P. 386 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38501-38600   *Has distinct variable group(s)

Type	Label	Description
Statement
21.33.13.10  Associative algebras (extension)
Theorem	ascl0 38501	The scalar 0 embedded into a left module corresponds to the 0 of the left module if the left module is also a ring. (Contributed by AV, 31-Jul-2019.)
|- A = (algSc ` W )   &    |- F = (Scalar ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> W e. Ring )   =>    |- ( ph -> ( A ` ( 0g ` F ) ) = ( 0g ` W ) )
Theorem	ascl1 38502	The scalar 1 embedded into a left module corresponds to the 1 of the left module if the left module is also a ring. (Contributed by AV, 31-Jul-2019.)
|- A = (algSc ` W )   &    |- F = (Scalar ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> W e. Ring )   =>    |- ( ph -> ( A ` ( 1r ` F ) ) = ( 1r ` W ) )
Theorem	assaascl0 38503	The scalar 0 embedded into an associative algebra corresponds to the 0 of the associative algebra. (Contributed by AV, 31-Jul-2019.)
|- A = (algSc ` W )   &    |- F = (Scalar ` W )   &    |- ( ph -> W e. AssAlg )   =>    |- ( ph -> ( A ` ( 0g ` F ) ) = ( 0g ` W ) )
Theorem	assaascl1 38504	The scalar 1 embedded into an associative algebra corresponds to the 1 of the an associative algebra. (Contributed by AV, 31-Jul-2019.)
|- A = (algSc ` W )   &    |- F = (Scalar ` W )   &    |- ( ph -> W e. AssAlg )   =>    |- ( ph -> ( A ` ( 1r ` F ) ) = ( 1r ` W ) )
21.33.13.11  Univariate polynomials (extension)
Theorem	ply1vr1smo 38505	The variable in a polynomial expressed as scaled monomial. (Contributed by AV, 12-Aug-2019.)
|- P = (Poly1 ` R )   &    |- .1. = ( 1r ` R )   &    |- .x. = ( .s ` P )   &    |- G = (mulGrp ` P )   &    |- .^ = (.g ` G )   &    |- X = (var1 ` R )   =>    |- ( R e. Ring -> ( .1. .x. ( 1 .^ X ) ) = X )
Theorem	ply1ass23l 38506	Associative identity with scalar and ring multiplication for the polynomial ring. (Contributed by AV, 14-Aug-2019.)
|- P = (Poly1 ` R )   &    |- .X. = ( .r ` P )   &    |- B = ( Base ` P )   &    |- K = ( Base ` R )   &    |- .x. = ( .s ` P )   =>    |- ( ( R e. Ring /\ ( A e. K /\ X e. B /\ Y e. B ) ) -> ( ( A .x. X ) .X. Y ) = ( A .x. ( X .X. Y ) ) )
Theorem	ply1sclrmsm 38507	The ring multiplication of a polynomial with a scalar polynomial is equal to the scalar multiplication of the polynomial with the corresponding scalar. (Contributed by AV, 14-Aug-2019.)
|- K = ( Base ` R )   &    |- P = (Poly1 ` R )   &    |- E = ( Base ` P )   &    |- X = (var1 ` R )   &    |- .x. = ( .s ` P )   &    |- .X. = ( .r ` P )   &    |- N = (mulGrp ` P )   &    |- .^ = (.g ` N )   &    |- A = (algSc ` P )   =>    |- ( ( R e. Ring /\ F e. K /\ Z e. E ) -> ( ( A ` F ) .X. Z ) = ( F .x. Z ) )
Theorem	coe1id 38508*	Coefficient vector of the unit polynomial. (Contributed by AV, 9-Aug-2019.)
|- P = (Poly1 ` R )   &    |- I = ( 1r ` P )   &    |- .0. = ( 0g ` R )   &    |- .1. = ( 1r ` R )   =>    |- ( R e. Ring -> (coe1 ` I ) = ( x e. NN0 |-> if ( x = 0 , .1. , .0. ) ) )
Theorem	coe1sclmulval 38509	The value of the coefficient vector of a polynomial multiplied on the left by a scalar. (Contributed by AV, 14-Aug-2019.)
|- P = (Poly1 ` R )   &    |- B = ( Base ` P )   &    |- K = ( Base ` R )   &    |- A = (algSc ` P )   &    |- S = ( .s ` P )   &    |- .xb = ( .r ` P )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Ring /\ ( Y e. K /\ Z e. B ) /\ N e. NN0 ) -> ( (coe1 ` ( Y S Z ) ) ` N ) = ( Y .x. ( (coe1 ` Z ) ` N ) ) )
Theorem	ply1mulgsumlem1 38510*	Lemma 1 for ply1mulgsum 38514. (Contributed by AV, 19-Oct-2019.)
|- P = (Poly1 ` R )   &    |- B = ( Base ` P )   &    |- A = (coe1 ` K )   &    |- C = (coe1 ` L )   &    |- X = (var1 ` R )   &    |- .X. = ( .r ` P )   &    |- .x. = ( .s ` P )   &    |- .* = ( .r ` R )   &    |- M = (mulGrp ` P )   &    |- .^ = (.g ` M )   =>    |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> E. s e. NN0 A. n e. NN0 ( s < n -> ( ( A ` n ) = ( 0g ` R ) /\ ( C ` n ) = ( 0g ` R ) ) ) )
Theorem	ply1mulgsumlem2 38511*	Lemma 2 for ply1mulgsum 38514. (Contributed by AV, 19-Oct-2019.)
|- P = (Poly1 ` R )   &    |- B = ( Base ` P )   &    |- A = (coe1 ` K )   &    |- C = (coe1 ` L )   &    |- X = (var1 ` R )   &    |- .X. = ( .r ` P )   &    |- .x. = ( .s ` P )   &    |- .* = ( .r ` R )   &    |- M = (mulGrp ` P )   &    |- .^ = (.g ` M )   =>    |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> E. s e. NN0 A. n e. NN0 ( s < n -> ( R gsumg ( l e. ( 0 ... n ) |-> ( ( A ` l ) .* ( C ` ( n - l ) ) ) ) ) = ( 0g ` R ) ) )
Theorem	ply1mulgsumlem3 38512*	Lemma 3 for ply1mulgsum 38514. (Contributed by AV, 20-Oct-2019.)
