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Theorem rpvmasum2 27450

Description: A partial result along the lines of rpvmasum 27464. The sum of the von Mangoldt function over those integers 𝑛≡𝐴 (mod 𝑁) is asymptotic to (1 − 𝑀)(log𝑥 / ϕ(𝑥)) + 𝑂(1), where 𝑀 is the number of non-principal Dirichlet characters with Σ𝑛 ∈ ℕ, 𝑋(𝑛) / 𝑛 = 0. Our goal is to show this set is empty. Equation 9.4.3 of [Shapiro], p. 375. (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 = (0_g‘𝐺)
rpvmasum2.w	⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
rpvmasum2.u	⊢ 𝑈 = (Unit‘𝑍)
rpvmasum2.b	⊢ (𝜑 → 𝐴 ∈ 𝑈)
rpvmasum2.t	⊢ 𝑇 = (^◡𝐿 “ {𝐴})
rpvmasum2.z1	⊢ ((𝜑 ∧ 𝑓 ∈ 𝑊) → 𝐴 = (1_r‘𝑍))

**Assertion**
Ref	Expression
rpvmasum2	⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊))))) ∈ 𝑂(1))

Distinct variable groups: 𝑚,𝑛,𝑥,𝑦,𝑓, 1 𝐴,𝑓,𝑚,𝑥,𝑦 𝑓,𝐺 𝑓,𝑁,𝑚,𝑛,𝑥,𝑦 𝜑,𝑓,𝑚,𝑛,𝑥 𝑇,𝑚,𝑛,𝑥,𝑦 𝑈,𝑚,𝑛,𝑥 𝑓,𝑊,𝑥 𝑓,𝑍,𝑚,𝑛,𝑥,𝑦 𝐷,𝑓,𝑚,𝑛,𝑥,𝑦 𝑓,𝐿,𝑚,𝑛,𝑥,𝑦 𝐴,𝑛 Allowed substitution hints: 𝜑(𝑦) 𝑇(𝑓) 𝑈(𝑦,𝑓) 𝐺(𝑥,𝑦,𝑚,𝑛) 𝑊(𝑦,𝑚,𝑛)

Proof of Theorem rpvmasum2 Dummy variables 𝑐 𝑡 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpvmasum.a	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑁 ∈ ℕ)
3		rpvmasum2.g	. . . . . . 7 ⊢ 𝐺 = (DChr‘𝑁)
4		rpvmasum2.d	. . . . . . 7 ⊢ 𝐷 = (Base‘𝐺)
5	3, 4	dchrfi 27193	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin)
6	2, 5	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐷 ∈ Fin)
7		fzfid 13880	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → (1...(⌊‘𝑥)) ∈ Fin)
8		rpvmasum.z	. . . . . . . . . . . . 13 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
9		eqid 2731	. . . . . . . . . . . . 13 ⊢ (Base‘𝑍) = (Base‘𝑍)
10		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝑓 ∈ 𝐷)
11	3, 8, 4, 9, 10	dchrf 27180	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝑓:(Base‘𝑍)⟶ℂ)
12		rpvmasum2.u	. . . . . . . . . . . . . . 15 ⊢ 𝑈 = (Unit‘𝑍)
13	9, 12	unitss 20294	. . . . . . . . . . . . . 14 ⊢ 𝑈 ⊆ (Base‘𝑍)
14		rpvmasum2.b	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
15	13, 14	sselid 3927	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑍))
16	15	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝐴 ∈ (Base‘𝑍))
17	11, 16	ffvelcdmd 7018	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → (𝑓‘𝐴) ∈ ℂ)
18	17	cjcld 15103	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → (∗‘(𝑓‘𝐴)) ∈ ℂ)
19	18	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → (∗‘(𝑓‘𝐴)) ∈ ℂ)
20	19	adantrl 716	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑓 ∈ 𝐷)) → (∗‘(𝑓‘𝐴)) ∈ ℂ)
21	11	ad4ant14 752	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑓 ∈ 𝐷) → 𝑓:(Base‘𝑍)⟶ℂ)
22	1	nnnn0d 12442	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
23		rpvmasum.l	. . . . . . . . . . . . . . . 16 ⊢ 𝐿 = (ℤRHom‘𝑍)
24	8, 9, 23	znzrhfo 21484	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑍))
25		fof 6735	. . . . . . . . . . . . . . 15 ⊢ (𝐿:ℤ–onto→(Base‘𝑍) → 𝐿:ℤ⟶(Base‘𝑍))
26	22, 24, 25	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍))
27	26	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐿:ℤ⟶(Base‘𝑍))
28		elfzelz 13424	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℤ)
29		ffvelcdm 7014	. . . . . . . . . . . . 13 ⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑛 ∈ ℤ) → (𝐿‘𝑛) ∈ (Base‘𝑍))
30	27, 28, 29	syl2an 596	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐿‘𝑛) ∈ (Base‘𝑍))
31	30	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑓 ∈ 𝐷) → (𝐿‘𝑛) ∈ (Base‘𝑍))
32	21, 31	ffvelcdmd 7018	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑓 ∈ 𝐷) → (𝑓‘(𝐿‘𝑛)) ∈ ℂ)
33	32	anasss 466	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑓 ∈ 𝐷)) → (𝑓‘(𝐿‘𝑛)) ∈ ℂ)
34		elfznn 13453	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
35	34	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
36		vmacl 27055	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
37	35, 36	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
38	37, 35	nndivred 12179	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
39	38	recnd 11140	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
40	39	adantrr 717	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑓 ∈ 𝐷)) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
41	33, 40	mulcld 11132	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑓 ∈ 𝐷)) → ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
42	20, 41	mulcld 11132	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑓 ∈ 𝐷)) → ((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) ∈ ℂ)
43	42	anass1rs 655	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) ∈ ℂ)
44	7, 43	fsumcl 15640	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → Σ𝑛 ∈ (1...(⌊‘𝑥))((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) ∈ ℂ)
45		relogcl 26511	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
46	45	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
47	46	recnd 11140	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
48	47	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → (log‘𝑥) ∈ ℂ)
49		ax-1cn 11064	. . . . . . 7 ⊢ 1 ∈ ℂ
50		neg1cn 12110	. . . . . . . 8 ⊢ -1 ∈ ℂ
51		0cn 11104	. . . . . . . 8 ⊢ 0 ∈ ℂ
52	50, 51	ifcli 4520	. . . . . . 7 ⊢ if(𝑓 ∈ 𝑊, -1, 0) ∈ ℂ
