P. 333 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 33201-33300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	zlmodzxzadd 33201	The addition of the ZZ-module ZZ X. ZZ. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
|- Z = (ℤring freeLMod { 0 , 1 } )   &    |- .+ = ( +g ` Z )   =>    |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. ZZ /\ D e. ZZ ) ) -> ( { <. 0 , A >. , <. 1 , C >. } .+ { <. 0 , B >. , <. 1 , D >. } ) = { <. 0 , ( A + B ) >. , <. 1 , ( C + D ) >. } )
Theorem	zlmodzxzsubm 33202	The subtraction of the ZZ-module ZZ X. ZZ expressed as addition. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
|- Z = (ℤring freeLMod { 0 , 1 } )   &    |- .- = ( -g ` Z )   =>    |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. ZZ /\ D e. ZZ ) ) -> ( { <. 0 , A >. , <. 1 , C >. } .- { <. 0 , B >. , <. 1 , D >. } ) = ( { <. 0 , A >. , <. 1 , C >. } ( +g ` Z ) ( -u 1 ( .s ` Z ) { <. 0 , B >. , <. 1 , D >. } ) ) )
Theorem	zlmodzxzsub 33203	The subtraction of the ZZ-module ZZ X. ZZ. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
|- Z = (ℤring freeLMod { 0 , 1 } )   &    |- .- = ( -g ` Z )   =>    |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. ZZ /\ D e. ZZ ) ) -> ( { <. 0 , A >. , <. 1 , C >. } .- { <. 0 , B >. , <. 1 , D >. } ) = { <. 0 , ( A - B ) >. , <. 1 , ( C - D ) >. } )
21.24.11.4  Ordered group sum operation (extension)
Theorem	gsumpr 33204*	Group sum of a pair. (Contributed by AV, 6-Dec-2018.) (Proof shortened by AV, 28-Jul-2019.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( k = M -> A = C )   &    |- ( k = N -> A = D )   =>    |- ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> ( G gsumg ( k e. { M , N } |-> A ) ) = ( C .+ D ) )
Theorem	mgpsumunsn 33205*	Extract a summand/factor from the group sum for the multiplicative group of a unital ring. (Contributed by AV, 29-Dec-2018.)
|- M = (mulGrp ` R )   &    |- .x. = ( .r ` R )   &    |- ( ph -> R e. CRing )   &    |- ( ph -> N e. Fin )   &    |- ( ph -> I e. N )   &    |- ( ( ph /\ k e. N ) -> A e. ( Base ` R ) )   &    |- ( ph -> X e. ( Base ` R ) )   &    |- ( k = I -> A = X )   =>    |- ( ph -> ( M gsumg ( k e. N |-> A ) ) = ( ( M gsumg ( k e. ( N \ { I } ) |-> A ) ) .x. X ) )
Theorem	mgpsumz 33206*	If the group sum for the multiplicative group of a unital ring contains a summand/factor that is the zero of the ring, the group sum itself is zero. (Contributed by AV, 29-Dec-2018.)
|- M = (mulGrp ` R )   &    |- .x. = ( .r ` R )   &    |- ( ph -> R e. CRing )   &    |- ( ph -> N e. Fin )   &    |- ( ph -> I e. N )   &    |- ( ( ph /\ k e. N ) -> A e. ( Base ` R ) )   &    |- .0. = ( 0g ` R )   &    |- ( k = I -> A = .0. )   =>    |- ( ph -> ( M gsumg ( k e. N |-> A ) ) = .0. )
Theorem	mgpsumn 33207*	If the group sum for the multiplicative group of a unital ring contains a summand/factor that is the one of the ring, this summand/ factor can be removed from the group sum. (Contributed by AV, 29-Dec-2018.)
|- M = (mulGrp ` R )   &    |- .x. = ( .r ` R )   &    |- ( ph -> R e. CRing )   &    |- ( ph -> N e. Fin )   &    |- ( ph -> I e. N )   &    |- ( ( ph /\ k e. N ) -> A e. ( Base ` R ) )   &    |- .1. = ( 1r ` R )   &    |- ( k = I -> A = .1. )   =>    |- ( ph -> ( M gsumg ( k e. N |-> A ) ) = ( M gsumg ( k e. ( N \ { I } ) |-> A ) ) )
Theorem	gsumsplit2f 33208*	Split a group sum into two parts. (Contributed by AV, 4-Sep-2019.)
