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Theorem List for Metamath Proof Explorer - 40401-40500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem2t6m3t4e0 40401 2 times 6 minus 3 times 4 equals 0. (Contributed by AV, 24-May-2019.)
 |-  (
 ( 2  x.  6
 )  -  ( 3  x.  4 ) )  =  0
 
Theoremssnn0ssfz 40402* For any finite subset of  NN0, find a superset in the form of a set of sequential integers, analogous to ssnnssfz 28415. (Contributed by AV, 30-Sep-2019.)
 |-  ( A  e.  ( ~P NN0 
 i^i  Fin )  ->  E. n  e.  NN0  A  C_  (
 0 ... n ) )
 
Theoremnn0sumltlt 40403 If the sum of two nonnegative integers is less than a third integer, then one of the summands is already less than this third integer. (Contributed by AV, 19-Oct-2019.)
 |-  (
 ( a  e.  NN0  /\  b  e.  NN0  /\  c  e.  NN0 )  ->  (
 ( a  +  b
 )  <  c  ->  b  <  c ) )
 
21.33.14.2  The binomial coefficient operation (extension)
 
Theorembcpascm1 40404 Pascal's rule for the binomial coefficient, generalized to all integers  K, shifted down by 1. (Contributed by AV, 8-Sep-2019.)
 |-  (
 ( N  e.  NN  /\  K  e.  ZZ )  ->  ( ( ( N  -  1 )  _C  K )  +  (
 ( N  -  1
 )  _C  ( K  -  1 ) ) )  =  ( N  _C  K ) )
 
Theoremaltgsumbc 40405* The sum of binomial coefficients for a fixed positive  N with alternating signs is zero. Notice that this is not valid for  N  =  0 (since  ( ( -u
1 ^ 0 )  x.  ( 0  _C  0 ) )  =  ( 1  x.  1 )  =  1). For a proof using Pascal's rule (bcpascm1 40404) instead of the binomial theorem (binom 13936) , see altgsumbcALT 40406. (Contributed by AV, 13-Sep-2019.)
 |-  ( N  e.  NN  ->  sum_
 k  e.  ( 0
 ... N ) ( ( -u 1 ^ k
 )  x.  ( N  _C  k ) )  =  0 )
 
TheoremaltgsumbcALT 40406* Alternate proof of altgsumbc 40405, using Pascal's rule (bcpascm1 40404) instead of the binomial theorem (binom 13936). (Contributed by AV, 8-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( N  e.  NN  ->  sum_
 k  e.  ( 0
 ... N ) ( ( -u 1 ^ k
 )  x.  ( N  _C  k ) )  =  0 )
 
21.33.14.3  The ` ZZ `-module ` ZZ X. ZZ `
 
Theoremzlmodzxzlmod 40407 The  ZZ-module  ZZ  X.  ZZ is a (left) module with the ring of integers as base set. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   =>    |-  ( Z  e.  LMod  /\ring  =  (Scalar `  Z ) )
 
Theoremzlmodzxzel 40408 An element of the (base set of the) 
ZZ-module  ZZ  X.  ZZ. (Contributed by AV, 21-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   =>    |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  { <. 0 ,  A >. ,  <. 1 ,  B >. }  e.  ( Base `  Z ) )
 
Theoremzlmodzxz0 40409 The  0 of the  ZZ-module  ZZ  X.  ZZ. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  .0.  =  { <. 0 ,  0 >. ,  <. 1 ,  0
 >. }   =>    |- 
 .0.  =  ( 0g `  Z )
 
Theoremzlmodzxzscm 40410 The scalar multiplication of the 
ZZ-module  ZZ  X.  ZZ. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  .xb  =  ( .s `  Z )   =>    |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A 
 .xb  { <. 0 ,  B >. ,  <. 1 ,  C >. } )  =  { <. 0 ,  ( A  x.  B ) >. , 
 <. 1 ,  ( A  x.  C ) >. } )
 
Theoremzlmodzxzadd 40411 The addition of the  ZZ-module  ZZ  X.  ZZ. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  .+  =  ( +g  `  Z )   =>    |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )
 )  ->  ( { <. 0 ,  A >. , 
 <. 1 ,  C >. } 
 .+  { <. 0 ,  B >. ,  <. 1 ,  D >. } )  =  { <. 0 ,  ( A  +  B ) >. , 
 <. 1 ,  ( C  +  D ) >. } )
 