53	49, 52	ifcli 4520	. . . . . 6 ⊢ if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)) ∈ ℂ
54		mulcl 11090	. . . . . 6 ⊢ (((log‘𝑥) ∈ ℂ ∧ if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)) ∈ ℂ) → ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))) ∈ ℂ)
55	48, 53, 54	sylancl 586	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))) ∈ ℂ)
56	6, 44, 55	fsumsub 15695	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑓 ∈ 𝐷 (Σ𝑛 ∈ (1...(⌊‘𝑥))((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))) = (Σ𝑓 ∈ 𝐷 Σ𝑛 ∈ (1...(⌊‘𝑥))((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) − Σ𝑓 ∈ 𝐷 ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))))
57	41	anass1rs 655	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
58	7, 57	fsumcl 15640	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
59	19, 58, 55	subdid 11573	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → ((∗‘(𝑓‘𝐴)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))))) = (((∗‘(𝑓‘𝐴)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) − ((∗‘(𝑓‘𝐴)) · ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))))))
60	7, 19, 57	fsummulc2 15691	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → ((∗‘(𝑓‘𝐴)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))))
61	53	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)) ∈ ℂ)
62	19, 48, 61	mul12d 11322	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → ((∗‘(𝑓‘𝐴)) · ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))) = ((log‘𝑥) · ((∗‘(𝑓‘𝐴)) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))))
63		ovif2 7445	. . . . . . . . . 10 ⊢ ((∗‘(𝑓‘𝐴)) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))) = if(𝑓 = 1 , ((∗‘(𝑓‘𝐴)) · 1), ((∗‘(𝑓‘𝐴)) · if(𝑓 ∈ 𝑊, -1, 0)))
64		fveq1 6821	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 1 → (𝑓‘𝐴) = ( 1 ‘𝐴))
65		rpvmasum2.1	. . . . . . . . . . . . . . . . 17 ⊢ 1 = (0_g‘𝐺)
66	1	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → 𝑁 ∈ ℕ)
67	14	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → 𝐴 ∈ 𝑈)
68	3, 8, 65, 12, 66, 67	dchr1 27195	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → ( 1 ‘𝐴) = 1)
69	64, 68	sylan9eqr 2788	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 = 1 ) → (𝑓‘𝐴) = 1)
70	69	fveq2d 6826	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 = 1 ) → (∗‘(𝑓‘𝐴)) = (∗‘1))
71		1re 11112	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
72		cjre 15046	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℝ → (∗‘1) = 1)
73	71, 72	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (∗‘1) = 1
74	70, 73	eqtrdi 2782	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 = 1 ) → (∗‘(𝑓‘𝐴)) = 1)
75	74	oveq1d 7361	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 = 1 ) → ((∗‘(𝑓‘𝐴)) · 1) = (1 · 1))
76		1t1e1 12282	. . . . . . . . . . . 12 ⊢ (1 · 1) = 1
77	75, 76	eqtrdi 2782	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 = 1 ) → ((∗‘(𝑓‘𝐴)) · 1) = 1)
78		df-ne 2929	. . . . . . . . . . . 12 ⊢ (𝑓 ≠ 1 ↔ ¬ 𝑓 = 1 )
79		ovif2 7445	. . . . . . . . . . . . 13 ⊢ ((∗‘(𝑓‘𝐴)) · if(𝑓 ∈ 𝑊, -1, 0)) = if(𝑓 ∈ 𝑊, ((∗‘(𝑓‘𝐴)) · -1), ((∗‘(𝑓‘𝐴)) · 0))
80		rpvmasum2.z1	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑊) → 𝐴 = (1_r‘𝑍))
81	80	fveq2d 6826	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑊) → (𝑓‘𝐴) = (𝑓‘(1_r‘𝑍)))
82	81	ad5ant15 758	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ 𝑓 ∈ 𝑊) → (𝑓‘𝐴) = (𝑓‘(1_r‘𝑍)))
83	3, 8, 4	dchrmhm 27179	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
84		simpr 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → 𝑓 ∈ 𝐷)
85	83, 84	sselid 3927	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → 𝑓 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
86		eqid 2731	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
87		eqid 2731	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
88	86, 87	ringidval 20101	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1_r‘𝑍) = (0_g‘(mulGrp‘𝑍))
89		eqid 2731	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
90		cnfld1 21330	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 = (1_r‘ℂ_fld)
91	89, 90	ringidval 20101	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
92	88, 91	mhm0 18702	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) → (𝑓‘(1_r‘𝑍)) = 1)
93	85, 92	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → (𝑓‘(1_r‘𝑍)) = 1)
94	93	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ 𝑓 ∈ 𝑊) → (𝑓‘(1_r‘𝑍)) = 1)
95	82, 94	eqtrd 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ 𝑓 ∈ 𝑊) → (𝑓‘𝐴) = 1)
96	95	fveq2d 6826	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ 𝑓 ∈ 𝑊) → (∗‘(𝑓‘𝐴)) = (∗‘1))
97	96, 73	eqtrdi 2782	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ 𝑓 ∈ 𝑊) → (∗‘(𝑓‘𝐴)) = 1)
98	97	oveq1d 7361	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ 𝑓 ∈ 𝑊) → ((∗‘(𝑓‘𝐴)) · -1) = (1 · -1))
99	50	mullidi 11117	. . . . . . . . . . . . . . . 16 ⊢ (1 · -1) = -1
100	98, 99	eqtrdi 2782	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ 𝑓 ∈ 𝑊) → ((∗‘(𝑓‘𝐴)) · -1) = -1)
101	100	ifeq1da 4504	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → if(𝑓 ∈ 𝑊, ((∗‘(𝑓‘𝐴)) · -1), ((∗‘(𝑓‘𝐴)) · 0)) = if(𝑓 ∈ 𝑊, -1, ((∗‘(𝑓‘𝐴)) · 0)))
