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Theorem List for Metamath Proof Explorer - 19101-19200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfrlmbas3 19101 An element of the base set of a finite free module with a Cartesian product as index set as operation value. (Contributed by AV, 14-Feb-2019.)
 |-  F  =  ( R freeLMod  ( N  X.  M ) )   &    |-  B  =  (
 Base `  R )   &    |-  V  =  ( Base `  F )   =>    |-  (
 ( ( R  e.  W  /\  X  e.  V )  /\  ( N  e.  Fin  /\  M  e.  Fin )  /\  ( I  e.  N  /\  J  e.  M ) )  ->  ( I X J )  e.  B )
 
Theoremmpt2frlmd 19102* Elements of the free module are mappings with two arguments defined by their operation values. (Contributed by AV, 20-Feb-2019.)
 |-  F  =  ( R freeLMod  ( N  X.  M ) )   &    |-  V  =  (
 Base `  F )   &    |-  (
 ( i  =  a 
 /\  j  =  b )  ->  A  =  B )   &    |-  ( ( ph  /\  i  e.  N  /\  j  e.  M )  ->  A  e.  X )   &    |-  ( ( ph  /\  a  e.  N  /\  b  e.  M )  ->  B  e.  Y )   &    |-  ( ph  ->  ( N  e.  U  /\  M  e.  W  /\  Z  e.  V )
 )   =>    |-  ( ph  ->  ( Z  =  ( a  e.  N ,  b  e.  M  |->  B )  <->  A. i  e.  N  A. j  e.  M  ( i Z j )  =  A ) )
 
Theoremfrlmip 19103* The inner product of a free module. (Contributed by Thierry Arnoux, 20-Jun-2019.)
 |-  Y  =  ( R freeLMod  I )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( f  e.  ( B  ^m  I ) ,  g  e.  ( B  ^m  I
 )  |->  ( R  gsumg  ( x  e.  I  |->  ( ( f `  x ) 
 .x.  ( g `  x ) ) ) ) )  =  ( .i `  Y ) )
 
Theoremfrlmipval 19104 The inner product of a free module. (Contributed by Thierry Arnoux, 21-Jun-2019.)
 |-  Y  =  ( R freeLMod  I )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  V  =  ( Base `  Y )   &    |-  .,  =  ( .i `  Y )   =>    |-  ( ( ( I  e.  W  /\  R  e.  X )  /\  ( F  e.  V  /\  G  e.  V ) )  ->  ( F  .,  G )  =  ( R  gsumg  ( F  oF  .x.  G ) ) )
 
Theoremfrlmphllem 19105* Lemma for frlmphl 19106. (Contributed by AV, 21-Jul-2019.)
 |-  Y  =  ( R freeLMod  I )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  V  =  ( Base `  Y )   &    |-  .,  =  ( .i `  Y )   &    |-  O  =  ( 0g `  Y )   &    |- 
 .0.  =  ( 0g `  R )   &    |-  .*  =  ( *r `  R )   &    |-  ( ph  ->  R  e. Field )   &    |-  ( ( ph  /\  g  e.  V  /\  ( g  .,  g )  =  .0.  )  ->  g  =  O )   &    |-  (
 ( ph  /\  x  e.  B )  ->  (  .*  `  x )  =  x )   &    |-  ( ph  ->  I  e.  W )   =>    |-  ( ( ph  /\  g  e.  V  /\  h  e.  V )  ->  ( x  e.  I  |->  ( ( g `  x )  .x.  ( h `
  x ) ) ) finSupp  .0.  )
 
Theoremfrlmphl 19106* Conditions for a free module to be a pre-Hilbert space. (Contributed by Thierry Arnoux, 21-Jun-2019.) (Proof shortened by AV, 21-Jul-2019.)
 |-  Y  =  ( R freeLMod  I )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  V  =  ( Base `  Y )   &    |-  .,  =  ( .i `  Y )   &    |-  O  =  ( 0g `  Y )   &    |- 
 .0.  =  ( 0g `  R )   &    |-  .*  =  ( *r `  R )   &    |-  ( ph  ->  R  e. Field )   &    |-  ( ( ph  /\  g  e.  V  /\  ( g  .,  g )  =  .0.  )  ->  g  =  O )   &    |-  (
 ( ph  /\  x  e.  B )  ->  (  .*  `  x )  =  x )   &    |-  ( ph  ->  I  e.  W )   =>    |-  ( ph  ->  Y  e.  PreHil )
 
11.1.3  Standard basis (unit vectors)

According to Wikipedia ("Standard basis", 16-Mar-2019, https://en.wikipedia.org/wiki/Standard_basis) "In mathematics, the standard basis (also called natural basis) for a Euclidean space is the set of unit vectors pointing in the direction of the axes of a Cartesian coordinate system.", and ("Unit vector", 16-Mar-2019, https://en.wikipedia.org/wiki/Unit_vector) "In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1.". In the following, the term "unit vector" (or more specific "basic unit vector") is used for the (special) unit vectors forming the standard basis of free modules. However, the length of the unit vectors is not considered here, so it is not required to regard normed spaces.

 
Syntaxcuvc 19107 Class of basic unit vectors for an explicit free module.
 class unitVec
 
Definitiondf-uvc 19108*  ( ( R unitVec  I ) `  j
) is the unit vector in 
( R freeLMod  I ) along the  j axis. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |- unitVec  =  ( r  e.  _V ,  i  e.  _V  |->  ( j  e.  i  |->  ( k  e.  i  |->  if ( k  =  j ,  ( 1r
 `  r ) ,  ( 0g `  r
 ) ) ) ) )
 
Theoremuvcfval 19109* Value of the unit-vector generator for a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  V  /\  I  e.  W )  ->  U  =  ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) ) ) )
 
Theoremuvcval 19110* Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I ) 
 ->  ( U `  J )  =  ( k  e.  I  |->  if (
 k  =  J ,  .1.  ,  .0.  ) ) )
 