|- F/ k ph   &    |- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> A e. V )   &    |- ( ( ph /\ k e. A ) -> X e. B )   &    |- ( ph -> ( k e. A |-> X ) finSupp .0. )   &    |- ( ph -> ( C i^i D ) = (/) )   &    |- ( ph -> A = ( C u. D ) )   =>    |- ( ph -> ( G gsumg ( k e. A |-> X ) ) = ( ( G gsumg ( k e. C |-> X ) ) .+ ( G gsumg ( k e. D |-> X ) ) ) )
Theorem	gsumdifsndf 33209*	Extract a summand from a finitely supported group sum. (Contributed by AV, 4-Sep-2019.)
|- F/_ k Y   &    |- F/ k ph   &    |- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> A e. W )   &    |- ( ph -> ( k e. A |-> X ) finSupp ( 0g ` G ) )   &    |- ( ( ph /\ k e. A ) -> X e. B )   &    |- ( ph -> M e. A )   &    |- ( ph -> Y e. B )   &    |- ( ( ph /\ k = M ) -> X = Y )   =>    |- ( ph -> ( G gsumg ( k e. A |-> X ) ) = ( ( G gsumg ( k e. ( A \ { M } ) |-> X ) ) .+ Y ) )
21.24.11.5  Symmetric groups (extension)
Theorem	nn0le2is012 33210	A nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 16-Mar-2019.)
|- ( ( N e. NN0 /\ N <_ 2 ) -> ( N = 0 \/ N = 1 \/ N = 2 ) )
Theorem	exple2lt6 33211	A nonnegative integer to the power of itself is less than 6 if it is less than or equal to 2. (Contributed by AV, 16-Mar-2019.)
|- ( ( N e. NN0 /\ N <_ 2 ) -> ( N ^ N ) < 6 )
Theorem	pgrple2abl 33212	Every symmetric group on a set with at most 2 elements is abelian. (Contributed by AV, 16-Mar-2019.)
|- G = ( SymGrp ` A )   =>    |- ( ( A e. V /\ ( # ` A ) <_ 2 ) -> G e. Abel )
Theorem	pgrpgt2nabl 33213	Every symmetric group on a set with more than 2 elements is not abelian, see also the remark in [Rotman] p. 28. (Contributed by AV, 21-Mar-2019.)
|- G = ( SymGrp ` A )   =>    |- ( ( A e. V /\ 2 < ( # ` A ) ) -> G e/ Abel )
21.24.11.6  Divisibility (extension)
Theorem	invginvrid 33214	Identity for a multiplication with additive and multiplicative inverses in a ring. (Contributed by AV, 18-May-2018.)
|- B = ( Base ` R )   &    |- U = (Unit ` R )   &    |- N = ( invg ` R )   &    |- I = ( invr ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Ring /\ X e. B /\ Y e. U ) -> ( ( N ` Y ) .x. ( ( I ` ( N ` Y ) ) .x. X ) ) = X )
21.24.11.7  The support of functions (extension)
Theorem	rmsupp0 33215*	The support of a mapping of a multiplication of zero with a function into a ring is empty. (Contributed by AV, 10-Apr-2019.)
|- R = ( Base ` M )   =>    |- ( ( ( M e. Ring /\ V e. X /\ C = ( 0g ` M ) ) /\ A e. ( R ^m V ) ) -> ( ( v e. V |-> ( C ( .r ` M ) ( A ` v ) ) ) supp ( 0g ` M ) ) = (/) )
Theorem	domnmsuppn0 33216*	The support of a mapping of a multiplication of a nonzero constant with a function into a (ring theoretic) domain equals the support of the function. (Contributed by AV, 11-Apr-2019.)
|- R = ( Base ` M )   =>    |- ( ( ( M e. Domn /\ V e. X ) /\ ( C e. R /\ C =/= ( 0g ` M ) ) /\ A e. ( R ^m V ) ) -> ( ( v e. V |-> ( C ( .r ` M ) ( A ` v ) ) ) supp ( 0g ` M ) ) = ( A supp ( 0g ` M ) ) )
Theorem	rmsuppss 33217*	The support of a mapping of a multiplication of a constant with a function into a ring is a subset of the support of the function. (Contributed by AV, 11-Apr-2019.)
|- R = ( Base ` M )   =>    |- ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) -> ( ( v e. V |-> ( C ( .r ` M ) ( A ` v ) ) ) supp ( 0g ` M ) ) C_ ( A supp ( 0g ` M ) ) )
Theorem	mndpsuppss 33218	The support of a mapping of a scalar multiplication with a function of scalars is a subset of the support of the function of scalars. (Contributed by AV, 5-Apr-2019.)
|- R = ( Base ` M )   =>    |- ( ( ( M e. Mnd /\ V e. X ) /\ ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) ) -> ( ( A oF ( +g ` M ) B ) supp ( 0g ` M ) ) C_ ( ( A supp ( 0g ` M ) ) u. ( B supp ( 0g ` M ) ) ) )
Theorem	scmsuppss 33219*	The support of a mapping of a scalar multiplication with a function of scalars is a subset of the support of the function of scalars. (Contributed by AV, 5-Apr-2019.)
