P. 143 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14201-14300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nzrunit 14201	A unit is nonzero in any nonzero ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
|- U = (Unit ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. NzRing /\ A e. U ) -> A =/= .0. )
Theorem	01eq0ring 14202	If the zero and the identity element of a ring are the same, the ring is the zero ring. (Contributed by AV, 16-Apr-2019.) (Proof shortened by SN, 23-Feb-2025.)
|- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- .1. = ( 1r ` R )   =>    |- ( ( R e. Ring /\ .0. = .1. ) -> B = { .0. } )
7.3.10  Local rings
Syntax	clring 14203	Extend class notation with class of all local rings.
class LRing
Definition	df-lring 14204*	A local ring is a nonzero ring where for any two elements summing to one, at least one is invertible. Any field is a local ring; the ring of integers is an example of a ring which is not a local ring. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.)
|- LRing = { r e. NzRing | A. x e. ( Base ` r ) A. y e. ( Base ` r ) ( ( x ( +g ` r ) y ) = ( 1r ` r ) -> ( x e. (Unit ` r ) \/ y e. (Unit ` r ) ) ) }
Theorem	islring 14205*	The predicate "is a local ring". (Contributed by SN, 23-Feb-2025.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .1. = ( 1r ` R )   &    |- U = (Unit ` R )   =>    |- ( R e. LRing <-> ( R e. NzRing /\ A. x e. B A. y e. B ( ( x .+ y ) = .1. -> ( x e. U \/ y e. U ) ) ) )
Theorem	lringnzr 14206	A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.)
|- ( R e. LRing -> R e. NzRing )
Theorem	lringring 14207	A local ring is a ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)
|- ( R e. LRing -> R e. Ring )
Theorem	lringnz 14208	A local ring is a nonzero ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)
|- .1. = ( 1r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. LRing -> .1. =/= .0. )
Theorem	lringuplu 14209	If the sum of two elements of a local ring is invertible, then at least one of the summands must be invertible. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> U = (Unit ` R ) )   &    |- ( ph -> .+ = ( +g ` R ) )   &    |- ( ph -> R e. LRing )   &    |- ( ph -> ( X .+ Y ) e. U )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X e. U \/ Y e. U ) )
7.3.11  Subrings
7.3.11.1  Subrings of non-unital rings
Syntax	csubrng 14210	Extend class notation with all subrings of a non-unital ring.
class SubRng
Definition	df-subrng 14211*	Define a subring of a non-unital ring as a set of elements that is a non-unital ring in its own right. In this section, a subring of a non-unital ring is simply called "subring", unless it causes any ambiguity with SubRing. (Contributed by AV, 14-Feb-2025.)
|- SubRng = ( w e. Rng |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Rng } )
Theorem	issubrng 14212	The subring of non-unital ring predicate. (Contributed by AV, 14-Feb-2025.)
|- B = ( Base ` R )   =>    |- ( A e. (SubRng ` R ) <-> ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ B ) )
Theorem	subrngss 14213	A subring is a subset. (Contributed by AV, 14-Feb-2025.)
|- B = ( Base ` R )   =>    |- ( A e. (SubRng ` R ) -> A C_ B )
Theorem	subrngid 14214	Every non-unital ring is a subring of itself. (Contributed by AV, 14-Feb-2025.)
|- B = ( Base ` R )   =>    |- ( R e. Rng -> B e. (SubRng ` R ) )
Theorem	subrngrng 14215	A subring is a non-unital ring. (Contributed by AV, 14-Feb-2025.)
|- S = ( R ↾s A )   =>    |- ( A e. (SubRng ` R ) -> S e. Rng )
Theorem	subrngrcl 14216	Reverse closure for a subring predicate. (Contributed by AV, 14-Feb-2025.)
|- ( A e. (SubRng ` R ) -> R e. Rng )
Theorem	subrngsubg 14217	A subring is a subgroup. (Contributed by AV, 14-Feb-2025.)
|- ( A e. (SubRng ` R ) -> A e. (SubGrp ` R ) )
Theorem	subrngringnsg 14218	A subring is a normal subgroup. (Contributed by AV, 25-Feb-2025.)