Theoremzlmodzxzsubm 40412 The subtraction of the  ZZ-module  ZZ  X.  ZZ expressed as addition. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  .-  =  ( -g `  Z )   =>    |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )
 )  ->  ( { <. 0 ,  A >. , 
 <. 1 ,  C >. } 
 .-  { <. 0 ,  B >. ,  <. 1 ,  D >. } )  =  ( { <. 0 ,  A >. ,  <. 1 ,  C >. }  ( +g  `  Z ) ( -u 1
 ( .s `  Z ) { <. 0 ,  B >. ,  <. 1 ,  D >. } ) ) )
 
Theoremzlmodzxzsub 40413 The subtraction of the  ZZ-module  ZZ  X.  ZZ. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  .-  =  ( -g `  Z )   =>    |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )
 )  ->  ( { <. 0 ,  A >. , 
 <. 1 ,  C >. } 
 .-  { <. 0 ,  B >. ,  <. 1 ,  D >. } )  =  { <. 0 ,  ( A  -  B ) >. , 
 <. 1 ,  ( C  -  D ) >. } )
 
21.33.14.4  Ordered group sum operation (extension)
 
Theoremgsumpr 40414* Group sum of a pair. (Contributed by AV, 6-Dec-2018.) (Proof shortened by AV, 28-Jul-2019.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  (
 k  =  M  ->  A  =  C )   &    |-  (
 k  =  N  ->  A  =  D )   =>    |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B ) )  ->  ( G 
 gsumg  ( k  e.  { M ,  N }  |->  A ) )  =  ( C 
 .+  D ) )
 
Theoremmgpsumunsn 40415* Extract a summand/factor from the group sum for the multiplicative group of a unital ring. (Contributed by AV, 29-Dec-2018.)
 |-  M  =  (mulGrp `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  CRing
 )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  I  e.  N )   &    |-  (
 ( ph  /\  k  e.  N )  ->  A  e.  ( Base `  R )
 )   &    |-  ( ph  ->  X  e.  ( Base `  R )
 )   &    |-  ( k  =  I 
 ->  A  =  X )   =>    |-  ( ph  ->  ( M  gsumg  (
 k  e.  N  |->  A ) )  =  ( ( M  gsumg  ( k  e.  ( N  \  { I }
 )  |->  A ) ) 
 .x.  X ) )
 
Theoremmgpsumz 40416* If the group sum for the multiplicative group of a unital ring contains a summand/factor that is the zero of the ring, the group sum itself is zero. (Contributed by AV, 29-Dec-2018.)
 |-  M  =  (mulGrp `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  CRing
 )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  I  e.  N )   &    |-  (
 ( ph  /\  k  e.  N )  ->  A  e.  ( Base `  R )
 )   &    |- 
 .0.  =  ( 0g `  R )   &    |-  ( k  =  I  ->  A  =  .0.  )   =>    |-  ( ph  ->  ( M  gsumg  ( k  e.  N  |->  A ) )  =  .0.  )
 
Theoremmgpsumn 40417* If the group sum for the multiplicative group of a unital ring contains a summand/factor that is the one of the ring, this summand/ factor can be removed from the group sum. (Contributed by AV, 29-Dec-2018.)
 |-  M  =  (mulGrp `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  CRing
 )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  I  e.  N )   &    |-  (
 ( ph  /\  k  e.  N )  ->  A  e.  ( Base `  R )
 )   &    |- 
 .1.  =  ( 1r `  R )   &    |-  ( k  =  I  ->  A  =  .1.  )   =>    |-  ( ph  ->  ( M  gsumg  ( k  e.  N  |->  A ) )  =  ( M  gsumg  ( k  e.  ( N  \  { I }
 )  |->  A ) ) )
 
Theoremgsumsplit2f 40418* Split a group sum into two parts. (Contributed by AV, 4-Sep-2019.)
 |-  F/ k ph   &    |-  B  =  (
 Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |- 
 .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  (
 ( ph  /\  k  e.  A )  ->  X  e.  B )   &    |-  ( ph  ->  ( k  e.  A  |->  X ) finSupp  .0.  )   &    |-  ( ph  ->  ( C  i^i  D )  =  (/) )   &    |-  ( ph  ->  A  =  ( C  u.  D ) )   =>    |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  X ) )  =  ( ( G  gsumg  ( k  e.  C  |->  X ) )  .+  ( G 
 gsumg  ( k  e.  D  |->  X ) ) ) )
 
