P. 160 of Theorem List - Metamath Proof Explorer

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Theorem List for Metamath Proof Explorer - 15901-16000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	srngmul 15901	The involution function in a star ring distributes over multiplication, with a change in the order of the factors. (Contributed by Mario Carneiro, 6-Oct-2015.)
|- .* = ( * r ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. *Ring /\ X e. B /\ Y e. B ) -> ( .* ` ( X .x. Y ) ) = ( ( .* ` Y ) .x. ( .* ` X ) ) )
Theorem	srng1 15902	The conjugate of the ring identity is the identity. (This is sometimes taken as an axiom, and indeed the proof here follows because we defined * r to be a ring homomorphism, which preserves 1; nevertheless, it is redundant, as can be seen from the proof of issrngd 15904.) (Contributed by Mario Carneiro, 6-Oct-2015.)
|- .* = ( * r ` R )   &    |- .1. = ( 1r ` R )   =>    |- ( R e. *Ring -> ( .* ` .1. ) = .1. )
Theorem	srng0 15903	The conjugate of the ring zero is zero. (Contributed by Mario Carneiro, 7-Oct-2015.)
|- .* = ( * r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. *Ring -> ( .* ` .0. ) = .0. )
Theorem	issrngd 15904*	Properties that determine a star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2015.)
|- ( ph -> K = ( Base ` R ) )   &    |- ( ph -> .+ = ( +g ` R ) )   &    |- ( ph -> .x. = ( .r ` R ) )   &    |- ( ph -> .* = ( * r ` R ) )   &    |- ( ph -> R e. Ring )   &    |- ( ( ph /\ x e. K ) -> ( .* ` x ) e. K )   &    |- ( ( ph /\ x e. K /\ y e. K ) -> ( .* ` ( x .+ y ) ) = ( ( .* ` x ) .+ ( .* ` y ) ) )   &    |- ( ( ph /\ x e. K /\ y e. K ) -> ( .* ` ( x .x. y ) ) = ( ( .* ` y ) .x. ( .* ` x ) ) )   &    |- ( ( ph /\ x e. K ) -> ( .* ` ( .* ` x ) ) = x )   =>    |- ( ph -> R e. *Ring )
10.6  Left modules
10.6.1  Definition and basic properties
Syntax	clmod 15905	Extend class notation with class of all left modules.
class LMod
Syntax	cscaf 15906	The functionalization of the scalar multiplication operation.
class .s f
Definition	df-lmod 15907*	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 15908*	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.)
|- .s f = ( g e. _V |-> ( x e. ( Base ` (Scalar ` g ) ) , y e. ( Base ` g ) |-> ( x ( .s ` g ) y ) ) )
Theorem	islmod 15909*	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 15910	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 15911*	Properties that determine a left module. See note in isgrpd2 14783 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 15912	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	lmodrng 15913	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 15914	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	lmodbn0 15915	The base set of a left module is nonempty. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- B = ( Base ` W )   =>    |- ( W e. LMod -> B =/= (/) )
Theorem	lmodacl 15916	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 15917	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 15918	The set of scalars in a left module is nonempty. (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 15919	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 15920	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 15921	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 15922	Closure of scalar product for a left module. (hvmulcl 22469 analog.) (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	scaffval 15923*	The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- B = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .xb = ( .s f ` W )   &    |- .x. = ( .s ` W )   =>    |- .xb = ( x e. K , y e. B |-> ( x .x. y ) )
Theorem	scafval 15924	The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- B = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .xb = ( .s f ` W )   &    |- .x. = ( .s ` W )   =>    |- ( ( X e. K /\ Y e. B ) -> ( X .xb Y ) = ( X .x. Y ) )
Theorem	scafeq 15925	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 = ( .s f ` W )   &    |- .x. = ( .s ` W )   =>    |- ( .x. Fn ( K X. B ) -> .xb = .x. )
Theorem	scaffn 15926	The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- B = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .xb = ( .s f ` W )   =>    |- .xb Fn ( K X. B )
Theorem	lmodscaf 15927	The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- B = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .xb = ( .s f ` W )   =>    |- ( W e. LMod -> .xb : ( K X. B ) --> B )