|- ( A e. (SubRng ` R ) -> A e. (NrmSGrp ` R ) )
Theorem	subrngbas 14219	Base set of a subring structure. (Contributed by AV, 14-Feb-2025.)
|- S = ( R ↾s A )   =>    |- ( A e. (SubRng ` R ) -> A = ( Base ` S ) )
Theorem	subrng0 14220	A subring always has the same additive identity. (Contributed by AV, 14-Feb-2025.)
|- S = ( R ↾s A )   &    |- .0. = ( 0g ` R )   =>    |- ( A e. (SubRng ` R ) -> .0. = ( 0g ` S ) )
Theorem	subrngacl 14221	A subring is closed under addition. (Contributed by AV, 14-Feb-2025.)
|- .+ = ( +g ` R )   =>    |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> ( X .+ Y ) e. A )
Theorem	subrngmcl 14222	A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) Generalization of subrgmcl 14246. (Revised by AV, 14-Feb-2025.)
|- .x. = ( .r ` R )   =>    |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> ( X .x. Y ) e. A )
Theorem	issubrng2 14223*	Characterize the subrings of a ring by closure properties. (Contributed by AV, 15-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. Rng -> ( A e. (SubRng ` R ) <-> ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) )
Theorem	opprsubrngg 14224	Being a subring is a symmetric property. (Contributed by AV, 15-Feb-2025.)
|- O = (oppr ` R )   =>    |- ( R e. V -> (SubRng ` R ) = (SubRng ` O ) )
Theorem	subrngintm 14225*	The intersection of a nonempty collection of subrings is a subring. (Contributed by AV, 15-Feb-2025.)
|- ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) -> |^| S e. (SubRng ` R ) )
Theorem	subrngin 14226	The intersection of two subrings is a subring. (Contributed by AV, 15-Feb-2025.)
|- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` R ) ) -> ( A i^i B ) e. (SubRng ` R ) )
Theorem	subsubrng 14227	A subring of a subring is a subring. (Contributed by AV, 15-Feb-2025.)
|- S = ( R ↾s A )   =>    |- ( A e. (SubRng ` R ) -> ( B e. (SubRng ` S ) <-> ( B e. (SubRng ` R ) /\ B C_ A ) ) )
Theorem	subsubrng2 14228	The set of subrings of a subring are the smaller subrings. (Contributed by AV, 15-Feb-2025.)
|- S = ( R ↾s A )   =>    |- ( A e. (SubRng ` R ) -> (SubRng ` S ) = ( (SubRng ` R ) i^i ~P A ) )
Theorem	subrngpropd 14229*	If two structures have the same ring components (properties), they have the same set of subrings. (Contributed by AV, 17-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 -> (SubRng ` K ) = (SubRng ` L ) )
7.3.11.2  Subrings of unital rings
Syntax	csubrg 14230	Extend class notation with all subrings of a ring.
class SubRing
Syntax	crgspn 14231	Extend class notation with span of a set of elements over a ring.
class RingSpan
Definition	df-subrg 14232*	Define a subring of a ring as a set of elements that is a ring in its own right and contains the multiplicative identity. The additional constraint is necessary because the multiplicative identity of a ring, unlike the additive identity of a ring/group or the multiplicative identity of a field, cannot be identified by a local property. Thus, it is possible for a subset of a ring to be a ring while not containing the true identity if it contains a false identity. For instance, the subset ( ZZ X. { 0 } ) of ( ZZ X. ZZ ) (where multiplication is componentwise) contains the false identity <. 1 , 0 >. which preserves every element of the subset and thus appears to be the identity of the subset, but is not the identity of the larger ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
|- SubRing = ( w e. Ring |-> { s e. ~P ( Base ` w ) | ( ( w ↾s s ) e. Ring /\ ( 1r ` w ) e. s ) } )
Definition	df-rgspn 14233*	The ring-span of a set of elements in a ring is the smallest subring which contains all of them. (Contributed by Stefan O'Rear, 7-Dec-2014.)
