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	subrgintm 14201*	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 14202	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 14203	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 14204	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 14205	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 14206	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 14207	Restriction of the codomain of a (ring) homomorphism. resghm2b 13794 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 14208	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 14209	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 14210	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 14211*	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 14212*	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 14213	Set of left-regular elements in a ring.
class RLReg
Syntax	cdomn 14214	Class of (ring theoretic) domains.
class Domn
Syntax	cidom 14215	Class of integral domains.
class IDomn
Definition	df-rlreg 14216*	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 14217*	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 14218	An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
|- IDomn = ( CRing i^i Domn )
Theorem	rrgmex 14219	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 14220*	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 14221*	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 14222	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 14223	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 14224	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 14225	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 14226	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 14227*	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 14228	A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
|- ( R e. Domn -> R e. NzRing )
Theorem	domnring 14229	A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
|- ( R e. Domn -> R e. Ring )
Theorem	domneq0 14230	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 14231	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 14232	A class is a domain if and only if its opposite is a domain, biconditional form of opprdomn 14233. (Contributed by SN, 15-Jun-2015.)
|- O = (oppr ` R )   =>    |- ( R e. V -> ( R e. Domn <-> O e. Domn ) )
Theorem	opprdomn 14233	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 14234	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 14235	An integral domain is a domain. (Contributed by Thierry Arnoux, 22-Mar-2025.)
|- ( ph -> R e. IDomn )   =>    |- ( ph -> R e. Domn )
Theorem	idomcringd 14236	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 14237	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 14238	Extend class notation with ring apartness.
class #r
Definition	df-apr 14239*	The relation between elements whose difference is invertible, which for a local ring is an apartness relation by aprap 14244. (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 14240	Expand Definition df-apr 14239. (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 14241	The apartness relation given by df-apr 14239 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 14242	The apartness relation given by df-apr 14239 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 14243	The apartness relation given by df-apr 14239 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 14244	The relation given by df-apr 14239 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 14245	Extend class notation with class of all left modules.
class LMod
Syntax	cscaf 14246	The functionalization of the scalar multiplication operation.
class .sf
Definition	df-lmod 14247*	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 14248*	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 14249*	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 ) ) ) )
Theorem	lmodlema 14250	Lemma for properties of a left module. (Contributed by NM, 8-Dec-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 /\ ( Q e. K /\ R e. K ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( R .x. Y ) e. V /\ ( R .x. ( Y .+ X ) ) = ( ( R .x. Y ) .+ ( R .x. X ) ) /\ ( ( Q .+^ R ) .x. Y ) = ( ( Q .x. Y ) .+ ( R .x. Y ) ) ) /\ ( ( ( Q .X. R ) .x. Y ) = ( Q .x. ( R .x. Y ) ) /\ ( .1. .x. Y ) = Y ) ) )
Theorem	islmodd 14251*	Properties that determine a left module. See note in isgrpd2 13549 regarding the ph on hypotheses that name structure components. (Contributed by Mario Carneiro, 22-Jun-2014.)
|- ( ph -> V = ( Base ` W ) )   &    |- ( ph -> .+ = ( +g ` W ) )   &    |- ( ph -> F = (Scalar ` W ) )   &    |- ( ph -> .x. = ( .s ` W ) )   &    |- ( ph -> B = ( Base ` F ) )   &    |- ( ph -> .+^ = ( +g ` F ) )   &    |- ( ph -> .X. = ( .r ` F ) )   &    |- ( ph -> .1. = ( 1r ` F ) )   &    |- ( ph -> F e. Ring )   &    |- ( ph -> W e. Grp )   &    |- ( ( ph /\ x e. B /\ y e. V ) -> ( x .x. y ) e. V )   &    |- ( ( ph /\ ( x e. B /\ y e. V /\ z e. V ) ) -> ( x .x. ( y .+ z ) ) = ( ( x .x. y ) .+ ( x .x. z ) ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. V ) ) -> ( ( x .+^ y ) .x. z ) = ( ( x .x. z ) .+ ( y .x. z ) ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. V ) ) -> ( ( x .X. y ) .x. z ) = ( x .x. ( y .x. z ) ) )   &    |- ( ( ph /\ x e. V ) -> ( .1. .x. x ) = x )   =>    |- ( ph -> W e. LMod )
Theorem	lmodgrp 14252	A left module is a group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.)