Theoremuvcvval 19111 Value of a unit vector coordinate in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I )  /\  K  e.  I )  ->  ( ( U `  J ) `
  K )  =  if ( K  =  J ,  .1.  ,  .0.  ) )
 
Theoremuvcvvcl 19112 A coodinate of a unit vector is either 0 or 1. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I )  /\  K  e.  I )  ->  ( ( U `  J ) `
  K )  e. 
 {  .0.  ,  .1.  } )
 
Theoremuvcvvcl2 19113 A unit vector coordinate is a ring element. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  B  =  (
 Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  J  e.  I )   &    |-  ( ph  ->  K  e.  I
 )   =>    |-  ( ph  ->  (
 ( U `  J ) `  K )  e.  B )
 
Theoremuvcvv1 19114 The unit vector is one at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  J  e.  I )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ph  ->  ( ( U `  J ) `  J )  =  .1.  )
 
Theoremuvcvv0 19115 The unit vector is zero at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  J  e.  I )   &    |-  ( ph  ->  K  e.  I )   &    |-  ( ph  ->  J  =/=  K )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ph  ->  (
 ( U `  J ) `  K )  =  .0.  )
 
Theoremuvcff 19116 Domain and range of the unit vector generator; ring condition required to be sure 1 and 0 are actually in the ring. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
 |-  U  =  ( R unitVec  I )   &    |-  Y  =  ( R freeLMod  I )   &    |-  B  =  (
 Base `  Y )   =>    |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  U : I --> B )
 
Theoremuvcf1 19117 In a nonzero ring, each unit vector is different. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  Y  =  ( R freeLMod  I )   &    |-  B  =  (
 Base `  Y )   =>    |-  ( ( R  e. NzRing  /\  I  e.  W )  ->  U : I -1-1-> B )
 
Theoremuvcresum 19118 Any element of a free module can be expressed as a finite linear combination of unit vectors. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by Mario Carneiro, 5-Jul-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  Y  =  ( R freeLMod  I )   &    |-  B  =  (
 Base `  Y )   &    |-  .x.  =  ( .s `  Y )   =>    |-  ( ( R  e.  Ring  /\  I  e.  W  /\  X  e.  B ) 
 ->  X  =  ( Y 
 gsumg  ( X  oF  .x.  U ) ) )
 
Theoremfrlmssuvc1 19119* A scalar multiple of a unit vector included in a support-restriction subspace is included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 24-Jun-2019.)
 |-  F  =  ( R freeLMod  I )   &    |-  U  =  ( R unitVec  I )   &    |-  B  =  (
 Base `  F )   &    |-  K  =  ( Base `  R )   &    |-  .x.  =  ( .s `  F )   &    |-  .0.  =  ( 0g `  R )   &    |-  C  =  { x  e.  B  |  ( x supp  .0.  )  C_  J }   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  J  C_  I
 )   &    |-  ( ph  ->  L  e.  J )   &    |-  ( ph  ->  X  e.  K )   =>    |-  ( ph  ->  ( X  .x.  ( U `  L ) )  e.  C )
 
Theoremfrlmssuvc2 19120* A nonzero scalar multiple of a unit vector not included in a support-restriction subspace is not included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 24-Jun-2019.)
 |-  F  =  ( R freeLMod  I )   &    |-  U  =  ( R unitVec  I )   &    |-  B  =  (
 Base `  F )   &    |-  K  =  ( Base `  R )   &    |-  .x.  =  ( .s `  F )   &    |-  .0.  =  ( 0g `  R )   &    |-  C  =  { x  e.  B  |  ( x supp  .0.  )  C_  J }   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  J  C_  I
 )   &    |-  ( ph  ->  L  e.  ( I  \  J ) )   &    |-  ( ph  ->  X  e.  ( K  \  {  .0.  } ) )   =>    |-  ( ph  ->  -.  ( X  .x.  ( U `  L ) )  e.  C )
 
Theoremfrlmsslsp 19121* A subset of a free module obtained by restricting the support set is spanned by the relevant unit vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by AV, 24-Jun-2019.)
 |-  Y  =  ( R freeLMod  I )   &    |-  U  =  ( R unitVec  I )   &    |-  K  =  (
 LSpan `  Y )   &    |-  B  =  ( Base `  Y )   &    |-  .0.  =  ( 0g `  R )   &    |-  C  =  { x  e.  B  |  ( x supp 
 .0.  )  C_  J }   =>    |-  ( ( R  e.  Ring  /\  I  e.  V  /\  J  C_  I )  ->  ( K `  ( U " J ) )  =  C )
 
Theoremfrlmlbs 19122 The unit vectors comprise a basis for a free module. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
 |-  F  =  ( R freeLMod  I )   &    |-  U  =  ( R unitVec  I )   &    |-  J  =  (LBasis `  F )   =>    |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ran  U  e.  J )
 
Theoremfrlmup1 19123* Any assignment of unit vectors to target vectors can be extended (uniquely) to a homomorphism from a free module to an arbitrary other module on the same base ring. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
 |-  F  =  ( R freeLMod  I )   &    |-  B  =  (
 Base `  F )   &    |-  C  =  ( Base `  T )   &    |-  .x.  =  ( .s `  T )   &    |-  E  =  ( x  e.  B  |->  ( T  gsumg  ( x  oF  .x.  A ) ) )   &    |-  ( ph  ->  T  e.  LMod )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ph  ->  R  =  (Scalar `  T ) )   &    |-  ( ph  ->  A : I --> C )   =>    |-  ( ph  ->  E  e.  ( F LMHom  T ) )
 
Theoremfrlmup2 19124* The evaluation map has the intended behavior on the unit vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
 |-  F  =  ( R freeLMod  I )   &    |-  B  =  (
 Base `  F )   &    |-  C  =  ( Base `  T )   &    |-  .x.  =  ( .s `  T )   &    |-  E  =  ( x  e.  B  |->  ( T  gsumg  ( x  oF  .x.  A ) ) )   &    |-  ( ph  ->  T  e.  LMod )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ph  ->  R  =  (Scalar `  T ) )   &    |-  ( ph  ->  A : I --> C )   &    |-  ( ph  ->  Y  e.  I )   &    |-  U  =  ( R unitVec  I )   =>    |-  ( ph  ->  ( E `  ( U `  Y ) )  =  ( A `  Y ) )
 