|- RingSpan = ( w e. _V |-> ( s e. ~P ( Base ` w ) |-> |^| { t e. (SubRing ` w ) | s C_ t } ) )
Theorem	issubrg 14234	The subring predicate. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Proof shortened by AV, 12-Oct-2020.)
|- B = ( Base ` R )   &    |- .1. = ( 1r ` R )   =>    |- ( A e. (SubRing ` R ) <-> ( ( R e. Ring /\ ( R ↾s A ) e. Ring ) /\ ( A C_ B /\ .1. e. A ) ) )
Theorem	subrgss 14235	A subring is a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
|- B = ( Base ` R )   =>    |- ( A e. (SubRing ` R ) -> A C_ B )
Theorem	subrgid 14236	Every ring is a subring of itself. (Contributed by Stefan O'Rear, 30-Nov-2014.)
|- B = ( Base ` R )   =>    |- ( R e. Ring -> B e. (SubRing ` R ) )
Theorem	subrgring 14237	A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
|- S = ( R ↾s A )   =>    |- ( A e. (SubRing ` R ) -> S e. Ring )
Theorem	subrgcrng 14238	A subring of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
|- S = ( R ↾s A )   =>    |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> S e. CRing )
Theorem	subrgrcl 14239	Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.)
|- ( A e. (SubRing ` R ) -> R e. Ring )
Theorem	subrgsubg 14240	A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.)
|- ( A e. (SubRing ` R ) -> A e. (SubGrp ` R ) )
Theorem	subrg0 14241	A subring always has the same additive identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
|- S = ( R ↾s A )   &    |- .0. = ( 0g ` R )   =>    |- ( A e. (SubRing ` R ) -> .0. = ( 0g ` S ) )
Theorem	subrg1cl 14242	A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
|- .1. = ( 1r ` R )   =>    |- ( A e. (SubRing ` R ) -> .1. e. A )
Theorem	subrgbas 14243	Base set of a subring structure. (Contributed by Stefan O'Rear, 27-Nov-2014.)
|- S = ( R ↾s A )   =>    |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
Theorem	subrg1 14244	A subring always has the same multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
|- S = ( R ↾s A )   &    |- .1. = ( 1r ` R )   =>    |- ( A e. (SubRing ` R ) -> .1. = ( 1r ` S ) )
Theorem	subrgacl 14245	A subring is closed under addition. (Contributed by Mario Carneiro, 2-Dec-2014.)
|- .+ = ( +g ` R )   =>    |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> ( X .+ Y ) e. A )
Theorem	subrgmcl 14246	A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
|- .x. = ( .r ` R )   =>    |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. A ) -> ( X .x. Y ) e. A )
Theorem	subrgsubm 14247	A subring is a submonoid of the multiplicative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
|- M = (mulGrp ` R )   =>    |- ( A e. (SubRing ` R ) -> A e. (SubMnd ` M ) )
Theorem	subrgdvds 14248	If an element divides another in a subring, then it also divides the other in the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
|- S = ( R ↾s A )   &    |- .|| = ( ||r ` R )   &    |- E = ( ||r ` S )   =>    |- ( A e. (SubRing ` R ) -> E C_ .|| )
Theorem	subrguss 14249	A unit of a subring is a unit of the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
|- S = ( R ↾s A )   &    |- U = (Unit ` R )   &    |- V = (Unit ` S )   =>    |- ( A e. (SubRing ` R ) -> V C_ U )
Theorem	subrginv 14250	A subring always has the same inversion function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
|- S = ( R ↾s A )   &    |- I = ( invr ` R )   &    |- U = (Unit ` S )   &    |- J = ( invr ` S )   =>    |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( I ` X ) = ( J ` X ) )
Theorem	subrgdv 14251	A subring always has the same division function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
|- S = ( R ↾s A )   &    |- ./ = (/r ` R )   &    |- U = (Unit ` S )   &    |- E = (/r ` S )   =>    |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( X ./ Y ) = ( X E Y ) )
Theorem	subrgunit 14252	An element of a ring is a unit of a subring iff it is a unit of the parent ring and both it and its inverse are in the subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
|- S = ( R ↾s A )   &    |- U = (Unit ` R )   &    |- V = (Unit ` S )   &    |- I = ( invr ` R )   =>    |- ( A e. (SubRing ` R ) -> ( X e. V <-> ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) )
Theorem	subrgugrp 14253	The units of a subring form a subgroup of the unit group of the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
|- S = ( R ↾s A )   &    |- U = (Unit ` R )   &    |- V = (Unit ` S )   &    |- G = ( (mulGrp ` R ) ↾s U )   =>    |- ( A e. (SubRing ` R ) -> V e. (SubGrp ` G ) )
Theorem	issubrg2 14254*	Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.)
