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Theorem List for Metamath Proof Explorer - 27301-27400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremf1omvdco2 27301 If exactly one of two permutations is limited to a set of points, then the composition will not be. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  (
 ( F : A -1-1-onto-> A  /\  G : A -1-1-onto-> A  /\  ( dom  ( F  \  _I  )  C_  X  \/_  dom  ( G  \  _I  )  C_  X ) )  ->  -.  dom  ( ( F  o.  G )  \  _I  )  C_  X )
 
Theoremf1omvdco3 27302 If a point is moved by exactly one of two permutations, then it will be moved by their composite. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  (
 ( F : A -1-1-onto-> A  /\  G : A -1-1-onto-> A  /\  ( X  e.  dom  ( F  \  _I  )  \/_  X  e.  dom  ( G  \  _I  ) ) )  ->  X  e.  dom  ( ( F  o.  G ) 
 \  _I  ) )
 
Theorempmtrfval 27303* The function generating transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   =>    |-  ( D  e.  V  ->  T  =  ( p  e. 
 { y  e.  ~P D  |  y  ~~  2o }  |->  ( z  e.  D  |->  if ( z  e.  p ,  U. ( p  \  { z }
 ) ,  z ) ) ) )
 
Theorempmtrval 27304* A generated transposition, expressed in a symmetric form. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   =>    |-  (
 ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  =  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z }
 ) ,  z ) ) )
 
Theorempmtrfv 27305 General value of mapping a point under a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   =>    |-  (
 ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D ) 
 ->  ( ( T `  P ) `  Z )  =  if ( Z  e.  P ,  U. ( P  \  { Z } ) ,  Z ) )
 
Theorempmtrprfv 27306 In a transposition of two given points, each maps to the other. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   =>    |-  (
 ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  X  =/=  Y ) ) 
 ->  ( ( T `  { X ,  Y }
 ) `  X )  =  Y )
 
Theorempmtrf 27307 Functionality of a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   =>    |-  (
 ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P ) : D --> D )
 
Theorempmtrmvd 27308 A transposition moves precisely the transposed points. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   =>    |-  (
 ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  dom  ( ( T `  P )  \  _I  )  =  P )
 
Theorempmtrrn 27309 Transposing two points gives a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  e.  R )
 
Theorempmtrfrn 27310 A transposition (as a kind of function) is the function transposing the two points it moves. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   &    |-  P  =  dom  ( F  \  _I  )   =>    |-  ( F  e.  R  ->  ( ( D  e.  _V  /\  P  C_  D  /\  P  ~~  2o )  /\  F  =  ( T `  P ) ) )
 
Theorempmtrffv 27311 Mapping of a point under a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   &    |-  P  =  dom  ( F  \  _I  )   =>    |-  (
 ( F  e.  R  /\  Z  e.  D ) 
 ->  ( F `  Z )  =  if ( Z  e.  P ,  U. ( P  \  { Z } ) ,  Z ) )
 
Theorempmtrfinv 27312 A transposition function is an involution. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  ( F  e.  R  ->  ( F  o.  F )  =  (  _I  |`  D ) )
 
Theorempmtrfmvdn0 27313 A transpositon moves at least one point. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  ( F  e.  R  ->  dom  ( F  \  _I  )  =/=  (/) )
 
Theorempmtrff1o 27314 A transposition function is a permutation. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  ( F  e.  R  ->  F : D -1-1-onto-> D )
 
Theorempmtrfcnv 27315 A transposition function is its own inverse. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  ( F  e.  R  ->  `' F  =  F )
 
Theorempmtrfb 27316 An intrinsic characterization of the transposition permutations. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  ( F  e.  R  <->  ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o ) )
 
Theorempmtrfconj 27317 Any conjugate of a transposition is a transposition. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  ( ( F  e.  R  /\  G : D -1-1-onto-> D )  ->  ( ( G  o.  F )  o.  `' G )  e.  R )
 
Theoremsymgsssg 27318* The symmetric group has subgroups restricting the set of non-fixed points. (Contributed by Stefan O'Rear, 24-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  B  =  ( Base `  G )   =>    |-  ( D  e.  V  ->  { x  e.  B  |  dom  ( x  \  _I  )  C_  X }  e.  (SubGrp `  G ) )
 