|- P = (Poly1 ` R )   &    |- B = ( Base ` P )   &    |- A = (coe1 ` K )   &    |- C = (coe1 ` L )   &    |- X = (var1 ` R )   &    |- .X. = ( .r ` P )   &    |- .x. = ( .s ` P )   &    |- .* = ( .r ` R )   &    |- M = (mulGrp ` P )   &    |- .^ = (.g ` M )   =>    |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> ( k e. NN0 |-> ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) ) finSupp ( 0g ` R ) )
Theorem	ply1mulgsumlem4 38513*	Lemma 4 for ply1mulgsum 38514. (Contributed by AV, 19-Oct-2019.)
|- P = (Poly1 ` R )   &    |- B = ( Base ` P )   &    |- A = (coe1 ` K )   &    |- C = (coe1 ` L )   &    |- X = (var1 ` R )   &    |- .X. = ( .r ` P )   &    |- .x. = ( .s ` P )   &    |- .* = ( .r ` R )   &    |- M = (mulGrp ` P )   &    |- .^ = (.g ` M )   =>    |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> ( k e. NN0 |-> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) ) finSupp ( 0g ` P ) )
Theorem	ply1mulgsum 38514*	The product of two polynomials expressed as group sum of scaled monomials. (Contributed by AV, 20-Oct-2019.)
|- P = (Poly1 ` R )   &    |- B = ( Base ` P )   &    |- A = (coe1 ` K )   &    |- C = (coe1 ` L )   &    |- X = (var1 ` R )   &    |- .X. = ( .r ` P )   &    |- .x. = ( .s ` P )   &    |- .* = ( .r ` R )   &    |- M = (mulGrp ` P )   &    |- .^ = (.g ` M )   =>    |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> ( K .X. L ) = ( P gsumg ( k e. NN0 |-> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) ) ) )
Theorem	evl1at0 38515	Polynomial evaluation for the 0 scalar. (Contributed by AV, 10-Aug-2019.)
|- O = (eval1 ` R )   &    |- P = (Poly1 ` R )   &    |- .0. = ( 0g ` R )   &    |- Z = ( 0g ` P )   =>    |- ( R e. CRing -> ( ( O ` Z ) ` .0. ) = .0. )
Theorem	evl1at1 38516	Polynomial evaluation for the 1 scalar. (Contributed by AV, 10-Aug-2019.)
|- O = (eval1 ` R )   &    |- P = (Poly1 ` R )   &    |- .1. = ( 1r ` R )   &    |- I = ( 1r ` P )   =>    |- ( R e. CRing -> ( ( O ` I ) ` .1. ) = .1. )
21.33.13.12  Univariate polynomials (examples)
Theorem	linply1 38517	A term of the form x - C is a (univariate) polynomial, also called "linear polynomial". (Part of ply1remlem 22857). (Contributed by AV, 3-Jul-2019.)
|- P = (Poly1 ` R )   &    |- B = ( Base ` P )   &    |- K = ( Base ` R )   &    |- X = (var1 ` R )   &    |- .- = ( -g ` P )   &    |- A = (algSc ` P )   &    |- G = ( X .- ( A ` C ) )   &    |- ( ph -> C e. K )   &    |- ( ph -> R e. Ring )   =>    |- ( ph -> G e. B )
Theorem	lineval 38518	A term of the form x - C evaluated for x = V results in V - C (part of ply1remlem 22857). (Contributed by AV, 3-Jul-2019.)
|- P = (Poly1 ` R )   &    |- B = ( Base ` P )   &    |- K = ( Base ` R )   &    |- X = (var1 ` R )   &    |- .- = ( -g ` P )   &    |- A = (algSc ` P )   &    |- G = ( X .- ( A ` C ) )   &    |- ( ph -> C e. K )   &    |- O = (eval1 ` R )   &    |- ( ph -> R e. CRing )   &    |- ( ph -> V e. K )   =>    |- ( ph -> ( ( O ` G ) ` V ) = ( V ( -g ` R ) C ) )
Theorem	zringsubgval 38519	Subtraction in the ring of integers. (Contributed by AV, 3-Aug-2019.)
|- .- = ( -g ` ℤring )   =>    |- ( ( X e. ZZ /\ Y e. ZZ ) -> ( X - Y ) = ( X .- Y ) )
Theorem	linevalexample 38520	The polynomial x - 3 over ZZ evaluated for x = 5 results in 2. (Contributed by AV, 3-Jul-2019.)
|- P = (Poly1 ` ℤring )   &    |- B = ( Base ` P )   &    |- X = (var1 ` ℤring )   &    |- .- = ( -g ` P )   &    |- A = (algSc ` P )   &    |- G = ( X .- ( A ` 3 ) )   &    |- O = (eval1 ` ℤring )   =>    |- ( ( O ` ( X .- ( A ` 3 ) ) ) ` 5 ) = 2
21.33.14  Linear algebra (extension)
21.33.14.1  The subalgebras of diagonal and scalar matrices (extension) In the following, alternative definitions for diagonal and scalar matrices are provided. These definitions define diagonal and scalar matrices as extensible structures, whereas the definitions df-dmat 19286 and df-scmat 19287 define diagonal and scalar matrices as sets.