|- ( W e. LMod -> W e. Grp )
Theorem	lmodring 14253	The scalar component of a left module is a ring. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- F = (Scalar ` W )   =>    |- ( W e. LMod -> F e. Ring )
Theorem	lmodfgrp 14254	The scalar component of a left module is an additive group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- F = (Scalar ` W )   =>    |- ( W e. LMod -> F e. Grp )
Theorem	lmodgrpd 14255	A left module is a group. (Contributed by SN, 16-May-2024.)
|- ( ph -> W e. LMod )   =>    |- ( ph -> W e. Grp )
Theorem	lmodbn0 14256	The base set of a left module is nonempty. It is also inhabited (by lmod0vcl 14275). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- B = ( Base ` W )   =>    |- ( W e. LMod -> B =/= (/) )
Theorem	lmodacl 14257	Closure of ring addition for a left module. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .+ = ( +g ` F )   =>    |- ( ( W e. LMod /\ X e. K /\ Y e. K ) -> ( X .+ Y ) e. K )
Theorem	lmodmcl 14258	Closure of ring multiplication for a left module. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .x. = ( .r ` F )   =>    |- ( ( W e. LMod /\ X e. K /\ Y e. K ) -> ( X .x. Y ) e. K )
Theorem	lmodsn0 14259	The set of scalars in a left module is nonempty. It is also inhabited, by lmod0cl 14272. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- F = (Scalar ` W )   &    |- B = ( Base ` F )   =>    |- ( W e. LMod -> B =/= (/) )
Theorem	lmodvacl 14260	Closure of vector addition for a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   =>    |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( X .+ Y ) e. V )
Theorem	lmodass 14261	Left module vector sum is associative. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   =>    |- ( ( W e. LMod /\ ( X e. V /\ Y e. V /\ Z e. V ) ) -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )
Theorem	lmodlcan 14262	Left cancellation law for vector sum. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   =>    |- ( ( W e. LMod /\ ( X e. V /\ Y e. V /\ Z e. V ) ) -> ( ( Z .+ X ) = ( Z .+ Y ) <-> X = Y ) )
Theorem	lmodvscl 14263	Closure of scalar product for a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- .x. = ( .s ` W )   &    |- K = ( Base ` F )   =>    |- ( ( W e. LMod /\ R e. K /\ X e. V ) -> ( R .x. X ) e. V )
Theorem	scaffvalg 14264*	The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)
|- B = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .xb = ( .sf ` W )   &    |- .x. = ( .s ` W )   =>    |- ( W e. V -> .xb = ( x e. K , y e. B |-> ( x .x. y ) ) )
Theorem	scafvalg 14265	The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- B = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .xb = ( .sf ` W )   &    |- .x. = ( .s ` W )   =>    |- ( ( W e. V /\ X e. K /\ Y e. B ) -> ( X .xb Y ) = ( X .x. Y ) )
Theorem	scafeqg 14266	If the scalar multiplication operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- B = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .xb = ( .sf ` W )   &    |- .x. = ( .s ` W )   =>    |- ( ( W e. V /\ .x. Fn ( K X. B ) ) -> .xb = .x. )
Theorem	scaffng 14267	The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- B = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .xb = ( .sf ` W )   =>    |- ( W e. V -> .xb Fn ( K X. B ) )
Theorem	lmodscaf 14268	The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- B = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .xb = ( .sf ` W )   =>    |- ( W e. LMod -> .xb : ( K X. B ) --> B )
Theorem	lmodvsdi 14269	Distributive law for scalar product (left-distributivity). (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- F = (Scalar ` W )   &    |- .x. = ( .s ` W )   &    |- K = ( Base ` F )   =>    |- ( ( W e. LMod /\ ( R e. K /\ X e. V /\ Y e. V ) ) -> ( R .x. ( X .+ Y ) ) = ( ( R .x. X ) .+ ( R .x. Y ) ) )
