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Theorem List for Metamath Proof Explorer - 20301-20400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremperfectlem2 20301 Lemma for perfect 20302. (Contributed by Mario Carneiro, 17-May-2016.)
 |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  -.  2  ||  B )   &    |-  ( ph  ->  ( 1  sigma  ( (
 2 ^ A )  x.  B ) )  =  ( 2  x.  ( ( 2 ^ A )  x.  B ) ) )   =>    |-  ( ph  ->  ( B  e.  Prime  /\  B  =  ( ( 2 ^
 ( A  +  1 ) )  -  1
 ) ) )
 
Theoremperfect 20302* The Euclid-Euler theorem, or Perfect Number theorem. A positive even integer  N is a perfect number (that is, its divisor sum is  2 N) if and only if it is of the form  2 ^ ( p  - 
1 )  x.  (
2 ^ p  - 
1 ), where  2 ^ p  -  1 is prime (a Mersenne prime). (It follows from this that  p is also prime.) (Contributed by Mario Carneiro, 17-May-2016.)
 |-  ( ( N  e.  NN  /\  2  ||  N )  ->  ( ( 1 
 sigma  N )  =  ( 2  x.  N )  <->  E. p  e.  ZZ  ( ( ( 2 ^ p )  -  1 )  e.  Prime  /\  N  =  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  -  1 ) ) ) ) )
 
13.4.6  Characters of Z/nZ
 
Syntaxcdchr 20303 Extend class notation with the group of Dirichlet characters.
 class DChr
 
Definitiondf-dchr 20304* The group of Dirichlet characters 
mod  n is the set of monoid homomorphisms from  ZZ  /  n ZZ to the multiplicative monoid of the complexes, equipped with the group operation of pointwise multiplication. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |- DChr  =  ( n  e.  NN  |->  [_ (ℤ/n `  n )  /  z ]_ [_ { x  e.  ( (mulGrp `  z
 ) MndHom  (mulGrp ` fld ) )  |  ( ( ( Base `  z
 )  \  (Unit `  z
 ) )  X.  {
 0 } )  C_  x }  /  b ]_ { <. ( Base `  ndx ) ,  b >. , 
 <. ( +g  `  ndx ) ,  (  o F  x.  |`  ( b  X.  b ) ) >. } )
 
Theoremdchrval 20305* Value of the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  B  =  (
 Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  D  =  { x  e.  (
 (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U )  X.  { 0 } )  C_  x }
 )   =>    |-  ( ph  ->  G  =  { <. ( Base `  ndx ) ,  D >. , 
 <. ( +g  `  ndx ) ,  (  o F  x.  |`  ( D  X.  D ) ) >. } )
 
Theoremdchrbas 20306* Base set of the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  B  =  (
 Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  D  =  ( Base `  G )   =>    |-  ( ph  ->  D  =  { x  e.  (
 (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U )  X.  { 0 } )  C_  x }
 )
 
Theoremdchrelbas 20307 A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of  CC, which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  B  =  (
 Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  D  =  ( Base `  G )   =>    |-  ( ph  ->  ( X  e.  D  <->  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  /\  ( ( B  \  U )  X.  { 0 } )  C_  X )
 ) )
 
Theoremdchrelbas2 20308* A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of  CC, which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  B  =  (
 Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  D  =  ( Base `  G )   =>    |-  ( ph  ->  ( X  e.  D  <->  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  /\  A. x  e.  B  ( ( X `
  x )  =/=  0  ->  x  e.  U ) ) ) )
 
Theoremdchrelbas3 20309* A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of  CC, which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  B  =  (
 Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  D  =  ( Base `  G )   =>    |-  ( ph  ->  ( X  e.  D  <->  ( X : B
 --> CC  /\  ( A. x  e.  U  A. y  e.  U  ( X `  ( x ( .r `  Z ) y ) )  =  ( ( X `  x )  x.  ( X `  y ) )  /\  ( X `  ( 1r
 `  Z ) )  =  1  /\  A. x  e.  B  (
 ( X `  x )  =/=  0  ->  x  e.  U ) ) ) ) )
 
Theoremdchrelbasd 20310* A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of  CC, which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  B  =  (
 Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  D  =  ( Base `  G )   &    |-  ( k  =  x  ->  X  =  A )   &    |-  ( k  =  y  ->  X  =  C )   &    |-  ( k  =  ( x ( .r
 `  Z ) y )  ->  X  =  E )   &    |-  ( k  =  ( 1r `  Z )  ->  X  =  Y )   &    |-  ( ( ph  /\  k  e.  U )  ->  X  e.  CC )   &    |-  ( ( ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  E  =  ( A  x.  C ) )   &    |-  ( ph  ->  Y  =  1 )   =>    |-  ( ph  ->  ( k  e.  B  |->  if ( k  e.  U ,  X ,  0 ) )  e.  D )
 
Theoremdchrrcl 20311 Reverse closure for a Dirichlet character. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  G  =  (DChr `  N )   &    |-  D  =  (
 Base `  G )   =>    |-  ( X  e.  D  ->  N  e.  NN )
 
Theoremdchrmhm 20312 A Dirichlet character is a monoid homomorphism. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   =>    |-  D  C_  (
 (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
 
Theoremdchrf 20313 A Dirichlet character is a function. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  B  =  ( Base `  Z )   &    |-  ( ph  ->  X  e.  D )   =>    |-  ( ph  ->  X : B --> CC )
 
Theoremdchrelbas4 20314* A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of  CC, which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  L  =  ( ZRHom `  Z )   =>    |-  ( X  e.  D  <->  ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  /\  A. x  e.  ZZ  (
 1  <  ( x  gcd  N )  ->  ( X `  ( L `  x ) )  =  0 ) ) )
 