Syntax	cdmatalt 38521	Alternative notation for the algebra of diagonal matrices.
class DMatALT
Syntax	cscmatalt 38522	Alternative notation for the algebra of scalar matrices.
class ScMatALT
Definition	df-dmatalt 38523*	Define the set of n x n diagonal (square) matrices over a set (usually a ring) r, see definition in [Roman] p. 4 or Definition 3.12 in [Hefferon] p. 240. (Contributed by AV, 8-Dec-2019.)
|- DMatALT = ( n e. Fin , r e. _V |-> [_ ( n Mat r ) / a ]_ ( a ↾s { m e. ( Base ` a ) | A. i e. n A. j e. n ( i =/= j -> ( i m j ) = ( 0g ` r ) ) } ) )
Definition	df-scmatalt 38524*	Define the algebra of n x n scalar matrices over a set (usually a ring) r, see definition in [Connell] p. 57: "A scalar matrix is a diagonal matrix for which all the diagonal terms are equal, i.e., a matrix of the form cIn";. (Contributed by AV, 8-Dec-2019.)
|- ScMatALT = ( n e. Fin , r e. _V |-> [_ ( n Mat r ) / a ]_ ( a ↾s { m e. ( Base ` a ) | E. c e. ( Base ` r ) A. i e. n A. j e. n ( i m j ) = if ( i = j , c , ( 0g ` r ) ) } ) )
Theorem	dmatALTval 38525*	The algebra of N x N diagonal matrices over a ring R. (Contributed by AV, 8-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .0. = ( 0g ` R )   &    |- D = ( N DMatALT R )   =>    |- ( ( N e. Fin /\ R e. _V ) -> D = ( A ↾s { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } ) )
Theorem	dmatALTbas 38526*	The base set of the algebra of N x N diagonal matrices over a ring R, i.e. the set of all N x N diagonal matrices over the ring R. (Contributed by AV, 8-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .0. = ( 0g ` R )   &    |- D = ( N DMatALT R )   =>    |- ( ( N e. Fin /\ R e. _V ) -> ( Base ` D ) = { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } )
Theorem	dmatALTbasel 38527*	An element of the base set of the algebra of N x N diagonal matrices over a ring R, i.e. an N x N diagonal matrix over the ring R. (Contributed by AV, 8-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .0. = ( 0g ` R )   &    |- D = ( N DMatALT R )   =>    |- ( ( N e. Fin /\ R e. _V ) -> ( M e. ( Base ` D ) <-> ( M e. B /\ A. i e. N A. j e. N ( i =/= j -> ( i M j ) = .0. ) ) ) )
Theorem	dmatbas 38528	The set of all N x N diagonal matrices over (the ring) R is the base set of the algebra of N x N diagonal matrices over (the ring) R. (Contributed by AV, 8-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .0. = ( 0g ` R )   &    |- D = ( N DMat R )   =>    |- ( ( N e. Fin /\ R e. V ) -> D = ( Base ` ( N DMatALT R ) ) )
21.33.14.2  Linear combinations According to Wikipedia ("Linear combination", 29-Mar-2019, https://en.wikipedia.org/wiki/Linear_combination) "In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics." In linear algebra, these "terms" are "vectors" (elements from vector spaces or left modules), and the constants are elements of the underlying field resp. ring. This corresponds to the definition in [Lang] p. 129: "Let M be a module over a ring A and let S be a subset of M. By a linear combination of elements of S (with coefficients in A) one means a sum ∑x ∈S axx where {ax} is a set of elements of A, ...". In the definition in [Lang] p. 129, it is additionally claimed that "..., almost all of which [elements of A] are equal to 0.". This is not necessarily required in the following definition df-linc 38531, but it is essential if additions and scalar multiplications of linear combinations are considered. Therefore, we define the set of all linear combinations with finite support in df-lco 38532, so that we can show that such sets are submodules of the corresponding modules, see lincolss 38559. Remark:According to Wikipedia ("Linear span", 28-Apr-2019, https://en.wikipedia.org/wiki/Linear_span) "In linear algebra, the linear span (also called the linear hull or just span) of a set of vectors in a vector space [or module] is the intersection of all linear subspaces which each contain every vector in that set.", and "Alternatively, the span of [a set] S may be defined as the set of all finite linear combinations of elements (vectors) of S". Whereas spans are defined according to the first approach in df-lsp 17940, the set of all linear combinations as defined by df-lco 38532 follows the alternative approach. That both definitions are equivalent is shown by lspeqlco 38564.
Syntax	clinc 38529	Extend class notation with the operation constructing a linear combination (of vectors from a left module).
class linC
Syntax	clinco 38530	Extend class notation with the operation constructing a set of linear combinations (of vectors from a left module) with finite support.
class LinCo
Definition	df-linc 38531*	Define the operation constructing a linear combination. Although this definition is taylored for linear combinations of vectors from left modules, it can be used for any structure having a Base, Scalar s and a scalar multiplication .s. (Contributed by AV, 29-Mar-2019.)
|- linC = ( m e. _V |-> ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) , v e. ~P ( Base ` m ) |-> ( m gsumg ( x e. v |-> ( ( s ` x ) ( .s ` m ) x ) ) ) ) )
Definition	df-lco 38532*	Define the operation constructing the set of all linear combinations for a set of vectors. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 28-Jul-2019.)
|- LinCo = ( m e. _V , v e. ~P ( Base ` m ) |-> { c e. ( Base ` m ) | E. s e. ( ( Base ` (Scalar ` m ) ) ^m v ) ( s finSupp ( 0g ` (Scalar ` m ) ) /\ c = ( s ( linC ` m ) v ) ) } )
Theorem	lincop 38533*	A linear combination as operation. (Contributed by AV, 30-Mar-2019.)
|- ( M e. X -> ( linC ` M ) = ( s e. ( ( Base ` (Scalar ` M ) ) ^m v ) , v e. ~P ( Base ` M ) |-> ( M gsumg ( x e. v |-> ( ( s ` x ) ( .s ` M ) x ) ) ) ) )
Theorem	lincval 38534*	The value of a linear combination. (Contributed by AV, 30-Mar-2019.)
|- ( ( M e. X /\ S e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ V e. ~P ( Base ` M ) ) -> ( S ( linC ` M ) V ) = ( M gsumg ( x e. V |-> ( ( S ` x ) ( .s ` M ) x ) ) ) )
Theorem	dflinc2 38535*	Alternative definition of linear combinations using the function operation. (Contributed by AV, 1-Apr-2019.)
|- linC = ( m e. _V |-> ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) , v e. ~P ( Base ` m ) |-> ( m gsumg ( s oF ( .s ` m ) ( _I |` v ) ) ) ) )
Theorem	lcoop 38536*	A linear combination as operation. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.)
|- B = ( Base ` M )   &    |- S = (Scalar ` M )   &    |- R = ( Base ` S )   =>    |- ( ( M e. X /\ V e. ~P B ) -> ( M LinCo V ) = { c e. B | E. s e. ( R ^m V ) ( s finSupp ( 0g ` S ) /\ c = ( s ( linC ` M ) V ) ) } )
Theorem	lcoval 38537*	The value of a linear combination. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.)