Theoremgsumdifsndf 40419* Extract a summand from a finitely supported group sum. (Contributed by AV, 4-Sep-2019.)
 |-  F/_ k Y   &    |- 
 F/ k ph   &    |-  B  =  (
 Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  A  e.  W )   &    |-  ( ph  ->  (
 k  e.  A  |->  X ) finSupp  ( 0g `  G ) )   &    |-  (
 ( ph  /\  k  e.  A )  ->  X  e.  B )   &    |-  ( ph  ->  M  e.  A )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ( ph  /\  k  =  M )  ->  X  =  Y )   =>    |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  X ) )  =  ( ( G  gsumg  ( k  e.  ( A  \  { M } )  |->  X ) )  .+  Y ) )
 
21.33.14.5  Symmetric groups (extension)
 
Theoremnn0le2is012 40420 A nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 16-Mar-2019.)
 |-  (
 ( N  e.  NN0  /\  N  <_  2 )  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) )
 
Theoremexple2lt6 40421 A nonnegative integer to the power of itself is less than 6 if it is less than or equal to 2. (Contributed by AV, 16-Mar-2019.)
 |-  (
 ( N  e.  NN0  /\  N  <_  2 )  ->  ( N ^ N )  <  6 )
 
Theorempgrple2abl 40422 Every symmetric group on a set with at most 2 elements is abelian. (Contributed by AV, 16-Mar-2019.)
 |-  G  =  ( SymGrp `  A )   =>    |-  (
 ( A  e.  V  /\  ( # `  A )  <_  2 )  ->  G  e.  Abel )
 
Theorempgrpgt2nabl 40423 Every symmetric group on a set with more than 2 elements is not abelian, see also the remark in [Rotman] p. 28. (Contributed by AV, 21-Mar-2019.)
 |-  G  =  ( SymGrp `  A )   =>    |-  (
 ( A  e.  V  /\  2  <  ( # `  A ) )  ->  G  e/  Abel )
 
21.33.14.6  Divisibility (extension)
 
Theoreminvginvrid 40424 Identity for a multiplication with additive and multiplicative inverses in a ring. (Contributed by AV, 18-May-2018.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  N  =  ( invg `  R )   &    |-  I  =  (
 invr `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  U )  ->  ( ( N `  Y )  .x.  ( ( I `  ( N `
  Y ) ) 
 .x.  X ) )  =  X )
 
21.33.14.7  The support of functions (extension)
 
Theoremrmsupp0 40425* The support of a mapping of a multiplication of zero with a function into a ring is empty. (Contributed by AV, 10-Apr-2019.)
 |-  R  =  ( Base `  M )   =>    |-  (
 ( ( M  e.  Ring  /\  V  e.  X  /\  C  =  ( 0g `  M ) )  /\  A  e.  ( R  ^m  V ) )  ->  ( ( v  e.  V  |->  ( C ( .r `  M ) ( A `  v
 ) ) ) supp  ( 0g `  M ) )  =  (/) )
 
Theoremdomnmsuppn0 40426* The support of a mapping of a multiplication of a nonzero constant with a function into a (ring theoretic) domain equals the support of the function. (Contributed by AV, 11-Apr-2019.)
 |-  R  =  ( Base `  M )   =>    |-  (
 ( ( M  e. Domn  /\  V  e.  X ) 
 /\  ( C  e.  R  /\  C  =/=  ( 0g `  M ) ) 
 /\  A  e.  ( R  ^m  V ) ) 
 ->  ( ( v  e.  V  |->  ( C ( .r `  M ) ( A `  v
 ) ) ) supp  ( 0g `  M ) )  =  ( A supp  ( 0g `  M ) ) )
 
Theoremrmsuppss 40427* The support of a mapping of a multiplication of a constant with a function into a ring is a subset of the support of the function. (Contributed by AV, 11-Apr-2019.)
 |-  R  =  ( Base `  M )   =>    |-  (
 ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R 
 ^m  V ) ) 
 ->  ( ( v  e.  V  |->  ( C ( .r `  M ) ( A `  v
 ) ) ) supp  ( 0g `  M ) ) 
 C_  ( A supp  ( 0g `  M ) ) )
 
Theoremmndpsuppss 40428 The support of a mapping of a scalar multiplication with a function of scalars is a subset of the support of the function of scalars. (Contributed by AV, 5-Apr-2019.)
 |-  R  =  ( Base `  M )   =>    |-  (
 ( ( M  e.  Mnd  /\  V  e.  X ) 
 /\  ( A  e.  ( R  ^m  V ) 
 /\  B  e.  ( R  ^m  V ) ) )  ->  ( ( A  oF ( +g  `  M ) B ) supp 
 ( 0g `  M ) )  C_  ( ( A supp  ( 0g `  M ) )  u.  ( B supp  ( 0g
 `  M ) ) ) )
 