Theorem	lmodvsdi 15928	Distributive law for scalar product. (ax-hvdistr1 22464 analog.) (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 15929	Distributive law for scalar product. (ax-hvdistr1 22464 analog.) (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 15930	Associative law for scalar product. (ax-hvmulass 22463 analog.) (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 15931	The ring zero in a left module belongs to the ring base set. (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 15932	The ring unit in a left module belongs to the ring base set. (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 15933	Scalar product with ring unit. (ax-hvmulid 22462 analog.) (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 15934	The zero vector is a vector. (ax-hv0cl 22459 analog.) (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 15935	Left identity law for the zero vector. (hvaddid2 22478 analog.) (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 15936	Right identity law for the zero vector. (ax-hvaddid 22460 analog.) (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 15937	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 15938	Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 22466 analog.) (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 15939	Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 22479 analog.) (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	lmodvnegcl 15940	Closure of vector negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- N = ( inv g ` W )   =>    |- ( ( W e. LMod /\ X e. V ) -> ( N ` X ) e. V )
Theorem	lmodvnegid 15941	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 = ( inv g ` W )   =>    |- ( ( W e. LMod /\ X e. V ) -> ( X .+ ( N ` X ) ) = .0. )
Theorem	lmodvneg1 15942	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 = ( inv g ` W )   &    |- F = (Scalar ` W )   &    |- .x. = ( .s ` W )   &    |- .1. = ( 1r ` F )   &    |- M = ( inv g ` F )   =>    |- ( ( W e. LMod /\ X e. V ) -> ( ( M ` .1. ) .x. X ) = ( N ` X ) )
Theorem	lmodvsneg 15943	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 = ( inv g ` W )   &    |- K = ( Base ` F )   &    |- M = ( inv g ` F )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> X e. B )   &    |- ( ph -> R e. K )   =>    |- ( ph -> ( N ` ( R .x. X ) ) = ( ( M ` R ) .x. X ) )
Theorem	lmodvsubcl 15944	Closure of vector subtraction. (hvsubcl 22473 analog.) (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 15945	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 15946	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 15947	A left module is a commutative monoid under addition. (Contributed by NM, 7-Jan-2015.)
|- ( W e. LMod -> W e. CMnd )
Theorem	lmodnegadd 15948	Distribute negation through addition of scalar products. (Contributed by NM, 9-Apr-2015.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .x. = ( .s ` W )   &    |- N = ( inv g ` W )   &    |- R = (Scalar ` W )   &    |- K = ( Base ` R )   &    |- I = ( inv g ` 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 15949	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 15950	Relationship between vector subtraction and addition. (hvsubadd 22532 analog.) (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 15951	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 15952	Addition/subtraction cancellation law for vectors. (hvpncan 22494 analog.) (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 15953	Cancellation law for vector subtraction (npcan 9270 analog). (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 15954	Value of vector subtraction in terms of addition. (hvsubval 22472 analog.) (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 = ( inv g ` F )   &    |- .1. = ( 1r ` F )   =>    |- ( ( W e. LMod /\ A e. V /\ B e. V ) -> ( A .- B ) = ( A .+ ( ( N ` .1. ) .x. B ) ) )
Theorem	lmodsubvs 15955	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 = ( inv g ` 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 15956	Scalar multiplication distributive law for subtraction. (hvsubdistr1 22504 analog, with longer proof since our scalar multiplication is not commutative.) (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 15957	Scalar multiplication distributive law for subtraction. (hvsubdistr2 22505 analog.) (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 15958	If the difference between two vectors is zero, they are equal. (hvsubeq0 22523 analog.) (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 15959	Subtraction of a vector from itself. (hvsubid 22481 analog.) (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	lmodvsghm 15960*	Scalar multiplication of the vector space by a fixed scalar is an automorphism of the addiive group of vectors. (Contributed by Mario Carneiro, 5-May-2015.)
|- 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. ( W GrpHom W ) )
Theorem	lmodprop2d 15961*	If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 15962 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 15962*	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 ) )
10.6.2  Subspaces and spans in a left module
Syntax	clss 15963	Extend class notation with linear subspaces of a left module or left vector space.