|- B = ( Base ` M )   &    |- S = (Scalar ` M )   &    |- R = ( Base ` S )   =>    |- ( ( M e. X /\ V e. ~P B ) -> ( C e. ( M LinCo V ) <-> ( C e. B /\ E. s e. ( R ^m V ) ( s finSupp ( 0g ` S ) /\ C = ( s ( linC ` M ) V ) ) ) ) )
Theorem	lincfsuppcl 38538	A linear combination of vectors (with finite support) is a vector. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 28-Jul-2019.)
|- B = ( Base ` M )   &    |- R = (Scalar ` M )   &    |- S = ( Base ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( M e. LMod /\ ( V e. W /\ V C_ B ) /\ ( F e. ( S ^m V ) /\ F finSupp .0. ) ) -> ( F ( linC ` M ) V ) e. B )
Theorem	linccl 38539	A linear combination of vectors is a vector. (Contributed by AV, 31-Mar-2019.)
|- B = ( Base ` M )   &    |- R = ( Base ` (Scalar ` M ) )   =>    |- ( ( M e. LMod /\ ( V e. Fin /\ V C_ B /\ S e. ( R ^m V ) ) ) -> ( S ( linC ` M ) V ) e. B )
Theorem	lincval0 38540	The value of an empty linear combination. (Contributed by AV, 12-Apr-2019.)
|- ( M e. X -> ( (/) ( linC ` M ) (/) ) = ( 0g ` M ) )
Theorem	lincvalsng 38541	The linear combination over a singleton. (Contributed by AV, 25-May-2019.)
|- B = ( Base ` M )   &    |- S = (Scalar ` M )   &    |- R = ( Base ` S )   &    |- .x. = ( .s ` M )   =>    |- ( ( M e. LMod /\ V e. B /\ Y e. R ) -> ( { <. V , Y >. } ( linC ` M ) { V } ) = ( Y .x. V ) )
Theorem	lincvalsn 38542	The linear combination over a singleton. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 25-May-2019.)
|- B = ( Base ` M )   &    |- S = (Scalar ` M )   &    |- R = ( Base ` S )   &    |- .x. = ( .s ` M )   &    |- F = { <. V , Y >. }   =>    |- ( ( M e. LMod /\ V e. B /\ Y e. R ) -> ( F ( linC ` M ) { V } ) = ( Y .x. V ) )
Theorem	lincvalpr 38543	The linear combination over an unordered pair. (Contributed by AV, 16-Apr-2019.)
|- B = ( Base ` M )   &    |- S = (Scalar ` M )   &    |- R = ( Base ` S )   &    |- .x. = ( .s ` M )   &    |- .+ = ( +g ` M )   &    |- F = { <. V , X >. , <. W , Y >. }   =>    |- ( ( ( M e. LMod /\ V =/= W ) /\ ( V e. B /\ X e. R ) /\ ( W e. B /\ Y e. R ) ) -> ( F ( linC ` M ) { V , W } ) = ( ( X .x. V ) .+ ( Y .x. W ) ) )
Theorem	lincval1 38544	The linear combination over a singleton mapping to 0. (Contributed by AV, 12-Apr-2019.)
|- B = ( Base ` M )   &    |- S = (Scalar ` M )   &    |- R = ( Base ` S )   &    |- F = { <. V , ( 0g ` S ) >. }   =>    |- ( ( M e. LMod /\ V e. B ) -> ( F ( linC ` M ) { V } ) = ( 0g ` M ) )
Theorem	lcosn0 38545	Properties of a linear combination over a singleton mapping to 0. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 28-Jul-2019.)
|- B = ( Base ` M )   &    |- S = (Scalar ` M )   &    |- R = ( Base ` S )   &    |- F = { <. V , ( 0g ` S ) >. }   =>    |- ( ( M e. LMod /\ V e. B ) -> ( F e. ( R ^m { V } ) /\ F finSupp ( 0g ` S ) /\ ( F ( linC ` M ) { V } ) = ( 0g ` M ) ) )
Theorem	lincvalsc0 38546*	The linear combination where all scalars are 0. (Contributed by AV, 12-Apr-2019.)
|- B = ( Base ` M )   &    |- S = (Scalar ` M )   &    |- .0. = ( 0g ` S )   &    |- Z = ( 0g ` M )   &    |- F = ( x e. V |-> .0. )   =>    |- ( ( M e. LMod /\ V e. ~P B ) -> ( F ( linC ` M ) V ) = Z )
Theorem	lcoc0 38547*	Properties of a linear combination where all scalars are 0. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 28-Jul-2019.)
|- B = ( Base ` M )   &    |- S = (Scalar ` M )   &    |- .0. = ( 0g ` S )   &    |- Z = ( 0g ` M )   &    |- F = ( x e. V |-> .0. )   &    |- R = ( Base ` S )   =>    |- ( ( M e. LMod /\ V e. ~P B ) -> ( F e. ( R ^m V ) /\ F finSupp .0. /\ ( F ( linC ` M ) V ) = Z ) )
Theorem	linc0scn0 38548*	If a set contains the zero element of a module, there is a linear combination being 0 where not all scalars are 0. (Contributed by AV, 13-Apr-2019.)
|- B = ( Base ` M )   &    |- S = (Scalar ` M )   &    |- .0. = ( 0g ` S )   &    |- .1. = ( 1r ` S )   &    |- Z = ( 0g ` M )   &    |- F = ( x e. V |-> if ( x = Z , .1. , .0. ) )   =>    |- ( ( M e. LMod /\ V e. ~P B ) -> ( F ( linC ` M ) V ) = Z )
Theorem	lincdifsn 38549	A vector is a linear combination of a set containing this vector. (Contributed by AV, 21-Apr-2019.) (Revised by AV, 28-Jul-2019.)