Theorem	lmodvsdir 14270	Distributive law for scalar product (right-distributivity). (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- F = (Scalar ` W )   &    |- .x. = ( .s ` W )   &    |- K = ( Base ` F )   &    |- .+^ = ( +g ` F )   =>    |- ( ( W e. LMod /\ ( Q e. K /\ R e. K /\ X e. V ) ) -> ( ( Q .+^ R ) .x. X ) = ( ( Q .x. X ) .+ ( R .x. X ) ) )
Theorem	lmodvsass 14271	Associative law for scalar product. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
|- V = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- .x. = ( .s ` W )   &    |- K = ( Base ` F )   &    |- .X. = ( .r ` F )   =>    |- ( ( W e. LMod /\ ( Q e. K /\ R e. K /\ X e. V ) ) -> ( ( Q .X. R ) .x. X ) = ( Q .x. ( R .x. X ) ) )
Theorem	lmod0cl 14272	The ring zero in a left module belongs to the set of scalars. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .0. = ( 0g ` F )   =>    |- ( W e. LMod -> .0. e. K )
Theorem	lmod1cl 14273	The ring unity in a left module belongs to the set of scalars. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .1. = ( 1r ` F )   =>    |- ( W e. LMod -> .1. e. K )
Theorem	lmodvs1 14274	Scalar product with the ring unity. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- .x. = ( .s ` W )   &    |- .1. = ( 1r ` F )   =>    |- ( ( W e. LMod /\ X e. V ) -> ( .1. .x. X ) = X )
Theorem	lmod0vcl 14275	The zero vector is a vector. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   =>    |- ( W e. LMod -> .0. e. V )
Theorem	lmod0vlid 14276	Left identity law for the zero vector. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .0. = ( 0g ` W )   =>    |- ( ( W e. LMod /\ X e. V ) -> ( .0. .+ X ) = X )
Theorem	lmod0vrid 14277	Right identity law for the zero vector. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .0. = ( 0g ` W )   =>    |- ( ( W e. LMod /\ X e. V ) -> ( X .+ .0. ) = X )
Theorem	lmod0vid 14278	Identity equivalent to the value of the zero vector. Provides a convenient way to compute the value. (Contributed by NM, 9-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .0. = ( 0g ` W )   =>    |- ( ( W e. LMod /\ X e. V ) -> ( ( X .+ X ) = X <-> .0. = X ) )
Theorem	lmod0vs 14279	Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- .x. = ( .s ` W )   &    |- O = ( 0g ` F )   &    |- .0. = ( 0g ` W )   =>    |- ( ( W e. LMod /\ X e. V ) -> ( O .x. X ) = .0. )
Theorem	lmodvs0 14280	Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- F = (Scalar ` W )   &    |- .x. = ( .s ` W )   &    |- K = ( Base ` F )   &    |- .0. = ( 0g ` W )   =>    |- ( ( W e. LMod /\ X e. K ) -> ( X .x. .0. ) = .0. )
Theorem	lmodvsmmulgdi 14281	Distributive law for a group multiple of a scalar multiplication. (Contributed by AV, 2-Sep-2019.)
|- V = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- .x. = ( .s ` W )   &    |- K = ( Base ` F )   &    |- .^ = (.g ` W )   &    |- E = (.g ` F )   =>    |- ( ( W e. LMod /\ ( C e. K /\ N e. NN0 /\ X e. V ) ) -> ( N .^ ( C .x. X ) ) = ( ( N E C ) .x. X ) )
Theorem	lmodfopnelem1 14282	Lemma 1 for lmodfopne 14284. (Contributed by AV, 2-Oct-2021.)
|- .x. = ( .sf ` W )   &    |- .+ = ( +f ` W )   &    |- V = ( Base ` W )   &    |- S = (Scalar ` W )   &    |- K = ( Base ` S )   =>    |- ( ( W e. LMod /\ .+ = .x. ) -> V = K )
Theorem	lmodfopnelem2 14283	Lemma 2 for lmodfopne 14284. (Contributed by AV, 2-Oct-2021.)
|- .x. = ( .sf ` W )   &    |- .+ = ( +f ` W )   &    |- V = ( Base ` W )   &    |- S = (Scalar ` W )   &    |- K = ( Base ` S )   &    |- .0. = ( 0g ` S )   &    |- .1. = ( 1r ` S )   =>    |- ( ( W e. LMod /\ .+ = .x. ) -> ( .0. e. V /\ .1. e. V ) )
Theorem	lmodfopne 14284	The (functionalized) operations of a left module (over a nonzero ring) cannot be identical. (Contributed by NM, 31-May-2008.) (Revised by AV, 2-Oct-2021.)