Theoremfrlmup3 19125* The range of such an evaluation map is the finite linear combinations of the target vectors and also the span of the target vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  B  =  (
 Base `  F )   &    |-  C  =  ( Base `  T )   &    |-  .x.  =  ( .s `  T )   &    |-  E  =  ( x  e.  B  |->  ( T  gsumg  ( x  oF  .x.  A ) ) )   &    |-  ( ph  ->  T  e.  LMod )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ph  ->  R  =  (Scalar `  T ) )   &    |-  ( ph  ->  A : I --> C )   &    |-  K  =  ( LSpan `  T )   =>    |-  ( ph  ->  ran  E  =  ( K `  ran  A ) )
 
Theoremfrlmup4 19126* Universal property of the free module by existential uniquenes. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  R  =  (Scalar `  T )   &    |-  F  =  ( R freeLMod  I )   &    |-  U  =  ( R unitVec  I )   &    |-  C  =  (
 Base `  T )   =>    |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A : I
 --> C )  ->  E! m  e.  ( F LMHom  T ) ( m  o.  U )  =  A )
 
Theoremellspd 19127* The elements of the span of an indexed collection of basic vectors are those vectors which can be written as finite linear combinations of basic vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by AV, 24-Jun-2019.)
 |-  N  =  ( LSpan `  M )   &    |-  B  =  (
 Base `  M )   &    |-  K  =  ( Base `  S )   &    |-  S  =  (Scalar `  M )   &    |-  .0.  =  ( 0g `  S )   &    |- 
 .x.  =  ( .s `  M )   &    |-  ( ph  ->  F : I --> B )   &    |-  ( ph  ->  M  e.  LMod
 )   &    |-  ( ph  ->  I  e.  _V )   =>    |-  ( ph  ->  ( X  e.  ( N `  ( F " I
 ) )  <->  E. f  e.  ( K  ^m  I ) ( f finSupp  .0.  /\  X  =  ( M  gsumg  ( f  oF  .x.  F ) ) ) ) )
 
Theoremelfilspd 19128* Simplified version of ellspd 19127 when the spanning set is finite: all linear combinations are then acceptable. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
 |-  N  =  ( LSpan `  M )   &    |-  B  =  (
 Base `  M )   &    |-  K  =  ( Base `  S )   &    |-  S  =  (Scalar `  M )   &    |-  .0.  =  ( 0g `  S )   &    |- 
 .x.  =  ( .s `  M )   &    |-  ( ph  ->  F : I --> B )   &    |-  ( ph  ->  M  e.  LMod
 )   &    |-  ( ph  ->  I  e.  Fin )   =>    |-  ( ph  ->  ( X  e.  ( N `  ( F " I
 ) )  <->  E. f  e.  ( K  ^m  I ) X  =  ( M  gsumg  ( f  oF  .x.  F ) ) ) )
 
11.1.4  Independent sets and families

According to the definition in [Lang] p. 129: "A subset S of a module M is said to be linearly independent (over A) if whenever we have a linear combination ∑x ∈ S axx which is equal to 0, then ax = 0 for all x ∈ S.", and according to the Definition in [Lang] p. 130: "a familiy {xi}i ∈ I of elements of M is said to be linearly independent (over A) if whenever we have a linear combination ∑i ∈ I aixi = 0, then ai = 0 for all i.". These definitions correspond to the definitions df-linds 19132 resp. df-lindf 19131, where it is claimed that a nonzero summand can be extracted ( ∑i ∈ {I \ { j } }aixi = -ajxj ) and be represented as a linear combination of the remaining elements of the family.
TODO: After introducing a definition of "linear combination", it should be shown that these definitions are actually equivalent.

 
Syntaxclindf 19129 The class relationship of independent families in a module.
 class LIndF
 
Syntaxclinds 19130 The class generator of independent sets in a module.
 class LIndS
 
Definitiondf-lindf 19131* An independent family is a family of vectors, no nonzero multiple of which can be expressed as a linear combination of other elements of the family. This is almost, but not quite, the same as a function into an independent set.

This is a defined concept because it matters in many cases whether independence is taken at a set or family level. For instance, a number is transcedental iff its nonzero powers are linearly independent. Is 1 transcedental? It has only one nonzero power.

We can almost define family independence as a family of unequal elements with independent range, as islindf3 19151, but taking that as primitive would lead to unpleasant corner case behavior with the zero ring.

This is equivalent to the common definition of having no nontrivial representations of zero (islindf4 19163) and only one representation for each element of the range (islindf5 19164). (Contributed by Stefan O'Rear, 24-Feb-2015.)

 |- LIndF  =  { <. f ,  w >.  |  ( f : dom  f --> ( Base `  w )  /\  [. (Scalar `  w )  /  s ]. A. x  e.  dom  f A. k  e.  (
 ( Base `  s )  \  { ( 0g `  s ) } )  -.  ( k ( .s
 `  w ) ( f `  x ) )  e.  ( (
 LSpan `  w ) `  ( f " ( dom  f  \  { x } ) ) ) ) }
 
Definitiondf-linds 19132* An independent set is a set which is independent as a family. See also islinds3 19159 and islinds4 19160. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |- LIndS  =  ( w  e.  _V  |->  { s  e.  ~P ( Base `  w )  |  (  _I  |`  s ) LIndF 
 w } )
 
Theoremrellindf 19133 The independent-family predicate is a proper relation and can be used with brrelexi 4863. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |- 
 Rel LIndF
 
Theoremislinds 19134 Property of an independent set of vectors in terms of an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  W )   =>    |-  ( W  e.  V  ->  ( X  e.  (LIndS `  W )  <->  ( X  C_  B  /\  (  _I  |`  X ) LIndF  W ) ) )
 
