P. 144 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14301-14400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	prds0g 14301	The identity in a product of monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
|- Y = ( S X_s R )   &    |- ( ph -> I e. W )   &    |- ( ph -> S e. V )   &    |- ( ph -> R : I --> Mnd )   =>    |- ( ph -> ( 0g o. R ) = ( 0g ` Y ) )
Theorem	prdsinvlem 14302*	Characterization of inverses in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
|- Y = ( S X_s R )   &    |- B = ( Base ` Y )   &    |- .+ = ( +g ` Y )   &    |- ( ph -> S e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> R : I --> Grp )   &    |- ( ph -> F e. B )   &    |- .0. = ( 0g o. R )   &    |- N = ( y e. I |-> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) )   =>    |- ( ph -> ( N e. B /\ ( N .+ F ) = .0. ) )
Theorem	prdsgrpd 14303	The product of a family of groups is a group. (Contributed by Stefan O'Rear, 10-Jan-2015.)
|- Y = ( S X_s R )   &    |- ( ph -> I e. W )   &    |- ( ph -> S e. V )   &    |- ( ph -> R : I --> Grp )   =>    |- ( ph -> Y e. Grp )
Theorem	prdsinvgd 14304*	Negation in a product of groups. (Contributed by Stefan O'Rear, 10-Jan-2015.)
|- Y = ( S X_s R )   &    |- ( ph -> I e. W )   &    |- ( ph -> S e. V )   &    |- ( ph -> R : I --> Grp )   &    |- B = ( Base ` Y )   &    |- N = ( invg ` Y )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( N ` X ) = ( x e. I |-> ( ( invg ` ( R ` x ) ) ` ( X ` x ) ) ) )
7.2.9  Binary product on structures
Syntax	cxps 14305	Binary product structure function.
class X.s
Definition	df-xps 14306*	Define a binary product on structures. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
|- X.s = ( r e. _V , s e. _V |-> ( `' ( x e. ( Base ` r ) , y e. ( Base ` s ) |-> { <. (/) , x >. , <. 1o , y >. } ) "s ( (Scalar ` r ) X_s { <. (/) , r >. , <. 1o , s >. } ) ) )
Theorem	xpsval 14307*	Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
|- T = ( R X.s S )   &    |- X = ( Base ` R )   &    |- Y = ( Base ` S )   &    |- ( ph -> R e. V )   &    |- ( ph -> S e. W )   &    |- F = ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } )   &    |- G = (Scalar ` R )   &    |- U = ( G X_s { <. (/) , R >. , <. 1o , S >. } )   =>    |- ( ph -> T = ( `' F "s U ) )
7.2.10  Structure power
Syntax	cpws 14308	The function constructing structure powers.
class ^s
Definition	df-pws 14309*	Define a structure power, which is just a structure product where all the factors are the same. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- ^s = ( r e. _V , i e. _V |-> ( (Scalar ` r ) X_s ( i X. { r } ) ) )
Theorem	pwsval 14310	Value of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( R ^s I )   &    |- F = (Scalar ` R )   =>    |- ( ( R e. V /\ I e. W ) -> Y = ( F X_s ( I X. { R } ) ) )
Theorem	pwsbas 14311	Base set of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( R ^s I )   &    |- B = ( Base ` R )   =>    |- ( ( R e. V /\ I e. W ) -> ( B ^m I ) = ( Base ` Y ) )
Theorem	pwselbasb 14312	Membership in the base set of a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.)
|- Y = ( R ^s I )   &    |- B = ( Base ` R )   &    |- V = ( Base ` Y )   =>    |- ( ( R e. W /\ I e. Z ) -> ( X e. V <-> X : I --> B ) )
Theorem	pwselbas 14313	An element of a structure power is a function from the index set to the base set of the structure. (Contributed by Mario Carneiro, 11-Jan-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)
|- Y = ( R ^s I )   &    |- B = ( Base ` R )   &    |- V = ( Base ` Y )   &    |- ( ph -> R e. W )   &    |- ( ph -> I e. Z )   &    |- ( ph -> X e. V )   =>    |- ( ph -> X : I --> B )
Theorem	pwsplusgval 14314	Value of addition in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( R ^s I )   &    |- B = ( Base ` Y )   &    |- ( ph -> R e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> F e. B )   &    |- ( ph -> G e. B )   &    |- .+ = ( +g ` R )   &    |- .+b = ( +g ` Y )   =>    |- ( ph -> ( F .+b G ) = ( F oF .+ G ) )
Theorem	pwsmulrval 14315	Value of multiplication in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( R ^s I )   &    |- B = ( Base ` Y )   &    |- ( ph -> R e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> F e. B )   &    |- ( ph -> G e. B )   &    |- .x. = ( .r ` R )   &    |- .xb = ( .r ` Y )   =>    |- ( ph -> ( F .xb G ) = ( F oF .x. G ) )
Theorem	pwsdiagel 14316	Membership of diagonal elements in the structure power base set. (Contributed by Stefan O'Rear, 24-Jan-2015.)