Theoremdchrzrh1 20315 Value of a Dirichlet character at one. (Contributed by Mario Carneiro, 4-May-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  X  e.  D )   =>    |-  ( ph  ->  ( X `  ( L `  1 ) )  =  1 )
 
Theoremdchrzrhcl 20316 A Dirichlet character takes values in the complexes. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  A  e.  ZZ )   =>    |-  ( ph  ->  ( X `  ( L `
  A ) )  e.  CC )
 
Theoremdchrzrhmul 20317 A Dirichlet character is completely multiplicative. (Contributed by Mario Carneiro, 4-May-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  L  =  ( ZRHom `  Z )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  C  e.  ZZ )   =>    |-  ( ph  ->  ( X `  ( L `  ( A  x.  C ) ) )  =  ( ( X `  ( L `  A ) )  x.  ( X `
  ( L `  C ) ) ) )
 
Theoremdchrplusg 20318 Group operation on the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  .x.  =  ( +g  `  G )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  .x.  =  (  o F  x.  |`  ( D  X.  D ) ) )
 
Theoremdchrmul 20319 Group operation on the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  .x.  =  ( +g  `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  Y  e.  D )   =>    |-  ( ph  ->  ( X  .x.  Y )  =  ( X  o F  x.  Y ) )
 
Theoremdchrmulcl 20320 Closure of the group operation on Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  .x.  =  ( +g  `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  Y  e.  D )   =>    |-  ( ph  ->  ( X  .x.  Y )  e.  D )
 
Theoremdchrn0 20321 A Dirichlet character is nonzero on the units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  B  =  ( Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  (
 ( X `  A )  =/=  0  <->  A  e.  U ) )
 
Theoremdchr1cl 20322* Closure of the principal Dirichlet character. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  B  =  ( Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  .1.  =  ( k  e.  B  |->  if ( k  e.  U ,  1 ,  0 ) )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  .1. 
 e.  D )
 
Theoremdchrmulid2 20323* Left identity for the principal Dirichlet character. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  B  =  ( Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  .1.  =  ( k  e.  B  |->  if ( k  e.  U ,  1 ,  0 ) )   &    |-  .x.  =  ( +g  `  G )   &    |-  ( ph  ->  X  e.  D )   =>    |-  ( ph  ->  (  .1.  .x.  X )  =  X )
 
Theoremdchrinvcl 20324* Closure of the group inverse operation on Dirichlet characters. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  B  =  ( Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  .1.  =  ( k  e.  B  |->  if ( k  e.  U ,  1 ,  0 ) )   &    |-  .x.  =  ( +g  `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  K  =  ( k  e.  B  |->  if (
 k  e.  U ,  ( 1  /  ( X `  k ) ) ,  0 ) )   =>    |-  ( ph  ->  ( K  e.  D  /\  ( K 
 .x.  X )  =  .1.  ) )
 
Theoremdchrabl 20325 The set of Dirichlet characters is an Abelian group. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  G  =  (DChr `  N )   =>    |-  ( N  e.  NN  ->  G  e.  Abel )
 
Theoremdchrfi 20326 The group of Dirichlet characters is a finite group. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  D  =  (
 Base `  G )   =>    |-  ( N  e.  NN  ->  D  e.  Fin )
 
Theoremdchrghm 20327 A Dirichlet character restricted to the unit group of ℤ/nℤ is a group homomorphism into the multiplicative group of nonzero complex numbers. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  U  =  (Unit `  Z )   &    |-  H  =  ( (mulGrp `  Z )s  U )   &    |-  M  =  ( (mulGrp ` fld )s  ( CC  \  {
 0 } ) )   &    |-  ( ph  ->  X  e.  D )   =>    |-  ( ph  ->  ( X  |`  U )  e.  ( H  GrpHom  M ) )
 
Theoremdchr1 20328 Value of the principal Dirichlet character. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  .1.  =  ( 0g `  G )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  (  .1.  `  A )  =  1 )
 
Theoremdchreq 20329* A Dirichlet character is determined by its values on the unit group. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  Y  e.  D )   =>    |-  ( ph  ->  ( X  =  Y  <->  A. k  e.  U  ( X `  k )  =  ( Y `  k ) ) )
 
Theoremdchrresb 20330 A Dirichlet character is determined by its values on the unit group. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  Y  e.  D )   =>    |-  ( ph  ->  (
 ( X  |`  U )  =  ( Y  |`  U )  <->  X  =  Y )
 )
 
Theoremdchrabs 20331 A Dirichlet character takes values on the unit circle. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  D  =  (
 Base `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  Z  =  (ℤ/n `  N )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  ( abs `  ( X `  A ) )  =  1 )
 
Theoremdchrinv 20332 The inverse of a Dirichlet character is the conjugate (which is also the multiplicative inverse, because the values of  X are unimodular). (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  D  =  (
 Base `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  I  =  ( inv
 g `  G )   =>    |-  ( ph  ->  ( I `  X )  =  ( *  o.  X ) )
 
Theoremdchrabs2 20333 A Dirichlet character takes values inside the unit circle. (Contributed by Mario Carneiro, 3-May-2016.)
 |-  G  =  (DChr `  N )   &    |-  D  =  (
 Base `  G )   &    |-  Z  =  (ℤ/n `  N )   &    |-  B  =  (
 Base `  Z )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  ( abs `  ( X `  A ) )  <_ 
 1 )
 
Theoremdchr1re 20334 The principal Dirichlet character is a real character. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  .1.  =  ( 0g `  G )   &    |-  B  =  ( Base `  Z )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  .1. 
 : B --> RR )
 
