MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dchrisum0fval Structured version   Unicode version

Theorem dchrisum0fval 21191
Description: Value of the function  F, the divisor sum of a Dirichlet character. (Contributed by Mario Carneiro, 5-May-2016.)
Hypotheses
Ref Expression
rpvmasum.z  |-  Z  =  (ℤ/n `  N )
rpvmasum.l  |-  L  =  ( ZRHom `  Z
)
rpvmasum.a  |-  ( ph  ->  N  e.  NN )
rpvmasum2.g  |-  G  =  (DChr `  N )
rpvmasum2.d  |-  D  =  ( Base `  G
)
rpvmasum2.1  |-  .1.  =  ( 0g `  G )
dchrisum0f.f  |-  F  =  ( b  e.  NN  |->  sum_ v  e.  { q  e.  NN  |  q 
||  b }  ( X `  ( L `  v ) ) )
Assertion
Ref Expression
dchrisum0fval  |-  ( A  e.  NN  ->  ( F `  A )  =  sum_ t  e.  {
q  e.  NN  | 
q  ||  A } 
( X `  ( L `  t )
) )
Distinct variable groups:    t,  .1.    t, F    q, b, t, v, A    N, q,
t    ph, t    t, D    L, b, t, v    X, b, t, v
Allowed substitution hints:    ph( v, q, b)    D( v, q, b)    .1. ( v, q, b)    F( v, q, b)    G( v, t, q, b)    L( q)    N( v, b)    X( q)    Z( v, t, q, b)

Proof of Theorem dchrisum0fval
StepHypRef Expression
1 breq2 4208 . . . . 5  |-  ( b  =  A  ->  (
q  ||  b  <->  q  ||  A ) )
21rabbidv 2940 . . . 4  |-  ( b  =  A  ->  { q  e.  NN  |  q 
||  b }  =  { q  e.  NN  |  q  ||  A }
)
32sumeq1d 12487 . . 3  |-  ( b  =  A  ->  sum_ v  e.  { q  e.  NN  |  q  ||  b }  ( X `  ( L `  v )
)  =  sum_ v  e.  { q  e.  NN  |  q  ||  A } 
( X `  ( L `  v )
) )
4 fveq2 5720 . . . . 5  |-  ( v  =  t  ->  ( L `  v )  =  ( L `  t ) )
54fveq2d 5724 . . . 4  |-  ( v  =  t  ->  ( X `  ( L `  v ) )  =  ( X `  ( L `  t )
) )
65cbvsumv 12482 . . 3  |-  sum_ v  e.  { q  e.  NN  |  q  ||  A } 
( X `  ( L `  v )
)  =  sum_ t  e.  { q  e.  NN  |  q  ||  A } 
( X `  ( L `  t )
)
73, 6syl6eq 2483 . 2  |-  ( b  =  A  ->  sum_ v  e.  { q  e.  NN  |  q  ||  b }  ( X `  ( L `  v )
)  =  sum_ t  e.  { q  e.  NN  |  q  ||  A } 
( X `  ( L `  t )
) )
8 dchrisum0f.f . 2  |-  F  =  ( b  e.  NN  |->  sum_ v  e.  { q  e.  NN  |  q 
||  b }  ( X `  ( L `  v ) ) )
9 sumex 12473 . 2  |-  sum_ t  e.  { q  e.  NN  |  q  ||  A } 
( X `  ( L `  t )
)  e.  _V
107, 8, 9fvmpt 5798 1  |-  ( A  e.  NN  ->  ( F `  A )  =  sum_ t  e.  {
q  e.  NN  | 
q  ||  A } 
( X `  ( L `  t )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {crab 2701   class class class wbr 4204    e. cmpt 4258   ` cfv 5446   NNcn 9992   sum_csu 12471    || cdivides 12844   Basecbs 13461   0gc0g 13715   ZRHomczrh 16770  ℤ/nczn 16773  DChrcdchr 21008
This theorem is referenced by:  dchrisum0fmul  21192  dchrisum0flblem1  21194  dchrisum0  21206
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-seq 11316  df-sum 12472
  Copyright terms: Public domain W3C validator