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Theorem rpvmasum2 26860

Description: A partial result along the lines of rpvmasum 26874. 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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑁 ∈ ℕ)
3		rpvmasum2.g	. . . . . . 7 ⊢ 𝐺 = (DChr‘𝑁)
4		rpvmasum2.d	. . . . . . 7 ⊢ 𝐷 = (Base‘𝐺)
5	3, 4	dchrfi 26603	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin)
6	2, 5	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐷 ∈ Fin)
7		fzfid 13878	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → (1...(⌊‘𝑥)) ∈ Fin)
8		rpvmasum.z	. . . . . . . . . . . . 13 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
9		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Base‘𝑍) = (Base‘𝑍)
10		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝑓 ∈ 𝐷)
11	3, 8, 4, 9, 10	dchrf 26590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝑓:(Base‘𝑍)⟶ℂ)
12		rpvmasum2.u	. . . . . . . . . . . . . . 15 ⊢ 𝑈 = (Unit‘𝑍)
13	9, 12	unitss 20089	. . . . . . . . . . . . . 14 ⊢ 𝑈 ⊆ (Base‘𝑍)
14		rpvmasum2.b	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
15	13, 14	sselid 3942	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑍))
16	15	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝐴 ∈ (Base‘𝑍))
17	11, 16	ffvelcdmd 7036	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → (𝑓‘𝐴) ∈ ℂ)
18	17	cjcld 15081	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → (∗‘(𝑓‘𝐴)) ∈ ℂ)
19	18	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → (∗‘(𝑓‘𝐴)) ∈ ℂ)
20	19	adantrl 714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑓 ∈ 𝐷)) → (∗‘(𝑓‘𝐴)) ∈ ℂ)
21	11	ad4ant14 750	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑓 ∈ 𝐷) → 𝑓:(Base‘𝑍)⟶ℂ)
22	1	nnnn0d 12473	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
23		rpvmasum.l	. . . . . . . . . . . . . . . 16 ⊢ 𝐿 = (ℤRHom‘𝑍)
24	8, 9, 23	znzrhfo 20954	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑍))
25		fof 6756	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐿:ℤ⟶(Base‘𝑍))
28		elfzelz 13441	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℤ)
29		ffvelcdm 7032	. . . . . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑓 ∈ 𝐷) → (𝐿‘𝑛) ∈ (Base‘𝑍))
32	21, 31	ffvelcdmd 7036	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑓 ∈ 𝐷) → (𝑓‘(𝐿‘𝑛)) ∈ ℂ)
33	32	anasss 467	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑓 ∈ 𝐷)) → (𝑓‘(𝐿‘𝑛)) ∈ ℂ)
34		elfznn 13470	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
35	34	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
36		vmacl 26467	. . . . . . . . . . . . 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 12207	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
39	38	recnd 11183	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
40	39	adantrr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑓 ∈ 𝐷)) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
41	33, 40	mulcld 11175	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑓 ∈ 𝐷)) → ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
42	20, 41	mulcld 11175	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑓 ∈ 𝐷)) → ((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) ∈ ℂ)
43	42	anass1rs 653	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) ∈ ℂ)
44	7, 43	fsumcl 15618	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → Σ𝑛 ∈ (1...(⌊‘𝑥))((∗‘(𝑓‘𝐴)) · ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) ∈ ℂ)
45		relogcl 25931	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
46	45	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
47	46	recnd 11183	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
48	47	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → (log‘𝑥) ∈ ℂ)
49		ax-1cn 11109	. . . . . . 7 ⊢ 1 ∈ ℂ
50		neg1cn 12267	. . . . . . . 8 ⊢ -1 ∈ ℂ
51		0cn 11147	. . . . . . . 8 ⊢ 0 ∈ ℂ
52	50, 51	ifcli 4533	. . . . . . 7 ⊢ if(𝑓 ∈ 𝑊, -1, 0) ∈ ℂ
53	49, 52	ifcli 4533	. . . . . 6 ⊢ if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)) ∈ ℂ