102	19	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → (∗‘(𝑓‘𝐴)) ∈ ℂ)
103	102	mul01d 11312	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → ((∗‘(𝑓‘𝐴)) · 0) = 0)
104	103	ifeq2d 4493	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → if(𝑓 ∈ 𝑊, -1, ((∗‘(𝑓‘𝐴)) · 0)) = if(𝑓 ∈ 𝑊, -1, 0))
105	101, 104	eqtrd 2766	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → if(𝑓 ∈ 𝑊, ((∗‘(𝑓‘𝐴)) · -1), ((∗‘(𝑓‘𝐴)) · 0)) = if(𝑓 ∈ 𝑊, -1, 0))
106	79, 105	eqtrid 2778	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → ((∗‘(𝑓‘𝐴)) · if(𝑓 ∈ 𝑊, -1, 0)) = if(𝑓 ∈ 𝑊, -1, 0))
107	78, 106	sylan2br 595	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ ¬ 𝑓 = 1 ) → ((∗‘(𝑓‘𝐴)) · if(𝑓 ∈ 𝑊, -1, 0)) = if(𝑓 ∈ 𝑊, -1, 0))
108	77, 107	ifeq12da 4506	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → if(𝑓 = 1 , ((∗‘(𝑓‘𝐴)) · 1), ((∗‘(𝑓‘𝐴)) · if(𝑓 ∈ 𝑊, -1, 0))) = if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))
109	63, 108	eqtrid 2778	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → ((∗‘(𝑓‘𝐴)) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))) = if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))
110	109	oveq2d 7362	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → ((log‘𝑥) · ((∗‘(𝑓‘𝐴)) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))) = ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))))
111	62, 110	eqtrd 2766	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → ((∗‘(𝑓‘𝐴)) · ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))) = ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))))
112	60, 111	oveq12d 7364	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → (((∗‘(𝑓‘𝐴)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) − ((∗‘(𝑓‘𝐴)) · ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))))
113	59, 112	eqtrd 2766	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → ((∗‘(𝑓‘𝐴)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))))
114	113	sumeq2dv 15609	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑓 ∈ 𝐷 ((∗‘(𝑓‘𝐴)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))))) = Σ𝑓 ∈ 𝐷 (Σ𝑛 ∈ (1...(⌊‘𝑥))((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))))
115		fzfid 13880	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
116		inss1 4184	. . . . . . . . 9 ⊢ ((1...(⌊‘𝑥)) ∩ 𝑇) ⊆ (1...(⌊‘𝑥))
117		ssfi 9082	. . . . . . . . 9 ⊢ (((1...(⌊‘𝑥)) ∈ Fin ∧ ((1...(⌊‘𝑥)) ∩ 𝑇) ⊆ (1...(⌊‘𝑥))) → ((1...(⌊‘𝑥)) ∩ 𝑇) ∈ Fin)
118	115, 116, 117	sylancl 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1...(⌊‘𝑥)) ∩ 𝑇) ∈ Fin)
119	2	phicld 16683	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (ϕ‘𝑁) ∈ ℕ)
120	119	nncnd 12141	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (ϕ‘𝑁) ∈ ℂ)
121	116	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1...(⌊‘𝑥)) ∩ 𝑇) ⊆ (1...(⌊‘𝑥)))
122	121	sselda 3929	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)) → 𝑛 ∈ (1...(⌊‘𝑥)))
123	122, 39	syldan 591	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
124	118, 120, 123	fsummulc2 15691	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) = Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((ϕ‘𝑁) · ((Λ‘𝑛) / 𝑛)))
125	120	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ϕ‘𝑁) ∈ ℂ)
126	125, 39	mulcld 11132	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ϕ‘𝑁) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
127	122, 126	syldan 591	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)) → ((ϕ‘𝑁) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
128	127	ralrimiva 3124	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∀𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((ϕ‘𝑁) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
129	115	olcd 874	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1...(⌊‘𝑥)) ⊆ (ℤ_≥‘1) ∨ (1...(⌊‘𝑥)) ∈ Fin))
130		sumss2 15633	. . . . . . . 8 ⊢ (((((1...(⌊‘𝑥)) ∩ 𝑇) ⊆ (1...(⌊‘𝑥)) ∧ ∀𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((ϕ‘𝑁) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ) ∧ ((1...(⌊‘𝑥)) ⊆ (ℤ_≥‘1) ∨ (1...(⌊‘𝑥)) ∈ Fin)) → Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((ϕ‘𝑁) · ((Λ‘𝑛) / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝑥))if(𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇), ((ϕ‘𝑁) · ((Λ‘𝑛) / 𝑛)), 0))
131	121, 128, 129, 130	syl21anc 837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((ϕ‘𝑁) · ((Λ‘𝑛) / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝑥))if(𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇), ((ϕ‘𝑁) · ((Λ‘𝑛) / 𝑛)), 0))
132		elin 3913	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇) ↔ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ 𝑇))
133	132	baib 535	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → (𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇) ↔ 𝑛 ∈ 𝑇))
134	133	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇) ↔ 𝑛 ∈ 𝑇))
135		rpvmasum2.t	. . . . . . . . . . . . 13 ⊢ 𝑇 = (^◡𝐿 “ {𝐴})
136	135	eleq2i 2823	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑇 ↔ 𝑛 ∈ (^◡𝐿 “ {𝐴}))
137	27	ffnd 6652	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐿 Fn ℤ)
138		fniniseg 6993	. . . . . . . . . . . . . 14 ⊢ (𝐿 Fn ℤ → (𝑛 ∈ (^◡𝐿 “ {𝐴}) ↔ (𝑛 ∈ ℤ ∧ (𝐿‘𝑛) = 𝐴)))
