P. 146 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14501-14600   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-rgspn 14501*	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 14502	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 14503	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 14504	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 14505	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 14506	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 14507	Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.)
|- ( A e. (SubRing ` R ) -> R e. Ring )
Theorem	subrgsubg 14508	A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.)
|- ( A e. (SubRing ` R ) -> A e. (SubGrp ` R ) )
Theorem	subrg0 14509	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 14510	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 14511	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 14512	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 14513	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 14514	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 14515	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 14516	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 14517	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 14518	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 14519	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 14520	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 14521	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 14522*	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 14523	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 14524*	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 14525	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 14526	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 14527	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 14528	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 14529	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 14530	Restriction of the codomain of a (ring) homomorphism. resghm2b 14042 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 14531	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 14532	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 14533	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 14534*	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 14535*	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 14536	Set of left-regular elements in a ring.
class RLReg
Syntax	cdomn 14537	Class of (ring theoretic) domains.
class Domn
Syntax	cidom 14538	Class of integral domains.
class IDomn
Definition	df-rlreg 14539*	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 14540*	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 14541	An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
|- IDomn = ( CRing i^i Domn )
Theorem	rrgmex 14542	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 14543*	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 14544*	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 14545	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 14546	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	rrgsupp 14547	Left multiplication by a left regular element does not change the support set of a vector. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Revised by AV, 20-Jul-2019.)
|- E = (RLReg ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. E )   &    |- ( ph -> Y : I --> B )   =>    |- ( ph -> ( ( ( I X. { X } ) oF .x. Y ) supp .0. ) = ( Y supp .0. ) )
Theorem	rrgss 14548	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 14549	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 14550	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 14551*	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 14552	A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
|- ( R e. Domn -> R e. NzRing )
Theorem	domnring 14553	A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
|- ( R e. Domn -> R e. Ring )
Theorem	domneq0 14554	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 14555	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 14556	A class is a domain if and only if its opposite is a domain, biconditional form of opprdomn 14557. (Contributed by SN, 15-Jun-2015.)
|- O = (oppr ` R )   =>    |- ( R e. V -> ( R e. Domn <-> O e. Domn ) )
Theorem	opprdomn 14557	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 14558	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 14559	An integral domain is a domain. (Contributed by Thierry Arnoux, 22-Mar-2025.)
|- ( ph -> R e. IDomn )   =>    |- ( ph -> R e. Domn )
Theorem	idomcringd 14560	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 14561	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 14562	Extend class notation with ring apartness.
class #r
Definition	df-apr 14563*	The relation between elements whose difference is invertible, which for a local ring is an apartness relation by aprap 14571. (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 14564	Expand Definition df-apr 14563. (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	aprunit 14565	The df-apr 14563 relation with zero expresses whether a ring element is a unit. That is, the difference of an element of a ring and zero is invertible iff the element is a unit. (Contributed by Jim Kingdon, 29-May-2026.)
|- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- U = (Unit ` R )   &    |- # = (#r ` R )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( X # .0. <-> X e. U ) )
Theorem	ringunitap 14566	Elementhood in the set of units. (Contributed by Jim Kingdon, 30-May-2026.)
|- B = ( Base ` R )   &    |- U = (Unit ` R )   &    |- .0. = ( 0g ` R )   &    |- # = (#r ` R )   =>    |- ( R e. Ring -> ( X e. U <-> ( X e. B /\ X # .0. ) ) )
Theorem	ringunitsap0 14567*	The set of units of a ring. If R is a local ring, # is an apartness and this theorem states that the units of a ring are those elements apart from zero (see aprlring 14573). Given the definition of #r this theorem holds even if # is not an apartness, however. (Contributed by Jim Kingdon, 31-May-2026.)
|- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- # = (#r ` R )   =>    |- ( R e. Ring -> { x e. B | x # .0. } = (Unit ` R ) )
Theorem	aprirr 14568	The apartness relation given by df-apr 14563 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 14569	The apartness relation given by df-apr 14563 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 14570	The apartness relation given by df-apr 14563 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 14571	The relation given by df-apr 14563 for a local ring is an apartness relation. (Contributed by Jim Kingdon, 20-Feb-2025.)
