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Theorem dchrisum0fval 23807
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 4371 . . . . 5  |-  ( b  =  A  ->  (
q  ||  b  <->  q  ||  A ) )
21rabbidv 3026 . . . 4  |-  ( b  =  A  ->  { q  e.  NN  |  q 
||  b }  =  { q  e.  NN  |  q  ||  A }
)
32sumeq1d 13525 . . 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 5774 . . . . 5  |-  ( v  =  t  ->  ( L `  v )  =  ( L `  t ) )
54fveq2d 5778 . . . 4  |-  ( v  =  t  ->  ( X `  ( L `  v ) )  =  ( X `  ( L `  t )
) )
65cbvsumv 13520 . . 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 2439 . 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 13512 . 2  |-  sum_ t  e.  { q  e.  NN  |  q  ||  A } 
( X `  ( L `  t )
)  e.  _V
107, 8, 9fvmpt 5857 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 1399    e. wcel 1826   {crab 2736   class class class wbr 4367    |-> cmpt 4425   ` cfv 5496   NNcn 10452   sum_csu 13510    || cdvds 13988   Basecbs 14634   0gc0g 14847   ZRHomczrh 18630  ℤ/nczn 18633  DChrcdchr 23624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-recs 6960  df-rdg 6994  df-seq 12011  df-sum 13511
This theorem is referenced by:  dchrisum0fmul  23808  dchrisum0flblem1  23810  dchrisum0  23822
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