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Theorem dchrisum0fval 20654
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 4027 . . . . 5  |-  ( b  =  A  ->  (
q  ||  b  <->  q  ||  A ) )
21rabbidv 2780 . . . 4  |-  ( b  =  A  ->  { q  e.  NN  |  q 
||  b }  =  { q  e.  NN  |  q  ||  A }
)
32sumeq1d 12174 . . 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 5525 . . . . 5  |-  ( v  =  t  ->  ( L `  v )  =  ( L `  t ) )
54fveq2d 5529 . . . 4  |-  ( v  =  t  ->  ( X `  ( L `  v ) )  =  ( X `  ( L `  t )
) )
65cbvsumv 12169 . . 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 2331 . 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 12160 . 2  |-  sum_ t  e.  { q  e.  NN  |  q  ||  A } 
( X `  ( L `  t )
)  e.  _V
107, 8, 9fvmpt 5602 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 1623    e. wcel 1684   {crab 2547   class class class wbr 4023    e. cmpt 4077   ` cfv 5255   NNcn 9746   sum_csu 12158    || cdivides 12531   Basecbs 13148   0gc0g 13400   ZRHomczrh 16451  ℤ/nczn 16454  DChrcdchr 20471
This theorem is referenced by:  dchrisum0fmul  20655  dchrisum0flblem1  20657  dchrisum0  20669
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-seq 11047  df-sum 12159
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