|- B = ( Base ` M )   &    |- R = (Scalar ` M )   &    |- S = ( Base ` R )   &    |- .x. = ( .s ` M )   &    |- .+ = ( +g ` M )   &    |- .0. = ( 0g ` R )   =>    |- ( ( ( M e. LMod /\ V e. ~P B /\ X e. V ) /\ ( F e. ( S ^m V ) /\ F finSupp .0. ) /\ G = ( F |` ( V \ { X } ) ) ) -> ( F ( linC ` M ) V ) = ( ( G ( linC ` M ) ( V \ { X } ) ) .+ ( ( F ` X ) .x. X ) ) )
Theorem	linc1 38550*	A vector is a linear combination of a set containing this vector. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.)
|- B = ( Base ` M )   &    |- S = (Scalar ` M )   &    |- .0. = ( 0g ` S )   &    |- .1. = ( 1r ` S )   &    |- F = ( x e. V |-> if ( x = X , .1. , .0. ) )   =>    |- ( ( M e. LMod /\ V e. ~P B /\ X e. V ) -> ( F ( linC ` M ) V ) = X )
Theorem	lincellss 38551	A linear combination of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 28-Jul-2019.)
|- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> ( ( F e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ F finSupp ( 0g ` (Scalar ` M ) ) ) -> ( F ( linC ` M ) V ) e. S ) )
Theorem	lco0 38552	The set of empty linear combinations over a monoid is the singleton with the identity element of the monoid. (Contributed by AV, 12-Apr-2019.)
|- ( M e. Mnd -> ( M LinCo (/) ) = { ( 0g ` M ) } )
Theorem	lcoel0 38553	The zero vector is always a linear combination. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
|- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( 0g ` M ) e. ( M LinCo V ) )
Theorem	lincsum 38554	The sum of two linear combinations is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 4-Apr-2019.) (Revised by AV, 28-Jul-2019.)
|- .+ = ( +g ` M )   &    |- X = ( A ( linC ` M ) V )   &    |- Y = ( B ( linC ` M ) V )   &    |- S = (Scalar ` M )   &    |- R = ( Base ` S )   &    |- .+b = ( +g ` S )   =>    |- ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) /\ ( A finSupp ( 0g ` S ) /\ B finSupp ( 0g ` S ) ) ) -> ( X .+ Y ) = ( ( A oF .+b B ) ( linC ` M ) V ) )
Theorem	lincscm 38555*	A linear combinations multiplied with a scalar is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 9-Apr-2019.) (Revised by AV, 28-Jul-2019.)
|- .xb = ( .s ` M )   &    |- .x. = ( .r ` (Scalar ` M ) )   &    |- X = ( A ( linC ` M ) V )   &    |- R = ( Base ` (Scalar ` M ) )   &    |- F = ( x e. V |-> ( S .x. ( A ` x ) ) )   =>    |- ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ ( A e. ( R ^m V ) /\ S e. R ) /\ A finSupp ( 0g ` (Scalar ` M ) ) ) -> ( S .xb X ) = ( F ( linC ` M ) V ) )
Theorem	lincsumcl 38556	The sum of two linear combinations is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 4-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.)
|- .+ = ( +g ` M )   =>    |- ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ ( C e. ( M LinCo V ) /\ D e. ( M LinCo V ) ) ) -> ( C .+ D ) e. ( M LinCo V ) )
Theorem	lincscmcl 38557	The multiplication of a linear combination with a scalar is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 11-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.)
|- .x. = ( .s ` M )   &    |- R = ( Base ` (Scalar ` M ) )   =>    |- ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ C e. R /\ D e. ( M LinCo V ) ) -> ( C .x. D ) e. ( M LinCo V ) )
Theorem	lincsumscmcl 38558	The sum of a linear combination and a multiplication of a linear combination with a scalar is a linear combination. (Contributed by AV, 11-Apr-2019.)
|- .x. = ( .s ` M )   &    |- R = ( Base ` (Scalar ` M ) )   &    |- .+ = ( +g ` M )   =>    |- ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ ( C e. R /\ D e. ( M LinCo V ) /\ B e. ( M LinCo V ) ) ) -> ( ( C .x. D ) .+ B ) e. ( M LinCo V ) )
Theorem	lincolss 38559	According to the statement in [Lang] p. 129, the set ( LSubSp ` M ) of all linear combinations of a set of vectors V is a submodule (generated by V) of the module M. The elements of V are called generators of ( LSubSp ` M ). (Contributed by AV, 12-Apr-2019.)
|- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( M LinCo V ) e. ( LSubSp ` M ) )
Theorem	ellcoellss 38560*	Every linear combination of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
|- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> A. x e. ( M LinCo V ) x e. S )
Theorem	lcoss 38561	A set of vectors of a module is a subset of the set of all linear combinations of the set. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
|- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> V C_ ( M LinCo V ) )
Theorem	lspsslco 38562	Lemma for lspeqlco 38564. (Contributed by AV, 17-Apr-2019.)
|- B = ( Base ` M )   =>    |- ( ( M e. LMod /\ V e. ~P B ) -> ( ( LSpan ` M ) ` V ) C_ ( M LinCo V ) )
Theorem	lcosslsp 38563	Lemma for lspeqlco 38564. (Contributed by AV, 20-Apr-2019.)
|- B = ( Base ` M )   =>    |- ( ( M e. LMod /\ V e. ~P B ) -> ( M LinCo V ) C_ ( ( LSpan ` M ) ` V ) )
Theorem	lspeqlco 38564	Equivalence of a span of a set of vectors of a left module defined as the intersection of all linear subspaces which each contain every vector in that set ( see df-lsp 17940) and as the set of all linear combinations of the vectors of the set with finite support. (Contributed by AV, 20-Apr-2019.)