Theoremsymgfisg 27319* The symmetric group has a subgroup of permutations that move finitely many points. (Contributed by Stefan O'Rear, 24-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  B  =  ( Base `  G )   =>    |-  ( D  e.  V  ->  { x  e.  B  |  dom  ( x  \  _I  )  e.  Fin }  e.  (SubGrp `  G ) )
 
Theoremsymgtrf 27320 Transpositions are elements of the symmetric group. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  T  =  ran  (pmTrsp `  D )   &    |-  G  =  ( SymGrp `  D )   &    |-  B  =  (
 Base `  G )   =>    |-  T  C_  B
 
Theoremsymggen 27321* The span of the transpositions is the subgroup that moves finitely many points. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  T  =  ran  (pmTrsp `  D )   &    |-  G  =  ( SymGrp `  D )   &    |-  B  =  (
 Base `  G )   &    |-  K  =  (mrCls `  (SubMnd `  G ) )   =>    |-  ( D  e.  V  ->  ( K `  T )  =  { x  e.  B  |  dom  ( x  \  _I  )  e. 
 Fin } )
 
Theoremsymggen2 27322 A finite permutation group is generated by the transpositions. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  T  =  ran  (pmTrsp `  D )   &    |-  G  =  ( SymGrp `  D )   &    |-  B  =  (
 Base `  G )   &    |-  K  =  (mrCls `  (SubMnd `  G ) )   =>    |-  ( D  e.  Fin  ->  ( K `  T )  =  B )
 
Theoremsymgtrinv 27323 To invert a permutation represented as a sequence of transpositions, reverse the sequence. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  T  =  ran  (pmTrsp `  D )   &    |-  G  =  ( SymGrp `  D )   &    |-  I  =  ( inv g `  G )   =>    |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I `  ( G  gsumg 
 W ) )  =  ( G  gsumg  (reverse `  W )
 ) )
 
19.16.56  The sign of a permutation
 
Syntaxcpsgn 27324 Syntax for the sign of a permutation.
 class pmSgn
 
Definitiondf-psgn 27325* Define a function which takes the value  1 for even permutations and  -u 1 for odd. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |- pmSgn  =  ( d  e.  _V  |->  ( x  e.  { p  e.  ( Base `  ( SymGrp `  d
 ) )  |  dom  ( p  \  _I  )  e.  Fin }  |->  ( iota
 s E. w  e. Word  ran  (pmTrsp `  d )
 ( x  =  ( ( SymGrp `  d )  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `
  w ) ) ) ) ) )
 
Theorempsgnunilem1 27326* Lemma for psgnuni 27332. Given two consequtive transpositions in a representation of a permutation, either they are equal and therefore equivalent to the identity, or they are not and it is possible to commute them such that a chosen point in the left transposition is preserved in the right. By repeating this process, a point can be removed from a representation of the identity. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  ran  (pmTrsp `  D )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  P  e.  T )   &    |-  ( ph  ->  Q  e.  T )   &    |-  ( ph  ->  A  e.  dom  ( P  \  _I  ) )   =>    |-  ( ph  ->  (
 ( P  o.  Q )  =  (  _I  |`  D )  \/  E. r  e.  T  E. s  e.  T  ( ( P  o.  Q )  =  ( r  o.  s
 )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e.  dom  ( r  \  _I  ) ) ) )
 
Theorempsgnunilem5 27327* Lemma for psgnuni 27332. It is impossible to shift a transposition off the end because if the active transposition is at the right end, it is the only transposition moving  A in contradiction to this being a representation of the identity. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  W  e. Word  T )   &    |-  ( ph  ->  ( G  gsumg  W )  =  (  _I  |`  D ) )   &    |-  ( ph  ->  ( # `  W )  =  L )   &    |-  ( ph  ->  I  e.  ( 0..^ L ) )   &    |-  ( ph  ->  A  e.  dom  ( ( W `  I )  \  _I  ) )   &    |-  ( ph  ->  A. k  e.  ( 0..^ I )  -.  A  e.  dom  ( ( W `
  k )  \  _I  ) )   =>    |-  ( ph  ->  ( I  +  1 )  e.  ( 0..^ L ) )
 
