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Theorem dchrisum0fval 22888
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 4405 . . . . 5  |-  ( b  =  A  ->  (
q  ||  b  <->  q  ||  A ) )
21rabbidv 3070 . . . 4  |-  ( b  =  A  ->  { q  e.  NN  |  q 
||  b }  =  { q  e.  NN  |  q  ||  A }
)
32sumeq1d 13297 . . 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 5800 . . . . 5  |-  ( v  =  t  ->  ( L `  v )  =  ( L `  t ) )
54fveq2d 5804 . . . 4  |-  ( v  =  t  ->  ( X `  ( L `  v ) )  =  ( X `  ( L `  t )
) )
65cbvsumv 13292 . . 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 2511 . 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 13284 . 2  |-  sum_ t  e.  { q  e.  NN  |  q  ||  A } 
( X `  ( L `  t )
)  e.  _V
107, 8, 9fvmpt 5884 1  |-  ( A  e.  NN  ->  ( F `  A )  =  sum_ t  e.  {
q  e.  NN  | 
q  ||  A } 
( X `  ( L `  t )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   {crab 2803   class class class wbr 4401    |-> cmpt 4459   ` cfv 5527   NNcn 10434   sum_csu 13282    || cdivides 13654   Basecbs 14293   0gc0g 14498   ZRHomczrh 18057  ℤ/nczn 18060  DChrcdchr 22705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-recs 6943  df-rdg 6977  df-seq 11925  df-sum 13283
This theorem is referenced by:  dchrisum0fmul  22889  dchrisum0flblem1  22891  dchrisum0  22903
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