Theoremlinds1 19135 An independent set of vectors is a set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  W )   =>    |-  ( X  e.  (LIndS `  W )  ->  X  C_  B )
 
Theoremlinds2 19136 An independent set of vectors is independent as a family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  ( X  e.  (LIndS `  W )  ->  (  _I  |`  X ) LIndF  W )
 
Theoremislindf 19137* Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( LSpan `  W )   &    |-  S  =  (Scalar `  W )   &    |-  N  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  S )   =>    |-  ( ( W  e.  Y  /\  F  e.  X )  ->  ( F LIndF  W  <->  ( F : dom  F --> B  /\  A. x  e. 
 dom  F A. k  e.  ( N  \  {  .0.  } )  -.  (
 k  .x.  ( F `  x ) )  e.  ( K `  ( F " ( dom  F  \  { x } )
 ) ) ) ) )
 
Theoremislinds2 19138* Expanded property of an independent set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( LSpan `  W )   &    |-  S  =  (Scalar `  W )   &    |-  N  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  S )   =>    |-  ( W  e.  Y  ->  ( F  e.  (LIndS `  W )  <->  ( F  C_  B  /\  A. x  e.  F  A. k  e.  ( N  \  {  .0.  } )  -.  (
 k  .x.  x )  e.  ( K `  ( F  \  { x }
 ) ) ) ) )
 
Theoremislindf2 19139* Property of an independent family of vectors with prior constrained domain and codomain. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( LSpan `  W )   &    |-  S  =  (Scalar `  W )   &    |-  N  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  S )   =>    |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B ) 
 ->  ( F LIndF  W  <->  A. x  e.  I  A. k  e.  ( N  \  {  .0.  }
 )  -.  ( k  .x.  ( F `  x ) )  e.  ( K `  ( F "
 ( I  \  { x } ) ) ) ) )
 
Theoremlindff 19140 Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  W )   =>    |-  ( ( F LIndF  W  /\  W  e.  Y ) 
 ->  F : dom  F --> B )
 
Theoremlindfind 19141 A linearly independent family is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |- 
 .x.  =  ( .s `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  L  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  L )   &    |-  K  =  ( Base `  L )   =>    |-  ( ( ( F LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/=  .0.  ) )  ->  -.  ( A  .x.  ( F `  E ) )  e.  ( N `  ( F " ( dom 
 F  \  { E } ) ) ) )
 
Theoremlindsind 19142 A linearly independent set is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |- 
 .x.  =  ( .s `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  L  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  L )   &    |-  K  =  ( Base `  L )   =>    |-  ( ( ( F  e.  (LIndS `  W )  /\  E  e.  F )  /\  ( A  e.  K  /\  A  =/=  .0.  ) )  ->  -.  ( A  .x.  E )  e.  ( N `  ( F  \  { E }
 ) ) )
 
Theoremlindfind2 19143 In a linearly independent family in a module over a nonzero ring, no element is contained in the span of any non-containing set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  K  =  ( LSpan `  W )   &    |-  L  =  (Scalar `  W )   =>    |-  ( ( ( W  e.  LMod  /\  L  e. NzRing ) 
 /\  F LIndF  W  /\  E  e.  dom  F )  ->  -.  ( F `  E )  e.  ( K `  ( F " ( dom  F  \  { E } ) ) ) )
 
Theoremlindsind2 19144 In a linearly independent set in a module over a nonzero ring, no element is contained in the span of any non-containing set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  K  =  ( LSpan `  W )   &    |-  L  =  (Scalar `  W )   =>    |-  ( ( ( W  e.  LMod  /\  L  e. NzRing ) 
 /\  F  e.  (LIndS `  W )  /\  E  e.  F )  ->  -.  E  e.  ( K `  ( F  \  { E }
 ) ) )
 
Theoremlindff1 19145 A linearly independent family over a nonzero ring has no repeated elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  L  =  (Scalar `  W )   =>    |-  ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W )  ->  F : dom  F
 -1-1-> B )
 
Theoremlindfrn 19146 The range of an independent family is an independent set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  ( ( W  e.  LMod  /\  F LIndF  W )  ->  ran  F  e.  (LIndS `  W ) )
 
Theoremf1lindf 19147 Rearranging and deleting elements from an independent family gives an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  ( ( W  e.  LMod  /\  F LIndF  W  /\  G : K -1-1-> dom  F )  ->  ( F  o.  G ) LIndF  W )
 
Theoremlindfres 19148 Any restriction of an independent family is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  ( ( W  e.  LMod  /\  F LIndF  W )  ->  ( F  |`  X ) LIndF  W )
 
Theoremlindsss 19149 Any subset of an independent set is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  ( ( W  e.  LMod  /\  F  e.  (LIndS `  W )  /\  G  C_  F )  ->  G  e.  (LIndS `  W ) )
 
Theoremf1linds 19150 A family constructed from non-repeated elements of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  ( ( W  e.  LMod  /\  S  e.  (LIndS `  W )  /\  F : D -1-1-> S )  ->  F LIndF  W )
 
Theoremislindf3 19151 In a nonzero ring, independent families can be equivalently characterized as renamings of independent sets. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  L  =  (Scalar `  W )   =>    |-  ( ( W  e.  LMod  /\  L  e. NzRing )  ->  ( F LIndF  W  <->  ( F : dom  F -1-1-> _V  /\  ran  F  e.  (LIndS `  W )
 ) ) )
 
Theoremlindfmm 19152 Linear independence of a family is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  B  =  ( Base `  S )   &    |-  C  =  (
 Base `  T )   =>    |-  ( ( G  e.  ( S LMHom  T )  /\  G : B -1-1-> C 
 /\  F : I --> B )  ->  ( F LIndF  S 
 <->  ( G  o.  F ) LIndF  T ) )
 
