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Theorem List for Metamath Proof Explorer - 39801-39900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremply1mulgsumlem3 39801* Lemma 3 for ply1mulgsum 39803. (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 39802* Lemma 4 for ply1mulgsum 39803. (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 39803* 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 39804 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 39805 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.  )
 
21.33.14.12  Univariate polynomials (examples)
 
Theoremlinply1 39806 A term of the form  x  -  C is a (univariate) polynomial, also called "linear polynomial". (Part of ply1remlem 23111). (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 39807 A term of the form  x  -  C evaluated for  x  =  V results in  V  -  C (part of ply1remlem 23111). (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 39808 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 39809 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
 
21.33.15  Linear algebra (extension)
 
21.33.15.1  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 19513 and df-scmat 19514 define diagonal and scalar matrices as sets.

 
Syntaxcdmatalt 39810 Alternative notation for the algebra of diagonal matrices.
 class DMatALT
 
Syntaxcscmatalt 39811 Alternative notation for the algebra of scalar matrices.
 class ScMatALT
 
Definitiondf-dmatalt 39812* 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 39813* 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 39814* 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 39815* 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 39816* 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 39817 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 )
 ) )
 
21.33.15.2  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 39820, 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 39821, so that we can show that such sets are submodules of the corresponding modules, see lincolss 39848.
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 18194, the set of all linear combinations as defined by df-lco 39821 follows the alternative approach. That both definitions are equivalent is shown by lspeqlco 39853.

 
Syntaxclinc 39818 Extend class notation with the operation constructing a linear combination (of vectors from a left module).
 class linC
 
Syntaxclinco 39819 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 39820* 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 39821* 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 39822* 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 39823* 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 39824* 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 39825* 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 39826* 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 39827 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 39828 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 39829 The value of an empty linear combination. (Contributed by AV, 12-Apr-2019.)
 |-  ( M  e.  X  ->  ( (/) ( linC  `  M ) (/) )  =  ( 0g `  M ) )
 
Theoremlincvalsng 39830 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 39831 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 39832 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 39833 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 39834 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 39835* 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 39836* 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 39837* 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 39838 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 39839* 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 39840 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 39841 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 39842 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 39843 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 39844* 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 39845 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 39846 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 39847 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 39848 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 39849* 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 39850 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 39851 Lemma for lspeqlco 39853. (Contributed by AV, 17-Apr-2019.)
 |-  B  =  ( Base `  M )   =>    |-  (
 ( M  e.  LMod  /\  V  e.  ~P B )  ->  ( ( LSpan `  M ) `  V )  C_  ( M LinCo  V ) )
 
Theoremlcosslsp 39852 Lemma for lspeqlco 39853. (Contributed by AV, 20-Apr-2019.)
 |-  B  =  ( Base `  M )   =>    |-  (
 ( M  e.  LMod  /\  V  e.  ~P B )  ->  ( M LinCo  V )  C_  ( ( LSpan `  M ) `  V ) )
 
Theoremlspeqlco 39853 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 18194) 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 ) )
 
21.33.15.3  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 39856 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 19362) and (linearly) independent sets (df-linds 19363). 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 18303) "every element is not in the span of the remainder of the [set]". The equivalence of the definitions df-linds 19363 and df-lininds 39856 for (linear) independency for (left) modules is shown in lindslininds 39878.
Remark 2: Subsets of the base set of a (left) module are linearly dependent if they are not linearly indepent (see df-lindeps 39858) 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 39897. 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 39899) and not for (left) modules in general.

 
Syntaxclininds 39854 Extend class notation with the relation between a module and its linearly independent subsets.
 class linIndS
 
Syntaxclindeps 39855 Extend class notation with the relation between a module and its linearly dependent subsets.
 class linDepS
 
Definitiondf-lininds 39856* 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 39857 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 39858* Define the relation between a module and its linearly dependent subsets. (Contributed by AV, 26-Apr-2019.)
 |- linDepS  =  { <. s ,  m >.  |  -.  s linIndS  m }
 
Theoremlinindsv 39859 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 39860* 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 39861* 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 39862* 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 39863* 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 39864* 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 39865 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 39866 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 39867* 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 39868* 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 39869* 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 39870* 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 39871* Implication 1 for lindslininds 39878. (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 39872* Lemma 1 for lindslinindsimp2 39877. (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 39873* Lemma 2 for lindslinindsimp2 39877. (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 39874* Lemma 3 for lindslinindsimp2 39877. (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 39875* Lemma 4 for lindslinindsimp2 39877. (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 39876* Lemma 5 for lindslinindsimp2 39877. (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 39877* Implication 2 for lindslininds 39878. (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 39878 Equivalence of definitions df-linds 19363 and df-lininds 39856 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 39879 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 39880 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 39881 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 39882 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 18103), 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 39883 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 39884* Lemma for snlindsntor 39885. (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 39885* 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 39886 Lemma for ldepspr 39887. (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 39887 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 39888 Lemma 3 for lincresunit3 39895. (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 39889 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 )
 