|- Y = ( R ^s I )   &    |- B = ( Base ` R )   &    |- C = ( Base ` Y )   =>    |- ( ( ( R e. V /\ I e. W ) /\ A e. B ) -> ( I X. { A } ) e. C )
Theorem	pwssnf1o 14317*	Triviality of singleton powers: set equipollence. (Contributed by Stefan O'Rear, 24-Jan-2015.)
|- Y = ( R ^s { I } )   &    |- B = ( Base ` R )   &    |- F = ( x e. B |-> ( { I } X. { x } ) )   &    |- C = ( Base ` Y )   =>    |- ( ( R e. V /\ I e. W ) -> F : B -1-1-onto-> C )
Theorem	pwsmnd 14318	The structure power of a monoid is a monoid. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( R ^s I )   =>    |- ( ( R e. Mnd /\ I e. V ) -> Y e. Mnd )
Theorem	pws0g 14319	The identity in a structure power of a monoid. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( R ^s I )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Mnd /\ I e. V ) -> ( I X. { .0. } ) = ( 0g ` Y ) )
Theorem	pwsgrp 14320	A structure power of a group is a group. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( R ^s I )   =>    |- ( ( R e. Grp /\ I e. V ) -> Y e. Grp )
Theorem	pwsinvg 14321	Negation in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( R ^s I )   &    |- B = ( Base ` Y )   &    |- M = ( invg ` R )   &    |- N = ( invg ` Y )   =>    |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( N ` X ) = ( M o. X ) )
Theorem	pwssub 14322	Subtraction in a structure power. (Contributed by Mario Carneiro, 12-Jan-2015.)
|- Y = ( R ^s I )   &    |- B = ( Base ` Y )   &    |- M = ( -g ` R )   &    |- .- = ( -g ` Y )   =>    |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( F .- G ) = ( F oF M G ) )
7.3  Rings
7.3.1  Multiplicative Group
Syntax	cmgp 14323	Multiplicative group.
class mulGrp
Definition	df-mgp 14324	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 14369). (Contributed by Mario Carneiro, 21-Dec-2014.)
|- mulGrp = ( w e. _V |-> ( w sSet <. ( +g ` ndx ) , ( .r ` w ) >. ) )
Theorem	fnmgp 14325	The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
|- mulGrp Fn _V
Theorem	mgpvalg 14326	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 14327	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	mgpplusg 14328	Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.)
|- M = (mulGrp ` R )   &    |- .x. = ( .r ` R )   =>    |- .x. = ( +g ` M )
Theorem	mgpex 14329	Existence of the multiplication group. If R is known to be a semiring, see srgmgp 14378. (Contributed by Jim Kingdon, 10-Jan-2025.)
|- M = (mulGrp ` R )   =>    |- ( R e. V -> M e. _V )
Theorem	mgpbasg 14330	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	mgpbas 14331	Base set of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
|- M = (mulGrp ` R )   &    |- B = ( Base ` R )   =>    |- B = ( Base ` M )
Theorem	mgpscag 14332	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 14333	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 14334	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 14335	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 14336	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  Non-unital rings ("rngs") According to Wikipedia, "... in abstract algebra, a rng (or non-unital ring or pseudo-ring) is an algebraic structure satisfying the same properties as a [unital] ring, without assuming the existence of a multiplicative identity. The term "rng" (pronounced rung) is meant to suggest that it is a "ring" without "i", i.e. without the requirement for an "identity element"." (see https://en.wikipedia.org/wiki/Rng_(algebra), 28-Mar-2025).
Syntax	crng 14337	Extend class notation with class of all non-unital rings.
class Rng
Definition	df-rng 14338*	Define the class of all non-unital rings. A non-unital ring (or rng, or pseudoring) is a set equipped with two everywhere-defined internal operations, whose first one is an additive abelian group operation and the second one is a multiplicative semigroup operation, and where the addition is left- and right-distributive for the multiplication. Definition of a pseudo-ring in section I.8.1 of [BourbakiAlg1] p. 93 or the definition of a ring in part Preliminaries of [Roman] p. 18. As almost always in mathematics, "non-unital" means "not necessarily unital". Therefore, by talking about a ring (in general) or a non-unital ring the "unital" case is always included. In contrast to a unital ring, the commutativity of addition must be postulated and cannot be proven from the other conditions. (Contributed by AV, 6-Jan-2020.)