Theoremlindsmm 19153 Linear independence of a set is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  B  =  ( Base `  S )   &    |-  C  =  (
 Base `  T )   =>    |-  ( ( G  e.  ( S LMHom  T )  /\  G : B -1-1-> C 
 /\  F  C_  B )  ->  ( F  e.  (LIndS `  S )  <->  ( G " F )  e.  (LIndS `  T ) ) )
 
Theoremlindsmm2 19154 The monomorphic image of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  B  =  ( Base `  S )   &    |-  C  =  (
 Base `  T )   =>    |-  ( ( G  e.  ( S LMHom  T )  /\  G : B -1-1-> C 
 /\  F  e.  (LIndS `  S ) )  ->  ( G " F )  e.  (LIndS `  T ) )
 
Theoremlsslindf 19155 Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  U  =  ( LSubSp `  W )   &    |-  X  =  ( Ws  S )   =>    |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran 
 F  C_  S )  ->  ( F LIndF  X  <->  F LIndF  W ) )
 
Theoremlsslinds 19156 Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  U  =  ( LSubSp `  W )   &    |-  X  =  ( Ws  S )   =>    |-  ( ( W  e.  LMod  /\  S  e.  U  /\  F  C_  S )  ->  ( F  e.  (LIndS `  X )  <->  F  e.  (LIndS `  W ) ) )
 
Theoremislbs4 19157 A basis is an independent spanning set. This could have been used as alternative definition of a basis: LBasis  =  ( w  e.  _V  |->  { b  e.  ~P ( Base `  w )  |  ( ( ( LSpan `  w
)  `  b )  =  ( Base `  w
)  /\  b  e.  (LIndS `  w ) ) } ). (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  J  =  (LBasis `  W )   &    |-  K  =  (
 LSpan `  W )   =>    |-  ( X  e.  J 
 <->  ( X  e.  (LIndS `  W )  /\  ( K `  X )  =  B ) )
 
Theoremlbslinds 19158 A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  J  =  (LBasis `  W )   =>    |-  J  C_  (LIndS `  W )
 
Theoremislinds3 19159 A subset is linearly independent iff it is a basis of its span. (Contributed by Stefan O'Rear, 25-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  K  =  (
 LSpan `  W )   &    |-  X  =  ( Ws  ( K `  Y ) )   &    |-  J  =  (LBasis `  X )   =>    |-  ( W  e.  LMod  ->  ( Y  e.  (LIndS `  W )  <->  Y  e.  J ) )
 
Theoremislinds4 19160* A set is independent in a vector space iff it is a subset of some basis. (AC equivalent) (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  J  =  (LBasis `  W )   =>    |-  ( W  e.  LVec  ->  ( Y  e.  (LIndS `  W )  <->  E. b  e.  J  Y  C_  b ) )
 
11.1.5  Characterization of free modules
 
Theoremlmimlbs 19161 The isomorphic image of a basis is a basis. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  J  =  (LBasis `  S )   &    |-  K  =  (LBasis `  T )   =>    |-  ( ( F  e.  ( S LMIso  T )  /\  B  e.  J )  ->  ( F " B )  e.  K )
 
Theoremlmiclbs 19162 Having a basis is an isomorphism invariant. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  J  =  (LBasis `  S )   &    |-  K  =  (LBasis `  T )   =>    |-  ( S  ~=ph𝑚 
 T  ->  ( J  =/= 
 (/)  ->  K  =/=  (/) ) )
 
Theoremislindf4 19163* A family is independent iff it has no nontrivial representations of zero. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  Y  =  ( 0g
 `  R )   &    |-  L  =  ( Base `  ( R freeLMod  I ) )   =>    |-  ( ( W  e.  LMod  /\  I  e.  X  /\  F : I --> B ) 
 ->  ( F LIndF  W  <->  A. x  e.  L  ( ( W  gsumg  ( x  oF  .x.  F ) )  =  .0.  ->  x  =  ( I  X.  { Y }
 ) ) ) )
 
Theoremislindf5 19164* A family is independent iff the linear combinations homomorphism is injective. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  B  =  (
 Base `  F )   &    |-  C  =  ( Base `  T )   &    |-  .x.  =  ( .s `  T )   &    |-  E  =  ( x  e.  B  |->  ( T  gsumg  ( x  oF  .x.  A ) ) )   &    |-  ( ph  ->  T  e.  LMod )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ph  ->  R  =  (Scalar `  T ) )   &    |-  ( ph  ->  A : I --> C )   =>    |-  ( ph  ->  ( A LIndF  T  <->  E : B -1-1-> C ) )
 
Theoremindlcim 19165* An independent, spanning family extends to an isomorphism from a free module. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  B  =  (
 Base `  F )   &    |-  C  =  ( Base `  T )   &    |-  .x.  =  ( .s `  T )   &    |-  N  =  ( LSpan `  T )   &    |-  E  =  ( x  e.  B  |->  ( T  gsumg  ( x  oF  .x.  A ) ) )   &    |-  ( ph  ->  T  e.  LMod
 )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ph  ->  R  =  (Scalar `  T ) )   &    |-  ( ph  ->  A : I -onto-> J )   &    |-  ( ph  ->  A LIndF  T )   &    |-  ( ph  ->  ( N `  J )  =  C )   =>    |-  ( ph  ->  E  e.  ( F LMIso  T ) )
 
Theoremlbslcic 19166 A module with a basis is isomorphic to a free module with the same cardinality. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  F  =  (Scalar `  W )   &    |-  J  =  (LBasis `  W )   =>    |-  ( ( W  e.  LMod  /\  B  e.  J  /\  I  ~~  B )  ->  W  ~=ph𝑚  ( F freeLMod  I )
 )
 
Theoremlmisfree 19167* A module has a basis iff it is isomorphic to a free module. In settings where isomorphic objects are not distinguished, it is common to define "free module" as any module with a basis; thus for instance lbsex 18129 might be described as "every vector space is free." (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  J  =  (LBasis `  W )   &    |-  F  =  (Scalar `  W )   =>    |-  ( W  e.  LMod  ->  ( J  =/=  (/)  <->  E. k  W  ~=ph𝑚  ( F freeLMod  k ) ) )
 
