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Theorem dchrisum0fval 26935
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 (𝜑𝑁 ∈ ℕ)
rpvmasum2.g 𝐺 = (DChr‘𝑁)
rpvmasum2.d 𝐷 = (Base‘𝐺)
rpvmasum2.1 1 = (0g𝐺)
dchrisum0f.f 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)))
Assertion
Ref Expression
dchrisum0fval (𝐴 ∈ ℕ → (𝐹𝐴) = Σ𝑡 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝐴} (𝑋‘(𝐿𝑡)))
Distinct variable groups:   𝑡, 1   𝑡,𝐹   𝑞,𝑏,𝑡,𝑣,𝐴   𝑁,𝑞,𝑡   𝜑,𝑡   𝑡,𝐷   𝐿,𝑏,𝑡,𝑣   𝑋,𝑏,𝑡,𝑣
Allowed substitution hints:   𝜑(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 5145 . . . . 5 (𝑏 = 𝐴 → (𝑞𝑏𝑞𝐴))
21rabbidv 3439 . . . 4 (𝑏 = 𝐴 → {𝑞 ∈ ℕ ∣ 𝑞𝑏} = {𝑞 ∈ ℕ ∣ 𝑞𝐴})
32sumeq1d 15629 . . 3 (𝑏 = 𝐴 → Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)) = Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝐴} (𝑋‘(𝐿𝑣)))
4 2fveq3 6883 . . . 4 (𝑣 = 𝑡 → (𝑋‘(𝐿𝑣)) = (𝑋‘(𝐿𝑡)))
54cbvsumv 15624 . . 3 Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝐴} (𝑋‘(𝐿𝑣)) = Σ𝑡 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝐴} (𝑋‘(𝐿𝑡))
63, 5eqtrdi 2787 . 2 (𝑏 = 𝐴 → Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)) = Σ𝑡 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝐴} (𝑋‘(𝐿𝑡)))
7 dchrisum0f.f . 2 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)))
8 sumex 15616 . 2 Σ𝑡 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝐴} (𝑋‘(𝐿𝑡)) ∈ V
96, 7, 8fvmpt 6984 1 (𝐴 ∈ ℕ → (𝐹𝐴) = Σ𝑡 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝐴} (𝑋‘(𝐿𝑡)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  {crab 3431   class class class wbr 5141  cmpt 5224  cfv 6532  cn 12194  Σcsu 15614  cdvds 16179  Basecbs 17126  0gc0g 17367  ℤRHomczrh 20982  ℤ/nczn 20985  DChrcdchr 26662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-iota 6484  df-fun 6534  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-seq 13949  df-sum 15615
This theorem is referenced by:  dchrisum0fmul  26936  dchrisum0flblem1  26938  dchrisum0  26950
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