P. 132 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13101-13200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	iscmnd 13101*	Properties that determine a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
|- ( ph -> B = ( Base ` G ) )   &    |- ( ph -> .+ = ( +g ` G ) )   &    |- ( ph -> G e. Mnd )   &    |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) = ( y .+ x ) )   =>    |- ( ph -> G e. CMnd )
Theorem	isabld 13102*	Properties that determine an Abelian group. (Contributed by NM, 6-Aug-2013.)
|- ( ph -> B = ( Base ` G ) )   &    |- ( ph -> .+ = ( +g ` G ) )   &    |- ( ph -> G e. Grp )   &    |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) = ( y .+ x ) )   =>    |- ( ph -> G e. Abel )
Theorem	isabli 13103*	Properties that determine an Abelian group. (Contributed by NM, 4-Sep-2011.)
|- G e. Grp   &    |- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ( x e. B /\ y e. B ) -> ( x .+ y ) = ( y .+ x ) )   =>    |- G e. Abel
Theorem	cmnmnd 13104	A commutative monoid is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
|- ( G e. CMnd -> G e. Mnd )
Theorem	cmncom 13105	A commutative monoid is commutative. (Contributed by Mario Carneiro, 6-Jan-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. CMnd /\ X e. B /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) )
Theorem	ablcom 13106	An Abelian group operation is commutative. (Contributed by NM, 26-Aug-2011.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Abel /\ X e. B /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) )
Theorem	cmn32 13107	Commutative/associative law for commutative monoids. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .+ Z ) = ( ( X .+ Z ) .+ Y ) )
Theorem	cmn4 13108	Commutative/associative law for commutative monoids. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( ( X .+ Z ) .+ ( Y .+ W ) ) )
Theorem	cmn12 13109	Commutative/associative law for commutative monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .+ ( Y .+ Z ) ) = ( Y .+ ( X .+ Z ) ) )
Theorem	abl32 13110	Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. Abel )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( ( X .+ Y ) .+ Z ) = ( ( X .+ Z ) .+ Y ) )
Theorem	cmnmndd 13111	A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> G e. CMnd )   =>    |- ( ph -> G e. Mnd )
Theorem	rinvmod 13112*	Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6068. (Contributed by AV, 31-Dec-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> A e. B )   =>    |- ( ph -> E* w e. B ( A .+ w ) = .0. )
Theorem	ablinvadd 13113	The inverse of an Abelian group operation. (Contributed by NM, 31-Mar-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- N = ( invg ` G )   =>    |- ( ( G e. Abel /\ X e. B /\ Y e. B ) -> ( N ` ( X .+ Y ) ) = ( ( N ` X ) .+ ( N ` Y ) ) )
Theorem	ablsub2inv 13114	Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015.)
|- B = ( Base ` G )   &    |- .- = ( -g ` G )   &    |- N = ( invg ` G )   &    |- ( ph -> G e. Abel )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( ( N ` X ) .- ( N ` Y ) ) = ( Y .- X ) )
Theorem	ablsubadd 13115	Relationship between Abelian group subtraction and addition. (Contributed by NM, 31-Mar-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Abel /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .- Y ) = Z <-> ( Y .+ Z ) = X ) )
Theorem	ablsub4 13116	Commutative/associative subtraction law for Abelian groups. (Contributed by NM, 31-Mar-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Abel /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( ( X .+ Y ) .- ( Z .+ W ) ) = ( ( X .- Z ) .+ ( Y .- W ) ) )
Theorem	abladdsub4 13117	Abelian group addition/subtraction law. (Contributed by NM, 31-Mar-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Abel /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( ( X .+ Y ) = ( Z .+ W ) <-> ( X .- Z ) = ( W .- Y ) ) )
Theorem	abladdsub 13118	Associative-type law for group subtraction and addition. (Contributed by NM, 19-Apr-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Abel /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .- Z ) = ( ( X .- Z ) .+ Y ) )
Theorem	ablpncan2 13119	Cancellation law for subtraction in an Abelian group. (Contributed by NM, 2-Oct-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Abel /\ X e. B /\ Y e. B ) -> ( ( X .+ Y ) .- X ) = Y )
Theorem	ablpncan3 13120	A cancellation law for Abelian groups. (Contributed by NM, 23-Mar-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Abel /\ ( X e. B /\ Y e. B ) ) -> ( X .+ ( Y .- X ) ) = Y )
Theorem	ablsubsub 13121	Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   &    |- ( ph -> G e. Abel )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( X .- ( Y .- Z ) ) = ( ( X .- Y ) .+ Z ) )
Theorem	ablsubsub4 13122	Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   &    |- ( ph -> G e. Abel )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( ( X .- Y ) .- Z ) = ( X .- ( Y .+ Z ) ) )
Theorem	ablpnpcan 13123	Cancellation law for mixed addition and subtraction. (pnpcan 8196 analog.) (Contributed by NM, 29-May-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   &    |- ( ph -> G e. Abel )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   &    |- ( ph -> G e. Abel )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( ( X .+ Y ) .- ( X .+ Z ) ) = ( Y .- Z ) )
Theorem	ablnncan 13124	Cancellation law for group subtraction. (nncan 8186 analog.) (Contributed by NM, 7-Apr-2015.)