class LSubSp
Definition	df-lss 15964*	Define the set of linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)
|- LSubSp = ( w e. _V |-> { s e. ( ~P ( Base ` w ) \ { (/) } ) | A. x e. ( Base ` (Scalar ` w ) ) A. a e. s A. b e. s ( ( x ( .s ` w ) a ) ( +g ` w ) b ) e. s } )
Theorem	lssset 15965*	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 \ { (/) } ) | A. x e. B A. a e. s A. b e. s ( ( x .x. a ) .+ b ) e. s } )
Theorem	islss 15966*	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 /\ U =/= (/) /\ A. x e. B A. a e. U A. b e. U ( ( x .x. a ) .+ b ) e. U ) )
Theorem	islssd 15967*	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 -> U =/= (/) )   &    |- ( ( ph /\ ( x e. B /\ a e. U /\ b e. U ) ) -> ( ( x .x. a ) .+ b ) e. U )   =>    |- ( ph -> U e. S )
Theorem	lssss 15968	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 )   =>    |- ( U e. S -> U C_ V )
Theorem	lssel 15969	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 )   =>    |- ( ( U e. S /\ X e. U ) -> X e. V )
Theorem	lss1 15970	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 15971	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	lssn0 15972	A subspace is not empty. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
|- S = ( LSubSp ` W )   =>    |- ( U e. S -> U =/= (/) )
Theorem	00lss 15973	The empty structure has no subspaces (for use with fvco4i 5760). (Contributed by Stefan O'Rear, 31-Mar-2015.)
|- (/) = ( LSubSp ` (/) )
Theorem	lsscl 15974	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 )   =>    |- ( ( U e. S /\ ( Z e. B /\ X e. U /\ Y e. U ) ) -> ( ( Z .x. X ) .+ Y ) e. U )
Theorem	lssvsubcl 15975	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 15976	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. TODO: notice similarity to lspindp3 16163. Can it be used along with lspsnne1 16144, lspsnne2 16145 to shorten this proof? (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 15977	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 )
Theorem	lss0cl 15978	The zero vector belongs to every subspace. (Contributed by NM, 12-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
|- .0. = ( 0g ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( ( W e. LMod /\ U e. S ) -> .0. e. U )
Theorem	lsssn0 15979	The singleton of the zero vector is a subspace. (Contributed by NM, 13-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- .0. = ( 0g ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( W e. LMod -> { .0. } e. S )
Theorem	lss0ss 15980	The zero subspace is included in every subspace. (sh0le 22895 analog.) (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- .0. = ( 0g ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( ( W e. LMod /\ X e. S ) -> { .0. } C_ X )
Theorem	lssle0 15981	No subspace is smaller than the zero subspace. (shle0 22897 analog.) (Contributed by NM, 20-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- .0. = ( 0g ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( ( W e. LMod /\ X e. S ) -> ( X C_ { .0. } <-> X = { .0. } ) )
Theorem	lssne0 15982*	A nonzero subspace has a nonzero vector. (shne0i 22903 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)
|- .0. = ( 0g ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( X e. S -> ( X =/= { .0. } <-> E. y e. X y =/= .0. ) )
Theorem	lssneln0 15983	A vector which doesn't belong to a subspace is nonzero. (Contributed by NM, 14-May-2015.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- S = ( LSubSp ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. V )   &    |- ( ph -> -. X e. U )   =>    |- ( ph -> X e. ( V \ { .0. } ) )
Theorem	lssssr 15984*	Conclude subspace ordering from nonzero vector membership. (ssrdv 3314 analog.) (Contributed by NM, 17-Aug-2014.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- S = ( LSubSp ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> T C_ V )   &    |- ( ph -> U e. S )   &    |- ( ( ph /\ x e. ( V \ { .0. } ) ) -> ( x e. T -> x e. U ) )   =>    |- ( ph -> T C_ U )
Theorem	lssvacl 15985	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	lssvscl 15986	Closure of scalar product in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- F = (Scalar ` W )   &    |- .x. = ( .s ` W )   &    |- B = ( Base ` F )   &    |- S = ( LSubSp ` W )   =>    |- ( ( ( W e. LMod /\ U e. S ) /\ ( X e. B /\ Y e. U ) ) -> ( X .x. Y ) e. U )
Theorem	lssvnegcl 15987	Closure of negative vectors in a subspace. (Contributed by Stefan O'Rear, 11-Dec-2014.)
|- S = ( LSubSp ` W )   &    |- N = ( inv g ` W )   =>    |- ( ( W e. LMod /\ U e. S /\ X e. U ) -> ( N ` X ) e. U )
Theorem	lsssubg 15988	All subspaces are subgroups. (Contributed by Stefan O'Rear, 11-Dec-2014.)
