Theorem dchrisum0fval 25543

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	⊢ (𝜑 → 𝑁 ∈ ℕ)
rpvmasum2.g	⊢ 𝐺 = (DChr‘𝑁)
rpvmasum2.d	⊢ 𝐷 = (Base‘𝐺)
rpvmasum2.1	⊢ 1 = (0g‘𝐺)
dchrisum0f.f	⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣)))

Assertion

Ref	Expression
dchrisum0fval	⊢ (𝐴 ∈ ℕ → (𝐹‘𝐴) = Σ𝑡 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} (𝑋‘(𝐿‘𝑡)))

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

Step	Hyp	Ref	Expression
1		breq2 4846	. . . . 5 ⊢ (𝑏 = 𝐴 → (𝑞 ∥ 𝑏 ↔ 𝑞 ∥ 𝐴))
2	1	rabbidv 3372	. . . 4 ⊢ (𝑏 = 𝐴 → {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} = {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴})
3	2	sumeq1d 14769	. . 3 ⊢ (𝑏 = 𝐴 → Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣)) = Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} (𝑋‘(𝐿‘𝑣)))
4		2fveq3 6415	. . . 4 ⊢ (𝑣 = 𝑡 → (𝑋‘(𝐿‘𝑣)) = (𝑋‘(𝐿‘𝑡)))
5	4	cbvsumv 14764	. . 3 ⊢ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} (𝑋‘(𝐿‘𝑣)) = Σ𝑡 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} (𝑋‘(𝐿‘𝑡))
6	3, 5	syl6eq 2848	. 2 ⊢ (𝑏 = 𝐴 → Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣)) = Σ𝑡 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} (𝑋‘(𝐿‘𝑡)))
7		dchrisum0f.f	. 2 ⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣)))
8		sumex 14756	. 2 ⊢ Σ𝑡 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} (𝑋‘(𝐿‘𝑡)) ∈ V
9	6, 7, 8	fvmpt 6506	1 ⊢ (𝐴 ∈ ℕ → (𝐹‘𝐴) = Σ𝑡 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} (𝑋‘(𝐿‘𝑡)))

Syntax hints:   → wi 4   = wceq 1653   ∈ wcel 2157  {crab 3092   class class class wbr 4842   ↦ cmpt 4921  ‘cfv 6100  ℕcn 11311  Σcsu 14754   ∥ cdvds 15316  Basecbs 16181  0_gc0g 16412  ℤRHomczrh 20167  ℤ/nℤczn 20170  DChrcdchr 25306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-sep 4974  ax-nul 4982  ax-pr 5096

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3386  df-sbc 3633  df-csb 3728  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4628  df-br 4843  df-opab 4905  df-mpt 4922  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5897  df-iota 6063  df-fun 6102  df-f 6104  df-f1 6105  df-fo 6106  df-f1o 6107  df-fv 6108  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-wrecs 7644  df-recs 7706  df-rdg 7744  df-seq 13053  df-sum 14755

This theorem is referenced by:  dchrisum0fmul  25544  dchrisum0flblem1  25546  dchrisum0  25558