Theoremlvecisfrlm 19168* Every vector space is isomorphic to a free module. (Contributed by AV, 7-Mar-2019.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e.  LVec  ->  E. k  W  ~=ph𝑚  ( F freeLMod  k ) )
 
Theoremlmimco 19169 The composition of two isomorphisms of modules is an isomorphism of modules. (Contributed by AV, 10-Mar-2019.)
 |-  ( ( F  e.  ( S LMIso  T )  /\  G  e.  ( R LMIso  S ) )  ->  ( F  o.  G )  e.  ( R LMIso  T ) )
 
Theoremlmictra 19170 Module isomorphism is transitive. (Contributed by AV, 10-Mar-2019.)
 |-  ( ( R  ~=ph𝑚  S  /\  S  ~=ph𝑚 
 T )  ->  R  ~=ph𝑚  T )
 
Theoremuvcf1o 19171 In a nonzero ring, the mapping of the index set of a free module onto the unit vectors of the free module is a 1-1 onto function. (Contributed by AV, 10-Mar-2019.)
 |-  U  =  ( R unitVec  I )   =>    |-  ( ( R  e. NzRing  /\  I  e.  W ) 
 ->  U : I -1-1-onto-> ran  U )
 
Theoremuvcendim 19172 In a nonzero ring, the number of unit vectors of a free module corresponds to the dimension of the free module. (Contributed by AV, 10-Mar-2019.)
 |-  U  =  ( R unitVec  I )   =>    |-  ( ( R  e. NzRing  /\  I  e.  W ) 
 ->  I  ~~  ran  U )
 
Theoremfrlmisfrlm 19173 A free module is isomorphic to a free module over the same (nonzero) ring, with the same cardinality. (Contributed by AV, 10-Mar-2019.)
 |-  ( ( R  e. NzRing  /\  I  e.  Y  /\  I  ~~  J )  ->  ( R freeLMod  I )  ~=ph𝑚  ( R freeLMod  J ) )
 
Theoremfrlmiscvec 19174 Every free module is isomorphic to the free module of "column vectors" of the same dimension over the same (nonzero) ring. (Contributed by AV, 10-Mar-2019.)
 |-  ( ( R  e. NzRing  /\  I  e.  Y ) 
 ->  ( R freeLMod  I )  ~=ph𝑚  ( R freeLMod  ( I  X.  { (/)
 } ) ) )
 
11.2  Matrices

According to Wikipedia ("Matrix (mathemetics)", 02-Apr-2019, https://en.wikipedia.org/wiki/Matrix_(mathematics)) "A matrix is a rectangular array of numbers or other mathematical objects for which operations such as addition and multiplication are defined. Most commonly, a matrix over a field F is a rectangular array of scalars each of which is a member of F. The numbers, symbols or expressions in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively.", and in the definition of [Lang] p. 503 "By an m x n matrix in [a commutative ring] R one means a doubly indexed family of elements of R, (aij), (i= 1,..., m and j = 1,... n) ... We call the elements aij the coefficients or components of the matrix. A 1 x n matrix is called a row vector (of dimension, or size, n) and a m x 1 matrix is called a column vector (of dimension, or size, m). In general, we say that (m,n) is the size of the matrix, ...". In contrast to these definitions, we denote any free module over a (not necessarily commutative) ring (in the meaning of df-frlm 19074) with a Cartesian product as index set as "matrix". The two sets of the Cartesian product even need neither to be ordered or a range of (nonnegative/positive) integers nor finite. By this, the addition and scalar multiplication for matrices correspond to the addition (see frlmplusgval 19091) and scalar multiplication (see frlmvscafval 19093) for free modules. Actually, there isn't a definition for (arbitrary) matrices: Even the (general) matrix multiplication can be defined using functions from Cartesian products into a ring (which are elements of the base set of free modules), see df-mamu 19176. By this, a statement like "Then the set of m x n matrices in R is a module (i.e. an R-module)" as in [Lang] p. 504 follows immediatly from frlmlmod 19076.

However, for square matrices there is the definition df-mat 19200, defining the algebras of square matrices (of the same size over the same ring), extending the structure of the corresponding free module by the matrix multiplication as ring multiplication.

A "usual" matrix (aij), (i= 1,..., m and j = 1,... n) would be represented as element of (the base set of)  ( R freeLMod  ( (
1 ... m )  X.  ( 1 ... n
) ) ), and a square matrix (aij), (i= 1,..., n and j = 1,... n) would be represented as element of (the base set of)  ( ( 1 ... n ) Mat  R ).

Finally, it should be mentioned that our definitions of matrices include the zero-dimensional cases, which is excluded in the definition of many authors, e.g. in [Lang] p. 503. It is shown in mat0dimbas0 19258 that the empty set is the sole zero-dimensional matrix (also called "empty matrix", see Wikipedia https://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matrices). The determinant is also defined for such an empty matrix, see mdet0pr 19384.

 
11.2.1  The matrix multiplication

This section is about the multiplication of m x n matrices.

 
Syntaxcmmul 19175 Syntax for the matrix multiplication operator.
 class maMul
 
Definitiondf-mamu 19176* The operator which multiplies an m x n matrix with an n x p matrix, see also the definition in [Lang] p. 504. Note that it is not generally possible to recover the dimensions from the matrix, since all n x 0 and all 0 x n matrices are represented by the empty set. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |- maMul  =  ( r  e.  _V ,  o  e.  _V  |->  [_ ( 1st `  ( 1st `  o ) ) 
 /  m ]_ [_ ( 2nd `  ( 1st `  o
 ) )  /  n ]_
 [_ ( 2nd `  o
 )  /  p ]_ ( x  e.  ( ( Base `  r )  ^m  ( m  X.  n ) ) ,  y  e.  ( ( Base `  r
 )  ^m  ( n  X.  p ) )  |->  ( i  e.  m ,  k  e.  p  |->  ( r  gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r
 `  r ) ( j y k ) ) ) ) ) ) )
 
