Theorem dchrisum0fval 25194

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/nℤ ` 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

Step	Hyp	Ref	Expression
1		breq2 4657	. . . . 5 |- ( b = A -> ( q || b <-> q || A ) )
2	1	rabbidv 3189	. . . 4 |- ( b = A -> { q e. NN | q || b } = { q e. NN | q || A } )
3	2	sumeq1d 14431	. . 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 6191	. . . . 5 |- ( v = t -> ( L ` v ) = ( L ` t ) )
5	4	fveq2d 6195	. . . 4 |- ( v = t -> ( X ` ( L ` v ) ) = ( X ` ( L ` t ) ) )
6	5	cbvsumv 14426	. . 3 |- sum_ v e. { q e. NN | q || A } ( X ` ( L ` v ) ) = sum_ t e. { q e. NN | q || A } ( X ` ( L ` t ) )
7	3, 6	syl6eq 2672	. 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 14418	. 2 |- sum_ t e. { q e. NN | q || A } ( X ` ( L ` t ) ) e. _V
10	7, 8, 9	fvmpt 6282	1 |- ( A e. NN -> ( F ` A ) = sum_ t e. { q e. NN | q || A } ( X ` ( L ` t ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888   NNcn 11020   sum_csu 14416   || cdvds 14983   Basecbs 15857   0gc0g 16100   ZRHomczrh 19848  ℤ/nℤczn 19851  DChrcdchr 24957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fun 5890  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seq 12802  df-sum 14417

This theorem is referenced by:  dchrisum0fmul  25195  dchrisum0flblem1  25197  dchrisum0  25209