|- B = ( Base ` R )   &    |- .1. = ( 1r ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. Ring -> ( A e. (SubRing ` R ) <-> ( A e. (SubGrp ` R ) /\ .1. e. A /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) )
Theorem	subrgnzr 14255	A subring of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 15-Jun-2015.)
|- S = ( R ↾s A )   =>    |- ( ( R e. NzRing /\ A e. (SubRing ` R ) ) -> S e. NzRing )
Theorem	subrgintm 14256*	The intersection of an inhabited collection of subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
|- ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) -> |^| S e. (SubRing ` R ) )
Theorem	subrgin 14257	The intersection of two subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
|- ( ( A e. (SubRing ` R ) /\ B e. (SubRing ` R ) ) -> ( A i^i B ) e. (SubRing ` R ) )
Theorem	subsubrg 14258	A subring of a subring is a subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
|- S = ( R ↾s A )   =>    |- ( A e. (SubRing ` R ) -> ( B e. (SubRing ` S ) <-> ( B e. (SubRing ` R ) /\ B C_ A ) ) )
Theorem	subsubrg2 14259	The set of subrings of a subring are the smaller subrings. (Contributed by Stefan O'Rear, 9-Mar-2015.)
|- S = ( R ↾s A )   =>    |- ( A e. (SubRing ` R ) -> (SubRing ` S ) = ( (SubRing ` R ) i^i ~P A ) )
Theorem	issubrg3 14260	A subring is an additive subgroup which is also a multiplicative submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- M = (mulGrp ` R )   =>    |- ( R e. Ring -> ( S e. (SubRing ` R ) <-> ( S e. (SubGrp ` R ) /\ S e. (SubMnd ` M ) ) ) )
Theorem	resrhm 14261	Restriction of a ring homomorphism to a subring is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
|- U = ( S ↾s X )   =>    |- ( ( F e. ( S RingHom T ) /\ X e. (SubRing ` S ) ) -> ( F |` X ) e. ( U RingHom T ) )
Theorem	resrhm2b 14262	Restriction of the codomain of a (ring) homomorphism. resghm2b 13848 analog. (Contributed by SN, 7-Feb-2025.)
|- U = ( T ↾s X )   =>    |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> ( F e. ( S RingHom T ) <-> F e. ( S RingHom U ) ) )
Theorem	rhmeql 14263	The equalizer of two ring homomorphisms is a subring. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
|- ( ( F e. ( S RingHom T ) /\ G e. ( S RingHom T ) ) -> dom ( F i^i G ) e. (SubRing ` S ) )
Theorem	rhmima 14264	The homomorphic image of a subring is a subring. (Contributed by Stefan O'Rear, 10-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
|- ( ( F e. ( M RingHom N ) /\ X e. (SubRing ` M ) ) -> ( F " X ) e. (SubRing ` N ) )
Theorem	rnrhmsubrg 14265	The range of a ring homomorphism is a subring. (Contributed by SN, 18-Nov-2023.)
|- ( F e. ( M RingHom N ) -> ran F e. (SubRing ` N ) )
Theorem	subrgpropd 14266*	If two structures have the same group components (properties), they have the same set of subrings. (Contributed by Mario Carneiro, 9-Feb-2015.)
|- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )   =>    |- ( ph -> (SubRing ` K ) = (SubRing ` L ) )
Theorem	rhmpropd 14267*	Ring homomorphism depends only on the ring attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.)
|- ( ph -> B = ( Base ` J ) )   &    |- ( ph -> C = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ph -> C = ( Base ` M ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` J ) y ) = ( x ( +g ` L ) y ) )   &    |- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` M ) y ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` J ) y ) = ( x ( .r ` L ) y ) )   &    |- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` M ) y ) )   =>    |- ( ph -> ( J RingHom K ) = ( L RingHom M ) )
7.3.12  Left regular elements and domains
Syntax	crlreg 14268	Set of left-regular elements in a ring.
class RLReg
Syntax	cdomn 14269	Class of (ring theoretic) domains.
class Domn
Syntax	cidom 14270	Class of integral domains.
class IDomn
Definition	df-rlreg 14271*	Define the set of left-regular elements in a ring as those elements which are not left zero divisors, meaning that multiplying a nonzero element on the left by a left-regular element gives a nonzero product. (Contributed by Stefan O'Rear, 22-Mar-2015.)
|- RLReg = ( r e. _V |-> { x e. ( Base ` r ) | A. y e. ( Base ` r ) ( ( x ( .r ` r ) y ) = ( 0g ` r ) -> y = ( 0g ` r ) ) } )
Definition	df-domn 14272*	A domain is a nonzero ring in which there are no nontrivial zero divisors. (Contributed by Mario Carneiro, 28-Mar-2015.)
|- Domn = { r e. NzRing | [. ( Base ` r ) / b ]. [. ( 0g ` r ) / z ]. A. x e. b A. y e. b ( ( x ( .r ` r ) y ) = z -> ( x = z \/ y = z ) ) }
Definition	df-idom 14273	An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
|- IDomn = ( CRing i^i Domn )
Theorem	rrgmex 14274	A structure whose set of left-regular elements is inhabited is a set. (Contributed by Jim Kingdon, 12-Aug-2025.)
|- E = (RLReg ` R )   =>    |- ( A e. E -> R e. _V )
Theorem	rrgval 14275*	Value of the set or left-regular elements in a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
|- E = (RLReg ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   =>    |- E = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) }
Theorem	isrrg 14276*	Membership in the set of left-regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
|- E = (RLReg ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( X e. E <-> ( X e. B /\ A. y e. B ( ( X .x. y ) = .0. -> y = .0. ) ) )
Theorem	rrgeq0i 14277	Property of a left-regular element. (Contributed by Stefan O'Rear, 22-Mar-2015.)
|- E = (RLReg ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( X e. E /\ Y e. B ) -> ( ( X .x. Y ) = .0. -> Y = .0. ) )
Theorem	rrgeq0 14278	Left-multiplication by a left regular element does not change zeroness. (Contributed by Stefan O'Rear, 28-Mar-2015.)
|- E = (RLReg ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Ring /\ X e. E /\ Y e. B ) -> ( ( X .x. Y ) = .0. <-> Y = .0. ) )
Theorem	rrgss 14279	Left-regular elements are a subset of the base set. (Contributed by Stefan O'Rear, 22-Mar-2015.)
|- E = (RLReg ` R )   &    |- B = ( Base ` R )   =>    |- E C_ B
Theorem	unitrrg 14280	Units are regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
|- E = (RLReg ` R )   &    |- U = (Unit ` R )   =>    |- ( R e. Ring -> U C_ E )
Theorem	rrgnz 14281	In a nonzero ring, the zero is a left zero divisor (that is, not a left-regular element). (Contributed by Thierry Arnoux, 6-May-2025.)