139	138	baibd 539	. . . . . . . . . . . . 13 ⊢ ((𝐿 Fn ℤ ∧ 𝑛 ∈ ℤ) → (𝑛 ∈ (^◡𝐿 “ {𝐴}) ↔ (𝐿‘𝑛) = 𝐴))
140	137, 28, 139	syl2an 596	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ (^◡𝐿 “ {𝐴}) ↔ (𝐿‘𝑛) = 𝐴))
141	136, 140	bitrid 283	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ 𝑇 ↔ (𝐿‘𝑛) = 𝐴))
142	134, 141	bitr2d 280	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝐿‘𝑛) = 𝐴 ↔ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)))
143	39	mul02d 11311	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (0 · ((Λ‘𝑛) / 𝑛)) = 0)
144	142, 143	ifbieq2d 4499	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → if((𝐿‘𝑛) = 𝐴, ((ϕ‘𝑁) · ((Λ‘𝑛) / 𝑛)), (0 · ((Λ‘𝑛) / 𝑛))) = if(𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇), ((ϕ‘𝑁) · ((Λ‘𝑛) / 𝑛)), 0))
145		ovif 7444	. . . . . . . . . 10 ⊢ (if((𝐿‘𝑛) = 𝐴, (ϕ‘𝑁), 0) · ((Λ‘𝑛) / 𝑛)) = if((𝐿‘𝑛) = 𝐴, ((ϕ‘𝑁) · ((Λ‘𝑛) / 𝑛)), (0 · ((Λ‘𝑛) / 𝑛)))
146	1	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑁 ∈ ℕ)
147	146, 5	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐷 ∈ Fin)
148	18	ad4ant14 752	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑓 ∈ 𝐷) → (∗‘(𝑓‘𝐴)) ∈ ℂ)
149	32, 148	mulcld 11132	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑓 ∈ 𝐷) → ((𝑓‘(𝐿‘𝑛)) · (∗‘(𝑓‘𝐴))) ∈ ℂ)
150	147, 39, 149	fsummulc1 15692	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑓 ∈ 𝐷 ((𝑓‘(𝐿‘𝑛)) · (∗‘(𝑓‘𝐴))) · ((Λ‘𝑛) / 𝑛)) = Σ𝑓 ∈ 𝐷 (((𝑓‘(𝐿‘𝑛)) · (∗‘(𝑓‘𝐴))) · ((Λ‘𝑛) / 𝑛)))
151	14	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐴 ∈ 𝑈)
152	3, 4, 8, 9, 12, 146, 30, 151	sum2dchr 27212	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑓 ∈ 𝐷 ((𝑓‘(𝐿‘𝑛)) · (∗‘(𝑓‘𝐴))) = if((𝐿‘𝑛) = 𝐴, (ϕ‘𝑁), 0))
153	152	oveq1d 7361	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑓 ∈ 𝐷 ((𝑓‘(𝐿‘𝑛)) · (∗‘(𝑓‘𝐴))) · ((Λ‘𝑛) / 𝑛)) = (if((𝐿‘𝑛) = 𝐴, (ϕ‘𝑁), 0) · ((Λ‘𝑛) / 𝑛)))
154	39	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑓 ∈ 𝐷) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
155		mulass 11094	. . . . . . . . . . . . . 14 ⊢ (((𝑓‘(𝐿‘𝑛)) ∈ ℂ ∧ (∗‘(𝑓‘𝐴)) ∈ ℂ ∧ ((Λ‘𝑛) / 𝑛) ∈ ℂ) → (((𝑓‘(𝐿‘𝑛)) · (∗‘(𝑓‘𝐴))) · ((Λ‘𝑛) / 𝑛)) = ((𝑓‘(𝐿‘𝑛)) · ((∗‘(𝑓‘𝐴)) · ((Λ‘𝑛) / 𝑛))))
156		mul12 11278	. . . . . . . . . . . . . 14 ⊢ (((𝑓‘(𝐿‘𝑛)) ∈ ℂ ∧ (∗‘(𝑓‘𝐴)) ∈ ℂ ∧ ((Λ‘𝑛) / 𝑛) ∈ ℂ) → ((𝑓‘(𝐿‘𝑛)) · ((∗‘(𝑓‘𝐴)) · ((Λ‘𝑛) / 𝑛))) = ((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))))
157	155, 156	eqtrd 2766	. . . . . . . . . . . . 13 ⊢ (((𝑓‘(𝐿‘𝑛)) ∈ ℂ ∧ (∗‘(𝑓‘𝐴)) ∈ ℂ ∧ ((Λ‘𝑛) / 𝑛) ∈ ℂ) → (((𝑓‘(𝐿‘𝑛)) · (∗‘(𝑓‘𝐴))) · ((Λ‘𝑛) / 𝑛)) = ((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))))
158	32, 148, 154, 157	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑓 ∈ 𝐷) → (((𝑓‘(𝐿‘𝑛)) · (∗‘(𝑓‘𝐴))) · ((Λ‘𝑛) / 𝑛)) = ((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))))
159	158	sumeq2dv 15609	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑓 ∈ 𝐷 (((𝑓‘(𝐿‘𝑛)) · (∗‘(𝑓‘𝐴))) · ((Λ‘𝑛) / 𝑛)) = Σ𝑓 ∈ 𝐷 ((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))))
160	150, 153, 159	3eqtr3d 2774	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (if((𝐿‘𝑛) = 𝐴, (ϕ‘𝑁), 0) · ((Λ‘𝑛) / 𝑛)) = Σ𝑓 ∈ 𝐷 ((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))))
161	145, 160	eqtr3id 2780	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → if((𝐿‘𝑛) = 𝐴, ((ϕ‘𝑁) · ((Λ‘𝑛) / 𝑛)), (0 · ((Λ‘𝑛) / 𝑛))) = Σ𝑓 ∈ 𝐷 ((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))))
162	144, 161	eqtr3d 2768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → if(𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇), ((ϕ‘𝑁) · ((Λ‘𝑛) / 𝑛)), 0) = Σ𝑓 ∈ 𝐷 ((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))))
163	162	sumeq2dv 15609	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))if(𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇), ((ϕ‘𝑁) · ((Λ‘𝑛) / 𝑛)), 0) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑓 ∈ 𝐷 ((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))))
164	124, 131, 163	3eqtrd 2770	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑓 ∈ 𝐷 ((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))))
165	115, 6, 42	fsumcom 15682	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑓 ∈ 𝐷 ((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) = Σ𝑓 ∈ 𝐷 Σ𝑛 ∈ (1...(⌊‘𝑥))((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))))
166	164, 165	eqtrd 2766	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) = Σ𝑓 ∈ 𝐷 Σ𝑛 ∈ (1...(⌊‘𝑥))((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))))
167	3	dchrabl 27192	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel)
168		ablgrp 19697	. . . . . . . . . 10 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
169	4, 65	grpidcl 18878	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 1 ∈ 𝐷)
170	2, 167, 168, 169	4syl 19	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ 𝐷)
171	47	mulridd 11129	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · 1) = (log‘𝑥))
172	171, 47	eqeltrd 2831	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · 1) ∈ ℂ)
173		iftrue 4478	. . . . . . . . . . 11 ⊢ (𝑓 = 1 → if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)) = 1)
174	173	oveq2d 7362	. . . . . . . . . 10 ⊢ (𝑓 = 1 → ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))) = ((log‘𝑥) · 1))
