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Theorem dchrisum0fval 20670
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 4043 . . . . 5  |-  ( b  =  A  ->  (
q  ||  b  <->  q  ||  A ) )
21rabbidv 2793 . . . 4  |-  ( b  =  A  ->  { q  e.  NN  |  q 
||  b }  =  { q  e.  NN  |  q  ||  A }
)
32sumeq1d 12190 . . 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 5541 . . . . 5  |-  ( v  =  t  ->  ( L `  v )  =  ( L `  t ) )
54fveq2d 5545 . . . 4  |-  ( v  =  t  ->  ( X `  ( L `  v ) )  =  ( X `  ( L `  t )
) )
65cbvsumv 12185 . . 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 2344 . 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 12176 . 2  |-  sum_ t  e.  { q  e.  NN  |  q  ||  A } 
( X `  ( L `  t )
)  e.  _V
107, 8, 9fvmpt 5618 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 1632    e. wcel 1696   {crab 2560   class class class wbr 4039    e. cmpt 4093   ` cfv 5271   NNcn 9762   sum_csu 12174    || cdivides 12547   Basecbs 13164   0gc0g 13416   ZRHomczrh 16467  ℤ/nczn 16470  DChrcdchr 20487
This theorem is referenced by:  dchrisum0fmul  20671  dchrisum0flblem1  20673  dchrisum0  20685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-seq 11063  df-sum 12175
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