|- B = ( Base ` G )   &    |- .- = ( -g ` G )   &    |- ( ph -> G e. Abel )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .- ( X .- Y ) ) = Y )
Theorem	ablsub32 13125	Swap the second and third terms in a double group subtraction. (Contributed by NM, 7-Apr-2015.)
|- B = ( Base ` G )   &    |- .- = ( -g ` G )   &    |- ( ph -> G e. Abel )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( ( X .- Y ) .- Z ) = ( ( X .- Z ) .- Y ) )
Theorem	ablnnncan 13126	Cancellation law for group subtraction. (nnncan 8192 analog.) (Contributed by NM, 29-Feb-2008.) (Revised by AV, 27-Aug-2021.)
|- B = ( Base ` G )   &    |- .- = ( -g ` G )   &    |- ( ph -> G e. Abel )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( ( X .- ( Y .- Z ) ) .- Z ) = ( X .- Y ) )
Theorem	ablnnncan1 13127	Cancellation law for group subtraction. (nnncan1 8193 analog.) (Contributed by NM, 7-Apr-2015.)
|- B = ( Base ` G )   &    |- .- = ( -g ` G )   &    |- ( ph -> G e. Abel )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( ( X .- Y ) .- ( X .- Z ) ) = ( Z .- Y ) )
Theorem	ablsubsub23 13128	Swap subtrahend and result of group subtraction. (Contributed by NM, 14-Dec-2007.) (Revised by AV, 7-Oct-2021.)
|- V = ( Base ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Abel /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A .- B ) = C <-> ( A .- C ) = B ) )
Theorem	subcmnd 13129	A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 10-Jan-2015.)
|- ( ph -> H = ( G ↾s S ) )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> H e. Mnd )   &    |- ( ph -> S e. V )   =>    |- ( ph -> H e. CMnd )
7.3  Rings
7.3.1  Multiplicative Group
Syntax	cmgp 13130	Multiplicative group.
class mulGrp
Definition	df-mgp 13131	Define a structure that puts the multiplication operation of a ring in the addition slot. Note that this will not actually be a group for the average ring, or even for a field, but it will be a monoid, and we get a group if we restrict to the elements that have inverses. This allows us to formalize such notions as "the multiplication operation of a ring is a monoid" or "the multiplicative identity" in terms of the identity of a monoid (df-ur 13143). (Contributed by Mario Carneiro, 21-Dec-2014.)
|- mulGrp = ( w e. _V |-> ( w sSet <. ( +g ` ndx ) , ( .r ` w ) >. ) )
Theorem	fnmgp 13132	The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
|- mulGrp Fn _V
Theorem	mgpvalg 13133	Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.)
|- M = (mulGrp ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. V -> M = ( R sSet <. ( +g ` ndx ) , .x. >. ) )
Theorem	mgpplusgg 13134	Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.)
|- M = (mulGrp ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. V -> .x. = ( +g ` M ) )
Theorem	mgpex 13135	Existence of the multiplication group. If R is known to be a semiring, see srgmgp 13151. (Contributed by Jim Kingdon, 10-Jan-2025.)