Theoremscmsuppss 40429* The support of a mapping of a scalar multiplication with a function of scalars is a subset of the support of the function of scalars. (Contributed by AV, 5-Apr-2019.)
 |-  S  =  (Scalar `  M )   &    |-  R  =  ( Base `  S )   =>    |-  (
 ( M  e.  LMod  /\  V  e.  ~P ( Base `  M )  /\  A  e.  ( R  ^m  V ) )  ->  ( ( v  e.  V  |->  ( ( A `
  v ) ( .s `  M ) v ) ) supp  ( 0g `  M ) ) 
 C_  ( A supp  ( 0g `  S ) ) )
 
21.33.14.8  Finitely supported functions (extension)
 
Theoremrmsuppfi 40430* The support of a mapping of a multiplication of a constant with a function into a ring is finite if the support of the function is finite. (Contributed by AV, 11-Apr-2019.)
 |-  R  =  ( Base `  M )   =>    |-  (
 ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R 
 ^m  V )  /\  ( A supp  ( 0g `  M ) )  e. 
 Fin )  ->  (
 ( v  e.  V  |->  ( C ( .r `  M ) ( A `
  v ) ) ) supp  ( 0g `  M ) )  e. 
 Fin )
 
Theoremrmfsupp 40431* A mapping of a multiplication of a constant with a function into a ring is finitely supported if the function is finitely supported. (Contributed by AV, 9-Jun-2019.)
 |-  R  =  ( Base `  M )   =>    |-  (
 ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R 
 ^m  V )  /\  A finSupp  ( 0g `  M ) )  ->  ( v  e.  V  |->  ( C ( .r `  M ) ( A `  v ) ) ) finSupp  ( 0g `  M ) )
 
Theoremmndpsuppfi 40432 The support of a mapping of a scalar multiplication with a function of scalars is finite if the support of the function of scalars is finite. (Contributed by AV, 5-Apr-2019.)
 |-  R  =  ( Base `  M )   =>    |-  (
 ( ( M  e.  Mnd  /\  V  e.  X ) 
 /\  ( A  e.  ( R  ^m  V ) 
 /\  B  e.  ( R  ^m  V ) ) 
 /\  ( ( A supp 
 ( 0g `  M ) )  e.  Fin  /\  ( B supp  ( 0g
 `  M ) )  e.  Fin ) ) 
 ->  ( ( A  oF ( +g  `  M ) B ) supp  ( 0g
 `  M ) )  e.  Fin )
 
Theoremmndpfsupp 40433 A mapping of a scalar multiplication with a function of scalars is finitely supported if the function of scalars is finitely supported. (Contributed by AV, 9-Jun-2019.)
 |-  R  =  ( Base `  M )   =>    |-  (
 ( ( M  e.  Mnd  /\  V  e.  X ) 
 /\  ( A  e.  ( R  ^m  V ) 
 /\  B  e.  ( R  ^m  V ) ) 
 /\  ( A finSupp  ( 0g `  M )  /\  B finSupp  ( 0g `  M ) ) )  ->  ( A  oF
 ( +g  `  M ) B ) finSupp  ( 0g
 `  M ) )
 
Theoremscmsuppfi 40434* The support of a mapping of a scalar multiplication with a function of scalars is finite if the support of the function of scalars is finite. (Contributed by AV, 5-Apr-2019.)
 |-  S  =  (Scalar `  M )   &    |-  R  =  ( Base `  S )   =>    |-  (
 ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) ) 
 /\  A  e.  ( R  ^m  V )  /\  ( A supp  ( 0g `  S ) )  e. 
 Fin )  ->  (
 ( v  e.  V  |->  ( ( A `  v ) ( .s
 `  M ) v ) ) supp  ( 0g
 `  M ) )  e.  Fin )
 
Theoremscmfsupp 40435* A mapping of a scalar multiplication with a function of scalars is finitely supported if the function of scalars is finitely supported. (Contributed by AV, 9-Jun-2019.)
 |-  S  =  (Scalar `  M )   &    |-  R  =  ( Base `  S )   =>    |-  (
 ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) ) 
 /\  A  e.  ( R  ^m  V )  /\  A finSupp  ( 0g `  S ) )  ->  ( v  e.  V  |->  ( ( A `  v ) ( .s `  M ) v ) ) finSupp  ( 0g `  M ) )
 