|- B = ( Base ` M )   =>    |- ( ( M e. LMod /\ V e. ~P B ) -> ( M LinCo V ) = ( ( LSpan ` M ) ` V ) )
21.33.14.3  Linear independency According to the definition in [Lang] p. 129: "A subset S of a module M is said to be linearly independent (over [the ring] A) if whenever we have a linear combination ∑x ∈S axx which is equal to 0, then ax=0 for all x∈S.". This definition does not care for the finiteness of the set S (because the definition of a linear combination in [Lang] p.129 does already assure that only a finite number of coefficients can be 0 in the sum). Our definition df-lininds 38567 does also neither claim that the subset must be finite, nor that almost all coefficients within the linear combination are 0. If this is required, it must be explicitly stated as precondition in the corresponding theorems. Usually, the linear independency is defined for vector spaces, see Wikipedia ("Linear independence", 15-Apr-2019, https://en.wikipedia.org/wiki/Linear_independence): "In the theory of vector spaces, a set of vectors is said to be linearly dependent if at least one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be linearly independent.". Furthermore, "In order to allow the number of linearly independent vectors in a vector space to be countably infinite, it is useful to define linear dependence as follows. More generally, let V be a vector space over a field K, and let {vi | i∈I} be a family of elements of V. The family is linearly dependent over K if there exists a finite family {aj | j∈J} of elements of K, all non-zero, such that ∑j∈J ajvj=0. A set X of elements of V is linearly independent if the corresponding family{x}x∈X is linearly independent". Remark 1: There are already definitions of (linearly) independent families (df-lindf 19135) and (linearly) independent sets (df-linds 19136). These definitions are based on the principle "of vectors, no nonzero multiple of which can be expressed as a linear combination of other elements" or (see lbsind2 18049) "every element is not in the span of the remainder of the [set]". The equivalence of the definitions df-linds 19136 and df-lininds 38567 for (linear) independency for (left) modules is shown in lindslininds 38589. Remark 2: Subsets of the base set of a (left) module are linearly dependent if they are not linearly indepent (see df-lindeps 38569) or, according to Wikipedia, "if at least one of the vectors in the set can be defined as a linear combination of the others", see islindeps2 38608. The reversed implication is not valid for arbitrary modules (but for arbitrary vector spaces), because it requires a division by a coefficient. Therefore, the definition of Wikipedia is equivalent with our definition for (left) vector spaces (see isldepslvec2 38610) and not for (left) modules in general.
Syntax	clininds 38565	Extend class notation with the relation between a module and its linearly independent subsets.
class linIndS
Syntax	clindeps 38566	Extend class notation with the relation between a module and its linearly dependent subsets.
class linDepS
Definition	df-lininds 38567*	Define the relation between a module and its linearly independent subsets. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 24-Apr-2019.) (Revised by AV, 30-Jul-2019.)
|- linIndS = { <. s , m >. | ( s e. ~P ( Base ` m ) /\ A. f e. ( ( Base ` (Scalar ` m ) ) ^m s ) ( ( f finSupp ( 0g ` (Scalar ` m ) ) /\ ( f ( linC ` m ) s ) = ( 0g ` m ) ) -> A. x e. s ( f ` x ) = ( 0g ` (Scalar ` m ) ) ) ) }
Theorem	rellininds 38568	The class defining the relation between a module and its linearly independent subsets is a relation. (Contributed by AV, 13-Apr-2019.)
|- Rel linIndS
Definition	df-lindeps 38569*	Define the relation between a module and its linearly dependent subsets. (Contributed by AV, 26-Apr-2019.)
|- linDepS = { <. s , m >. | -. s linIndS m }
Theorem	linindsv 38570	The classes of the module and its linearly independent subsets are sets. (Contributed by AV, 13-Apr-2019.)
|- ( S linIndS M -> ( S e. _V /\ M e. _V ) )
Theorem	islininds 38571*	The property of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)
|- B = ( Base ` M )   &    |- Z = ( 0g ` M )   &    |- R = (Scalar ` M )   &    |- E = ( Base ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( S e. V /\ M e. W ) -> ( S linIndS M <-> ( S e. ~P B /\ A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) ) ) )
Theorem	linindsi 38572*	The implications of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)
|- B = ( Base ` M )   &    |- Z = ( 0g ` M )   &    |- R = (Scalar ` M )   &    |- E = ( Base ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( S linIndS M -> ( S e. ~P B /\ A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) ) )
Theorem	linindslinci 38573*	The implications of being a linearly independent subset and a linear combination of this subset being 0. (Contributed by AV, 24-Apr-2019.) (Revised by AV, 30-Jul-2019.)
|- B = ( Base ` M )   &    |- Z = ( 0g ` M )   &    |- R = (Scalar ` M )   &    |- E = ( Base ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( S linIndS M /\ ( F e. ( E ^m S ) /\ F finSupp .0. /\ ( F ( linC ` M ) S ) = Z ) ) -> A. x e. S ( F ` x ) = .0. )
Theorem	islinindfis 38574*	The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-2019.)
|- B = ( Base ` M )   &    |- Z = ( 0g ` M )   &    |- R = (Scalar ` M )   &    |- E = ( Base ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( S e. Fin /\ M e. W ) -> ( S linIndS M <-> ( S e. ~P B /\ A. f e. ( E ^m S ) ( ( f ( linC ` M ) S ) = Z -> A. x e. S ( f ` x ) = .0. ) ) ) )
Theorem	islinindfiss 38575*	The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-2019.)
|- B = ( Base ` M )   &    |- Z = ( 0g ` M )   &    |- R = (Scalar ` M )   &    |- E = ( Base ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( M e. W /\ S e. Fin /\ S e. ~P B ) -> ( S linIndS M <-> A. f e. ( E ^m S ) ( ( f ( linC ` M ) S ) = Z -> A. x e. S ( f ` x ) = .0. ) ) )
Theorem	linindscl 38576	A linearly independent set is a subset of (the base set of) a module. (Contributed by AV, 13-Apr-2019.)
|- ( S linIndS M -> S e. ~P ( Base ` M ) )
Theorem	lindepsnlininds 38577	A linearly dependent subset is not a linearly independent subset. (Contributed by AV, 26-Apr-2019.)
|- ( ( S e. V /\ M e. W ) -> ( S linDepS M <-> -. S linIndS M ) )
Theorem	islindeps 38578*	The property of being a linearly dependent subset. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.)
|- B = ( Base ` M )   &    |- Z = ( 0g ` M )   &    |- R = (Scalar ` M )   &    |- E = ( Base ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( M e. W /\ S e. ~P B ) -> ( S linDepS M <-> E. f e. ( E ^m S ) ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z /\ E. x e. S ( f ` x ) =/= .0. ) ) )
Theorem	lincext1 38579*	Property 1 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 29-Apr-2019.)
