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Theorem List for Metamath Proof Explorer - 38501-38600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
21.33.13.10  Associative algebras (extension)
 
Theoremascl0 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 ) )
 
Theoremascl1 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 ) )
 
Theoremassaascl0 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 ) )
 
Theoremassaascl1 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)
 
Theoremply1vr1smo 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 )
 
Theoremply1ass23l 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 ) ) )
 
Theoremply1sclrmsm 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 ) )
 
Theoremcoe1id 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.  ) ) )
 
Theoremcoe1sclmulval 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 ) ) )
 
Theoremply1mulgsumlem1 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 ) ) ) )
 
Theoremply1mulgsumlem2 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 ) ) )
 
Theoremply1mulgsumlem3 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 ) )
 
Theoremply1mulgsumlem4 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 ) )
 
Theoremply1mulgsum 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 ) ) ) ) )
 
Theoremevl1at0 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.  )
 
Theoremevl1at1 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)
 
Theoremlinply1 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 )
 
Theoremlineval 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 ) )
 
Theoremzringsubgval 38519 Subtraction in the ring of integers. (Contributed by AV, 3-Aug-2019.)
 |-  .-  =  ( -g ` ring )   =>    |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( X  -  Y )  =  ( X  .-  Y ) )
 
Theoremlinevalexample 38520 The polynomial  x  -  3 over 
ZZ evaluated for  x  =  5 results in 2. (Contributed by AV, 3-Jul-2019.)
 |-  P  =  (Poly1 ` ring )   &    |-  B  =  (
 Base `  P )   &    |-  X  =  (var1 ` ring )   &    |-  .-  =  ( -g `  P )   &    |-  A  =  (algSc `  P )   &    |-  G  =  ( X  .-  ( A `  3 ) )   &    |-  O  =  (eval1 ` ring )   =>    |-  ( ( 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.

 
Syntaxcdmatalt 38521 Alternative notation for the algebra of diagonal matrices.
 class DMatALT
 
Syntaxcscmatalt 38522 Alternative notation for the algebra of scalar matrices.
 class ScMatALT
 
Definitiondf-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 ]_ ( as  { m  e.  ( Base `  a )  | 
 A. i  e.  n  A. j  e.  n  ( i  =/=  j  ->  ( i m j )  =  ( 0g
 `  r ) ) } ) )
 
Definitiondf-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 ]_ ( as  { 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
 ) ) } )
 )
 
TheoremdmatALTval 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  =  ( As  { m  e.  B  |  A. i  e.  N  A. j  e.  N  ( i  =/=  j  ->  ( i m j )  =  .0.  ) } ) )
 
TheoremdmatALTbas 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.  ) } )
 
TheoremdmatALTbasel 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.  ) ) ) )
 
Theoremdmatbas 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.

 
Syntaxclinc 38529 Extend class notation with the operation constructing a linear combination (of vectors from a left module).
 class linC
 
Syntaxclinco 38530 Extend class notation with the operation constructing a set of linear combinations (of vectors from a left module) with finite support.
 class LinCo
 
Definitiondf-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 ) ) ) ) )
 
Definitiondf-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 ) ) } )
 
Theoremlincop 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 ) ) ) ) )
 
Theoremlincval 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 ) ) ) )
 
Theoremdflinc2 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 ) ) ) ) )
 
Theoremlcoop 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 ) ) } )
 
Theoremlcoval 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 ) ) ) ) )
 
Theoremlincfsuppcl 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 )
 
Theoremlinccl 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 )
 
Theoremlincval0 38540 The value of an empty linear combination. (Contributed by AV, 12-Apr-2019.)
 |-  ( M  e.  X  ->  ( (/) ( linC  `  M ) (/) )  =  ( 0g `  M ) )
 
Theoremlincvalsng 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 ) )
 
Theoremlincvalsn 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 ) )
 
Theoremlincvalpr 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 ) ) )
 
Theoremlincval1 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 ) )
 
Theoremlcosn0 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 ) ) )
 
Theoremlincvalsc0 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 )
 
Theoremlcoc0 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 ) )
 
Theoremlinc0scn0 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 )
 
Theoremlincdifsn 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 ) ) )
 
Theoremlinc1 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 )
 
Theoremlincellss 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 ) )
 
Theoremlco0 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 ) } )
 
Theoremlcoel0 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 ) )
 
Theoremlincsum 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 ) )
 
Theoremlincscm 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 ) )
 
Theoremlincsumcl 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 ) )
 
Theoremlincscmcl 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 ) )
 
Theoremlincsumscmcl 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 ) )
 
Theoremlincolss 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 ) )
 
Theoremellcoellss 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 )
 
Theoremlcoss 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 ) )
 
Theoremlspsslco 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 ) )
 
Theoremlcosslsp 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 ) )
 
Theoremlspeqlco 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.

 
Syntaxclininds 38565 Extend class notation with the relation between a module and its linearly independent subsets.
 class linIndS
 
Syntaxclindeps 38566 Extend class notation with the relation between a module and its linearly dependent subsets.
 class linDepS
 
Definitiondf-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 ) ) ) ) }
 
Theoremrellininds 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
 
Definitiondf-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 }
 
Theoremlinindsv 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 ) )
 
Theoremislininds 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.  )
 ) ) )
 
Theoremlinindsi 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.  )
 ) )
 
Theoremlinindslinci 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.  )
 
Theoremislinindfis 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.  )
 ) ) )
 
Theoremislinindfiss 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.  )
 ) )
 
Theoremlinindscl 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 )
 )
 
Theoremlindepsnlininds 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 )
 )
 
Theoremislindeps 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.  ) ) )
 
Theoremlincext1 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 ) )
 
Theoremlincext2 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.  )
 
Theoremlincext3 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 )
 
Theoremlindslinindsimp1 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 } )
 ) ) ) )
 
Theoremlindslinindimp2lem1 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 )
 
Theoremlindslinindimp2lem2 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 } ) ) )
 
Theoremlindslinindimp2lem3 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.  )
 
Theoremlindslinindimp2lem4 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 ) )
 
Theoremlindslinindsimp2lem5 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.  ) ) )
 
Theoremlindslinindsimp2 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.  )
 ) ) )
 
Theoremlindslininds 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 ) ) )
 
Theoremlinds0 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 )
 
Theoremel0ldep 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 )
 
Theoremel0ldepsnzr 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 )
 
Theoremlindsrng01 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 )
 
Theoremlindszr 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 )
 
Theoremsnlindsntorlem 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.  )
 ) )
 
Theoremsnlindsntor 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 )
 )
 
Theoremldepsprlem 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 ) )
 
Theoremldepspr 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 )
 )
 
Theoremlincresunit3lem3 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 )
 )
 
Theoremlincresunitlem1 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 )
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38873
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