|- S = (Scalar ` M )   &    |- R = ( Base ` S )   =>    |- ( ( M e. LMod /\ V e. ~P ( Base ` M ) /\ A e. ( R ^m V ) ) -> ( ( v e. V |-> ( ( A ` v ) ( .s ` M ) v ) ) supp ( 0g ` M ) ) C_ ( A supp ( 0g ` S ) ) )
21.24.11.8  Finitely supported functions (extension)
Theorem	rmsuppfi 33220*	The support of a mapping of a multiplication of a constant with a function into a ring is finite if the support of the function is finite. (Contributed by AV, 11-Apr-2019.)
|- R = ( Base ` M )   =>    |- ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) /\ ( A supp ( 0g ` M ) ) e. Fin ) -> ( ( v e. V |-> ( C ( .r ` M ) ( A ` v ) ) ) supp ( 0g ` M ) ) e. Fin )
Theorem	rmfsupp 33221*	A mapping of a multiplication of a constant with a function into a ring is finitely supported if the function is finitely supported. (Contributed by AV, 9-Jun-2019.)
|- R = ( Base ` M )   =>    |- ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) /\ A finSupp ( 0g ` M ) ) -> ( v e. V |-> ( C ( .r ` M ) ( A ` v ) ) ) finSupp ( 0g ` M ) )
Theorem	mndpsuppfi 33222	The support of a mapping of a scalar multiplication with a function of scalars is finite if the support of the function of scalars is finite. (Contributed by AV, 5-Apr-2019.)
|- R = ( Base ` M )   =>    |- ( ( ( M e. Mnd /\ V e. X ) /\ ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) /\ ( ( A supp ( 0g ` M ) ) e. Fin /\ ( B supp ( 0g ` M ) ) e. Fin ) ) -> ( ( A oF ( +g ` M ) B ) supp ( 0g ` M ) ) e. Fin )
Theorem	mndpfsupp 33223	A mapping of a scalar multiplication with a function of scalars is finitely supported if the function of scalars is finitely supported. (Contributed by AV, 9-Jun-2019.)
|- R = ( Base ` M )   =>    |- ( ( ( M e. Mnd /\ V e. X ) /\ ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) /\ ( A finSupp ( 0g ` M ) /\ B finSupp ( 0g ` M ) ) ) -> ( A oF ( +g ` M ) B ) finSupp ( 0g ` M ) )
Theorem	scmsuppfi 33224*	The support of a mapping of a scalar multiplication with a function of scalars is finite if the support of the function of scalars is finite. (Contributed by AV, 5-Apr-2019.)
|- S = (Scalar ` M )   &    |- R = ( Base ` S )   =>    |- ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ A e. ( R ^m V ) /\ ( A supp ( 0g ` S ) ) e. Fin ) -> ( ( v e. V |-> ( ( A ` v ) ( .s ` M ) v ) ) supp ( 0g ` M ) ) e. Fin )
Theorem	scmfsupp 33225*	A mapping of a scalar multiplication with a function of scalars is finitely supported if the function of scalars is finitely supported. (Contributed by AV, 9-Jun-2019.)
|- S = (Scalar ` M )   &    |- R = ( Base ` S )   =>    |- ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ A e. ( R ^m V ) /\ A finSupp ( 0g ` S ) ) -> ( v e. V |-> ( ( A ` v ) ( .s ` M ) v ) ) finSupp ( 0g ` M ) )
Theorem	suppmptcfin 33226*	The support of a mapping with value 0 except of one is finite. (Contributed by AV, 27-Apr-2019.)
|- B = ( Base ` M )   &    |- R = (Scalar ` M )   &    |- .0. = ( 0g ` R )   &    |- .1. = ( 1r ` R )   &    |- F = ( x e. V |-> if ( x = X , .1. , .0. ) )   =>    |- ( ( M e. LMod /\ V e. ~P B /\ X e. V ) -> ( F supp .0. ) e. Fin )
Theorem	mptcfsupp 33227*	A mapping with value 0 except of one is finitely supported. (Contributed by AV, 9-Jun-2019.)
|- B = ( Base ` M )   &    |- R = (Scalar ` M )   &    |- .0. = ( 0g ` R )   &    |- .1. = ( 1r ` R )   &    |- F = ( x e. V |-> if ( x = X , .1. , .0. ) )   =>    |- ( ( M e. LMod /\ V e. ~P B /\ X e. V ) -> F finSupp .0. )
Theorem	fsuppmptdmf 33228*	A mapping with a finite domain is finitely supported. (Contributed by AV, 4-Sep-2019.)