Theoremdchrptlem1 20335* Lemma for dchrpt 20338. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  B  =  ( Base `  Z )   &    |-  .1.  =  ( 1r `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  =/=  .1.  )   &    |-  U  =  (Unit `  Z )   &    |-  H  =  ( (mulGrp `  Z )s  U )   &    |-  .x.  =  (.g `  H )   &    |-  S  =  ( k  e.  dom  W  |->  ran  (  n  e.  ZZ  |->  ( n  .x.  ( W `
  k ) ) ) )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  W  e. Word  U )   &    |-  ( ph  ->  H dom DProd  S )   &    |-  ( ph  ->  ( H DProd  S )  =  U )   &    |-  P  =  ( HdProj S )   &    |-  O  =  ( od `  H )   &    |-  T  =  ( -u 1  ^ c  ( 2 
 /  ( O `  ( W `  I ) ) ) )   &    |-  ( ph  ->  I  e.  dom  W )   &    |-  ( ph  ->  ( ( P `  I
 ) `  A )  =/=  .1.  )   &    |-  X  =  ( u  e.  U  |->  (
 iota h E. m  e. 
 ZZ  ( ( ( P `  I ) `
  u )  =  ( m  .x.  ( W `  I ) ) 
 /\  h  =  ( T ^ m ) ) ) )   =>    |-  ( ( (
 ph  /\  C  e.  U )  /\  ( M  e.  ZZ  /\  (
 ( P `  I
 ) `  C )  =  ( M  .x.  ( W `  I ) ) ) )  ->  ( X `  C )  =  ( T ^ M ) )
 
Theoremdchrptlem2 20336* Lemma for dchrpt 20338. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  B  =  ( Base `  Z )   &    |-  .1.  =  ( 1r `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  =/=  .1.  )   &    |-  U  =  (Unit `  Z )   &    |-  H  =  ( (mulGrp `  Z )s  U )   &    |-  .x.  =  (.g `  H )   &    |-  S  =  ( k  e.  dom  W  |->  ran  (  n  e.  ZZ  |->  ( n  .x.  ( W `
  k ) ) ) )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  W  e. Word  U )   &    |-  ( ph  ->  H dom DProd  S )   &    |-  ( ph  ->  ( H DProd  S )  =  U )   &    |-  P  =  ( HdProj S )   &    |-  O  =  ( od `  H )   &    |-  T  =  ( -u 1  ^ c  ( 2 
 /  ( O `  ( W `  I ) ) ) )   &    |-  ( ph  ->  I  e.  dom  W )   &    |-  ( ph  ->  ( ( P `  I
 ) `  A )  =/=  .1.  )   &    |-  X  =  ( u  e.  U  |->  (
 iota h E. m  e. 
 ZZ  ( ( ( P `  I ) `
  u )  =  ( m  .x.  ( W `  I ) ) 
 /\  h  =  ( T ^ m ) ) ) )   =>    |-  ( ph  ->  E. x  e.  D  ( x `  A )  =/=  1 )
 
Theoremdchrptlem3 20337* Lemma for dchrpt 20338. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  B  =  ( Base `  Z )   &    |-  .1.  =  ( 1r `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  =/=  .1.  )   &    |-  U  =  (Unit `  Z )   &    |-  H  =  ( (mulGrp `  Z )s  U )   &    |-  .x.  =  (.g `  H )   &    |-  S  =  ( k  e.  dom  W  |->  ran  (  n  e.  ZZ  |->  ( n  .x.  ( W `
  k ) ) ) )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  W  e. Word  U )   &    |-  ( ph  ->  H dom DProd  S )   &    |-  ( ph  ->  ( H DProd  S )  =  U )   =>    |-  ( ph  ->  E. x  e.  D  ( x `  A )  =/=  1
 )
 
Theoremdchrpt 20338* For any element other than 1, there is a Dirichlet character that is not one at the given element. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  B  =  ( Base `  Z )   &    |-  .1.  =  ( 1r `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  =/=  .1.  )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  E. x  e.  D  ( x `  A )  =/=  1
 )
 
Theoremdchrsum2 20339* An orthogonality relation for Dirichlet characters: the sum of all the values of a Dirichlet character  X is  0 if  X is non-principal and  phi ( n ) otherwise. Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  U  =  (Unit `  Z )   =>    |-  ( ph  ->  sum_ a  e.  U  ( X `  a )  =  if ( X  =  .1.  ,  ( phi `  N ) ,  0 )
 )
 
Theoremdchrsum 20340* An orthogonality relation for Dirichlet characters: the sum of all the values of a Dirichlet character  X is  0 if  X is non-principal and  phi ( n ) otherwise. Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  .1.  =  ( 0g `  G )   &    |-  ( ph  ->  X  e.  D )   &    |-  B  =  (
 Base `  Z )   =>    |-  ( ph  ->  sum_
 a  e.  B  ( X `  a )  =  if ( X  =  .1.  ,  ( phi `  N ) ,  0 ) )
 
Theoremsumdchr2 20341* Lemma for sumdchr 20343. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  D  =  (
 Base `  G )   &    |-  Z  =  (ℤ/n `  N )   &    |-  .1.  =  ( 1r `  Z )   &    |-  B  =  ( Base `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  sum_ x  e.  D  ( x `  A )  =  if ( A  =  .1.  ,  ( # `  D ) ,  0 )
 )
 
Theoremdchrhash 20342 There are exactly  phi ( N ) Dirichlet characters modulo  N. Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  D  =  (
 Base `  G )   =>    |-  ( N  e.  NN  ->  ( # `  D )  =  ( phi `  N ) )
 
Theoremsumdchr 20343* An orthogonality relation for Dirichlet characters: the sum of  x ( A ) for fixed  A and all  x is  0 if  A  =  1 and  phi ( n ) otherwise. Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  D  =  (
 Base `  G )   &    |-  Z  =  (ℤ/n `  N )   &    |-  .1.  =  ( 1r `  Z )   &    |-  B  =  ( Base `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  sum_ x  e.  D  ( x `  A )  =  if ( A  =  .1.  ,  ( phi `  N ) ,  0 )
 )
 
