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Theorem dchrisum0fval 26869
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 = (β„€/nβ„€β€˜π‘)
rpvmasum.l 𝐿 = (β„€RHomβ€˜π‘)
rpvmasum.a (πœ‘ β†’ 𝑁 ∈ β„•)
rpvmasum2.g 𝐺 = (DChrβ€˜π‘)
rpvmasum2.d 𝐷 = (Baseβ€˜πΊ)
rpvmasum2.1 1 = (0gβ€˜πΊ)
dchrisum0f.f 𝐹 = (𝑏 ∈ β„• ↦ Σ𝑣 ∈ {π‘ž ∈ β„• ∣ π‘ž βˆ₯ 𝑏} (π‘‹β€˜(πΏβ€˜π‘£)))
Assertion
Ref Expression
dchrisum0fval (𝐴 ∈ β„• β†’ (πΉβ€˜π΄) = Σ𝑑 ∈ {π‘ž ∈ β„• ∣ π‘ž βˆ₯ 𝐴} (π‘‹β€˜(πΏβ€˜π‘‘)))
Distinct variable groups:   𝑑, 1   𝑑,𝐹   π‘ž,𝑏,𝑑,𝑣,𝐴   𝑁,π‘ž,𝑑   πœ‘,𝑑   𝑑,𝐷   𝐿,𝑏,𝑑,𝑣   𝑋,𝑏,𝑑,𝑣
Allowed substitution hints:   πœ‘(𝑣,π‘ž,𝑏)   𝐷(𝑣,π‘ž,𝑏)   1 (𝑣,π‘ž,𝑏)   𝐹(𝑣,π‘ž,𝑏)   𝐺(𝑣,𝑑,π‘ž,𝑏)   𝐿(π‘ž)   𝑁(𝑣,𝑏)   𝑋(π‘ž)   𝑍(𝑣,𝑑,π‘ž,𝑏)

Proof of Theorem dchrisum0fval
StepHypRef Expression
1 breq2 5114 . . . . 5 (𝑏 = 𝐴 β†’ (π‘ž βˆ₯ 𝑏 ↔ π‘ž βˆ₯ 𝐴))
21rabbidv 3418 . . . 4 (𝑏 = 𝐴 β†’ {π‘ž ∈ β„• ∣ π‘ž βˆ₯ 𝑏} = {π‘ž ∈ β„• ∣ π‘ž βˆ₯ 𝐴})
32sumeq1d 15593 . . 3 (𝑏 = 𝐴 β†’ Σ𝑣 ∈ {π‘ž ∈ β„• ∣ π‘ž βˆ₯ 𝑏} (π‘‹β€˜(πΏβ€˜π‘£)) = Σ𝑣 ∈ {π‘ž ∈ β„• ∣ π‘ž βˆ₯ 𝐴} (π‘‹β€˜(πΏβ€˜π‘£)))
4 2fveq3 6852 . . . 4 (𝑣 = 𝑑 β†’ (π‘‹β€˜(πΏβ€˜π‘£)) = (π‘‹β€˜(πΏβ€˜π‘‘)))
54cbvsumv 15588 . . 3 Σ𝑣 ∈ {π‘ž ∈ β„• ∣ π‘ž βˆ₯ 𝐴} (π‘‹β€˜(πΏβ€˜π‘£)) = Σ𝑑 ∈ {π‘ž ∈ β„• ∣ π‘ž βˆ₯ 𝐴} (π‘‹β€˜(πΏβ€˜π‘‘))
63, 5eqtrdi 2793 . 2 (𝑏 = 𝐴 β†’ Σ𝑣 ∈ {π‘ž ∈ β„• ∣ π‘ž βˆ₯ 𝑏} (π‘‹β€˜(πΏβ€˜π‘£)) = Σ𝑑 ∈ {π‘ž ∈ β„• ∣ π‘ž βˆ₯ 𝐴} (π‘‹β€˜(πΏβ€˜π‘‘)))
7 dchrisum0f.f . 2 𝐹 = (𝑏 ∈ β„• ↦ Σ𝑣 ∈ {π‘ž ∈ β„• ∣ π‘ž βˆ₯ 𝑏} (π‘‹β€˜(πΏβ€˜π‘£)))
8 sumex 15579 . 2 Σ𝑑 ∈ {π‘ž ∈ β„• ∣ π‘ž βˆ₯ 𝐴} (π‘‹β€˜(πΏβ€˜π‘‘)) ∈ V
96, 7, 8fvmpt 6953 1 (𝐴 ∈ β„• β†’ (πΉβ€˜π΄) = Σ𝑑 ∈ {π‘ž ∈ β„• ∣ π‘ž βˆ₯ 𝐴} (π‘‹β€˜(πΏβ€˜π‘‘)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  {crab 3410   class class class wbr 5110   ↦ cmpt 5193  β€˜cfv 6501  β„•cn 12160  Ξ£csu 15577   βˆ₯ cdvds 16143  Basecbs 17090  0gc0g 17328  β„€RHomczrh 20916  β„€/nβ„€czn 20919  DChrcdchr 26596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-iota 6453  df-fun 6503  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-seq 13914  df-sum 15578
This theorem is referenced by:  dchrisum0fmul  26870  dchrisum0flblem1  26872  dchrisum0  26884
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