Theoremmamufval 19177* Functional value of the matrix multiplication operator. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  F  =  ( R maMul  <. M ,  N ,  P >. )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  P  e.  Fin )   =>    |-  ( ph  ->  F  =  ( x  e.  ( B  ^m  ( M  X.  N ) ) ,  y  e.  ( B 
 ^m  ( N  X.  P ) )  |->  ( i  e.  M ,  k  e.  P  |->  ( R 
 gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
 j y k ) ) ) ) ) ) )
 
Theoremmamuval 19178* Multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  F  =  ( R maMul  <. M ,  N ,  P >. )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  P  e.  Fin )   &    |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Y  e.  ( B  ^m  ( N  X.  P ) ) )   =>    |-  ( ph  ->  ( X F Y )  =  ( i  e.  M ,  k  e.  P  |->  ( R  gsumg  ( j  e.  N  |->  ( ( i X j )  .x.  (
 j Y k ) ) ) ) ) )
 
Theoremmamufv 19179* A cell in the multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  F  =  ( R maMul  <. M ,  N ,  P >. )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  P  e.  Fin )   &    |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Y  e.  ( B  ^m  ( N  X.  P ) ) )   &    |-  ( ph  ->  I  e.  M )   &    |-  ( ph  ->  K  e.  P )   =>    |-  ( ph  ->  ( I ( X F Y ) K )  =  ( R  gsumg  ( j  e.  N  |->  ( ( I X j ) 
 .x.  ( j Y K ) ) ) ) )
 
Theoremmamudm 19180 The domain of the matrix multiplication function. (Contributed by AV, 10-Feb-2019.)
 |-  E  =  ( R freeLMod  ( M  X.  N ) )   &    |-  B  =  (
 Base `  E )   &    |-  F  =  ( R freeLMod  ( N  X.  P ) )   &    |-  C  =  ( Base `  F )   &    |-  .X.  =  ( R maMul  <. M ,  N ,  P >. )   =>    |-  ( ( R  e.  V  /\  ( M  e.  Fin  /\  N  e.  Fin  /\  P  e.  Fin ) ) 
 ->  dom  .X.  =  ( B  X.  C ) )
 
Theoremmamufacex 19181 Every solution of the equation  A * X  =  B for matrices  A and  B is a matrix. (Contributed by AV, 10-Feb-2019.)
 |-  E  =  ( R freeLMod  ( M  X.  N ) )   &    |-  B  =  (
 Base `  E )   &    |-  F  =  ( R freeLMod  ( N  X.  P ) )   &    |-  C  =  ( Base `  F )   &    |-  .X.  =  ( R maMul  <. M ,  N ,  P >. )   &    |-  G  =  ( R freeLMod  ( M  X.  P ) )   &    |-  D  =  (
 Base `  G )   =>    |-  ( ( ( M  =/=  (/)  /\  P  =/= 
 (/) )  /\  ( R  e.  V  /\  Y  e.  D )  /\  ( M  e.  Fin  /\  N  e.  Fin  /\  P  e.  Fin ) ) 
 ->  ( ( X  .X.  Z )  =  Y  ->  Z  e.  C ) )
 
Theoremmamures 19182 Rows in a matrix product are functions only of the corresponding rows in the left argument. (Contributed by SO, 9-Jul-2018.)
 |-  F  =  ( R maMul  <. M ,  N ,  P >. )   &    |-  G  =  ( R maMul  <. I ,  N ,  P >. )   &    |-  B  =  (
 Base `  R )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  P  e.  Fin )   &    |-  ( ph  ->  I  C_  M )   &    |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Y  e.  ( B  ^m  ( N  X.  P ) ) )   =>    |-  ( ph  ->  (
 ( X F Y )  |`  ( I  X.  P ) )  =  ( ( X  |`  ( I  X.  N ) ) G Y ) )
 
Theoremmndvcl 19183 Tuple-wise additive closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   =>    |-  ( ( M  e.  Mnd  /\  X  e.  ( B  ^m  I ) 
 /\  Y  e.  ( B  ^m  I ) ) 
 ->  ( X  oF  .+  Y )  e.  ( B  ^m  I ) )
 
Theoremmndvass 19184 Tuple-wise associativity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   =>    |-  ( ( M  e.  Mnd  /\  ( X  e.  ( B  ^m  I )  /\  Y  e.  ( B  ^m  I ) 
 /\  Z  e.  ( B  ^m  I ) ) )  ->  ( ( X  oF  .+  Y )  oF  .+  Z )  =  ( X  oF  .+  ( Y  oF  .+  Z ) ) )
 
Theoremmndvlid 19185 Tuple-wise left identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  .0.  =  ( 0g `  M )   =>    |-  ( ( M  e.  Mnd  /\  X  e.  ( B 
 ^m  I ) ) 
 ->  ( ( I  X.  {  .0.  } )  oF  .+  X )  =  X )
 
Theoremmndvrid 19186 Tuple-wise right identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  .0.  =  ( 0g `  M )   =>    |-  ( ( M  e.  Mnd  /\  X  e.  ( B 
 ^m  I ) ) 
 ->  ( X  oF  .+  ( I  X.  {  .0.  } ) )  =  X )
 
Theoremgrpvlinv 19187 Tuple-wise left inverse in groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  N  =  ( invg `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  ( B 
 ^m  I ) ) 
 ->  ( ( N  o.  X )  oF  .+  X )  =  ( I  X.  {  .0.  } ) )
 
Theoremgrpvrinv 19188 Tuple-wise right inverse in groups. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  N  =  ( invg `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  ( B 
 ^m  I ) ) 
 ->  ( X  oF  .+  ( N  o.  X ) )  =  ( I  X.  {  .0.  }
 ) )
 
Theoremmhmvlin 19189 Tuple extension of monoid homomorphisms. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  .+^  =  (
 +g  `  N )   =>    |-  (
 ( F  e.  ( M MndHom  N )  /\  X  e.  ( B  ^m  I
 )  /\  Y  e.  ( B  ^m  I ) )  ->  ( F  o.  ( X  oF  .+  Y ) )  =  ( ( F  o.  X )  oF  .+^  ( F  o.  Y ) ) )
 