Theoremdchr2sum 20344* An orthogonality relation for Dirichlet characters: the sum of  X ( a )  x.  * Y ( a ) over all  a is nonzero only when  X  =  Y. Part of Theorem 6.5.2 of [Shapiro] p. 232. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  Z  =  (ℤ/n `  N )   &    |-  D  =  (
 Base `  G )   &    |-  B  =  ( Base `  Z )   &    |-  ( ph  ->  X  e.  D )   &    |-  ( ph  ->  Y  e.  D )   =>    |-  ( ph  ->  sum_ a  e.  B  ( ( X `
  a )  x.  ( * `  ( Y `  a ) ) )  =  if ( X  =  Y ,  ( phi `  N ) ,  0 ) )
 
Theoremsum2dchr 20345* An orthogonality relation for Dirichlet characters: the sum of  x ( A ) for fixed  A and all  x is  0 if  A  =  1 and  phi ( n ) otherwise. Part of Theorem 6.5.2 of [Shapiro] p. 232. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  G  =  (DChr `  N )   &    |-  D  =  (
 Base `  G )   &    |-  Z  =  (ℤ/n `  N )   &    |-  B  =  (
 Base `  Z )   &    |-  U  =  (Unit `  Z )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  C  e.  U )   =>    |-  ( ph  ->  sum_
 x  e.  D  ( ( x `  A )  x.  ( * `  ( x `  C ) ) )  =  if ( A  =  C ,  ( phi `  N ) ,  0 )
 )
 
13.4.7  Bertrand's postulate
 
Theorembcctr 20346 Value of the central binomial coefficient. (Contributed by Mario Carneiro, 13-Mar-2014.)
 |-  ( N  e.  NN0  ->  ( ( 2  x.  N )  _C  N )  =  ( ( ! `  ( 2  x.  N ) )  /  ( ( ! `  N )  x.  ( ! `  N ) ) ) )
 
Theorempcbcctr 20347* Prime count of a central binomial coefficient. (Contributed by Mario Carneiro, 12-Mar-2014.)
 |-  ( ( N  e.  NN  /\  P  e.  Prime ) 
 ->  ( P  pCnt  (
 ( 2  x.  N )  _C  N ) )  =  sum_ k  e.  (
 1 ... ( 2  x.  N ) ) ( ( |_ `  (
 ( 2  x.  N )  /  ( P ^
 k ) ) )  -  ( 2  x.  ( |_ `  ( N  /  ( P ^
 k ) ) ) ) ) )
 
Theorembcmono 20348 The binomial coefficient is monotone in its second argument, up to the midway point. (Contributed by Mario Carneiro, 5-Mar-2014.)
 |-  ( ( N  e.  NN0  /\  B  e.  ( ZZ>= `  A )  /\  B  <_  ( N  /  2 ) )  ->  ( N  _C  A )  <_  ( N  _C  B ) )
 
Theorembcmax 20349 The binomial coefficient takes its maximum value at the center. (Contributed by Mario Carneiro, 5-Mar-2014.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( ( 2  x.  N )  _C  K )  <_  ( ( 2  x.  N )  _C  N ) )
 
Theorembcp1ctr 20350 Ratio of two central binomial coefficients. (Contributed by Mario Carneiro, 10-Mar-2014.)
 |-  ( N  e.  NN0  ->  ( ( 2  x.  ( N  +  1 ) )  _C  ( N  +  1 )
 )  =  ( ( ( 2  x.  N )  _C  N )  x.  ( 2  x.  (
 ( ( 2  x.  N )  +  1 )  /  ( N  +  1 ) ) ) ) )
 
Theorembclbnd 20351 A bound on the binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.)
 |-  ( N  e.  ( ZZ>=
 `  4 )  ->  ( ( 4 ^ N )  /  N )  <  ( ( 2  x.  N )  _C  N ) )
 
Theoremefexple 20352 Convert a bound on a power to a bound on the exponent. (Contributed by Mario Carneiro, 11-Mar-2014.)
 |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  ( ( A ^ N )  <_  B 
 <->  N  <_  ( |_ `  ( ( log `  B )  /  ( log `  A ) ) ) ) )
 
Theorembpos1lem 20353* Lemma for bpos1 20354. (Contributed by Mario Carneiro, 12-Mar-2014.)
 |-  ( E. p  e. 
 Prime  ( N  <  p  /\  p  <_  ( 2  x.  N ) ) 
 ->  ph )   &    |-  ( N  e.  ( ZZ>= `  P )  -> 
 ph )   &    |-  P  e.  Prime   &    |-  A  e.  NN0   &    |-  ( A  x.  2
 )  =  B   &    |-  A  <  P   &    |-  ( P  <  B  \/  P  =  B )   =>    |-  ( N  e.  ( ZZ>=
 `  A )  ->  ph )
 
Theorembpos1 20354* Bertrand's postulate, checked numerically for  N  <_  6 4, using the prime sequence  2 ,  3 ,  5 ,  7 ,  1 3 ,  2 3 ,  4 3 ,  8 3. (Contributed by Mario Carneiro, 12-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |-  ( ( N  e.  NN  /\  N  <_ ; 6 4 )  ->  E. p  e.  Prime  ( N  <  p  /\  p  <_  ( 2  x.  N ) ) )
 
Theorembposlem1 20355 An upper bound on the prime powers dividing a central binomial coefficient. (Contributed by Mario Carneiro, 9-Mar-2014.)
 |-  ( ( N  e.  NN  /\  P  e.  Prime ) 
 ->  ( P ^ ( P  pCnt  ( ( 2  x.  N )  _C  N ) ) ) 
 <_  ( 2  x.  N ) )
 