175	174	sumsn 15653	. . . . . . . . 9 ⊢ (( 1 ∈ 𝐷 ∧ ((log‘𝑥) · 1) ∈ ℂ) → Σ𝑓 ∈ { 1 } ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))) = ((log‘𝑥) · 1))
176	170, 172, 175	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑓 ∈ { 1 } ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))) = ((log‘𝑥) · 1))
177		eldifsn 4735	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝐷 ∖ { 1 }) ↔ (𝑓 ∈ 𝐷 ∧ 𝑓 ≠ 1 ))
178		ifnefalse 4484	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ≠ 1 → if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)) = if(𝑓 ∈ 𝑊, -1, 0))
179	178	ad2antll 729	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑓 ∈ 𝐷 ∧ 𝑓 ≠ 1 )) → if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)) = if(𝑓 ∈ 𝑊, -1, 0))
180		negeq 11352	. . . . . . . . . . . . . . 15 ⊢ (if(𝑓 ∈ 𝑊, 1, 0) = 1 → -if(𝑓 ∈ 𝑊, 1, 0) = -1)
181		negeq 11352	. . . . . . . . . . . . . . . 16 ⊢ (if(𝑓 ∈ 𝑊, 1, 0) = 0 → -if(𝑓 ∈ 𝑊, 1, 0) = -0)
182		neg0 11407	. . . . . . . . . . . . . . . 16 ⊢ -0 = 0
183	181, 182	eqtrdi 2782	. . . . . . . . . . . . . . 15 ⊢ (if(𝑓 ∈ 𝑊, 1, 0) = 0 → -if(𝑓 ∈ 𝑊, 1, 0) = 0)
184	180, 183	ifsb 4486	. . . . . . . . . . . . . 14 ⊢ -if(𝑓 ∈ 𝑊, 1, 0) = if(𝑓 ∈ 𝑊, -1, 0)
185	179, 184	eqtr4di 2784	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑓 ∈ 𝐷 ∧ 𝑓 ≠ 1 )) → if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)) = -if(𝑓 ∈ 𝑊, 1, 0))
186	185	oveq2d 7362	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑓 ∈ 𝐷 ∧ 𝑓 ≠ 1 )) → ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))) = ((log‘𝑥) · -if(𝑓 ∈ 𝑊, 1, 0)))
187	47	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑓 ∈ 𝐷 ∧ 𝑓 ≠ 1 )) → (log‘𝑥) ∈ ℂ)
188	49, 51	ifcli 4520	. . . . . . . . . . . . 13 ⊢ if(𝑓 ∈ 𝑊, 1, 0) ∈ ℂ
189		mulneg2 11554	. . . . . . . . . . . . 13 ⊢ (((log‘𝑥) ∈ ℂ ∧ if(𝑓 ∈ 𝑊, 1, 0) ∈ ℂ) → ((log‘𝑥) · -if(𝑓 ∈ 𝑊, 1, 0)) = -((log‘𝑥) · if(𝑓 ∈ 𝑊, 1, 0)))
190	187, 188, 189	sylancl 586	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑓 ∈ 𝐷 ∧ 𝑓 ≠ 1 )) → ((log‘𝑥) · -if(𝑓 ∈ 𝑊, 1, 0)) = -((log‘𝑥) · if(𝑓 ∈ 𝑊, 1, 0)))
191	186, 190	eqtrd 2766	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑓 ∈ 𝐷 ∧ 𝑓 ≠ 1 )) → ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))) = -((log‘𝑥) · if(𝑓 ∈ 𝑊, 1, 0)))
192	177, 191	sylan2b 594	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ (𝐷 ∖ { 1 })) → ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))) = -((log‘𝑥) · if(𝑓 ∈ 𝑊, 1, 0)))
193	192	sumeq2dv 15609	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑓 ∈ (𝐷 ∖ { 1 })((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))) = Σ𝑓 ∈ (𝐷 ∖ { 1 })-((log‘𝑥) · if(𝑓 ∈ 𝑊, 1, 0)))
194		diffi 9084	. . . . . . . . . . 11 ⊢ (𝐷 ∈ Fin → (𝐷 ∖ { 1 }) ∈ Fin)
195	6, 194	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝐷 ∖ { 1 }) ∈ Fin)
196	47	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ (𝐷 ∖ { 1 })) → (log‘𝑥) ∈ ℂ)
197		mulcl 11090	. . . . . . . . . . 11 ⊢ (((log‘𝑥) ∈ ℂ ∧ if(𝑓 ∈ 𝑊, 1, 0) ∈ ℂ) → ((log‘𝑥) · if(𝑓 ∈ 𝑊, 1, 0)) ∈ ℂ)
198	196, 188, 197	sylancl 586	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ (𝐷 ∖ { 1 })) → ((log‘𝑥) · if(𝑓 ∈ 𝑊, 1, 0)) ∈ ℂ)
199	195, 198	fsumneg 15694	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑓 ∈ (𝐷 ∖ { 1 })-((log‘𝑥) · if(𝑓 ∈ 𝑊, 1, 0)) = -Σ𝑓 ∈ (𝐷 ∖ { 1 })((log‘𝑥) · if(𝑓 ∈ 𝑊, 1, 0)))
200	188	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ (𝐷 ∖ { 1 })) → if(𝑓 ∈ 𝑊, 1, 0) ∈ ℂ)
201	195, 47, 200	fsummulc2 15691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · Σ𝑓 ∈ (𝐷 ∖ { 1 })if(𝑓 ∈ 𝑊, 1, 0)) = Σ𝑓 ∈ (𝐷 ∖ { 1 })((log‘𝑥) · if(𝑓 ∈ 𝑊, 1, 0)))
202		rpvmasum2.w	. . . . . . . . . . . . . . . . 17 ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
203	202	ssrab3 4029	. . . . . . . . . . . . . . . 16 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
204		difss 4083	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∖ { 1 }) ⊆ 𝐷
205	203, 204	sstri 3939	. . . . . . . . . . . . . . 15 ⊢ 𝑊 ⊆ 𝐷
206		ssfi 9082	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑊 ⊆ 𝐷) → 𝑊 ∈ Fin)
207	6, 205, 206	sylancl 586	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑊 ∈ Fin)
208		fsumconst 15697	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑓 ∈ 𝑊 1 = ((♯‘𝑊) · 1))
209	207, 49, 208	sylancl 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑓 ∈ 𝑊 1 = ((♯‘𝑊) · 1))
210	203	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑊 ⊆ (𝐷 ∖ { 1 }))
211	49	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
212	211	ralrimivw 3128	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∀𝑓 ∈ 𝑊 1 ∈ ℂ)
213	195	olcd 874	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝐷 ∖ { 1 }) ⊆ (ℤ_≥‘1) ∨ (𝐷 ∖ { 1 }) ∈ Fin))
214		sumss2 15633	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ⊆ (𝐷 ∖ { 1 }) ∧ ∀𝑓 ∈ 𝑊 1 ∈ ℂ) ∧ ((𝐷 ∖ { 1 }) ⊆ (ℤ_≥‘1) ∨ (𝐷 ∖ { 1 }) ∈ Fin)) → Σ𝑓 ∈ 𝑊 1 = Σ𝑓 ∈ (𝐷 ∖ { 1 })if(𝑓 ∈ 𝑊, 1, 0))
215	210, 212, 213, 214	syl21anc 837	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑓 ∈ 𝑊 1 = Σ𝑓 ∈ (𝐷 ∖ { 1 })if(𝑓 ∈ 𝑊, 1, 0))
216		hashcl 14263	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ Fin → (♯‘𝑊) ∈ ℕ₀)
217	207, 216	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (♯‘𝑊) ∈ ℕ₀)
218	217	nn0cnd 12444	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (♯‘𝑊) ∈ ℂ)