|- M = (mulGrp ` R )   =>    |- ( R e. V -> M e. _V )
Theorem	mgpbasg 13136	Base set of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
|- M = (mulGrp ` R )   &    |- B = ( Base ` R )   =>    |- ( R e. V -> B = ( Base ` M ) )
Theorem	mgpscag 13137	The multiplication monoid has the same (if any) scalars as the original ring. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)
|- M = (mulGrp ` R )   &    |- S = (Scalar ` R )   =>    |- ( R e. V -> S = (Scalar ` M ) )
Theorem	mgptsetg 13138	Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- M = (mulGrp ` R )   =>    |- ( R e. V -> (TopSet ` R ) = (TopSet ` M ) )
Theorem	mgptopng 13139	Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- M = (mulGrp ` R )   &    |- J = ( TopOpen ` R )   =>    |- ( R e. V -> J = ( TopOpen ` M ) )
Theorem	mgpdsg 13140	Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- M = (mulGrp ` R )   &    |- B = ( dist ` R )   =>    |- ( R e. V -> B = ( dist ` M ) )
Theorem	mgpress 13141	Subgroup commutes with the multiplicative group operator. (Contributed by Mario Carneiro, 10-Jan-2015.) (Proof shortened by AV, 18-Oct-2024.)
|- S = ( R ↾s A )   &    |- M = (mulGrp ` R )   =>    |- ( ( R e. V /\ A e. W ) -> ( M ↾s A ) = (mulGrp ` S ) )
7.3.2  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 13181). 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 13181 in set.mm), and we have "the ring unity is a unit", see https://us.metamath.org/mpeuni/1unit.html 13181.
Syntax	cur 13142	Extend class notation with ring unity.
class 1r
Definition	df-ur 13143	Define the multiplicative identity, i.e., the monoid identity (df-0g 12707) of the multiplicative monoid (df-mgp 13131) 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 13145, 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 13144	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 13145*	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 ) ) ) )
7.3.3  Semirings
Syntax	csrg 13146	Extend class notation with the class of all semirings.
class SRing
Definition	df-srg 13147*	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 13148*	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 13149	A semiring is a commutative monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
|- ( R e. SRing -> R e. CMnd )
Theorem	srgmnd 13150	A semiring is a monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
|- ( R e. SRing -> R e. Mnd )
Theorem	srgmgp 13151	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 13152	Lemma for srgdi 13157 and srgdir 13158. (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 13153	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 13154	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 13155*	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 13156	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 13157	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 13158	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 13159	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 13160	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 13161	Lemma for srglidm 13162 and srgridm 13163. (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 13162	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 13163	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 13164*	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 13165	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 13166	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 13167	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 13168	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 13169*	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 13170	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 13171	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 13172	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 13173	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 13174	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 13175	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 13176*	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 13177*	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 13178	The exponentiation (by a nonnegative integer) of the multiplicative identity of a semiring, analogous to mulgnn0z 13010. (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. )
7.3.4  Definition and basic properties of unital rings
Syntax	crg 13179	Extend class notation with class of all (unital) rings.
class Ring
Syntax	ccrg 13180	Extend class notation with class of all (unital) commutative rings.
class CRing
Definition	df-ring 13181*	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 13214), 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 13182	Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.)
|- CRing = { f e. Ring | (mulGrp ` f ) e. CMnd }
Theorem	isring 13183*	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 13184	A ring is a group. (Contributed by NM, 15-Sep-2011.)
|- ( R e. Ring -> R e. Grp )
Theorem	ringmgp 13185	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 13186	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 13187	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 13188	A ring is a group. (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. Ring )   =>    |- ( ph -> R e. Grp )
Theorem	ringmnd 13189	A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.)
|- ( R e. Ring -> R e. Mnd )
Theorem	ringmgm 13190	A ring is a magma. (Contributed by AV, 31-Jan-2020.)
|- ( R e. Ring -> R e. Mgm )
Theorem	crngring 13191	A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
|- ( R e. CRing -> R e. Ring )
Theorem	crngringd 13192	A commutative ring is a ring. (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. CRing )   =>    |- ( ph -> R e. Ring )
Theorem	crnggrpd 13193	A commutative ring is a group. (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. CRing )   =>    |- ( ph -> R e. Grp )
Theorem	mgpf 13194	Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.)
|- (mulGrp |` Ring ) : Ring --> Mnd
Theorem	ringdilem 13195	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 13196	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 13197	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 13198*	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 13199	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 13200*	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 ) )