P. 140 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13901-14000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	isdomn 13901*	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 13902	A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
|- ( R e. Domn -> R e. NzRing )
Theorem	domnring 13903	A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
|- ( R e. Domn -> R e. Ring )
Theorem	domneq0 13904	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 13905	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 13906	A class is a domain if and only if its opposite is a domain, biconditional form of opprdomn 13907. (Contributed by SN, 15-Jun-2015.)
|- O = (oppr ` R )   =>    |- ( R e. V -> ( R e. Domn <-> O e. Domn ) )
Theorem	opprdomn 13907	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 13908	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 13909	An integral domain is a domain. (Contributed by Thierry Arnoux, 22-Mar-2025.)
|- ( ph -> R e. IDomn )   =>    |- ( ph -> R e. Domn )
Theorem	idomcringd 13910	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 13911	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 13912	Extend class notation with ring apartness.
class #r
Definition	df-apr 13913*	The relation between elements whose difference is invertible, which for a local ring is an apartness relation by aprap 13918. (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 13914	Expand Definition df-apr 13913. (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 13915	The apartness relation given by df-apr 13913 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 13916	The apartness relation given by df-apr 13913 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 13917	The apartness relation given by df-apr 13913 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 13918	The relation given by df-apr 13913 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 13919	Extend class notation with class of all left modules.
class LMod
Syntax	cscaf 13920	The functionalization of the scalar multiplication operation.
class .sf
Definition	df-lmod 13921*	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 13922*	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 13923*	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 13924	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 13925*	Properties that determine a left module. See note in isgrpd2 13223 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 13926	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 13927	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 13928	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 13929	A left module is a group. (Contributed by SN, 16-May-2024.)
|- ( ph -> W e. LMod )   =>    |- ( ph -> W e. Grp )
Theorem	lmodbn0 13930	The base set of a left module is nonempty. It is also inhabited (by lmod0vcl 13949). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- B = ( Base ` W )   =>    |- ( W e. LMod -> B =/= (/) )
Theorem	lmodacl 13931	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 13932	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 13933	The set of scalars in a left module is nonempty. It is also inhabited, by lmod0cl 13946. (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 13934	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 13935	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 13936	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 13937	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 13938*	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 13939	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 13940	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 13941	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 13942	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 13943	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 13944	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 13945	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 13946	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 13947	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 13948	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 13949	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 13950	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 13951	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 13952	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 13953	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 13954	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 13955	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 13956	Lemma 1 for lmodfopne 13958. (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 13957	Lemma 2 for lmodfopne 13958. (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 13958	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 13959	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 13960	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 13961	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 13962	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 13963	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 13964	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 13965	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 13966	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 13967	A left module is a commutative monoid under addition. (Contributed by NM, 7-Jan-2015.)
|- ( W e. LMod -> W e. CMnd )
Theorem	lmodnegadd 13968	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 13969	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 13970	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 13971	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 13972	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 13973	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 13974	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 ) ) )
Theorem	lmodsubvs 13975	Subtraction of a scalar product in terms of addition. (Contributed by NM, 9-Apr-2015.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .- = ( -g ` W )   &    |- .x. = ( .s ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- N = ( invg ` F )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> A e. K )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   =>    |- ( ph -> ( X .- ( A .x. Y ) ) = ( X .+ ( ( N ` A ) .x. Y ) ) )
Theorem	lmodsubdi 13976	Scalar multiplication distributive law for subtraction. (Contributed by NM, 2-Jul-2014.)
|- V = ( Base ` W )   &    |- .x. = ( .s ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .- = ( -g ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> A e. K )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   =>    |- ( ph -> ( A .x. ( X .- Y ) ) = ( ( A .x. X ) .- ( A .x. Y ) ) )
Theorem	lmodsubdir 13977	Scalar multiplication distributive law for subtraction. (Contributed by NM, 2-Jul-2014.)
|- V = ( Base ` W )   &    |- .x. = ( .s ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .- = ( -g ` W )   &    |- S = ( -g ` F )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> A e. K )   &    |- ( ph -> B e. K )   &    |- ( ph -> X e. V )   =>    |- ( ph -> ( ( A S B ) .x. X ) = ( ( A .x. X ) .- ( B .x. X ) ) )
Theorem	lmodsubeq0 13978	If the difference between two vectors is zero, they are equal. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- .- = ( -g ` W )   =>    |- ( ( W e. LMod /\ A e. V /\ B e. V ) -> ( ( A .- B ) = .0. <-> A = B ) )
Theorem	lmodsubid 13979	Subtraction of a vector from itself. (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- .- = ( -g ` W )   =>    |- ( ( W e. LMod /\ A e. V ) -> ( A .- A ) = .0. )
Theorem	lmodprop2d 13980*	If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 13981 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.)
|- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- F = (Scalar ` K )   &    |- G = (Scalar ` L )   &    |- ( ph -> P = ( Base ` F ) )   &    |- ( ph -> P = ( Base ` G ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   &    |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> ( x ( +g ` F ) y ) = ( x ( +g ` G ) y ) )   &    |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> ( x ( .r ` F ) y ) = ( x ( .r ` G ) y ) )   &    |- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` K ) y ) = ( x ( .s ` L ) y ) )   =>    |- ( ph -> ( K e. LMod <-> L e. LMod ) )
Theorem	lmodpropd 13981*	If two structures have the same components (properties), one is a left module iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 27-Jun-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 -> F = (Scalar ` K ) )   &    |- ( ph -> F = (Scalar ` L ) )   &    |- P = ( Base ` F )   &    |- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` K ) y ) = ( x ( .s ` L ) y ) )   =>    |- ( ph -> ( K e. LMod <-> L e. LMod ) )
Theorem	rmodislmodlem 13982*	Lemma for rmodislmod 13983. This is the part of the proof of rmodislmod 13983 which requires the scalar ring to be commutative. (Contributed by AV, 3-Dec-2021.)
|- V = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .s ` R )   &    |- F = (Scalar ` R )   &    |- K = ( Base ` F )   &    |- .+^ = ( +g ` F )   &    |- .X. = ( .r ` F )   &    |- .1. = ( 1r ` F )   &    |- ( R e. Grp /\ F e. Ring /\ A. q e. K A. r e. K A. x e. V A. w e. V ( ( ( w .x. r ) e. V /\ ( ( w .+ x ) .x. r ) = ( ( w .x. r ) .+ ( x .x. r ) ) /\ ( w .x. ( q .+^ r ) ) = ( ( w .x. q ) .+ ( w .x. r ) ) ) /\ ( ( w .x. ( q .X. r ) ) = ( ( w .x. q ) .x. r ) /\ ( w .x. .1. ) = w ) ) )   &    |- .* = ( s e. K , v e. V |-> ( v .x. s ) )   &    |- L = ( R sSet <. ( .s ` ndx ) , .* >. )   =>    |- ( ( F e. CRing /\ ( a e. K /\ b e. K /\ c e. V ) ) -> ( ( a .X. b ) .* c ) = ( a .* ( b .* c ) ) )
Theorem	rmodislmod 13983*	The right module R induces a left module L by replacing the scalar multiplication with a reversed multiplication if the scalar ring is commutative. The hypothesis "rmodislmod.r" is a definition of a right module analogous to Definition df-lmod 13921 of a left module, see also islmod 13923. (Contributed by AV, 3-Dec-2021.) (Proof shortened by AV, 18-Oct-2024.)
|- V = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .s ` R )   &    |- F = (Scalar ` R )   &    |- K = ( Base ` F )   &    |- .+^ = ( +g ` F )   &    |- .X. = ( .r ` F )   &    |- .1. = ( 1r ` F )   &    |- ( R e. Grp /\ F e. Ring /\ A. q e. K A. r e. K A. x e. V A. w e. V ( ( ( w .x. r ) e. V /\ ( ( w .+ x ) .x. r ) = ( ( w .x. r ) .+ ( x .x. r ) ) /\ ( w .x. ( q .+^ r ) ) = ( ( w .x. q ) .+ ( w .x. r ) ) ) /\ ( ( w .x. ( q .X. r ) ) = ( ( w .x. q ) .x. r ) /\ ( w .x. .1. ) = w ) ) )   &    |- .* = ( s e. K , v e. V |-> ( v .x. s ) )   &    |- L = ( R sSet <. ( .s ` ndx ) , .* >. )   =>    |- ( F e. CRing -> L e. LMod )
7.5.2  Subspaces and spans in a left module
Syntax	clss 13984	Extend class notation with linear subspaces of a left module or left vector space.
class LSubSp
Definition	df-lssm 13985*	A linear subspace of a left module or left vector space is an inhabited (in contrast to non-empty for non-intuitionistic logic) subset of the base set of the left-module/vector space with a closure condition on vector addition and scalar multiplication. (Contributed by NM, 8-Dec-2013.)
|- LSubSp = ( w e. _V |-> { s e. ~P ( Base ` w ) | ( E. j j e. s /\ A. x e. ( Base ` (Scalar ` w ) ) A. a e. s A. b e. s ( ( x ( .s ` w ) a ) ( +g ` w ) b ) e. s ) } )
Theorem	lssex 13986	Existence of a linear subspace. (Contributed by Jim Kingdon, 27-Apr-2025.)