219	218	mulridd 11129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((♯‘𝑊) · 1) = (♯‘𝑊))
220	209, 215, 219	3eqtr3d 2774	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑓 ∈ (𝐷 ∖ { 1 })if(𝑓 ∈ 𝑊, 1, 0) = (♯‘𝑊))
221	220	oveq2d 7362	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · Σ𝑓 ∈ (𝐷 ∖ { 1 })if(𝑓 ∈ 𝑊, 1, 0)) = ((log‘𝑥) · (♯‘𝑊)))
222	201, 221	eqtr3d 2768	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑓 ∈ (𝐷 ∖ { 1 })((log‘𝑥) · if(𝑓 ∈ 𝑊, 1, 0)) = ((log‘𝑥) · (♯‘𝑊)))
223	222	negeqd 11354	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → -Σ𝑓 ∈ (𝐷 ∖ { 1 })((log‘𝑥) · if(𝑓 ∈ 𝑊, 1, 0)) = -((log‘𝑥) · (♯‘𝑊)))
224	193, 199, 223	3eqtrd 2770	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑓 ∈ (𝐷 ∖ { 1 })((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))) = -((log‘𝑥) · (♯‘𝑊)))
225	176, 224	oveq12d 7364	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑓 ∈ { 1 } ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))) + Σ𝑓 ∈ (𝐷 ∖ { 1 })((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))) = (((log‘𝑥) · 1) + -((log‘𝑥) · (♯‘𝑊))))
226	47, 218	mulcld 11132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · (♯‘𝑊)) ∈ ℂ)
227	172, 226	negsubd 11478	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((log‘𝑥) · 1) + -((log‘𝑥) · (♯‘𝑊))) = (((log‘𝑥) · 1) − ((log‘𝑥) · (♯‘𝑊))))
228	225, 227	eqtrd 2766	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑓 ∈ { 1 } ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))) + Σ𝑓 ∈ (𝐷 ∖ { 1 })((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))) = (((log‘𝑥) · 1) − ((log‘𝑥) · (♯‘𝑊))))
229		disjdif 4419	. . . . . . . 8 ⊢ ({ 1 } ∩ (𝐷 ∖ { 1 })) = ∅
230	229	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ({ 1 } ∩ (𝐷 ∖ { 1 })) = ∅)
231		undif2 4424	. . . . . . . 8 ⊢ ({ 1 } ∪ (𝐷 ∖ { 1 })) = ({ 1 } ∪ 𝐷)
232	170	snssd 4758	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → { 1 } ⊆ 𝐷)
233		ssequn1 4133	. . . . . . . . 9 ⊢ ({ 1 } ⊆ 𝐷 ↔ ({ 1 } ∪ 𝐷) = 𝐷)
234	232, 233	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ({ 1 } ∪ 𝐷) = 𝐷)
235	231, 234	eqtr2id 2779	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐷 = ({ 1 } ∪ (𝐷 ∖ { 1 })))
236	230, 235, 6, 55	fsumsplit 15648	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑓 ∈ 𝐷 ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))) = (Σ𝑓 ∈ { 1 } ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))) + Σ𝑓 ∈ (𝐷 ∖ { 1 })((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))))
237	47, 211, 218	subdid 11573	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · (1 − (♯‘𝑊))) = (((log‘𝑥) · 1) − ((log‘𝑥) · (♯‘𝑊))))
238	228, 236, 237	3eqtr4rd 2777	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · (1 − (♯‘𝑊))) = Σ𝑓 ∈ 𝐷 ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))))
239	166, 238	oveq12d 7364	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))) = (Σ𝑓 ∈ 𝐷 Σ𝑛 ∈ (1...(⌊‘𝑥))((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) − Σ𝑓 ∈ 𝐷 ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))))
240	56, 114, 239	3eqtr4d 2776	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑓 ∈ 𝐷 ((∗‘(𝑓‘𝐴)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))))) = (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))))
241	240	mpteq2dva 5182	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑓 ∈ 𝐷 ((∗‘(𝑓‘𝐴)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))))) = (𝑥 ∈ ℝ⁺ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊))))))
242		rpssre 12898	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
243	242	a1i 11	. . 3 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
244	1, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐷 ∈ Fin)
245	17	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → (𝑓‘𝐴) ∈ ℂ)
246	245	cjcld 15103	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → (∗‘(𝑓‘𝐴)) ∈ ℂ)
247	58, 55	subcld 11472	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))) ∈ ℂ)
248	246, 247	mulcld 11132	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → ((∗‘(𝑓‘𝐴)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))))) ∈ ℂ)
249	248	anasss 466	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑓 ∈ 𝐷)) → ((∗‘(𝑓‘𝐴)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))))) ∈ ℂ)
250	18	adantr 480	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → (∗‘(𝑓‘𝐴)) ∈ ℂ)
251	247	an32s 652	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))) ∈ ℂ)
252		o1const 15527	. . . . 5 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (∗‘(𝑓‘𝐴)) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (∗‘(𝑓‘𝐴))) ∈ 𝑂(1))
253	242, 18, 252	sylancr 587	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → (𝑥 ∈ ℝ⁺ ↦ (∗‘(𝑓‘𝐴))) ∈ 𝑂(1))
254		fveq1 6821	. . . . . . . . . . . 12 ⊢ (𝑓 = 1 → (𝑓‘(𝐿‘𝑛)) = ( 1 ‘(𝐿‘𝑛)))
255	254	oveq1d 7361	. . . . . . . . . . 11 ⊢ (𝑓 = 1 → ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) = (( 1 ‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)))
256	255	sumeq2sdv 15610	. . . . . . . . . 10 ⊢ (𝑓 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)))
257	256, 174	oveq12d 7364	. . . . . . . . 9 ⊢ (𝑓 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · 1)))