|- S = ( LSubSp ` W )   =>    |- ( ( W e. LMod /\ U e. S ) -> U e. (SubGrp ` W ) )
Theorem	lsssssubg 15989	All subspaces are subgroups. (Contributed by Mario Carneiro, 19-Apr-2016.)
|- S = ( LSubSp ` W )   =>    |- ( W e. LMod -> S C_ (SubGrp ` W ) )
Theorem	islss3 15990	A linear subspace of a module is a subset which is a module in its own right. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
|- X = ( W ↾s U )   &    |- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( W e. LMod -> ( U e. S <-> ( U C_ V /\ X e. LMod ) ) )
Theorem	lsslmod 15991	A submodule is a module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
|- X = ( W ↾s U )   &    |- S = ( LSubSp ` W )   =>    |- ( ( W e. LMod /\ U e. S ) -> X e. LMod )
Theorem	lsslss 15992	The subspaces of a subspace are the smaller subspaces. (Contributed by Stefan O'Rear, 12-Dec-2014.)
|- X = ( W ↾s U )   &    |- S = ( LSubSp ` W )   &    |- T = ( LSubSp ` X )   =>    |- ( ( W e. LMod /\ U e. S ) -> ( V e. T <-> ( V e. S /\ V C_ U ) ) )
Theorem	islss4 15993*	A linear subspace is a subgroup which respects scalar multiplication. (Contributed by Stefan O'Rear, 11-Dec-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
|- F = (Scalar ` W )   &    |- B = ( Base ` F )   &    |- V = ( Base ` W )   &    |- .x. = ( .s ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( W e. LMod -> ( U e. S <-> ( U e. (SubGrp ` W ) /\ A. a e. B A. b e. U ( a .x. b ) e. U ) ) )
Theorem	lss1d 15994*	One-dimensional subspace (or zero-dimensional if X is the zero vector). (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- .x. = ( .s ` W )   &    |- K = ( Base ` F )   &    |- S = ( LSubSp ` W )   =>    |- ( ( W e. LMod /\ X e. V ) -> { v | E. k e. K v = ( k .x. X ) } e. S )
Theorem	lssintcl 15995	The intersection of a nonempty set of subspaces is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- S = ( LSubSp ` W )   =>    |- ( ( W e. LMod /\ A C_ S /\ A =/= (/) ) -> |^| A e. S )
Theorem	lssincl 15996	The intersection of two subspaces is a subspace. (Contributed by NM, 7-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- S = ( LSubSp ` W )   =>    |- ( ( W e. LMod /\ T e. S /\ U e. S ) -> ( T i^i U ) e. S )
Theorem	lssmre 15997	The subspaces of a module comprise a Moore system on the vectors of the module. (Contributed by Stefan O'Rear, 31-Jan-2015.)
|- B = ( Base ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( W e. LMod -> S e. (Moore ` B ) )
Theorem	lssacs 15998	Submodules are an algebraic closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
|- B = ( Base ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( W e. LMod -> S e. (ACS ` B ) )
Theorem	prdsvscacl 15999*	Pointwise scalar multiplication is closed in products of modules. (Contributed by Stefan O'Rear, 10-Jan-2015.)
|- Y = ( S X_s R )   &    |- B = ( Base ` Y )   &    |- .x. = ( .s ` Y )   &    |- K = ( Base ` S )   &    |- ( ph -> S e. Ring )   &    |- ( ph -> I e. W )   &    |- ( ph -> R : I --> LMod )   &    |- ( ph -> F e. K )   &    |- ( ph -> G e. B )   &    |- ( ( ph /\ x e. I ) -> (Scalar ` ( R ` x ) ) = S )   =>    |- ( ph -> ( F .x. G ) e. B )
Theorem	prdslmodd 16000*	The product of a family of left modules is a left module. (Contributed by Stefan O'Rear, 10-Jan-2015.)
|- Y = ( S X_s R )   &    |- ( ph -> S e. Ring )   &    |- ( ph -> I e. V )   &    |- ( ph -> R : I --> LMod )   &    |- ( ( ph /\ y e. I ) -> (Scalar ` ( R ` y ) ) = S )   =>    |- ( ph -> Y e. LMod )