|- E = (RLReg ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. NzRing -> -. .0. e. E )
Theorem	isdomn 14282*	Expand definition of a domain. (Contributed by Mario Carneiro, 28-Mar-2015.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. Domn <-> ( R e. NzRing /\ A. x e. B A. y e. B ( ( x .x. y ) = .0. -> ( x = .0. \/ y = .0. ) ) ) )
Theorem	domnnzr 14283	A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
|- ( R e. Domn -> R e. NzRing )
Theorem	domnring 14284	A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
|- ( R e. Domn -> R e. Ring )
Theorem	domneq0 14285	In a domain, a product is zero iff it has a zero factor. (Contributed by Mario Carneiro, 28-Mar-2015.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> ( ( X .x. Y ) = .0. <-> ( X = .0. \/ Y = .0. ) ) )
Theorem	domnmuln0 14286	In a domain, a product of nonzero elements is nonzero. (Contributed by Mario Carneiro, 6-May-2015.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Domn /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> ( X .x. Y ) =/= .0. )
Theorem	opprdomnbg 14287	A class is a domain if and only if its opposite is a domain, biconditional form of opprdomn 14288. (Contributed by SN, 15-Jun-2015.)
|- O = (oppr ` R )   =>    |- ( R e. V -> ( R e. Domn <-> O e. Domn ) )
Theorem	opprdomn 14288	The opposite of a domain is also a domain. (Contributed by Mario Carneiro, 15-Jun-2015.)
|- O = (oppr ` R )   =>    |- ( R e. Domn -> O e. Domn )
Theorem	isidom 14289	An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
|- ( R e. IDomn <-> ( R e. CRing /\ R e. Domn ) )
Theorem	idomdomd 14290	An integral domain is a domain. (Contributed by Thierry Arnoux, 22-Mar-2025.)
|- ( ph -> R e. IDomn )   =>    |- ( ph -> R e. Domn )
Theorem	idomcringd 14291	An integral domain is a commutative ring with unity. (Contributed by Thierry Arnoux, 4-May-2025.) (Proof shortened by SN, 14-May-2025.)
|- ( ph -> R e. IDomn )   =>    |- ( ph -> R e. CRing )
Theorem	idomringd 14292	An integral domain is a ring. (Contributed by Thierry Arnoux, 22-Mar-2025.)
|- ( ph -> R e. IDomn )   =>    |- ( ph -> R e. Ring )
7.4  Division rings and fields
7.4.1  Ring apartness
Syntax	capr 14293	Extend class notation with ring apartness.
class #r
Definition	df-apr 14294*	The relation between elements whose difference is invertible, which for a local ring is an apartness relation by aprap 14299. (Contributed by Jim Kingdon, 13-Feb-2025.)
|- #r = ( w e. _V |-> { <. x , y >. | ( ( x e. ( Base ` w ) /\ y e. ( Base ` w ) ) /\ ( x ( -g ` w ) y ) e. (Unit ` w ) ) } )
Theorem	aprval 14295	Expand Definition df-apr 14294. (Contributed by Jim Kingdon, 17-Feb-2025.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> # = (#r ` R ) )   &    |- ( ph -> .- = ( -g ` R ) )   &    |- ( ph -> U = (Unit ` R ) )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X # Y <-> ( X .- Y ) e. U ) )
Theorem	aprirr 14296	The apartness relation given by df-apr 14294 for a nonzero ring is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2025.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> # = (#r ` R ) )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   &    |- ( ph -> ( 1r ` R ) =/= ( 0g ` R ) )   =>    |- ( ph -> -. X # X )
Theorem	aprsym 14297	The apartness relation given by df-apr 14294 for a ring is symmetric. (Contributed by Jim Kingdon, 17-Feb-2025.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> # = (#r ` R ) )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X # Y -> Y # X ) )
Theorem	aprcotr 14298	The apartness relation given by df-apr 14294 for a local ring is cotransitive. (Contributed by Jim Kingdon, 17-Feb-2025.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> # = (#r ` R ) )   &    |- ( ph -> R e. LRing )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( X # Y -> ( X # Z \/ Y # Z ) ) )
Theorem	aprap 14299	The relation given by df-apr 14294 for a local ring is an apartness relation. (Contributed by Jim Kingdon, 20-Feb-2025.)
|- ( R e. LRing -> (#r ` R ) Ap ( Base ` R ) )
7.5  Left modules
7.5.1  Definition and basic properties
Syntax	clmod 14300	Extend class notation with class of all left modules.
class LMod