Theoremsuppmptcfin 40436* The support of a mapping with value 0 except of one is finite. (Contributed by AV, 27-Apr-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .1.  =  ( 1r `  R )   &    |-  F  =  ( x  e.  V  |->  if ( x  =  X ,  .1.  ,  .0.  )
 )   =>    |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V ) 
 ->  ( F supp  .0.  )  e.  Fin )
 
Theoremmptcfsupp 40437* A mapping with value 0 except of one is finitely supported. (Contributed by AV, 9-Jun-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .1.  =  ( 1r `  R )   &    |-  F  =  ( x  e.  V  |->  if ( x  =  X ,  .1.  ,  .0.  )
 )   =>    |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V ) 
 ->  F finSupp  .0.  )
 
Theoremfsuppmptdmf 40438* A mapping with a finite domain is finitely supported. (Contributed by AV, 4-Sep-2019.)
 |-  F/ x ph   &    |-  F  =  ( x  e.  A  |->  Y )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  x  e.  A )  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  W )   =>    |-  ( ph  ->  F finSupp  Z )
 
21.33.14.9  Left modules (extension)
 
Theoremlmodvsmdi 40439 Multiple distributive law for scalar product (left-distributivity). (Contributed by AV, 5-Sep-2019.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   &    |-  .^  =  (.g `  W )   &    |-  E  =  (.g `  F )   =>    |-  ( ( W  e.  LMod  /\  ( R  e.  K  /\  N  e.  NN0  /\  X  e.  V ) )  ->  ( R  .x.  ( N 
 .^  X ) )  =  ( ( N E R )  .x.  X ) )
 
Theoremgsumlsscl 40440* Closure of a group sum in a linear subspace: A (finitely supported) sum of scalar multiplications of vectors of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 28-Jul-2019.)
 |-  S  =  ( LSubSp `  M )   &    |-  R  =  (Scalar `  M )   &    |-  B  =  ( Base `  R )   =>    |-  (
 ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  ( ( F  e.  ( B  ^m  V ) 
 /\  F finSupp  ( 0g `  R ) )  ->  ( M  gsumg  ( v  e.  V  |->  ( ( F `  v ) ( .s
 `  M ) v ) ) )  e.  Z ) )
 
21.33.14.10  Associative algebras (extension)
 
Theoremascl0 40441 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 40442 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 40443 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 40444 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 ) )
 
21.33.14.11  Univariate polynomials (extension)
 
Theoremply1vr1smo 40445 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 40446 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 40447 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 40448* 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 40449 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 40450* Lemma 1 for ply1mulgsum 40454. (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 40451* Lemma 2 for ply1mulgsum 40454. (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 40452* Lemma 3 for ply1mulgsum 40454. (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 40453* Lemma 4 for ply1mulgsum 40454. (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 40454* 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 40455 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 40456 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 40457 A term of the form  x  -  C is a (univariate) polynomial, also called "linear polynomial". (Part of ply1remlem 23161). (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 40458 A term of the form  x  -  C evaluated for  x  =  V results in  V  -  C (part of ply1remlem 23161). (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 40459 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 40460 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 19563 and df-scmat 19564 define diagonal and scalar matrices as sets.

 
Syntaxcdmatalt 40461 Alternative notation for the algebra of diagonal matrices.
 class DMatALT
 
Syntaxcscmatalt 40462 Alternative notation for the algebra of scalar matrices.
 class ScMatALT
 
Definitiondf-dmatalt 40463* 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 40464* 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 40465* 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 40466* 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 40467* 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 40468 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 40471, 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 40472, so that we can show that such sets are submodules of the corresponding modules, see lincolss 40499.
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 18243, the set of all linear combinations as defined by df-lco 40472 follows the alternative approach. That both definitions are equivalent is shown by lspeqlco 40504.

 
Syntaxclinc 40469 Extend class notation with the operation constructing a linear combination (of vectors from a left module).
 class linC
 
Syntaxclinco 40470 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 40471* 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 40472* 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 40473* 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 40474* 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 40475* 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 40476* 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 40477* 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 40478 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 40479 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 40480 The value of an empty linear combination. (Contributed by AV, 12-Apr-2019.)
 |-  ( M  e.  X  ->  ( (/) ( linC  `  M ) (/) )  =  ( 0g `  M ) )
 
Theoremlincvalsng 40481 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 40482 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 40483 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 40484 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 40485 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 40486* 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 40487* 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 40488* 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 40489 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 40490* 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 40491 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 40492 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 40493 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 40494 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 40495* 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 40496 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 40497 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 40498 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 40499 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 40500* 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 )
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