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Theorem List for Metamath Proof Explorer - 18601-18700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
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 18604 resp. df-lindf 18603, 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 18601 The class relationship of independent families in a module.
 class LIndF
 
Syntaxclinds 18602 The class generator of independent sets in a module.
 class LIndS
 
Definitiondf-lindf 18603* 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 18623, 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 18635) and only one representation for each element of the range (islindf5 18636). (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 18604* An independent set is a set which is independent as a family. See also islinds3 18631 and islinds4 18632. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |- LIndS  =  ( w  e.  _V  |->  { s  e.  ~P ( Base `  w )  |  (  _I  |`  s ) LIndF 
 w } )
 
Theoremrellindf 18605 The independent-family predicate is a proper relation and can be used with brrelexi 5034. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |- 
 Rel LIndF
 
Theoremislinds 18606 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 18607 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 18608 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 18609* 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 18610* 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 18611* 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 18612 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 18613 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 18614 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 18615 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 18616 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 18617 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 18618 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 18619 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 18620 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 18621 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 18622 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 18623 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 18624 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 18625 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 18626 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 18627 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 18628 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 18629 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 18630 A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  J  =  (LBasis `  W )   =>    |-  J  C_  (LIndS `  W )
 
Theoremislinds3 18631 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 18632* 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 18633 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 18634 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 18635* 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 18636* 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 18637* 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 18638 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 18639* 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 17589 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 18640* 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 18641 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 18642 Module isomorphism is transitive. (Contributed by AV, 10-Mar-2019.)
 |-  ( ( R  ~=ph𝑚  S  /\  S  ~=ph𝑚 
 T )  ->  R  ~=ph𝑚  T )
 
Theoremuvcf1o 18643 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 18644 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 18645 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 18646 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 18540) 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 18559) and scalar multiplication (see frlmvscafval 18561) 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 18648. 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 18542.

However, for square matrices there is the definition df-mat 18672, 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 18730 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 18856.

 
11.2.1  The matrix multiplication

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

 
Syntaxcmmul 18647 Syntax for the matrix multiplication operator.
 class maMul
 
Definitiondf-mamu 18648* 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 18649* 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 18650* 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 18651* 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 18652 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 18653 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 18654 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 18655 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 18656 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 18657 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 18658 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 18659 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 18660 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 18661 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 ) ) )
 
Theoremrngvcl 18662 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 18663* 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 18664* 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 18665 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 18666 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 18667 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 18668 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 18669 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 18670 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 18693. 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 18671 Syntax for the square matrix algebra.
 class Mat
 
Definitiondf-mat 18672* 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 >. ) >. ) )
 
Theoremmatbas0pc 18673 There is no matrix with a proper class either as dimension or as underlying ring. (Contributed by AV, 28-Dec-2018.)
 |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )
 
Theoremmatbas0 18674 There is no matrix for a not finite dimension or a proper class as the underlying ring. (Contributed by AV, 28-Dec-2018.)
 |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )
 
Theoremmatval 18675 Value of the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   &    |-  .x.  =  ( R maMul  <. N ,  N ,  N >. )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  A  =  ( G sSet  <. ( .r `  ndx ) ,  .x.  >. ) )
 
Theoremmatrcl 18676 Reverse closure for the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   =>    |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V )
 )
 
Theoremmatbas 18677 The matrix ring has the same base set as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( Base `  G )  =  ( Base `  A )
 )
 
Theoremmatplusg 18678 The matrix ring has the same addition as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( +g  `  G )  =  ( +g  `  A ) )
 
Theoremmatsca 18679 The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  (Scalar `  G )  =  (Scalar `  A )
 )
 
Theoremmatvsca 18680 The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( .s `  G )  =  ( .s `  A ) )
 
Theoremmat0 18681 The matrix ring has the same zero as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( 0g `  G )  =  ( 0g `  A ) )
 
Theoremmatinvg 18682 The matrix ring has the same additive inverse as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( invg `  G )  =  ( invg `  A ) )
 
Theoremmat0op 18683* Value of a zero matrix as operation. (Contributed by AV, 2-Dec-2018.)
 |-  A  =  ( N Mat 
 R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  ( 0g `  A )  =  ( i  e.  N ,  j  e.  N  |->  .0.  ) )
 
Theoremmatsca2 18684 The scalars of the matrix ring are the underlying ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  R  =  (Scalar `  A ) )
 
Theoremmatbas2 18685 The base set of the matrix ring as a set exponential. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 16-Dec-2018.)
 |-  A  =  ( N Mat 
 R )   &    |-  K  =  (
 Base `  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  ( K 
 ^m  ( N  X.  N ) )  =  ( Base `  A )
 )
 
Theoremmatbas2i 18686 A matrix is a function. (Contributed by Stefan O'Rear, 11-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  K  =  (
 Base `  R )   &    |-  B  =  ( Base `  A )   =>    |-  ( M  e.  B  ->  M  e.  ( K  ^m  ( N  X.  N ) ) )
 