|- F/ x ph   &    |- F = ( x e. A |-> Y )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ x e. A ) -> Y e. V )   &    |- ( ph -> Z e. W )   =>    |- ( ph -> F finSupp Z )
21.24.11.9  Left modules (extension)
Theorem	lmodvsmdi 33229	Multiple distributive law for scalar product (left-distributivity). (Contributed by AV, 5-Sep-2019.)
|- V = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- .x. = ( .s ` W )   &    |- K = ( Base ` F )   &    |- .^ = (.g ` W )   &    |- E = (.g ` F )   =>    |- ( ( W e. LMod /\ ( R e. K /\ N e. NN0 /\ X e. V ) ) -> ( R .x. ( N .^ X ) ) = ( ( N E R ) .x. X ) )
Theorem	gsumlsscl 33230*	Closure of a group sum in a linear subspace: A (finitely supported) sum of scalar multiplications of vectors 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.)
|- S = ( LSubSp ` M )   &    |- R = (Scalar ` M )   &    |- B = ( Base ` R )   =>    |- ( ( M e. LMod /\ Z e. S /\ V C_ Z ) -> ( ( F e. ( B ^m V ) /\ F finSupp ( 0g ` R ) ) -> ( M gsumg ( v e. V |-> ( ( F ` v ) ( .s ` M ) v ) ) ) e. Z ) )
21.24.11.10  Associative algebras (extension)
Theorem	ascl0 33231	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 33232	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 33233	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 33234	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 ) )
21.24.11.11  Univariate polynomials (extension)
Theorem	ply1vr1smo 33235	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 33236	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 33237	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 33238*	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 33239	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 33240*	Lemma 1 for ply1mulgsum 33244. (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 33241*	Lemma 2 for ply1mulgsum 33244. (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 33242*	Lemma 3 for ply1mulgsum 33244. (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 33243*	Lemma 4 for ply1mulgsum 33244. (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 33244*	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 33245	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 33246	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. )
21.24.11.12  Univariate polynomials (examples)
Theorem	linply1 33247	A term of the form x - C is a (univariate) polynomial, also called "linear polynomial". (Part of ply1remlem 22729). (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 33248	A term of the form x - C evaluated for x = V results in V - C (part of ply1remlem 22729). (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 33249	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 33250	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
21.24.12  Linear algebra (extension)
21.24.12.1  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 19159 and df-scmat 19160 define diagonal and scalar matrices as sets.
Syntax	cdmatalt 33251	Alternative notation for the algebra of diagonal matrices.
class DMatALT
Syntax	cscmatalt 33252	Alternative notation for the algebra of scalar matrices.
class ScMatALT
Definition	df-dmatalt 33253*	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 33254*	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 33255*	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 33256*	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 33257*	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 33258	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 ) ) )
21.24.12.2  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 33261, 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 33262, so that we can show that such sets are submodules of the corresponding modules, see lincolss 33289. 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 17813, the set of all linear combinations as defined by df-lco 33262 follows the alternative approach. That both definitions are equivalent is shown by lspeqlco 33294.
Syntax	clinc 33259	Extend class notation with the operation constructing a linear combination (of vectors from a left module).
class linC
Syntax	clinco 33260	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 33261*	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 33262*	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 33263*	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 33264*	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 33265*	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 33266*	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 33267*	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 33268	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 33269	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 33270	The value of an empty linear combination. (Contributed by AV, 12-Apr-2019.)
|- ( M e. X -> ( (/) ( linC ` M ) (/) ) = ( 0g ` M ) )
Theorem	lincvalsng 33271	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 33272	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 33273	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 33274	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 33275	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 33276*	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 33277*	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 33278*	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 33279	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 33280*	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 33281	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 33282	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 33283	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 33284	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 33285*	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 33286	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 33287	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 33288	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 33289	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 33290*	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 33291	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 33292	Lemma for lspeqlco 33294. (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 33293	Lemma for lspeqlco 33294. (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 33294	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 17813) 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 ) )
21.24.12.3  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 33297 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 19008) and (linearly) independent sets (df-linds 19009). 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 17922) "every element is not in the span of the remainder of the [set]". The equivalence of the definitions df-linds 19009 and df-lininds 33297 for (linear) independency for (left) modules is shown in lindslininds 33319. Remark 2: Subsets of the base set of a (left) module are linearly dependent if they are not linearly indepent (see df-lindeps 33299) 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 33338. 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 33340) and not for (left) modules in general.
Syntax	clininds 33295	Extend class notation with the relation between a module and its linearly independent subsets.
class linIndS
Syntax	clindeps 33296	Extend class notation with the relation between a module and its linearly dependent subsets.
class linDepS
Definition	df-lininds 33297*	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 33298	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 33299*	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 33300	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 ) )