|- Rng = { f e. Abel | ( (mulGrp ` f ) e. Smgrp /\ [. ( Base ` f ) / b ]. [. ( +g ` f ) / p ]. [. ( .r ` f ) / t ]. A. x e. b A. y e. b A. z e. b ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) /\ ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) ) ) ) }
Theorem	isrng 14339*	The predicate "is a non-unital ring." (Contributed by AV, 6-Jan-2020.)
|- B = ( Base ` R )   &    |- G = (mulGrp ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. Rng <-> ( R e. Abel /\ G e. Smgrp /\ 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	rngabl 14340	A non-unital ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)
|- ( R e. Rng -> R e. Abel )
Theorem	rngmgp 14341	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 14342	Restricted functionality of the multiplicative group on non-unital rings (mgpf 14421 analog). (Contributed by AV, 22-Feb-2025.)
|- (mulGrp |` Rng ) :Rng -->Smgrp
Theorem	rnggrp 14343	A non-unital ring is a (additive) group. (Contributed by AV, 16-Feb-2025.)
|- ( R e. Rng -> R e. Grp )
Theorem	rngass 14344	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 14345	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 14346	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 14347	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 14348	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 14349	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 14350	The zero of a non-unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringlz 14454. (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 14351	The zero of a non-unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringrz 14455. (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 14352	Negation of a product in a non-unital ring (mulneg1 8729 analog). In contrast to ringmneg1 14464, 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 14353	Negation of a product in a non-unital ring (mulneg2 8730 analog). In contrast to ringmneg2 14465, 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 14354	Double negation of a product in a non-unital ring (mul2neg 8732 analog). (Contributed by Mario Carneiro, 4-Dec-2014.) Generalization of ringm2neg 14466. (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 14355	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 14356	Ring multiplication distributes over subtraction. (subdi 8719 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdi 14467. (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 14357	Ring multiplication distributes over subtraction. (subdir 8720 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdir 14468. (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 14358*	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 14359	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 13483. (Contributed by Jim Kingdon, 5-May-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Rng -> ( G ↾s B ) e. Rng )
Theorem	rngpropd 14360*	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 14361*	The image structure of a non-unital ring is a non-unital ring (imasring 14477 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 14362	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 14363*	The quotient structure of a non-unital ring is a non-unital ring (qusring2 14479 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 )
Theorem	rng1zrlem 14364	Lemma for rng1zr 14365 and srg1zr 14397. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 18-Jun-2026.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .* = ( .r ` R )   =>    |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
Theorem	rng1zr 14365	The only ring 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, 18-Jun-2026.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .* = ( .r ` R )   =>    |- ( ( ( R e. Rng /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
Theorem	rngen1zr 14366	The only ring 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, 18-Jun-2026.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .* = ( .r ` R )   =>    |- ( ( ( R e. Rng /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B ~~ 1o <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
Theorem	rngen1zr0 14367	The only ring with one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 15-Feb-2010.) (Revised by AV, 18-Jun-2026.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .* = ( .r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Rng /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( B ~~ 1o <-> ( .+ = { <. <. .0. , .0. >. , .0. >. } /\ .* = { <. <. .0. , .0. >. , .0. >. } ) ) )
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 14408). 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 14408 in set.mm), and we have "the ring unity is a unit", see https://us.metamath.org/mpeuni/1unit.html 14408.
Syntax	cur 14368	Extend class notation with ring unity.
class 1r
Definition	df-ur 14369	Define the multiplicative identity, i.e., the monoid identity (df-0g 13670) of the multiplicative monoid (df-mgp 14324) 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 14372, 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 14370	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	ringidval 14371	The value of the unity element of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
|- G = (mulGrp ` R )   &    |- .1. = ( 1r ` R )   =>    |- .1. = ( 0g ` G )
Theorem	dfur2g 14372*	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 14373	Extend class notation with the class of all semirings.
class SRing
Definition	df-srg 14374*	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 14375*	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 14376	A semiring is a commutative monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
|- ( R e. SRing -> R e. CMnd )
Theorem	srgmnd 14377	A semiring is a monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
|- ( R e. SRing -> R e. Mnd )
Theorem	srgmgp 14378	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 14379	Lemma for srgdi 14384 and srgdir 14385. (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 14380	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 14381	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 14382*	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 14383	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 14384	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 14385	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 14386	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 14387	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 14388	Lemma for srglidm 14389 and srgridm 14390. (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 14389	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 14390	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 14391*	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 14392	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 14393	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 14394	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 14395	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 14396*	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 14397	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.) (Proof shortened by AV, 19-Jun-2026.)
|- 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	srgen1zr0 14398	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 14399	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 14400	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 ) ) )