|- ( R e. LRing -> (#r ` R ) Ap ( Base ` R ) )
Theorem	aprnzr 14572	If the relation given by df-apr 14563 on a ring is an apartness relation, then the ring is a nonzero ring. (Contributed by Jim Kingdon, 27-May-2026.)
|- ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) -> R e. NzRing )
Theorem	aprlring 14573	A ring is a local ring if and only if the relation given by df-apr 14563 is an apartness relation. (Contributed by Jim Kingdon, 28-May-2026.)
|- ( R e. Ring -> ( R e. LRing <-> (#r ` R ) Ap ( Base ` R ) ) )
Theorem	aprprop 14574	If two structures have the same ring components (properties), df-apr 14563 generates the same relation for both of them. (Contributed by Jim Kingdon, 31-May-2026.)
|- ( Base ` K ) = ( Base ` L )   &    |- ( +g ` K ) = ( +g ` L )   &    |- ( .r ` K ) = ( .r ` L )   =>    |- ( K e. Ring -> (#r ` K ) = (#r ` L ) )
7.4.2  Definition and basic properties
Syntax	cdr 14575	Extend class notation with class of all division rings.
class DivRing
Syntax	cfield 14576	Class of fields.
class Field
Definition	df-drngap 14577	Define class of all division rings. A division ring is a ring in which the relation given by df-apr 14563 is a tight apartness. (Contributed by Jim Kingdon, 29-May-2026.)
|- DivRing = { r e. Ring | (#r ` r ) TAp ( Base ` r ) }
Definition	df-field 14578	A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
|- Field = ( DivRing i^i CRing )
Theorem	isdrngtap 14579	The predicate "is a division ring". (Contributed by Jim Kingdon, 29-May-2026.)
|- B = ( Base ` R )   &    |- # = (#r ` R )   =>    |- ( R e. DivRing <-> ( R e. Ring /\ # TAp B ) )
Theorem	drnglring 14580	A division ring is a local ring. (Contributed by Jim Kingdon, 29-May-2026.)
|- ( R e. DivRing -> R e. LRing )
Theorem	drngunitap 14581	Elementhood in the set of units when R is a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
|- B = ( Base ` R )   &    |- U = (Unit ` R )   &    |- .0. = ( 0g ` R )   &    |- # = (#r ` R )   =>    |- ( R e. DivRing -> ( X e. U <-> ( X e. B /\ X # .0. ) ) )
Theorem	drnguiap 14582*	The set of units of a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
|- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- # = (#r ` R )   =>    |- ( R e. DivRing -> { x e. B | x # .0. } = (Unit ` R ) )
Theorem	drngring 14583	A division ring is a ring. (Contributed by NM, 8-Sep-2011.)
|- ( R e. DivRing -> R e. Ring )
Theorem	drngringd 14584	A division ring is a ring. (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. DivRing )   =>    |- ( ph -> R e. Ring )
Theorem	drnggrpd 14585	A division ring is a group (deduction form). (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. DivRing )   =>    |- ( ph -> R e. Grp )
Theorem	drnggrp 14586	A division ring is a group (closed form). (Contributed by NM, 8-Sep-2011.)
|- ( R e. DivRing -> R e. Grp )
Theorem	isfld 14587	A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
|- ( R e. Field <-> ( R e. DivRing /\ R e. CRing ) )
Theorem	flddrngd 14588	A field is a division ring. (Contributed by SN, 17-Jan-2025.)
|- ( ph -> R e. Field )   =>    |- ( ph -> R e. DivRing )
Theorem	fldcrngd 14589	A field is a commutative ring. (Contributed by SN, 23-Nov-2024.)
|- ( ph -> R e. Field )   =>    |- ( ph -> R e. CRing )
Theorem	drngprop 14590	If two structures have the same ring components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 28-Dec-2014.)