Theorembposlem2 20356 There are no odd primes in the range 
( 2 N  / 
3 ,  N ] dividing the  N-th central binomial coefficient. (Contributed by Mario Carneiro, 12-Mar-2014.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  2  <  P )   &    |-  ( ph  ->  ( ( 2  x.  N )  / 
 3 )  <  P )   &    |-  ( ph  ->  P  <_  N )   =>    |-  ( ph  ->  ( P  pCnt  ( ( 2  x.  N )  _C  N ) )  =  0 )
 
Theorembposlem3 20357* Lemma for bpos 20364. Since the binomial coefficient does not have any primes in the range  ( 2 N  / 
3 ,  N ] or  ( 2 N ,  +oo ) by bposlem2 20356 and prmfac1 12671, respectively, and it does not have any in the range  ( N , 
2 N ] by hypothesis, the product of the primes up through  2 N  / 
3 must be sufficient to compose the whole binomial coefficient. (Contributed by Mario Carneiro, 13-Mar-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  5 )
 )   &    |-  ( ph  ->  -.  E. p  e.  Prime  ( N  <  p  /\  p  <_  ( 2  x.  N ) ) )   &    |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ ( n  pCnt  ( ( 2  x.  N )  _C  N ) ) ) ,  1 ) )   &    |-  K  =  ( |_ `  ( ( 2  x.  N )  /  3
 ) )   =>    |-  ( ph  ->  (  seq  1 (  x.  ,  F ) `  K )  =  ( (
 2  x.  N )  _C  N ) )
 
Theorembposlem4 20358* Lemma for bpos 20364. (Contributed by Mario Carneiro, 13-Mar-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  5 )
 )   &    |-  ( ph  ->  -.  E. p  e.  Prime  ( N  <  p  /\  p  <_  ( 2  x.  N ) ) )   &    |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ ( n  pCnt  ( ( 2  x.  N )  _C  N ) ) ) ,  1 ) )   &    |-  K  =  ( |_ `  ( ( 2  x.  N )  /  3
 ) )   &    |-  M  =  ( |_ `  ( sqr `  ( 2  x.  N ) ) )   =>    |-  ( ph  ->  M  e.  ( 3 ...
 K ) )
 
Theorembposlem5 20359* Lemma for bpos 20364. Bound the product of all small primes in the binomial coefficient. (Contributed by Mario Carneiro, 15-Mar-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  5 )
 )   &    |-  ( ph  ->  -.  E. p  e.  Prime  ( N  <  p  /\  p  <_  ( 2  x.  N ) ) )   &    |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ ( n  pCnt  ( ( 2  x.  N )  _C  N ) ) ) ,  1 ) )   &    |-  K  =  ( |_ `  ( ( 2  x.  N )  /  3
 ) )   &    |-  M  =  ( |_ `  ( sqr `  ( 2  x.  N ) ) )   =>    |-  ( ph  ->  ( 
 seq  1 (  x. 
 ,  F ) `  M )  <_  ( ( 2  x.  N ) 
 ^ c  ( ( ( sqr `  (
 2  x.  N ) )  /  3 )  +  2 ) ) )
 
Theorembposlem6 20360* Lemma for bpos 20364. By using the various bounds at our disposal, arrive at an inequality that is false for  N large enough. (Contributed by Mario Carneiro, 14-Mar-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  5 )
 )   &    |-  ( ph  ->  -.  E. p  e.  Prime  ( N  <  p  /\  p  <_  ( 2  x.  N ) ) )   &    |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ ( n  pCnt  ( ( 2  x.  N )  _C  N ) ) ) ,  1 ) )   &    |-  K  =  ( |_ `  ( ( 2  x.  N )  /  3
 ) )   &    |-  M  =  ( |_ `  ( sqr `  ( 2  x.  N ) ) )   =>    |-  ( ph  ->  ( ( 4 ^ N )  /  N )  < 
 ( ( ( 2  x.  N )  ^ c  ( ( ( sqr `  ( 2  x.  N ) )  /  3
 )  +  2 ) )  x.  ( 2 
 ^ c  ( ( ( 4  x.  N )  /  3 )  -  5 ) ) ) )
 
Theorembposlem7 20361* Lemma for bpos 20364. The function  F is decreasing. (Contributed by Mario Carneiro, 13-Mar-2014.)
 |-  F  =  ( n  e.  NN  |->  ( ( ( ( sqr `  2
 )  x.  ( G `
  ( sqr `  n ) ) )  +  ( ( 9  / 
 4 )  x.  ( G `  ( n  / 
 2 ) ) ) )  +  ( ( log `  2 )  /  ( sqr `  (
 2  x.  n ) ) ) ) )   &    |-  G  =  ( x  e.  RR+  |->  ( ( log `  x )  /  x ) )   &    |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  ( _e ^ 2 )  <_  A )   &    |-  ( ph  ->  ( _e ^ 2 ) 
 <_  B )   =>    |-  ( ph  ->  ( A  <  B  ->  ( F `  B )  < 
 ( F `  A ) ) )
 
Theorembposlem8 20362 Lemma for bpos 20364. Evaluate  F ( 6 4 ) and show it is less than  log 2. (Contributed by Mario Carneiro, 14-Mar-2014.)
 |-  F  =  ( n  e.  NN  |->  ( ( ( ( sqr `  2
 )  x.  ( G `
  ( sqr `  n ) ) )  +  ( ( 9  / 
 4 )  x.  ( G `  ( n  / 
 2 ) ) ) )  +  ( ( log `  2 )  /  ( sqr `  (
 2  x.  n ) ) ) ) )   &    |-  G  =  ( x  e.  RR+  |->  ( ( log `  x )  /  x ) )   =>    |-  ( ( F ` ; 6 4 )  e.  RR  /\  ( F ` ; 6 4 )  < 
 ( log `  2 )
 )
 