|- .x. = ( .sf ` W )   &    |- .+ = ( +f ` W )   &    |- V = ( Base ` W )   &    |- S = (Scalar ` W )   &    |- K = ( Base ` S )   &    |- .0. = ( 0g ` S )   &    |- .1. = ( 1r ` S )   =>    |- ( ( W e. LMod /\ .1. =/= .0. ) -> .+ =/= .x. )
Theorem	lcomf 14285	A linear-combination sum is a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
|- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .x. = ( .s ` W )   &    |- B = ( Base ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> G : I --> K )   &    |- ( ph -> H : I --> B )   &    |- ( ph -> I e. V )   =>    |- ( ph -> ( G oF .x. H ) : I --> B )
Theorem	lmodvnegcl 14286	Closure of vector negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- N = ( invg ` W )   =>    |- ( ( W e. LMod /\ X e. V ) -> ( N ` X ) e. V )
Theorem	lmodvnegid 14287	Addition of a vector with its negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( invg ` W )   =>    |- ( ( W e. LMod /\ X e. V ) -> ( X .+ ( N ` X ) ) = .0. )
Theorem	lmodvneg1 14288	Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- N = ( invg ` W )   &    |- F = (Scalar ` W )   &    |- .x. = ( .s ` W )   &    |- .1. = ( 1r ` F )   &    |- M = ( invg ` F )   =>    |- ( ( W e. LMod /\ X e. V ) -> ( ( M ` .1. ) .x. X ) = ( N ` X ) )
Theorem	lmodvsneg 14289	Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.)
|- B = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- .x. = ( .s ` W )   &    |- N = ( invg ` W )   &    |- K = ( Base ` F )   &    |- M = ( invg ` F )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> X e. B )   &    |- ( ph -> R e. K )   =>    |- ( ph -> ( N ` ( R .x. X ) ) = ( ( M ` R ) .x. X ) )
Theorem	lmodvsubcl 14290	Closure of vector subtraction. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .- = ( -g ` W )   =>    |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( X .- Y ) e. V )
Theorem	lmodcom 14291	Left module vector sum is commutative. (Contributed by Gérard Lang, 25-Jun-2014.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   =>    |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( X .+ Y ) = ( Y .+ X ) )
Theorem	lmodabl 14292	A left module is an abelian group (of vectors, under addition). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.)
|- ( W e. LMod -> W e. Abel )
Theorem	lmodcmn 14293	A left module is a commutative monoid under addition. (Contributed by NM, 7-Jan-2015.)
|- ( W e. LMod -> W e. CMnd )
Theorem	lmodnegadd 14294	Distribute negation through addition of scalar products. (Contributed by NM, 9-Apr-2015.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .x. = ( .s ` W )   &    |- N = ( invg ` W )   &    |- R = (Scalar ` W )   &    |- K = ( Base ` R )   &    |- I = ( invg ` R )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> A e. K )   &    |- ( ph -> B e. K )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   =>    |- ( ph -> ( N ` ( ( A .x. X ) .+ ( B .x. Y ) ) ) = ( ( ( I ` A ) .x. X ) .+ ( ( I ` B ) .x. Y ) ) )
Theorem	lmod4 14295	Commutative/associative law for left module vector sum. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   =>    |- ( ( W e. LMod /\ ( X e. V /\ Y e. V ) /\ ( Z e. V /\ U e. V ) ) -> ( ( X .+ Y ) .+ ( Z .+ U ) ) = ( ( X .+ Z ) .+ ( Y .+ U ) ) )
Theorem	lmodvsubadd 14296	Relationship between vector subtraction and addition. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .- = ( -g ` W )   =>    |- ( ( W e. LMod /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A .- B ) = C <-> ( B .+ C ) = A ) )
Theorem	lmodvaddsub4 14297	Vector addition/subtraction law. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .- = ( -g ` W )   =>    |- ( ( W e. LMod /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( A .+ B ) = ( C .+ D ) <-> ( A .- C ) = ( D .- B ) ) )
Theorem	lmodvpncan 14298	Addition/subtraction cancellation law for vectors. (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .- = ( -g ` W )   =>    |- ( ( W e. LMod /\ A e. V /\ B e. V ) -> ( ( A .+ B ) .- B ) = A )
Theorem	lmodvnpcan 14299	Cancellation law for vector subtraction (Contributed by NM, 19-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .- = ( -g ` W )   =>    |- ( ( W e. LMod /\ A e. V /\ B e. V ) -> ( ( A .- B ) .+ B ) = A )
Theorem	lmodvsubval2 14300	Value of vector subtraction in terms of addition. (Contributed by NM, 31-Mar-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .- = ( -g ` W )   &    |- F = (Scalar ` W )   &    |- .x. = ( .s ` W )   &    |- N = ( invg ` F )   &    |- .1. = ( 1r ` F )   =>    |- ( ( W e. LMod /\ A e. V /\ B e. V ) -> ( A .- B ) = ( A .+ ( ( N ` .1. ) .x. B ) ) )