Theorempsgnunilem2 27328* Lemma for psgnuni 27332. Induction step for moving a transposition as far to the right as possible. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  W  e. Word  T )   &    |-  ( ph  ->  ( G  gsumg  W )  =  (  _I  |`  D ) )   &    |-  ( ph  ->  ( # `  W )  =  L )   &    |-  ( ph  ->  I  e.  ( 0..^ L ) )   &    |-  ( ph  ->  A  e.  dom  ( ( W `  I )  \  _I  ) )   &    |-  ( ph  ->  A. k  e.  ( 0..^ I )  -.  A  e.  dom  ( ( W `
  k )  \  _I  ) )   &    |-  ( ph  ->  -. 
 E. x  e. Word  T ( ( # `  x )  =  ( L  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
 ) )   =>    |-  ( ph  ->  E. w  e. Word  T ( ( ( G  gsumg  w )  =  (  _I  |`  D )  /\  ( # `  w )  =  L )  /\  ( ( I  +  1 )  e.  (
 0..^ L )  /\  A  e.  dom  ( ( w `  ( I  +  1 ) ) 
 \  _I  )  /\  A. j  e.  ( 0..^ ( I  +  1 ) )  -.  A  e.  dom  ( ( w `
  j )  \  _I  ) ) ) )
 
Theorempsgnunilem3 27329* Lemma for psgnuni 27332. Any nonempty representation of the identity can be incrementally transformed into a representation two shorter. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  W  e. Word  T )   &    |-  ( ph  ->  ( # `  W )  =  L )   &    |-  ( ph  ->  ( # `  W )  e.  NN )   &    |-  ( ph  ->  ( G  gsumg  W )  =  (  _I  |`  D ) )   &    |-  ( ph  ->  -. 
 E. x  e. Word  T ( ( # `  x )  =  ( L  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
 ) )   =>    |- 
 -.  ph
 
Theorempsgnunilem4 27330 Lemma for psgnuni 27332. An odd-length representation of the identity is impossible, as it could be repeatedly shortened to a length of 1, but a length 1 permutation must be a transposition. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  W  e. Word  T )   &    |-  ( ph  ->  ( G  gsumg  W )  =  (  _I  |`  D ) )   =>    |-  ( ph  ->  ( -u 1 ^ ( # `  W ) )  =  1 )
 
Theoremm1expaddsub 27331 Addition and subtraction of parities are the same. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  (
 ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( -u 1 ^ ( X  -  Y ) )  =  ( -u 1 ^ ( X  +  Y ) ) )
 
Theorempsgnuni 27332 If the same permutation can be written in more than one way as a product of transpositions, the parity of those products must agree; otherwise the product of one with the inverse of the other would be an odd representation of the identity. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  W  e. Word  T )   &    |-  ( ph  ->  X  e. Word  T )   &    |-  ( ph  ->  ( G  gsumg 
 W )  =  ( G  gsumg 
 X ) )   =>    |-  ( ph  ->  (
 -u 1 ^ ( # `
  W ) )  =  ( -u 1 ^ ( # `  X ) ) )
 
Theorempsgnfval 27333* Function definition of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  B  =  ( Base `  G )   &    |-  F  =  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  N  =  ( x  e.  F  |->  ( iota
 s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `
  w ) ) ) ) )
 
Theorempsgnfn 27334* Functionality and domain of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  B  =  ( Base `  G )   &    |-  F  =  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }   &    |-  N  =  (pmSgn `  D )   =>    |-  N  Fn  F
 
Theorempsgndmsubg 27335 The finitary permutations are a subgroup. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( D  e.  V  ->  dom 
 N  e.  (SubGrp `  G ) )
 
Theorempsgneldm 27336 Property of being a finitary permutation. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  N  =  (pmSgn `  D )   &    |-  B  =  ( Base `  G )   =>    |-  ( P  e.  dom  N  <->  ( P  e.  B  /\  dom  ( P  \  _I  )  e.  Fin ) )
 
Theorempsgneldm2 27337* The finitary permutations are the span of the transpositons. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( D  e.  V  ->  ( P  e.  dom  N  <->  E. w  e. Word  T P  =  ( G  gsumg  w ) ) )
 
