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Theorem dchrisum0fval 24906
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 4576 . . . . 5 (𝑏 = 𝐴 → (𝑞𝑏𝑞𝐴))
21rabbidv 3158 . . . 4 (𝑏 = 𝐴 → {𝑞 ∈ ℕ ∣ 𝑞𝑏} = {𝑞 ∈ ℕ ∣ 𝑞𝐴})
32sumeq1d 14220 . . 3 (𝑏 = 𝐴 → Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)) = Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝐴} (𝑋‘(𝐿𝑣)))
4 fveq2 6083 . . . . 5 (𝑣 = 𝑡 → (𝐿𝑣) = (𝐿𝑡))
54fveq2d 6087 . . . 4 (𝑣 = 𝑡 → (𝑋‘(𝐿𝑣)) = (𝑋‘(𝐿𝑡)))
65cbvsumv 14215 . . 3 Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝐴} (𝑋‘(𝐿𝑣)) = Σ𝑡 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝐴} (𝑋‘(𝐿𝑡))
73, 6syl6eq 2654 . 2 (𝑏 = 𝐴 → Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)) = Σ𝑡 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝐴} (𝑋‘(𝐿𝑡)))
8 dchrisum0f.f . 2 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)))
9 sumex 14207 . 2 Σ𝑡 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝐴} (𝑋‘(𝐿𝑡)) ∈ V
107, 8, 9fvmpt 6171 1 (𝐴 ∈ ℕ → (𝐹𝐴) = Σ𝑡 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝐴} (𝑋‘(𝐿𝑡)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  {crab 2894   class class class wbr 4572  cmpt 4632  cfv 5785  cn 10862  Σcsu 14205  cdvds 14762  Basecbs 15636  0gc0g 15864  ℤRHomczrh 19607  ℤ/nczn 19610  DChrcdchr 24669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pr 4823
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-iota 5749  df-fun 5787  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-wrecs 7266  df-recs 7327  df-rdg 7365  df-seq 12614  df-sum 14206
This theorem is referenced by:  dchrisum0fmul  24907  dchrisum0flblem1  24909  dchrisum0  24921
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