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Theorem List for Metamath Proof Explorer - 27401-27500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempmtrffv 27401 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 27402 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 27403 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 27404 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 27405 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 27406 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 27407 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 27408* 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 27409* 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 27410 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 27411* 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 27412 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 27413 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 )
 ) )
 
18.17.56  The sign of a permutation
 
Syntaxcpsgn 27414 Syntax for the sign of a permutation.
 class pmSgn
 
Definitiondf-psgn 27415* 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 27416* Lemma for psgnuni 27422. 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 27417* Lemma for psgnuni 27422. 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 27418* Lemma for psgnuni 27422. 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 27419* Lemma for psgnuni 27422. 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 27420 Lemma for psgnuni 27422. 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 27421 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 27422 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 27423* 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 27424* 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 27425 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 27426 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 27427* 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 27428 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 27429* 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 27430* 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 27431* 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 27432 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 27433 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 27434 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 27435 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 27436 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 27437 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 27438 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 ) )
 
18.17.57  The matrix algebra
 
Syntaxcmmul 27439 Syntax for the matrix multiplication operator.
 class maMul
 
Syntaxcmat 27440 Syntax for the square matrix algebra.
 class Mat
 
Definitiondf-mamu 27441* 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 27442* 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 27443* 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 27444* 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 27445* 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 27446 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 27447 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 27448 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 27449 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 27450 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 27451 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 27452 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 27453 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 27454* 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 27455* 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 27456 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 27457* 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 27458* 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 27459* 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 27460 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 27461 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 27462 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 27463 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 27464 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 27465 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 27466 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 27467 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 27468 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 27469 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 27470 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 27471 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 27472 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 27473 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 27474 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 27475 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 27476 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 27477 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 27478 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 27479 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 27480 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 27481 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 27482* 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.  ) ) )
 
18.17.58  The determinant
 
Syntaxcmdat 27483 Syntax for the matrix determinant function.
 class maDet
 
Syntaxcmadu 27484 Syntax for the matrix adjugate function.
 class maAdju
 
Definitiondf-mdet 27485* 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 27486* 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 27487* 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 27488* 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 ) ) ) ) ) ) )
 
18.17.59  Endomorphism algebra
 
Syntaxcmend 27489 Syntax for module endomorphism algebra.
 class MEndo
 
Definitiondf-mend 27490* 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 ) ) >. } ) )
 
Theoremmendval 27491* Value of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  B  =  ( M LMHom  M )   &    |-  .+  =  ( x  e.  B ,  y  e.  B  |->  ( x  o F ( +g  `  M ) y ) )   &    |-  .X. 
 =  ( x  e.  B ,  y  e.  B  |->  ( x  o.  y ) )   &    |-  S  =  (Scalar `  M )   &    |-  .x.  =  ( x  e.  ( Base `  S ) ,  y  e.  B  |->  ( ( ( Base `  M )  X.  { x }
 )  o F ( .s `  M ) y ) )   =>    |-  ( M  e.  X  ->  (MEndo `  M )  =  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } ) )
 
Theoremmendbas 27492 Base set of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  A  =  (MEndo `  M )   =>    |-  ( M LMHom  M )  =  (
 Base `  A )
 
Theoremmendplusgfval 27493* Addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  A  =  (MEndo `  M )   &    |-  B  =  ( Base `  A )   &    |-  .+  =  ( +g  `  M )   =>    |-  ( +g  `  A )  =  ( x  e.  B ,  y  e.  B  |->  ( x  o F  .+  y ) )
 
Theoremmendplusg 27494 A specific addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.)
 |-  A  =  (MEndo `  M )   &    |-  B  =  ( Base `  A )   &    |-  .+  =  ( +g  `  M )   &    |-  .+b  =  ( +g  `  A )   =>    |-  (
 ( X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .+b  Y )  =  ( X  o F  .+  Y ) )
 
Theoremmendmulrfval 27495* Multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  A  =  (MEndo `  M )   &    |-  B  =  ( Base `  A )   =>    |-  ( .r `  A )  =  ( x  e.  B ,  y  e.  B  |->  ( x  o.  y
 ) )
 
Theoremmendmulr 27496 A specific multiplication in the module endormoprhism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.)
 |-  A  =  (MEndo `  M )   &    |-  B  =  ( Base `  A )   &    |-  .x.  =  ( .r `  A )   =>    |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  Y )  =  ( X  o.  Y ) )
 
Theoremmendsca 27497 The module endomorphism algebra has the same scalars as the underlying module. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  A  =  (MEndo `  M )   &    |-  S  =  (Scalar `  M )   =>    |-  S  =  (Scalar `  A )
 
Theoremmendvscafval 27498* Scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  A  =  (MEndo `  M )   &    |-  .x.  =  ( .s `  M )   &    |-  B  =  ( Base `  A )   &    |-  S  =  (Scalar `  M )   &    |-  K  =  (
 Base `  S )   &    |-  E  =  ( Base `  M )   =>    |-  ( .s `  A )  =  ( x  e.  K ,  y  e.  B  |->  ( ( E  X.  { x } )  o F  .x.  y )
 )
 
Theoremmendvsca 27499 A specific scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.)
 |-  A  =  (MEndo `  M )   &    |-  .x.  =  ( .s `  M )   &    |-  B  =  ( Base `  A )   &    |-  S  =  (Scalar `  M )   &    |-  K  =  (
 Base `  S )   &    |-  E  =  ( Base `  M )   &    |-  .xb  =  ( .s `  A )   =>    |-  ( ( X  e.  K  /\  Y  e.  B )  ->  ( X  .xb  Y )  =  ( ( E  X.  { X } )  o F  .x.  Y ) )
 
Theoremmendrng 27500 The module endomorphism algebra is a ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  (MEndo `  M )   =>    |-  ( M  e.  LMod  ->  A  e.  Ring )
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