Theoremringvcl 19190 Tuple-wise multiplication closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  ( B  ^m  I ) 
 /\  Y  e.  ( B  ^m  I ) ) 
 ->  ( X  oF  .x.  Y )  e.  ( B  ^m  I ) )
 
Theoremgsumcom3 19191* A commutative law for finitely supported iterated sums. (Contributed by Stefan O'Rear, 2-Nov-2015.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  C  e.  W )   &    |-  (
 ( ph  /\  ( j  e.  A  /\  k  e.  C ) )  ->  X  e.  B )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  ( ( ph  /\  (
 ( j  e.  A  /\  k  e.  C )  /\  -.  j U k ) )  ->  X  =  .0.  )   =>    |-  ( ph  ->  ( G  gsumg  ( j  e.  A  |->  ( G 
 gsumg  ( k  e.  C  |->  X ) ) ) )  =  ( G 
 gsumg  ( k  e.  C  |->  ( G  gsumg  ( j  e.  A  |->  X ) ) ) ) )
 
Theoremgsumcom3fi 19192* A commutative law for finite iterated sums. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  C  e.  Fin )   &    |-  ( ( ph  /\  (
 j  e.  A  /\  k  e.  C )
 )  ->  X  e.  B )   =>    |-  ( ph  ->  ( G  gsumg  ( j  e.  A  |->  ( G  gsumg  ( k  e.  C  |->  X ) ) ) )  =  ( G 
 gsumg  ( k  e.  C  |->  ( G  gsumg  ( j  e.  A  |->  X ) ) ) ) )
 
Theoremmamucl 19193 Operation closure of matrix multiplication. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
 |-  B  =  ( Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  F  =  ( R maMul  <. M ,  N ,  P >. )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  P  e.  Fin )   &    |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Y  e.  ( B  ^m  ( N  X.  P ) ) )   =>    |-  ( ph  ->  ( X F Y )  e.  ( B  ^m  ( M  X.  P ) ) )
 
Theoremmamuass 19194 Matrix multiplication is associative, see also statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  O  e.  Fin )   &    |-  ( ph  ->  P  e.  Fin )   &    |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Y  e.  ( B  ^m  ( N  X.  O ) ) )   &    |-  ( ph  ->  Z  e.  ( B  ^m  ( O  X.  P ) ) )   &    |-  F  =  ( R maMul  <. M ,  N ,  O >. )   &    |-  G  =  ( R maMul  <. M ,  O ,  P >. )   &    |-  H  =  ( R maMul  <. M ,  N ,  P >. )   &    |-  I  =  ( R maMul  <. N ,  O ,  P >. )   =>    |-  ( ph  ->  (
 ( X F Y ) G Z )  =  ( X H ( Y I Z ) ) )
 
Theoremmamudi 19195 Matrix multiplication distributes over addition on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
 |-  B  =  ( Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  F  =  ( R maMul  <. M ,  N ,  O >. )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  O  e.  Fin )   &    |-  .+  =  ( +g  `  R )   &    |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Y  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Z  e.  ( B  ^m  ( N  X.  O ) ) )   =>    |-  ( ph  ->  (
 ( X  oF  .+  Y ) F Z )  =  ( ( X F Z )  oF  .+  ( Y F Z ) ) )
 
Theoremmamudir 19196 Matrix multiplication distributes over addition on the right. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
 |-  B  =  ( Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  F  =  ( R maMul  <. M ,  N ,  O >. )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  O  e.  Fin )   &    |-  .+  =  ( +g  `  R )   &    |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Y  e.  ( B  ^m  ( N  X.  O ) ) )   &    |-  ( ph  ->  Z  e.  ( B  ^m  ( N  X.  O ) ) )   =>    |-  ( ph  ->  ( X F ( Y  oF  .+  Z ) )  =  ( ( X F Y )  oF  .+  ( X F Z ) ) )
 
Theoremmamuvs1 19197 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  F  =  ( R maMul  <. M ,  N ,  O >. )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  O  e.  Fin )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Z  e.  ( B  ^m  ( N  X.  O ) ) )   =>    |-  ( ph  ->  (
 ( ( ( M  X.  N )  X.  { X } )  oF  .x.  Y ) F Z )  =  ( ( ( M  X.  O )  X.  { X } )  oF  .x.  ( Y F Z ) ) )
 
Theoremmamuvs2 19198 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
 |-  ( ph  ->  R  e.  CRing )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  F  =  ( R maMul  <. M ,  N ,  O >. )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  O  e.  Fin )   &    |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  ( B  ^m  ( N  X.  O ) ) )   =>    |-  ( ph  ->  ( X F ( ( ( N  X.  O )  X.  { Y }
 )  oF  .x.  Z ) )  =  ( ( ( M  X.  O )  X.  { Y } )  oF  .x.  ( X F Z ) ) )
 
11.2.2  Square matrices

In the following, the square matrix algebra is defined as extensible structure Mat. In this subsection, however, only square matrices and their basic properties are regarded. This includes showing that  ( N Mat  R ) is a left module, see matlmod 19221. That  ( N Mat  R ) is a ring and an associative algebra is shown in the next subsection, after theorems about the identity matrix are available. Nevertheless,  ( N Mat  R ) is called "matrix ring" or "matrix algebra" already in this subsection.

 
Syntaxcmat 19199 Syntax for the square matrix algebra.
 class Mat
 
Definitiondf-mat 19200* Define the algebra of n x n matrices over a ring r. (Contributed by Stefan O'Rear, 31-Aug-2015.)
 |- Mat 
 =  ( n  e. 
 Fin ,  r  e.  _V 
 |->  ( ( r freeLMod  ( n  X.  n ) ) sSet  <. ( .r `  ndx ) ,  ( r maMul  <. n ,  n ,  n >. ) >. ) )
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