258	257	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 = 1 ) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · 1)))
259	45	recnd 11140	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℂ)
260	259	mulridd 11129	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘𝑥) · 1) = (log‘𝑥))
261	260	oveq2d 7362	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · 1)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − (log‘𝑥)))
262	258, 261	sylan9eq 2786	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 = 1 ) ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − (log‘𝑥)))
263	262	mpteq2dva 5182	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 = 1 ) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − (log‘𝑥))))
264	8, 23, 1, 3, 4, 65	rpvmasumlem 27425	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − (log‘𝑥))) ∈ 𝑂(1))
265	264	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 = 1 ) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − (log‘𝑥))) ∈ 𝑂(1))
266	263, 265	eqeltrd 2831	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 = 1 ) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))))) ∈ 𝑂(1))
267	178	oveq2d 7362	. . . . . . . . . 10 ⊢ (𝑓 ≠ 1 → ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))) = ((log‘𝑥) · if(𝑓 ∈ 𝑊, -1, 0)))
268	267	oveq2d 7362	. . . . . . . . 9 ⊢ (𝑓 ≠ 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 ∈ 𝑊, -1, 0))))
269	47	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
270		mulcom 11092	. . . . . . . . . . . . . . 15 ⊢ (((log‘𝑥) ∈ ℂ ∧ -1 ∈ ℂ) → ((log‘𝑥) · -1) = (-1 · (log‘𝑥)))
271	269, 50, 270	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · -1) = (-1 · (log‘𝑥)))
272	269	mulm1d 11569	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → (-1 · (log‘𝑥)) = -(log‘𝑥))
273	271, 272	eqtrd 2766	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · -1) = -(log‘𝑥))
274	269	mul01d 11312	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · 0) = 0)
275	273, 274	ifeq12d 4494	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → if(𝑓 ∈ 𝑊, ((log‘𝑥) · -1), ((log‘𝑥) · 0)) = if(𝑓 ∈ 𝑊, -(log‘𝑥), 0))
276		ovif2 7445	. . . . . . . . . . . 12 ⊢ ((log‘𝑥) · if(𝑓 ∈ 𝑊, -1, 0)) = if(𝑓 ∈ 𝑊, ((log‘𝑥) · -1), ((log‘𝑥) · 0))
277		negeq 11352	. . . . . . . . . . . . 13 ⊢ (if(𝑓 ∈ 𝑊, (log‘𝑥), 0) = (log‘𝑥) → -if(𝑓 ∈ 𝑊, (log‘𝑥), 0) = -(log‘𝑥))
278		negeq 11352	. . . . . . . . . . . . . 14 ⊢ (if(𝑓 ∈ 𝑊, (log‘𝑥), 0) = 0 → -if(𝑓 ∈ 𝑊, (log‘𝑥), 0) = -0)
279	278, 182	eqtrdi 2782	. . . . . . . . . . . . 13 ⊢ (if(𝑓 ∈ 𝑊, (log‘𝑥), 0) = 0 → -if(𝑓 ∈ 𝑊, (log‘𝑥), 0) = 0)
280	277, 279	ifsb 4486	. . . . . . . . . . . 12 ⊢ -if(𝑓 ∈ 𝑊, (log‘𝑥), 0) = if(𝑓 ∈ 𝑊, -(log‘𝑥), 0)
281	275, 276, 280	3eqtr4g 2791	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · if(𝑓 ∈ 𝑊, -1, 0)) = -if(𝑓 ∈ 𝑊, (log‘𝑥), 0))
282	281	oveq2d 7362	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 ∈ 𝑊, -1, 0))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − -if(𝑓 ∈ 𝑊, (log‘𝑥), 0)))
283	58	an32s 652	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
284		ifcl 4518	. . . . . . . . . . . 12 ⊢ (((log‘𝑥) ∈ ℂ ∧ 0 ∈ ℂ) → if(𝑓 ∈ 𝑊, (log‘𝑥), 0) ∈ ℂ)
285	269, 51, 284	sylancl 586	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → if(𝑓 ∈ 𝑊, (log‘𝑥), 0) ∈ ℂ)
286	283, 285	subnegd 11479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − -if(𝑓 ∈ 𝑊, (log‘𝑥), 0)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑓 ∈ 𝑊, (log‘𝑥), 0)))
287	282, 286	eqtrd 2766	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 ∈ 𝑊, -1, 0))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑓 ∈ 𝑊, (log‘𝑥), 0)))
288	268, 287	sylan9eqr 2788	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ≠ 1 ) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑓 ∈ 𝑊, (log‘𝑥), 0)))
289	288	an32s 652	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑓 ∈ 𝑊, (log‘𝑥), 0)))
290	289	mpteq2dva 5182	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑓 ∈ 𝑊, (log‘𝑥), 0))))
291	1	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → 𝑁 ∈ ℕ)
292		simplr 768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → 𝑓 ∈ 𝐷)
293		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → 𝑓 ≠ 1 )
294		eqid 2731	. . . . . . . 8 ⊢ (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎)) = (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎))
295	8, 23, 291, 3, 4, 65, 292, 293, 294	dchrmusumlema 27431	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))
296	1	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝑁 ∈ ℕ)
297	296	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑁 ∈ ℕ)
298	292	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑓 ∈ 𝐷)
299		simplr 768	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑓 ≠ 1 )
300		simprl 770	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑐 ∈ (0[,)+∞))
301		simprrl 780	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡)
302		simprrr 781	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦))
303	8, 23, 297, 3, 4, 65, 298, 299, 294, 300, 301, 302, 202	dchrvmaeq0 27442	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → (𝑓 ∈ 𝑊 ↔ 𝑡 = 0))
304		ifbi 4495	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ 𝑊 ↔ 𝑡 = 0) → if(𝑓 ∈ 𝑊, (log‘𝑥), 0) = if(𝑡 = 0, (log‘𝑥), 0))