|- B = ( Base ` M )   &    |- R = (Scalar ` M )   &    |- E = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- Z = ( 0g ` M )   &    |- N = ( invg ` R )   &    |- F = ( z e. S |-> if ( z = X , ( N ` Y ) , ( G ` z ) ) )   =>    |- ( ( ( M e. LMod /\ S e. ~P B ) /\ ( Y e. E /\ X e. S /\ G e. ( E ^m ( S \ { X } ) ) ) ) -> F e. ( E ^m S ) )
Theorem	lincext2 38580*	Property 2 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 30-Jul-2019.)
|- B = ( Base ` M )   &    |- R = (Scalar ` M )   &    |- E = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- Z = ( 0g ` M )   &    |- N = ( invg ` R )   &    |- F = ( z e. S |-> if ( z = X , ( N ` Y ) , ( G ` z ) ) )   =>    |- ( ( ( M e. LMod /\ S e. ~P B ) /\ ( Y e. E /\ X e. S /\ G e. ( E ^m ( S \ { X } ) ) ) /\ G finSupp .0. ) -> F finSupp .0. )
Theorem	lincext3 38581*	Property 3 of an extension of a linear combination. (Contributed by AV, 23-Apr-2019.) (Revised by AV, 30-Jul-2019.)
|- B = ( Base ` M )   &    |- R = (Scalar ` M )   &    |- E = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- Z = ( 0g ` M )   &    |- N = ( invg ` R )   &    |- F = ( z e. S |-> if ( z = X , ( N ` Y ) , ( G ` z ) ) )   =>    |- ( ( ( M e. LMod /\ S e. ~P B ) /\ ( Y e. E /\ X e. S /\ G e. ( E ^m ( S \ { X } ) ) ) /\ ( G finSupp .0. /\ ( Y ( .s ` M ) X ) = ( G ( linC ` M ) ( S \ { X } ) ) ) ) -> ( F ( linC ` M ) S ) = Z )
Theorem	lindslinindsimp1 38582*	Implication 1 for lindslininds 38589. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
|- R = (Scalar ` M )   &    |- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- Z = ( 0g ` M )   =>    |- ( ( S e. V /\ M e. LMod ) -> ( ( S e. ~P ( Base ` M ) /\ A. f e. ( B ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) ) -> ( S C_ ( Base ` M ) /\ A. s e. S A. y e. ( B \ { .0. } ) -. ( y ( .s ` M ) s ) e. ( ( LSpan ` M ) ` ( S \ { s } ) ) ) ) )
Theorem	lindslinindimp2lem1 38583*	Lemma 1 for lindslinindsimp2 38588. (Contributed by AV, 25-Apr-2019.)
|- R = (Scalar ` M )   &    |- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- Z = ( 0g ` M )   &    |- Y = ( ( invg ` R ) ` ( f ` x ) )   &    |- G = ( f |` ( S \ { x } ) )   =>    |- ( ( ( S e. V /\ M e. LMod ) /\ ( S C_ ( Base ` M ) /\ x e. S /\ f e. ( B ^m S ) ) ) -> Y e. B )
Theorem	lindslinindimp2lem2 38584*	Lemma 2 for lindslinindsimp2 38588. (Contributed by AV, 25-Apr-2019.)
|- R = (Scalar ` M )   &    |- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- Z = ( 0g ` M )   &    |- Y = ( ( invg ` R ) ` ( f ` x ) )   &    |- G = ( f |` ( S \ { x } ) )   =>    |- ( ( ( S e. V /\ M e. LMod ) /\ ( S C_ ( Base ` M ) /\ x e. S /\ f e. ( B ^m S ) ) ) -> G e. ( B ^m ( S \ { x } ) ) )
Theorem	lindslinindimp2lem3 38585*	Lemma 3 for lindslinindsimp2 38588. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
|- R = (Scalar ` M )   &    |- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- Z = ( 0g ` M )   &    |- Y = ( ( invg ` R ) ` ( f ` x ) )   &    |- G = ( f |` ( S \ { x } ) )   =>    |- ( ( ( S e. V /\ M e. LMod ) /\ ( S C_ ( Base ` M ) /\ x e. S ) /\ ( f e. ( B ^m S ) /\ f finSupp .0. ) ) -> G finSupp .0. )
Theorem	lindslinindimp2lem4 38586*	Lemma 4 for lindslinindsimp2 38588. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
|- R = (Scalar ` M )   &    |- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- Z = ( 0g ` M )   &    |- Y = ( ( invg ` R ) ` ( f ` x ) )   &    |- G = ( f |` ( S \ { x } ) )   =>    |- ( ( ( S e. V /\ M e. LMod ) /\ ( S C_ ( Base ` M ) /\ x e. S ) /\ ( f e. ( B ^m S ) /\ f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) ) -> ( M gsumg ( y e. ( S \ { x } ) |-> ( ( f ` y ) ( .s ` M ) y ) ) ) = ( Y ( .s ` M ) x ) )
Theorem	lindslinindsimp2lem5 38587*	Lemma 5 for lindslinindsimp2 38588. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
|- R = (Scalar ` M )   &    |- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- Z = ( 0g ` M )   =>    |- ( ( ( S e. V /\ M e. LMod ) /\ ( S C_ ( Base ` M ) /\ x e. S ) ) -> ( ( f e. ( B ^m S ) /\ ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) ) -> ( A. y e. ( B \ { .0. } ) A. g e. ( B ^m ( S \ { x } ) ) ( -. g finSupp .0. \/ -. ( y ( .s ` M ) x ) = ( g ( linC ` M ) ( S \ { x } ) ) ) -> ( f ` x ) = .0. ) ) )
Theorem	lindslinindsimp2 38588*	Implication 2 for lindslininds 38589. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.)
