such as rings, integral domains, and fields. The multiplicative identity is often called unity in the latter context (a ring with unity). This should not be confused with a unit in ring theory, which is any element having a multiplicative inverse. By its own definition, unity itself is necessarily a unit." Calling the multiplicative identity of a ring a unity is taken from the definition of a ring with unity in section 17.3 of [BeauregardFraleigh] p. 135, "A ring ( R , + , . ) is a ring with unity if R is not the zero ring and ( R , . ) is a monoid. In this case, the identity element of ( R , . ) is denoted by 1 and is called the unity of R." This definition of a "ring with unity" corresponds to our definition of a unital ring (see df-ring 13564). Some authors call the multiplicative identity "unit" or "unit element" (for example in section I, 2.2 of [BourbakiAlg1] p. 14, definition in section 1.3 of [Hall] p. 4, or in section I, 1 of [Lang] p. 3), whereas other authors use the term "unit" for an element having a multiplicative inverse (for example in section 17.3 of [BeauregardFraleigh] p. 135, in definition in [Roman] p. 26, or even in section II, 1 of [Lang] p. 84). Sometimes, the multiplicative identity is simply called "one" (see, for example, chapter 8 in [Schechter] p. 180). To avoid this ambiguity of the term "unit", also mentioned in Wikipedia, we call the multiplicative identity of a structure with a multiplication (usually a ring) a "ring unity", or straightly "multiplicative identity". The term "unit" will be used for an element having a multiplicative inverse (see https://us.metamath.org/mpeuni/df-unit.html 13564 in set.mm), and we have "the ring unity is a unit", see https://us.metamath.org/mpeuni/1unit.html 13564.
Syntax	cur 13525	Extend class notation with ring unity.
class 1r
Definition	df-ur 13526	Define the multiplicative identity, i.e., the monoid identity (df-0g 12939) of the multiplicative monoid (df-mgp 13487) of a ring-like structure. This multiplicative identity is also called "ring unity" or "unity element". This definition works by transferring the multiplicative operation from the .r slot to the +g slot and then looking at the element which is then the 0g element, that is an identity with respect to the operation which started out in the .r slot. See also dfur2g 13528, 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 )
Theorem	ringidvalg 13527	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 )   =>    |- ( R e. V -> .1. = ( 0g ` G ) )
Theorem	dfur2g 13528*	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 )   =>    |- ( R e. V -> .1. = ( iota e ( e e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) ) )
7.3.4  Semirings
Syntax	csrg 13529	Extend class notation with the class of all semirings.
class SRing
Definition	df-srg 13530*	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 13531*	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 13532	A semiring is a commutative monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
|- ( R e. SRing -> R e. CMnd )
Theorem	srgmnd 13533	A semiring is a monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
|- ( R e. SRing -> R e. Mnd )
Theorem	srgmgp 13534	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 13535	Lemma for srgdi 13540 and srgdir 13541. (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 13536	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 13537	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 13538*	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 13539	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 13540	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 13541	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 13542	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 13543	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 13544	Lemma for srglidm 13545 and srgridm 13546. (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 13545	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 13546	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 13547*	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 13548	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 13549	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 13550	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 13551	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 13552*	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 13553	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 13554	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 13555	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 13556	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 13557	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 13558	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 13559*	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 13560*	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 13561	The exponentiation (by a nonnegative integer) of the multiplicative identity of a semiring, analogous to mulgnn0z 13289. (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 13562	Extend class notation with class of all (unital) rings.
class Ring
Syntax	ccrg 13563	Extend class notation with class of all (unital) commutative rings.
class CRing
Definition	df-ring 13564*	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 13597), 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 13565	Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.)
|- CRing = { f e. Ring | (mulGrp ` f ) e. CMnd }
Theorem	isring 13566*	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 13567	A ring is a group. (Contributed by NM, 15-Sep-2011.)
|- ( R e. Ring -> R e. Grp )
Theorem	ringmgp 13568	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 13569	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 13570	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 13571	A ring is a group. (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. Ring )   =>    |- ( ph -> R e. Grp )
Theorem	ringmnd 13572	A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.)
|- ( R e. Ring -> R e. Mnd )
Theorem	ringmgm 13573	A ring is a magma. (Contributed by AV, 31-Jan-2020.)
|- ( R e. Ring -> R e. Mgm )
Theorem	crngring 13574	A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
|- ( R e. CRing -> R e. Ring )
Theorem	crngringd 13575	A commutative ring is a ring. (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. CRing )   =>    |- ( ph -> R e. Ring )
Theorem	crnggrpd 13576	A commutative ring is a group. (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. CRing )   =>    |- ( ph -> R e. Grp )
Theorem	mgpf 13577	Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.)
|- (mulGrp |` Ring ) : Ring --> Mnd
Theorem	ringdilem 13578	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 13579	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 13580	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 13581*	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 13582	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 13583*	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 13584	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 13585	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 13586	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 13587	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 13588	Lemma for ringlidm 13589 and ringridm 13590. (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 13589	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 13590	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 13591*	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 13592*	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 13593*	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 13594	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 13595	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 13596	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 13597	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 13598	A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.)
|- ( R e. Ring -> R e. Abel )
Theorem	ringcmn 13599	A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
|- ( R e. Ring -> R e. CMnd )
Theorem	ringabld 13600	A ring is an Abelian group. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> R e. Ring )   =>    |- ( ph -> R e. Abel )