305	304	oveq2d 7362	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ 𝑊 ↔ 𝑡 = 0) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑓 ∈ 𝑊, (log‘𝑥), 0)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑡 = 0, (log‘𝑥), 0)))
306	305	mpteq2dv 5183	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ 𝑊 ↔ 𝑡 = 0) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑓 ∈ 𝑊, (log‘𝑥), 0))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑡 = 0, (log‘𝑥), 0))))
307	303, 306	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑓 ∈ 𝑊, (log‘𝑥), 0))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑡 = 0, (log‘𝑥), 0))))
308	8, 23, 297, 3, 4, 65, 298, 299, 294, 300, 301, 302	dchrvmasumif 27441	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑡 = 0, (log‘𝑥), 0))) ∈ 𝑂(1))
309	307, 308	eqeltrd 2831	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑓 ∈ 𝑊, (log‘𝑥), 0))) ∈ 𝑂(1))
310	309	rexlimdvaa 3134	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → (∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑓 ∈ 𝑊, (log‘𝑥), 0))) ∈ 𝑂(1)))
311	310	exlimdv 1934	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → (∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑓 ∈ 𝑊, (log‘𝑥), 0))) ∈ 𝑂(1)))
312	295, 311	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑓 ∈ 𝑊, (log‘𝑥), 0))) ∈ 𝑂(1))
313	290, 312	eqeltrd 2831	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))))) ∈ 𝑂(1))
314	266, 313	pm2.61dane 3015	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))))) ∈ 𝑂(1))
315	250, 251, 253, 314	o1mul2 15532	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → (𝑥 ∈ ℝ⁺ ↦ ((∗‘(𝑓‘𝐴)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))))) ∈ 𝑂(1))
316	243, 244, 249, 315	fsumo1 15719	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑓 ∈ 𝐷 ((∗‘(𝑓‘𝐴)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)))))) ∈ 𝑂(1))
317	241, 316	eqeltrrd 2832	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊))))) ∈ 𝑂(1))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∃wex 1780 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 {crab 3395 ∖ cdif 3894 ∪ cun 3895 ∩ cin 3896 ⊆ wss 3897 ∅c0 4280 ifcif 4472 {csn 4573 class class class wbr 5089 ↦ cmpt 5170 ^◡ccnv 5613 “ cima 5617 Fn wfn 6476 ⟶wf 6477 –onto→wfo 6479 ‘cfv 6481 (class class class)co 7346 Fincfn 8869 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 · cmul 11011 +∞cpnf 11143 ≤ cle 11147 − cmin 11344 -cneg 11345 / cdiv 11774 ℕcn 12125 ℕ₀cn0 12381 ℤcz 12468 ℤ_≥cuz 12732 ℝ⁺crp 12890 [,)cico 13247 ...cfz 13407 ⌊cfl 13694 seqcseq 13908 ♯chash 14237 ∗ccj 15003 abscabs 15141 ⇝ cli 15391 𝑂(1)co1 15393 Σcsu 15593 ϕcphi 16675 Basecbs 17120 0_gc0g 17343 MndHom cmhm 18689 Grpcgrp 18846 Abelcabl 19693 mulGrpcmgp 20058 1_rcur 20099 Unitcui 20273 ℂ_fldccnfld 21291 ℤRHomczrh 21436 ℤ/nℤczn 21439 logclog 26490 Λcvma 27029 DChrcdchr 27170

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085 ax-mulf 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-disj 5057 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-rpss 7656 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-omul 8390 df-er 8622 df-ec 8624 df-qs 8628 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-dju 9794 df-card 9832 df-acn 9835 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-xnn0 12455 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ioc 13250 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-word 14421 df-concat 14478 df-s1 14504 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-o1 15397 df-lo1 15398 df-sum 15594 df-ef 15974 df-e 15975 df-sin 15976 df-cos 15977 df-tan 15978 df-pi 15979 df-dvds 16164 df-gcd 16406 df-prm 16583 df-phi 16677 df-pc 16749 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-qus 17413 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-mulg 18981 df-subg 19036 df-nsg 19037 df-eqg 19038 df-ghm 19125 df-gim 19171 df-ga 19202 df-cntz 19229 df-oppg 19258 df-od 19440 df-gex 19441 df-pgp 19442 df-lsm 19548 df-pj1 19549 df-cmn 19694 df-abl 19695 df-cyg 19790 df-dprd 19909 df-dpj 19910 df-mgp 20059 df-rng 20071 df-ur 20100 df-ring 20153 df-cring 20154 df-oppr 20255 df-dvdsr 20275 df-unit 20276 df-invr 20306 df-dvr 20319 df-rhm 20390 df-subrng 20461 df-subrg 20485 df-drng 20646 df-lmod 20795 df-lss 20865 df-lsp 20905 df-sra 21107 df-rgmod 21108 df-lidl 21145 df-rsp 21146 df-2idl 21187 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-fbas 21288 df-fg 21289 df-cnfld 21292 df-zring 21384 df-zrh 21440 df-zn 21443 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-ntr 22935 df-cls 22936 df-nei 23013 df-lp 23051 df-perf 23052 df-cn 23142 df-cnp 23143 df-haus 23230 df-cmp 23302 df-tx 23477 df-hmeo 23670 df-fil 23761 df-fm 23853 df-flim 23854 df-flf 23855 df-xms 24235 df-ms 24236 df-tms 24237 df-cncf 24798 df-0p 25598 df-limc 25794 df-dv 25795 df-ply 26120 df-idp 26121 df-coe 26122 df-dgr 26123 df-quot 26226 df-ulm 26313 df-log 26492 df-cxp 26493 df-atan 26804 df-em 26930 df-cht 27034 df-vma 27035 df-chp 27036 df-ppi 27037 df-mu 27038 df-dchr 27171

This theorem is referenced by: dchrisum0re 27451 rpvmasum 27464

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