Theorempsgneldm2i 27338 A sequence of transpositions describes a finitary permutation. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( G  gsumg  W )  e.  dom  N )
 
Theorempsgneu 27339* A finitary permutation has exactly one parity. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( P  e.  dom  N 
 ->  E! s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `
  w ) ) ) )
 
Theorempsgnval 27340* Value of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( P  e.  dom  N 
 ->  ( N `  P )  =  ( iota s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `
  w ) ) ) ) )
 
Theorempsgnvali 27341* A finitary permutation has at least one representation for its parity. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( P  e.  dom  N 
 ->  E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  ( N `  P )  =  ( -u 1 ^ ( # `
  w ) ) ) )
 
Theorempsgnvalii 27342 Any representation of a permutation is length matching the permutation sign. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( N `  ( G  gsumg 
 W ) )  =  ( -u 1 ^ ( # `
  W ) ) )
 
Theorempsgnpmtr 27343 All transpositions are odd. (Contributed by Stefan O'Rear, 29-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( P  e.  T  ->  ( N `  P )  =  -u 1 )
 
Theoremcnmsgnsubg 27344 The signs form a multiplicative subgroup of the complex numbers. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  M  =  ( (mulGrp ` fld )s  ( CC  \  {
 0 } ) )   =>    |-  { 1 ,  -u 1 }  e.  (SubGrp `  M )
 
Theoremcnmsgnbas 27345 The base set of the sign subgroup of the complex numbers. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  U  =  ( (mulGrp ` fld )s  { 1 ,  -u 1 } )   =>    |- 
 { 1 ,  -u 1 }  =  ( Base `  U )
 
Theoremcnmsgngrp 27346 The group of signs under multiplication. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  U  =  ( (mulGrp ` fld )s  { 1 ,  -u 1 } )   =>    |-  U  e.  Grp
 
Theorempsgnghm 27347 The sign is a homomorphism from the finitary permutation group to the numeric signs. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  S  =  ( SymGrp `  D )   &    |-  N  =  (pmSgn `  D )   &    |-  F  =  ( Ss  dom  N )   &    |-  U  =  ( (mulGrp ` fld )s  { 1 ,  -u 1 } )   =>    |-  ( D  e.  V  ->  N  e.  ( F 
 GrpHom  U ) )
 
Theorempsgnghm2 27348 The sign is a homomorphism from the finite symmetric group to the numeric signs. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  S  =  ( SymGrp `  D )   &    |-  N  =  (pmSgn `  D )   &    |-  U  =  ( (mulGrp ` fld )s  { 1 ,  -u 1 } )   =>    |-  ( D  e.  Fin  ->  N  e.  ( S  GrpHom  U ) )
 
19.16.57  The matrix algebra
 
Syntaxcmmul 27349 Syntax for the matrix multiplication operator.
 class maMul
 
Syntaxcmat 27350 Syntax for the square matrix algebra.
 class Mat
 
Definitiondf-mamu 27351* The operator which multiplies an MxN matrix with an NxP matrix. Note that it is not generally possible to recover the dimensions from the matrix, since all Nx0 and all 0xN 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 ) ) ) ) ) ) )
 
Definitiondf-mat 27352* The algebra of NxN matrices over a ring... (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 >. ) >. ) )
 
Theoremmamufval 27353* 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 27354* 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 27355* 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 ) ) ) ) )
 
Theoremmndvcl 27356 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  o F  .+  Y )  e.  ( B  ^m  I ) )
 
Theoremmndvass 27357 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  o F  .+  Y )  o F  .+  Z )  =  ( X  o F  .+  ( Y  o F  .+  Z ) ) )
 
Theoremmndvlid 27358 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.  } )  o F  .+  X )  =  X )
 
Theoremmndvrid 27359 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  o F  .+  ( I  X.  {  .0.  } ) )  =  X )
 
Theoremgrpvlinv 27360 Tuple-wise left inverse in groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  N  =  ( inv g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  ( B 
 ^m  I ) ) 
 ->  ( ( N  o.  X )  o F  .+  X )  =  ( I  X.  {  .0.  } ) )
 