Theorembposlem9 20363* Lemma for bpos 20364. Derive a contradiction. (Contributed by Mario Carneiro, 14-Mar-2014.)
 |-  F  =  ( n  e.  NN  |->  ( ( ( ( sqr `  2
 )  x.  ( G `
  ( sqr `  n ) ) )  +  ( ( 9  / 
 4 )  x.  ( G `  ( n  / 
 2 ) ) ) )  +  ( ( log `  2 )  /  ( sqr `  (
 2  x.  n ) ) ) ) )   &    |-  G  =  ( x  e.  RR+  |->  ( ( log `  x )  /  x ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  -> ; 6 4  <  N )   &    |-  ( ph  ->  -.  E. p  e.  Prime  ( N  <  p 
 /\  p  <_  (
 2  x.  N ) ) )   =>    |-  ( ph  ->  ps )
 
Theorembpos 20364* Bertrand's postulate: there is a prime between  N and  2 N for every positive integer  N. This proof follows Erdős's method, for the most part, but with some refinements due to Shigenori Tochiori to save us some calculations of large primes. See http://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate for an overview of the proof strategy. (Contributed by Mario Carneiro, 14-Mar-2014.)
 |-  ( N  e.  NN  ->  E. p  e.  Prime  ( N  <  p  /\  p  <_  ( 2  x.  N ) ) )
 
13.4.8  Legendre symbol
 
Syntaxclgs 20365 Extend class notation with the Legendre symbol function.
 class  / L
 
Definitiondf-lgs 20366* Define the Legendre symbol (actually the Kronecker symbol, which extends the Legendre symbol to all integers). (Contributed by Mario Carneiro, 4-Feb-2015.)
 |- 
 / L  =  ( a  e.  ZZ ,  n  e.  ZZ  |->  if ( n  =  0 ,  if ( ( a ^
 2 )  =  1 ,  1 ,  0 ) ,  ( if ( ( n  < 
 0  /\  a  <  0 ) ,  -u 1 ,  1 )  x.  (  seq  1 (  x.  ,  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( if ( m  =  2 ,  if ( 2 
 ||  a ,  0 ,  if ( ( a  mod  8 )  e.  { 1 ,  7 } ,  1 ,  -u 1 ) ) ,  ( ( ( ( a ^ (
 ( m  -  1
 )  /  2 )
 )  +  1 ) 
 mod  m )  -  1 ) ) ^
 ( m  pCnt  n ) ) ,  1 ) ) ) `  ( abs `  n )
 ) ) ) )
 
Theoremlgslem1 20367 When  a is coprime to the prime  p,  a ^
( ( p  - 
1 )  /  2
) is equivalent  mod  p to  1 or  -u
1, and so adding  1 makes it equivalent to  0 or  2. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
 ( ( A ^
 ( ( P  -  1 )  /  2
 ) )  +  1 )  mod  P )  e.  { 0 ,  2 } )
 
Theoremlgslem2 20368 The set  Z of all integers with absolute value at most 
1 contains  { -u 1 ,  0 ,  1 }. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  Z  =  { x  e.  ZZ  |  ( abs `  x )  <_  1 }   =>    |-  ( -u 1  e.  Z  /\  0  e.  Z  /\  1  e.  Z )
 
Theoremlgslem3 20369* The set  Z of all integers with absolute value at most 
1 is closed under multiplication. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  Z  =  { x  e.  ZZ  |  ( abs `  x )  <_  1 }   =>    |-  ( ( A  e.  Z  /\  B  e.  Z )  ->  ( A  x.  B )  e.  Z )
 
Theoremlgslem4 20370* The function  F is closed in integers with absolute value less than  1 (namely  { -u
1 ,  0 ,  1 } although this representation is less useful to us). (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  Z  =  { x  e.  ZZ  |  ( abs `  x )  <_  1 }   =>    |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  (
 ( ( ( A ^ ( ( P  -  1 )  / 
 2 ) )  +  1 )  mod  P )  -  1 )  e.  Z )
 
Theoremlgsval 20371* Value of the Legendre symbol at an arbitrary integer. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( if ( n  =  2 ,  if ( 2 
 ||  A ,  0 ,  if ( ( A  mod  8 )  e.  { 1 ,  7 } ,  1 ,  -u 1 ) ) ,  ( ( ( ( A ^ (
 ( n  -  1
 )  /  2 )
 )  +  1 ) 
 mod  n )  -  1 ) ) ^
 ( n  pCnt  N ) ) ,  1 ) )   =>    |-  ( ( A  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L N )  =  if ( N  =  0 ,  if ( ( A ^ 2 )  =  1 ,  1 ,  0 ) ,  ( if ( ( N  <  0 
 /\  A  <  0
 ) ,  -u 1 ,  1 )  x.  (  seq  1 (  x.  ,  F ) `
  ( abs `  N ) ) ) ) )
 
Theoremlgsfval 20372* Value of the function  F which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( if ( n  =  2 ,  if ( 2 
 ||  A ,  0 ,  if ( ( A  mod  8 )  e.  { 1 ,  7 } ,  1 ,  -u 1 ) ) ,  ( ( ( ( A ^ (
 ( n  -  1
 )  /  2 )
 )  +  1 ) 
 mod  n )  -  1 ) ) ^
 ( n  pCnt  N ) ) ,  1 ) )   =>    |-  ( M  e.  NN  ->  ( F `  M )  =  if ( M  e.  Prime ,  ( if ( M  =  2 ,  if ( 2 
 ||  A ,  0 ,  if ( ( A  mod  8 )  e.  { 1 ,  7 } ,  1 ,  -u 1 ) ) ,  ( ( ( ( A ^ (
 ( M  -  1
 )  /  2 )
 )  +  1 ) 
 mod  M )  -  1
 ) ) ^ ( M  pCnt  N ) ) ,  1 ) )
 