|- R = (Scalar ` M )   &    |- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- Z = ( 0g ` M )   =>    |- ( ( S e. V /\ M e. LMod ) -> ( ( S C_ ( Base ` M ) /\ A. s e. S A. y e. ( B \ { .0. } ) -. ( y ( .s ` M ) s ) e. ( ( LSpan ` M ) ` ( S \ { s } ) ) ) -> ( S e. ~P ( Base ` M ) /\ A. f e. ( B ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) ) ) )
Theorem	lindslininds 38589	Equivalence of definitions df-linds 19136 and df-lininds 38567 for (linear) independency for (left) modules. (Contributed by AV, 26-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
|- ( ( S e. V /\ M e. LMod ) -> ( S linIndS M <-> S e. (LIndS ` M ) ) )
Theorem	linds0 38590	The empty set is always a linearly independet subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
|- ( M e. V -> (/) linIndS M )
Theorem	el0ldep 38591	A set containing the zero element of a module is always linearly dependent, if the underlying ring has at least two elements. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
|- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> S linDepS M )
Theorem	el0ldepsnzr 38592	A set containing the zero element of a module over a nonzero ring is always linearly dependent. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)
|- ( ( ( M e. LMod /\ (Scalar ` M ) e. NzRing ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> S linDepS M )
Theorem	lindsrng01 38593	Any subset of a module is always linearly independent if the underlying ring has at most one element. Since the underlying ring cannot be the empty set (see lmodsn0 17847), this means that the underlying ring has only one element, so it is a zero ring. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)
|- B = ( Base ` M )   &    |- R = (Scalar ` M )   &    |- E = ( Base ` R )   =>    |- ( ( M e. LMod /\ ( ( # ` E ) = 0 \/ ( # ` E ) = 1 ) /\ S e. ~P B ) -> S linIndS M )
Theorem	lindszr 38594	Any subset of a module over a zero ring is always linearly independent. (Contributed by AV, 27-Apr-2019.)
|- ( ( M e. LMod /\ -. (Scalar ` M ) e. NzRing /\ S e. ~P ( Base ` M ) ) -> S linIndS M )
Theorem	snlindsntorlem 38595*	Lemma for snlindsntor 38596. (Contributed by AV, 15-Apr-2019.)
|- B = ( Base ` M )   &    |- R = (Scalar ` M )   &    |- S = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- Z = ( 0g ` M )   &    |- .x. = ( .s ` M )   =>    |- ( ( M e. LMod /\ X e. B ) -> ( A. f e. ( S ^m { X } ) ( ( f ( linC ` M ) { X } ) = Z -> ( f ` X ) = .0. ) -> A. s e. S ( ( s .x. X ) = Z -> s = .0. ) ) )
Theorem	snlindsntor 38596*	A singleton is linearly independent iff it does not contain a torsion element. According to Wikipedia ("Torsion (algebra)", 15-Apr-2019, https://en.wikipedia.org/wiki/Torsion_(algebra)): "An element m of a module M over a ring R is called a torsion element of the module if there exists a regular element r of the ring (an element that is neither a left nor a right zero divisor) that annihilates m, i.e., ( r .x. m ) = 0. In an integral domain (a commutative ring without zero divisors), every non-zero element is regular, so a torsion element of a module over an integral domain is one annihilated by a non-zero element of the integral domain." Analogously, the definition in [Lang] p. 147 states that "An element x of [a module] E [over a ring R] is called a torsion element if there exists a e. R, a =/= 0, such that a .x. x = 0. This definition includes the zero element of the module. Some authors, however, exclude the zero element from the definition of torsion elements. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)
|- B = ( Base ` M )   &    |- R = (Scalar ` M )   &    |- S = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- Z = ( 0g ` M )   &    |- .x. = ( .s ` M )   =>    |- ( ( M e. LMod /\ X e. B ) -> ( A. s e. ( S \ { .0. } ) ( s .x. X ) =/= Z <-> { X } linIndS M ) )
Theorem	ldepsprlem 38597	Lemma for ldepspr 38598. (Contributed by AV, 16-Apr-2019.)
|- B = ( Base ` M )   &    |- R = (Scalar ` M )   &    |- S = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- Z = ( 0g ` M )   &    |- .x. = ( .s ` M )   &    |- .1. = ( 1r ` R )   &    |- N = ( invg ` R )   =>    |- ( ( M e. LMod /\ ( X e. B /\ Y e. B /\ A e. S ) ) -> ( X = ( A .x. Y ) -> ( ( .1. .x. X ) ( +g ` M ) ( ( N ` A ) .x. Y ) ) = Z ) )
Theorem	ldepspr 38598	If a vector is a scalar multiple of another vector, the (unordered pair containing the) two vectors are linearly dependent. (Contributed by AV, 16-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
|- B = ( Base ` M )   &    |- R = (Scalar ` M )   &    |- S = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- Z = ( 0g ` M )   &    |- .x. = ( .s ` M )   =>    |- ( ( M e. LMod /\ ( X e. B /\ Y e. B /\ X =/= Y ) ) -> ( ( A e. S /\ X = ( A .x. Y ) ) -> { X , Y } linDepS M ) )
Theorem	lincresunit3lem3 38599	Lemma 3 for lincresunit3 38606. (Contributed by AV, 18-May-2019.)
|- B = ( Base ` M )   &    |- R = (Scalar ` M )   &    |- E = ( Base ` R )   &    |- U = (Unit ` R )   &    |- N = ( invg ` R )   &    |- .x. = ( .s ` M )   =>    |- ( ( ( M e. LMod /\ X e. B /\ Y e. B ) /\ A e. U ) -> ( ( ( N ` A ) .x. X ) = ( ( N ` A ) .x. Y ) <-> X = Y ) )
Theorem	lincresunitlem1 38600	Lemma 1 for properties of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
|- B = ( Base ` M )   &    |- R = (Scalar ` M )   &    |- E = ( Base ` R )   &    |- U = (Unit ` R )   &    |- .0. = ( 0g ` R )   &    |- Z = ( 0g ` M )   &    |- N = ( invg ` R )   &    |- I = ( invr ` R )   &    |- .x. = ( .r ` R )   &    |- G = ( s e. ( S \ { X } ) |-> ( ( I ` ( N ` ( F ` X ) ) ) .x. ( F ` s ) ) )   =>    |- ( ( ( S e. ~P B /\ M e. LMod /\ X e. S ) /\ ( F e. ( E ^m S ) /\ ( F ` X ) e. U ) ) -> ( I ` ( N ` ( F ` X ) ) ) e. E )