Theoremgrpvrinv 27361 Tuple-wise right inverse in groups. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  N  =  ( inv g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  ( B 
 ^m  I ) ) 
 ->  ( X  o F  .+  ( N  o.  X ) )  =  ( I  X.  {  .0.  }
 ) )
 
Theoremmhmvlin 27362 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  o F  .+  Y ) )  =  ( ( F  o.  X )  o F  .+^  ( F  o.  Y ) ) )
 
Theoremrngvcl 27363 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  o F  .x.  Y )  e.  ( B  ^m  I ) )
 
Theoremgsumcom3 27364* 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 27365* 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 27366 Operation closure of matrix multiplication. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  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 ) ) )
 
Theoremmamudiagcl 27367* Diagonal matrices are matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  B  =  ( Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  I  =  ( i  e.  M ,  j  e.  M  |->  if ( i  =  j ,  .1.  ,  .0.  ) )   &    |-  ( ph  ->  M  e.  Fin )   =>    |-  ( ph  ->  I  e.  ( B  ^m  ( M  X.  M ) ) )
 
Theoremmamulid 27368* Diagonal matrices are left identities. (Contributed by Stefan O'Rear, 3-Sep-2015.)
 |-  B  =  ( Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  I  =  ( i  e.  M ,  j  e.  M  |->  if ( i  =  j ,  .1.  ,  .0.  ) )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  F  =  ( R maMul  <. M ,  M ,  N >. )   &    |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )   =>    |-  ( ph  ->  ( I F X )  =  X )
 
Theoremmamurid 27369* Diagonal matrices are right identities. (Contributed by Stefan O'Rear, 3-Sep-2015.)
 |-  B  =  ( Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  I  =  ( i  e.  M ,  j  e.  M  |->  if ( i  =  j ,  .1.  ,  .0.  ) )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  F  =  ( R maMul  <. N ,  M ,  M >. )   &    |-  ( ph  ->  X  e.  ( B  ^m  ( N  X.  M ) ) )   =>    |-  ( ph  ->  ( X F I )  =  X )
 
Theoremmamuass 27370 Matrix multiplication is associative. (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 27371 Matrix multiplication distributes over addition 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 )   &    |-  .+  =  ( +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  o F  .+  Y ) F Z )  =  ( ( X F Z )  o F  .+  ( Y F Z ) ) )
 
Theoremmamudir 27372 Matrix multiplication distributes over addition on the right. (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 )   &    |-  .+  =  ( +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  o F  .+  Z ) )  =  ( ( X F Y )  o F  .+  ( X F Z ) ) )
 
Theoremmamuvs1 27373 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 } )  o F  .x.  Y ) F Z )  =  ( ( ( M  X.  O )  X.  { X } )  o F  .x.  ( Y F Z ) ) )
 
Theoremmamuvs2 27374 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  ( 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 }
 )  o F  .x.  Z ) )  =  ( ( ( M  X.  O )  X.  { Y } )  o F  .x.  ( X F Z ) ) )
 
Theoremmatval 27375 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 27376 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 )
 )
 
Theoremmatmulr 27377 Multiplication in the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat  R )   &    |-  .x. 
 =  ( R maMul  <. N ,  N ,  N >. )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  .x.  =  ( .r
 `  A ) )
 
Theoremmatbas 27378 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 27379 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 27380 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 27381 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 27382 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 27383 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 ) 
 ->  ( inv g `  G )  =  ( inv g `  A ) )
 
Theoremmatsca2 27384 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 27385 The base set of the matrix ring as a set exponential. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  ( N Mat  R )   &    |-  K  =  ( Base `  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( K  ^m  ( N  X.  N ) )  =  ( Base `  A ) )
 
Theoremmatbas2i 27386 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 ) ) )
 
Theoremmatplusg2 27387 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  o F  .+  Y ) )
 
Theoremmatvsca2 27388 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 } )  o F  .X.  Y ) )
 
Theoremmatlmod 27389 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 )
 
Theoremmatrng 27390 Existence of the matrix ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  ( N Mat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Ring )
 
Theoremmatassa 27391 Existence of the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  ( N Mat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e. AssAlg )
 