|- ( W e. V -> ( LSubSp ` W ) e. _V )
Theorem	lssmex 13987	If a linear subspace is inhabited, the class it is built from is a set. (Contributed by Jim Kingdon, 28-Apr-2025.)
|- S = ( LSubSp ` W )   =>    |- ( U e. S -> W e. _V )
Theorem	lsssetm 13988*	The set of all (not necessarily closed) linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 15-Jul-2014.)
|- F = (Scalar ` W )   &    |- B = ( Base ` F )   &    |- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .x. = ( .s ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( W e. X -> S = { s e. ~P V | ( E. j j e. s /\ A. x e. B A. a e. s A. b e. s ( ( x .x. a ) .+ b ) e. s ) } )
Theorem	islssm 13989*	The predicate "is a subspace" (of a left module or left vector space). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
|- F = (Scalar ` W )   &    |- B = ( Base ` F )   &    |- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .x. = ( .s ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( U e. S <-> ( U C_ V /\ E. j j e. U /\ A. x e. B A. a e. U A. b e. U ( ( x .x. a ) .+ b ) e. U ) )
Theorem	islssmg 13990*	The predicate "is a subspace" (of a left module or left vector space). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.) Use islssm 13989 instead. (New usage is discouraged.)
|- F = (Scalar ` W )   &    |- B = ( Base ` F )   &    |- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .x. = ( .s ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( W e. X -> ( U e. S <-> ( U C_ V /\ E. j j e. U /\ A. x e. B A. a e. U A. b e. U ( ( x .x. a ) .+ b ) e. U ) ) )
Theorem	islssmd 13991*	Properties that determine a subspace of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
|- ( ph -> F = (Scalar ` W ) )   &    |- ( ph -> B = ( Base ` F ) )   &    |- ( ph -> V = ( Base ` W ) )   &    |- ( ph -> .+ = ( +g ` W ) )   &    |- ( ph -> .x. = ( .s ` W ) )   &    |- ( ph -> S = ( LSubSp ` W ) )   &    |- ( ph -> U C_ V )   &    |- ( ph -> E. j j e. U )   &    |- ( ( ph /\ ( x e. B /\ a e. U /\ b e. U ) ) -> ( ( x .x. a ) .+ b ) e. U )   &    |- ( ph -> W e. X )   =>    |- ( ph -> U e. S )
Theorem	lssssg 13992	A subspace is a set of vectors. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( ( W e. X /\ U e. S ) -> U C_ V )
Theorem	lsselg 13993	A subspace member is a vector. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( ( W e. C /\ U e. S /\ X e. U ) -> X e. V )
Theorem	lss1 13994	The set of vectors in a left module is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( W e. LMod -> V e. S )
Theorem	lssuni 13995	The union of all subspaces is the vector space. (Contributed by NM, 13-Mar-2015.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- ( ph -> W e. LMod )   =>    |- ( ph -> U. S = V )
Theorem	lssclg 13996	Closure property of a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
|- F = (Scalar ` W )   &    |- B = ( Base ` F )   &    |- .+ = ( +g ` W )   &    |- .x. = ( .s ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( ( W e. C /\ U e. S /\ ( Z e. B /\ X e. U /\ Y e. U ) ) -> ( ( Z .x. X ) .+ Y ) e. U )
Theorem	lssvacl 13997	Closure of vector addition in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- .+ = ( +g ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( ( ( W e. LMod /\ U e. S ) /\ ( X e. U /\ Y e. U ) ) -> ( X .+ Y ) e. U )
Theorem	lssvsubcl 13998	Closure of vector subtraction in a subspace. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- .- = ( -g ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( ( ( W e. LMod /\ U e. S ) /\ ( X e. U /\ Y e. U ) ) -> ( X .- Y ) e. U )
Theorem	lssvancl1 13999	Non-closure: if one vector belongs to a subspace but another does not, their sum does not belong. Useful for obtaining a new vector not in a subspace. (Contributed by NM, 14-May-2015.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- S = ( LSubSp ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. V )   &    |- ( ph -> -. Y e. U )   =>    |- ( ph -> -. ( X .+ Y ) e. U )
Theorem	lssvancl2 14000	Non-closure: if one vector belongs to a subspace but another does not, their sum does not belong. Useful for obtaining a new vector not in a subspace. (Contributed by NM, 20-May-2015.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- S = ( LSubSp ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. V )   &    |- ( ph -> -. Y e. U )   =>    |- ( ph -> -. ( Y .+ X ) e. U )