|- ( Base ` K ) = ( Base ` L )   &    |- ( +g ` K ) = ( +g ` L )   &    |- ( .r ` K ) = ( .r ` L )   =>    |- ( K e. DivRing <-> L e. DivRing )
Theorem	drngunz 14591	A division ring's unity is different from its zero. (Contributed by NM, 8-Sep-2011.)
|- .0. = ( 0g ` R )   &    |- .1. = ( 1r ` R )   =>    |- ( R e. DivRing -> .1. =/= .0. )
Theorem	drngnzr 14592	A division ring is a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
|- ( R e. DivRing -> R e. NzRing )
Theorem	opprdrng 14593	The opposite of a division ring is also a division ring. (Contributed by NM, 18-Oct-2014.)
|- O = (oppr ` R )   =>    |- ( R e. DivRing <-> O e. DivRing )
Theorem	ring1zr 14594	The only unital ring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). This holds already for nonunital rings, see rng1zr 14234, and semirings, see srg1zr 14265. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) (Proof shortened by AV, 7-Feb-2020.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .* = ( .r ` R )   =>    |- ( ( ( R e. Ring /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
Theorem	ringen1zr0 14595	The only unital ring with one element is the zero ring (at least if its operations are internal binary operations). This holds already for nonunital rings, see rngen1zr0 14236, and semirings, see srgen1zr0 14266. (Contributed by FL, 15-Feb-2010.) (Revised by AV, 25-Jan-2020.) (Proof shortened by AV, 19-Jun-2026.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .* = ( .r ` R )   &    |- Z = ( 0g ` R )   =>    |- ( ( R e. Ring /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( B ~~ 1o <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
7.5  Left modules
7.5.1  Definition and basic properties
Syntax	clmod 14596	Extend class notation with class of all left modules.
class LMod
Syntax	cscaf 14597	The functionalization of the scalar multiplication operation.
class .sf
Definition	df-lmod 14598*	Define the class of all left modules, which are generalizations of left vector spaces. A left module over a ring is an (Abelian) group (vectors) together with a ring (scalars) and a left scalar product connecting them. (Contributed by NM, 4-Nov-2013.)
|- LMod = { g e. Grp | [. ( Base ` g ) / v ]. [. ( +g ` g ) / a ]. [. (Scalar ` g ) / f ]. [. ( .s ` g ) / s ]. [. ( Base ` f ) / k ]. [. ( +g ` f ) / p ]. [. ( .r ` f ) / t ]. ( f e. Ring /\ A. q e. k A. r e. k A. x e. v A. w e. v ( ( ( r s w ) e. v /\ ( r s ( w a x ) ) = ( ( r s w ) a ( r s x ) ) /\ ( ( q p r ) s w ) = ( ( q s w ) a ( r s w ) ) ) /\ ( ( ( q t r ) s w ) = ( q s ( r s w ) ) /\ ( ( 1r ` f ) s w ) = w ) ) ) }
Definition	df-scaf 14599*	Define the functionalization of the .s operator. This restricts the value of .s to the stated domain, which is necessary when working with restricted structures, whose operations may be defined on a larger set than the true base. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- .sf = ( g e. _V |-> ( x e. ( Base ` (Scalar ` g ) ) , y e. ( Base ` g ) |-> ( x ( .s ` g ) y ) ) )
Theorem	islmod 14600*	The predicate "is a left module". (Contributed by NM, 4-Nov-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .x. = ( .s ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .+^ = ( +g ` F )   &    |- .X. = ( .r ` F )   &    |- .1. = ( 1r ` F )   =>    |- ( W e. LMod <-> ( W e. Grp /\ F e. Ring /\ A. q e. K A. r e. K A. x e. V A. w e. V ( ( ( r .x. w ) e. V /\ ( r .x. ( w .+ x ) ) = ( ( r .x. w ) .+ ( r .x. x ) ) /\ ( ( q .+^ r ) .x. w ) = ( ( q .x. w ) .+ ( r .x. w ) ) ) /\ ( ( ( q .X. r ) .x. w ) = ( q .x. ( r .x. w ) ) /\ ( .1. .x. w ) = w ) ) ) )