Theoremmat1 27392* Value of an identity matrix. (Contributed by Stefan O'Rear, 7-Sep-2015.)
 |-  A  =  ( N Mat  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  ( 1r `  A )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  j ,  .1.  ,  .0.  ) ) )
 
19.16.58  The determinant
 
Syntaxcmdat 27393 Syntax for the matrix determinant function.
 class maDet
 
Syntaxcmadu 27394 Syntax for the matrix adjugate function.
 class maAdju
 
Definitiondf-mdet 27395* Determinant of a square matrix... (Contributed by Stefan O'Rear, 9-Sep-2015.)
 |- maDet  =  ( n  e.  _V ,  r  e.  _V  |->  ( m  e.  ( Base `  ( n Mat  r ) )  |->  ( r  gsumg  ( p  e.  ( Base `  ( SymGrp `  n ) )  |->  ( ( ( ZRHom `  r
 ) `  ( (pmSgn `  n ) `  p ) ) ( .r
 `  r ) ( (mulGrp `  r )  gsumg  ( x  e.  n  |->  ( ( p `  x ) m x ) ) ) ) ) ) ) )
 
Definitiondf-madu 27396* Define the adjunct (matrix of cofactors) of a square matrix. This definition gives the standard cofactors, however the internal minors are not the standard minors. (Contributed by Stefan O'Rear, 7-Sep-2015.)
 |- maAdju  =  ( n  e.  _V ,  r  e.  _V  |->  ( m  e.  ( Base `  ( n Mat  r ) )  |->  ( i  e.  n ,  j  e.  n  |->  ( if ( i  =  j ,  ( 1r
 `  r ) ,  ( ( inv g `  r ) `  ( 1r `  r ) ) ) ( .r `  r ) ( ( ( n  \  {
 i } ) maDet  r
 ) `  ( k  e.  ( n  \  {
 i } ) ,  l  e.  ( n 
 \  { i }
 )  |->  ( if (
 k  =  j ,  i ,  k ) m l ) ) ) ) ) ) )
 
Theoremmdetfval 27397* First substitution for the determinant definition. (Contributed by Stefan O'Rear, 9-Sep-2015.)
 |-  D  =  ( N maDet  R )   &    |-  A  =  ( N Mat  R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  ( Base `  ( SymGrp `  N ) )   &    |-  Y  =  ( ZRHom `  R )   &    |-  S  =  (pmSgn `  N )   &    |-  .x.  =  ( .r `  R )   &    |-  U  =  (mulGrp `  R )   =>    |-  D  =  ( m  e.  B  |->  ( R 
 gsumg  ( p  e.  P  |->  ( ( Y `  ( S `  p ) )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )
 
Theoremmdetleib 27398* Full substitution of our determinant definition (also known as Leibniz' Formula). (Contributed by Stefan O'Rear, 3-Oct-2015.)
 |-  D  =  ( N maDet  R )   &    |-  A  =  ( N Mat  R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  ( Base `  ( SymGrp `  N ) )   &    |-  Y  =  ( ZRHom `  R )   &    |-  S  =  (pmSgn `  N )   &    |-  .x.  =  ( .r `  R )   &    |-  U  =  (mulGrp `  R )   =>    |-  ( M  e.  B  ->  ( D `  M )  =  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `
  p ) ) 
 .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) ) ) ) )
 
19.16.59  Endomorphism algebra
 
Syntaxcmend 27399 Syntax for module endomorphism algebra.
 class MEndo
 
Definitiondf-mend 27400* Define the endomorphism algebra of a module. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |- MEndo  =  ( m  e.  _V  |->  [_ ( m LMHom  m )  /  b ]_ ( { <. (
 Base `  ndx ) ,  b >. ,  <. ( +g  ` 
 ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  o F (
 +g  `  m )
 y ) ) >. , 
 <. ( .r `  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  o.  y ) ) >. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  m ) >. ,  <. ( .s
 `  ndx ) ,  ( x  e.  ( Base `  (Scalar `  m )
 ) ,  y  e.  b  |->  ( ( (
 Base `  m )  X.  { x } )  o F ( .s `  m ) y ) ) >. } ) )
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