Theoremlgsfcl2 20373* The function  F is closed in integers with absolute value less than  1 (namely  { -u
1 ,  0 ,  1 } although this representation is less useful to us). (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( if ( n  =  2 ,  if ( 2 
 ||  A ,  0 ,  if ( ( A  mod  8 )  e.  { 1 ,  7 } ,  1 ,  -u 1 ) ) ,  ( ( ( ( A ^ (
 ( n  -  1
 )  /  2 )
 )  +  1 ) 
 mod  n )  -  1 ) ) ^
 ( n  pCnt  N ) ) ,  1 ) )   &    |-  Z  =  { x  e.  ZZ  |  ( abs `  x )  <_  1 }   =>    |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 ) 
 ->  F : NN --> Z )
 
Theoremlgscllem 20374* The Legendre symbol is an element of  Z. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( if ( n  =  2 ,  if ( 2 
 ||  A ,  0 ,  if ( ( A  mod  8 )  e.  { 1 ,  7 } ,  1 ,  -u 1 ) ) ,  ( ( ( ( A ^ (
 ( n  -  1
 )  /  2 )
 )  +  1 ) 
 mod  n )  -  1 ) ) ^
 ( n  pCnt  N ) ) ,  1 ) )   &    |-  Z  =  { x  e.  ZZ  |  ( abs `  x )  <_  1 }   =>    |-  ( ( A  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L N )  e.  Z )
 
Theoremlgsfcl 20375* Closure of the function  F which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( if ( n  =  2 ,  if ( 2 
 ||  A ,  0 ,  if ( ( A  mod  8 )  e.  { 1 ,  7 } ,  1 ,  -u 1 ) ) ,  ( ( ( ( A ^ (
 ( n  -  1
 )  /  2 )
 )  +  1 ) 
 mod  n )  -  1 ) ) ^
 ( n  pCnt  N ) ) ,  1 ) )   =>    |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 ) 
 ->  F : NN --> ZZ )
 
Theoremlgsfle1 20376* The function  F has magnitude less or equal to  1. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( if ( n  =  2 ,  if ( 2 
 ||  A ,  0 ,  if ( ( A  mod  8 )  e.  { 1 ,  7 } ,  1 ,  -u 1 ) ) ,  ( ( ( ( A ^ (
 ( n  -  1
 )  /  2 )
 )  +  1 ) 
 mod  n )  -  1 ) ) ^
 ( n  pCnt  N ) ) ,  1 ) )   =>    |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  M  e.  NN )  ->  ( abs `  ( F `  M ) )  <_  1 )
 
Theoremlgsval2lem 20377* Lemma for lgsval2 20383. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( if ( n  =  2 ,  if ( 2 
 ||  A ,  0 ,  if ( ( A  mod  8 )  e.  { 1 ,  7 } ,  1 ,  -u 1 ) ) ,  ( ( ( ( A ^ (
 ( n  -  1
 )  /  2 )
 )  +  1 ) 
 mod  n )  -  1 ) ) ^
 ( n  pCnt  N ) ) ,  1 ) )   =>    |-  ( ( A  e.  ZZ  /\  N  e.  Prime ) 
 ->  ( A  / L N )  =  if ( N  =  2 ,  if ( 2  ||  A ,  0 ,  if ( ( A  mod  8 )  e.  { 1 ,  7 } , 
 1 ,  -u 1
 ) ) ,  (
 ( ( ( A ^ ( ( N  -  1 )  / 
 2 ) )  +  1 )  mod  N )  -  1 ) ) )
 
Theoremlgsval4lem 20378* Lemma for lgsval4 20387. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( if ( n  =  2 ,  if ( 2 
 ||  A ,  0 ,  if ( ( A  mod  8 )  e.  { 1 ,  7 } ,  1 ,  -u 1 ) ) ,  ( ( ( ( A ^ (
 ( n  -  1
 )  /  2 )
 )  +  1 ) 
 mod  n )  -  1 ) ) ^
 ( n  pCnt  N ) ) ,  1 ) )   =>    |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 ) 
 ->  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  (
 ( A  / L n ) ^ ( n  pCnt  N ) ) ,  1 ) ) )
 
Theoremlgscl2 20379* The Legendre symbol is an integer with absolute value less or equal to  1. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  Z  =  { x  e.  ZZ  |  ( abs `  x )  <_  1 }   =>    |-  ( ( A  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L N )  e.  Z )
 
Theoremlgs0 20380 The Legendre symbol when the second argument is zero. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( A  e.  ZZ  ->  ( A  / L
 0 )  =  if ( ( A ^
 2 )  =  1 ,  1 ,  0 ) )
 
Theoremlgscl 20381 The Legendre symbol is an integer. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L N )  e.  ZZ )
 
Theoremlgsle1 20382 The Legendre symbol has absolute value less or equal to  1. Together with lgscl 20381 this implies that it takes values in  { -u 1 ,  0 ,  1 }. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  ( A  / L N ) )  <_  1 )
 
Theoremlgsval2 20383 The Legendre symbol at a prime (this is the traditional domain of the Legendre symbol, except for the addition of prime  2). (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  P  e.  Prime ) 
 ->  ( A  / L P )  =  if ( P  =  2 ,  if ( 2  ||  A ,  0 ,  if ( ( A  mod  8 )  e.  { 1 ,  7 } , 
 1 ,  -u 1
 ) ) ,  (
 ( ( ( A ^ ( ( P  -  1 )  / 
 2 ) )  +  1 )  mod  P )  -  1 ) ) )
 