Theoremlincresunitlem2 39890 Lemma 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 ) )  /\  Y  e.  S )  ->  ( ( I `  ( N `  ( F `
  X ) ) )  .x.  ( F `  Y ) )  e.  E )
 
Theoremlincresunit1 39891* Property 1 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 ) )  ->  G  e.  ( E  ^m  ( S  \  { X } ) ) )
 
Theoremlincresunit2 39892* Property 2 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  /\  F finSupp  .0.  )
 )  ->  G finSupp  .0.  )
 
Theoremlincresunit3lem1 39893* Lemma 1 for lincresunit3 39895. (Contributed by AV, 17-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  /\  z  e.  ( S  \  { X } ) ) ) 
 ->  ( ( N `  ( F `  X ) ) ( .s `  M ) ( ( G `  z ) ( .s `  M ) z ) )  =  ( ( F `
  z ) ( .s `  M ) z ) )
 
Theoremlincresunit3lem2 39894* Lemma 2 for lincresunit3 39895. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-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  /\  F finSupp  .0.  )
 )  ->  ( ( N `  ( F `  X ) ) ( .s `  M ) ( M  gsumg  ( z  e.  ( S  \  { X }
 )  |->  ( ( G `
  z ) ( .s `  M ) z ) ) ) )  =  ( ( F  |`  ( S  \  { X } )
 ) ( linC  `  M ) ( S  \  { X } ) ) )
 
Theoremlincresunit3 39895* Property 3 of a specially modified restriction of a linear combination in a vector space. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-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  /\  F finSupp  .0.  )  /\  ( F ( linC  `  M ) S )  =  Z )  ->  ( G ( linC  `  M ) ( S 
 \  { X }
 ) )  =  X )
 
Theoremlincreslvec3 39896* Property 3 of a specially modified restriction of a linear combination in a vector space. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-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.  LVec  /\  X  e.  S )  /\  ( F  e.  ( E  ^m  S )  /\  ( F `  X )  =/= 
 .0.  /\  F finSupp  .0.  )  /\  ( F ( linC  `  M ) S )  =  Z )  ->  ( G ( linC  `  M ) ( S 
 \  { X }
 ) )  =  X )
 
Theoremislindeps2 39897* Conditions for being a linearly dependent subset of a (left) module over a nonzero ring. (Contributed by AV, 29-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  ( 0g `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( M  e.  LMod  /\  S  e.  ~P B  /\  R  e. NzRing )  ->  ( E. s  e.  S  E. f  e.  ( E  ^m  ( S  \  { s } )
 ) ( f finSupp  .0.  /\  ( f ( linC  `  M ) ( S  \  { s } )
 )  =  s ) 
 ->  S linDepS  M ) )
 
Theoremislininds2 39898* Implication of being a linearly independent subset of a (left) module over a nonzero ring. (Contributed by AV, 29-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  ( 0g `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( M  e.  LMod  /\  S  e.  ~P B  /\  R  e. NzRing )  ->  ( S linIndS  M  ->  A. s  e.  S  A. f  e.  ( E  ^m  ( S  \  { s }
 ) ) ( -.  f finSupp  .0.  \/  ( f ( linC  `  M )
 ( S  \  {
 s } ) )  =/=  s ) ) )
 
Theoremisldepslvec2 39899* Alternative definition of being a linearly dependent subset of a (left) vector space. In this case, the reverse implication of islindeps2 39897 holds, so that both definitions are equivalent (see theorem 1.6 in [Roman] p. 46 and the note in [Roman] p. 112: if a nontrivial linear combination of elements (where not all of the coefficients are 0) in an R-vector space is 0, then and only then each of the elements is a linear combination of the others. (Contributed by AV, 30-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  ( 0g `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( M  e.  LVec  /\  S  e.  ~P B )  ->  ( E. s  e.  S  E. f  e.  ( E  ^m  ( S  \  { s }
 ) ) ( f finSupp  .0.  /\  ( f ( linC  `  M ) ( S 
 \  { s }
 ) )  =  s )  <->  S linDepS  M ) )
 
Theoremlindssnlvec 39900 A singleton not containing the zero element of a vector space is always linearly independent. (Contributed by AV, 16-Apr-2019.) (Revised by AV, 28-Apr-2019.)
 |-  (
 ( M  e.  LVec  /\  S  e.  ( Base `  M )  /\  S  =/=  ( 0g `  M ) )  ->  { S } linIndS  M )
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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 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