P. 145 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14401-14500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rrgeq0i 14401	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 14402	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 14403	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 14404	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 14405	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 14406	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 14407*	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 14408	A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
|- ( R e. Domn -> R e. NzRing )
Theorem	domnring 14409	A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
|- ( R e. Domn -> R e. Ring )
Theorem	domneq0 14410	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 14411	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 14412	A class is a domain if and only if its opposite is a domain, biconditional form of opprdomn 14413. (Contributed by SN, 15-Jun-2015.)
|- O = (oppr ` R )   =>    |- ( R e. V -> ( R e. Domn <-> O e. Domn ) )
Theorem	opprdomn 14413	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 14414	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 14415	An integral domain is a domain. (Contributed by Thierry Arnoux, 22-Mar-2025.)
|- ( ph -> R e. IDomn )   =>    |- ( ph -> R e. Domn )
Theorem	idomcringd 14416	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 14417	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 14418	Extend class notation with ring apartness.
class #r
Definition	df-apr 14419*	The relation between elements whose difference is invertible, which for a local ring is an apartness relation by aprap 14424. (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 14420	Expand Definition df-apr 14419. (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 14421	The apartness relation given by df-apr 14419 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 14422	The apartness relation given by df-apr 14419 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 14423	The apartness relation given by df-apr 14419 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 14424	The relation given by df-apr 14419 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 14425	If the relation given by df-apr 14419 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 14426	A ring is a local ring if and only if the relation given by df-apr 14419 is an apartness relation. (Contributed by Jim Kingdon, 28-May-2026.)
|- ( R e. Ring -> ( R e. LRing <-> (#r ` R ) Ap ( Base ` R ) ) )
7.5  Left modules
7.5.1  Definition and basic properties
Syntax	clmod 14427	Extend class notation with class of all left modules.
class LMod
Syntax	cscaf 14428	The functionalization of the scalar multiplication operation.
class .sf
Definition	df-lmod 14429*	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 14430*	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 14431*	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 14432	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 14433*	Properties that determine a left module. See note in isgrpd2 13726 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 14434	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 14435	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 14436	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 14437	A left module is a group. (Contributed by SN, 16-May-2024.)
|- ( ph -> W e. LMod )   =>    |- ( ph -> W e. Grp )
Theorem	lmodbn0 14438	The base set of a left module is nonempty. It is also inhabited (by lmod0vcl 14457). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- B = ( Base ` W )   =>    |- ( W e. LMod -> B =/= (/) )
Theorem	lmodacl 14439	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 14440	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 14441	The set of scalars in a left module is nonempty. It is also inhabited, by lmod0cl 14454. (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 14442	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 14443	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 14444	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 14445	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 14446*	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 14447	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 14448	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 14449	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 14450	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 14451	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 14452	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 14453	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 14454	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 14455	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 14456	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 14457	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 14458	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 14459	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 14460	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 14461	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 14462	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 14463	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 14464	Lemma 1 for lmodfopne 14466. (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 14465	Lemma 2 for lmodfopne 14466. (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 14466	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 14467	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 14468	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 14469	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 14470	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 14471	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 14472	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 14473	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 14474	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 14475	A left module is a commutative monoid under addition. (Contributed by NM, 7-Jan-2015.)
|- ( W e. LMod -> W e. CMnd )
Theorem	lmodnegadd 14476	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 14477	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 14478	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 14479	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 14480	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 14481	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 14482	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 14483	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 14484	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 14485	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 14486	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 14487	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 14488*	If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 14489 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 14489*	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 14490*	Lemma for rmodislmod 14491. This is the part of the proof of rmodislmod 14491 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 14491*	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 14429 of a left module, see also islmod 14431. (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 14492	Extend class notation with linear subspaces of a left module or left vector space.
class LSubSp
Definition	df-lssm 14493*	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 14494	Existence of a linear subspace. (Contributed by Jim Kingdon, 27-Apr-2025.)
|- ( W e. V -> ( LSubSp ` W ) e. _V )
Theorem	lssmex 14495	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 14496*	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 14497*	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 14498*	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 14497 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 14499*	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 14500	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 )