Theoremmatbas2d 18687* The base set of the matrix ring as a mapping operation. (Contributed by Stefan O'Rear, 11-Jul-2018.)
 |-  A  =  ( N Mat 
 R )   &    |-  K  =  (
 Base `  R )   &    |-  B  =  ( Base `  A )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ( ph  /\  x  e.  N  /\  y  e.  N )  ->  C  e.  K )   =>    |-  ( ph  ->  ( x  e.  N ,  y  e.  N  |->  C )  e.  B )
 
Theoremeqmat 18688* Two square matrices of the same dimension are equal if they have the same entries. (Contributed by AV, 25-Sep-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   =>    |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  =  Y  <->  A. i  e.  N  A. j  e.  N  ( i X j )  =  ( i Y j ) ) )
 
Theoremmatecl 18689 Each entry (according to Wikipedia "Matrix (mathematics)", 30-Dec-2018, https://en.wikipedia.org/wiki/Matrix_(mathematics)#Definition (or element or component or coefficient or cell) of a matrix is an element of the underlying ring. (Contributed by AV, 16-Dec-2018.)
 |-  A  =  ( N Mat 
 R )   &    |-  K  =  (
 Base `  R )   =>    |-  ( ( I  e.  N  /\  J  e.  N  /\  M  e.  ( Base `  A )
 )  ->  ( I M J )  e.  K )
 
Theoremmatecld 18690 Each entry (according to Wikipedia "Matrix (mathematics)", 30-Dec-2018, https://en.wikipedia.org/wiki/Matrix_(mathematics)#Definition (or element or component or coefficient or cell) of a matrix is an element of the underlying ring, deduction form. (Contributed by AV, 27-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  K  =  (
 Base `  R )   &    |-  B  =  ( Base `  A )   &    |-  ( ph  ->  I  e.  N )   &    |-  ( ph  ->  J  e.  N )   &    |-  ( ph  ->  M  e.  B )   =>    |-  ( ph  ->  ( I M J )  e.  K )
 
Theoremmatplusg2 18691 Addition in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .+b  =  ( +g  `  A )   &    |-  .+  =  ( +g  `  R )   =>    |-  (
 ( X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .+b  Y )  =  ( X  oF  .+  Y ) )
 
Theoremmatvsca2 18692 Scalar multiplication in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  K  =  ( Base `  R )   &    |-  .x.  =  ( .s `  A )   &    |-  .X. 
 =  ( .r `  R )   &    |-  C  =  ( N  X.  N )   =>    |-  ( ( X  e.  K  /\  Y  e.  B )  ->  ( X  .x.  Y )  =  ( ( C  X.  { X } )  oF  .X.  Y ) )
 
Theoremmatlmod 18693 The matrix ring is a linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  LMod )
 
Theoremmatgrp 18694 The matrix ring is a group. (Contributed by AV, 21-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Grp )
 
Theoremmatvscl 18695 Closure of the scalar multiplication in the matrix ring. (lmodvscl 17307 analog.) (Contributed by AV, 27-Nov-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( N Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  .x.  =  ( .s `  A )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  Ring
 )  /\  ( C  e.  K  /\  X  e.  B ) )  ->  ( C  .x.  X )  e.  B )
 
Theoremmatsubg 18696 The matrix ring has the same addition as its underlying group. (Contributed by AV, 2-Aug-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( -g `  G )  =  ( -g `  A ) )
 
Theoremmatplusgcell 18697 Addition in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .+b  =  ( +g  `  A )   &    |-  .+  =  ( +g  `  R )   =>    |-  (
 ( ( X  e.  B  /\  Y  e.  B )  /\  ( I  e.  N  /\  J  e.  N ) )  ->  ( I ( X  .+b  Y ) J )  =  ( ( I X J )  .+  ( I Y J ) ) )
 
Theoremmatsubgcell 18698 Subtraction in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  S  =  ( -g `  A )   &    |-  .-  =  ( -g `  R )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B ) 
 /\  ( I  e.  N  /\  J  e.  N ) )  ->  ( I ( X S Y ) J )  =  ( ( I X J )  .-  ( I Y J ) ) )
 
Theoremmatinvgcell 18699 Additive inversion in the matrix ring is cell-wise. (Contributed by AV, 17-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  V  =  ( invg `  R )   &    |-  W  =  ( invg `  A )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B  /\  ( I  e.  N  /\  J  e.  N ) )  ->  ( I
 ( W `  X ) J )  =  ( V `  ( I X J ) ) )
 
Theoremmatvscacell 18700 Scalar multiplication in the matrix ring is cell-wise. (Contributed by AV, 7-Aug-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  K  =  ( Base `  R )   &    |-  .x.  =  ( .s `  A )   &    |-  .X. 
 =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  K  /\  Y  e.  B ) 
 /\  ( I  e.  N  /\  J  e.  N ) )  ->  ( I ( X  .x.  Y ) J )  =  ( X  .X.  ( I Y J ) ) )
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