Theoremlgs2 20384 The Legendre symbol at  2. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( A  e.  ZZ  ->  ( A  / L
 2 )  =  if ( 2  ||  A ,  0 ,  if ( ( A  mod  8 )  e.  { 1 ,  7 } , 
 1 ,  -u 1
 ) ) )
 
Theoremlgsval3 20385 The Legendre symbol at an odd prime (this is the traditional domain of the Legendre symbol). (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( A  / L P )  =  ( ( ( ( A ^ (
 ( P  -  1
 )  /  2 )
 )  +  1 ) 
 mod  P )  -  1
 ) )
 
Theoremlgsvalmod 20386 The Legendre symbol is equivalent to 
a ^ ( ( p  -  1 )  /  2 ),  mod  p. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  (
 ( A  / L P )  mod  P )  =  ( ( A ^ ( ( P  -  1 )  / 
 2 ) )  mod  P ) )
 
Theoremlgsval4 20387* Restate lgsval 20371 for nonzero  N, where the function  F has been abbreviated into a self-referential expression taking the value of  / L on the primes as given. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  (
 ( A  / L n ) ^ ( n  pCnt  N ) ) ,  1 ) )   =>    |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 ) 
 ->  ( A  / L N )  =  ( if ( ( N  <  0 
 /\  A  <  0
 ) ,  -u 1 ,  1 )  x.  (  seq  1 (  x.  ,  F ) `
  ( abs `  N ) ) ) )
 
Theoremlgsfcl3 20388* Closure of the function  F which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  (
 ( A  / L n ) ^ ( n  pCnt  N ) ) ,  1 ) )   =>    |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 ) 
 ->  F : NN --> ZZ )
 
Theoremlgsval4a 20389* Same as lgsval4 20387 for positive  N. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  (
 ( A  / L n ) ^ ( n  pCnt  N ) ) ,  1 ) )   =>    |-  ( ( A  e.  ZZ  /\  N  e.  NN )  ->  ( A  / L N )  =  ( 
 seq  1 (  x. 
 ,  F ) `  N ) )
 
Theoremlgsneg 20390 The Legendre symbol is either even or odd under negation with respect to the second parameter according to the sign of the first. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 ) 
 ->  ( A  / L -u N )  =  ( if ( A  <  0 ,  -u 1 ,  1 )  x.  ( A 
 / L N ) ) )
 
Theoremlgsneg1 20391 The Legendre symbol for nonnegative first parameter is unchanged by negation of the second. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( A  e.  NN0  /\  N  e.  ZZ )  ->  ( A  / L -u N )  =  ( A  / L N ) )
 
Theoremlgsmod 20392 The Legendre (Jacobi) symbol is preserved under reduction  mod  n when  n is odd. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\ 
 -.  2  ||  N )  ->  ( ( A 
 mod  N )  / L N )  =  ( A  / L N ) )
 
Theoremlgsdilem 20393 Lemma for lgsdi 20403 and lgsdir 20401: the sign part of the Legendre symbol is multiplicative. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  if ( ( N  <  0 
 /\  ( A  x.  B )  <  0 ) ,  -u 1 ,  1 )  =  ( if ( ( N  <  0 
 /\  A  <  0
 ) ,  -u 1 ,  1 )  x. 
 if ( ( N  <  0  /\  B  <  0 ) ,  -u 1 ,  1 ) ) )
 
Theoremlgsdir2lem1 20394 Lemma for lgsdir2 20399. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( ( 1 
 mod  8 )  =  1  /\  ( -u 1  mod  8 )  =  7 )  /\  (
 ( 3  mod  8
 )  =  3  /\  ( -u 3  mod  8
 )  =  5 ) )
 
Theoremlgsdir2lem2 20395 Lemma for lgsdir2 20399. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( K  e.  ZZ  /\  2  ||  ( K  +  1 )  /\  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  e.  ( 0 ...
 K )  ->  ( A  mod  8 )  e.  S ) ) )   &    |-  M  =  ( K  +  1 )   &    |-  N  =  ( M  +  1 )   &    |-  N  e.  S   =>    |-  ( N  e.  ZZ  /\  2  ||  ( N  +  1 )  /\  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( ( A  mod  8 )  e.  (
 0 ... N )  ->  ( A  mod  8 )  e.  S ) ) )
 
Theoremlgsdir2lem3 20396 Lemma for lgsdir2 20399. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  -.  2  ||  A )  ->  ( A 
 mod  8 )  e.  ( { 1 ,  7 }  u.  {
 3 ,  5 } ) )
 
Theoremlgsdir2lem4 20397 Lemma for lgsdir2 20399. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  e. 
 { 1 ,  7 } )  ->  (
 ( ( A  x.  B )  mod  8 )  e.  { 1 ,  7 }  <->  ( B  mod  8 )  e.  { 1 ,  7 } )
 )
 
Theoremlgsdir2lem5 20398 Lemma for lgsdir2 20399. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  (
 ( A  mod  8
 )  e.  { 3 ,  5 }  /\  ( B  mod  8 )  e.  { 3 ,  5 } ) ) 
 ->  ( ( A  x.  B )  mod  8 )  e.  { 1 ,  7 } )
 
Theoremlgsdir2 20399 The Legendre symbol is completely multiplicative at  2. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  x.  B )  / L 2 )  =  ( ( A  / L 2 )  x.  ( B  / L
 2 ) ) )
 
Theoremlgsdirprm 20400 The Legendre symbol is completely multiplicative at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  ->  ( ( A  x.  B )  / L P )  =  ( ( A  / L P )  x.  ( B  / L P ) ) )
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