54		mulcl 11135	. . . . . 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 15673	. . . 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 653	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
58	7, 57	fsumcl 15618	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
59	19, 58, 55	subdid 11611	. . . . . 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 15669	. . . . . . 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 11364	. . . . . . . 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 7455	. . . . . . . . . 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 6841	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 1 → (𝑓‘𝐴) = ( 1 ‘𝐴))
65		rpvmasum2.1	. . . . . . . . . . . . . . . . 17 ⊢ 1 = (0_g‘𝐺)
66	1	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → 𝑁 ∈ ℕ)
67	14	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → 𝐴 ∈ 𝑈)
68	3, 8, 65, 12, 66, 67	dchr1 26605	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → ( 1 ‘𝐴) = 1)
69	64, 68	sylan9eqr 2798	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 = 1 ) → (𝑓‘𝐴) = 1)
70	69	fveq2d 6846	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 = 1 ) → (∗‘(𝑓‘𝐴)) = (∗‘1))
71		1re 11155	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
72		cjre 15024	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℝ → (∗‘1) = 1)
73	71, 72	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (∗‘1) = 1
74	70, 73	eqtrdi 2792	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 = 1 ) → (∗‘(𝑓‘𝐴)) = 1)
75	74	oveq1d 7372	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 = 1 ) → ((∗‘(𝑓‘𝐴)) · 1) = (1 · 1))
76		1t1e1 12315	. . . . . . . . . . . 12 ⊢ (1 · 1) = 1
77	75, 76	eqtrdi 2792	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 = 1 ) → ((∗‘(𝑓‘𝐴)) · 1) = 1)
78		df-ne 2944	. . . . . . . . . . . 12 ⊢ (𝑓 ≠ 1 ↔ ¬ 𝑓 = 1 )
79		ovif2 7455	. . . . . . . . . . . . 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 6846	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑊) → (𝑓‘𝐴) = (𝑓‘(1_r‘𝑍)))
82	81	ad5ant15 757	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ 𝑓 ∈ 𝑊) → (𝑓‘𝐴) = (𝑓‘(1_r‘𝑍)))
83	3, 8, 4	dchrmhm 26589	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
84		simpr 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → 𝑓 ∈ 𝐷)
85	83, 84	sselid 3942	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → 𝑓 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
86		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
87		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
88	86, 87	ringidval 19915	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1_r‘𝑍) = (0_g‘(mulGrp‘𝑍))
89		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
90		cnfld1 20822	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 = (1_r‘ℂ_fld)
91	89, 90	ringidval 19915	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
92	88, 91	mhm0 18610	. . . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ 𝑓 ∈ 𝑊) → (𝑓‘(1_r‘𝑍)) = 1)
95	82, 94	eqtrd 2776	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ 𝑓 ∈ 𝑊) → (𝑓‘𝐴) = 1)
96	95	fveq2d 6846	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ 𝑓 ∈ 𝑊) → (∗‘(𝑓‘𝐴)) = (∗‘1))
97	96, 73	eqtrdi 2792	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ 𝑓 ∈ 𝑊) → (∗‘(𝑓‘𝐴)) = 1)
98	97	oveq1d 7372	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ 𝑓 ∈ 𝑊) → ((∗‘(𝑓‘𝐴)) · -1) = (1 · -1))
99	50	mulid2i 11160	. . . . . . . . . . . . . . . 16 ⊢ (1 · -1) = -1
100	98, 99	eqtrdi 2792	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) ∧ 𝑓 ∈ 𝑊) → ((∗‘(𝑓‘𝐴)) · -1) = -1)
101	100	ifeq1da 4517	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → (∗‘(𝑓‘𝐴)) ∈ ℂ)
103	102	mul01d 11354	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → ((∗‘(𝑓‘𝐴)) · 0) = 0)
104	103	ifeq2d 4506	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → if(𝑓 ∈ 𝑊, -1, ((∗‘(𝑓‘𝐴)) · 0)) = if(𝑓 ∈ 𝑊, -1, 0))
105	101, 104	eqtrd 2776	. . . . . . . . . . . . 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 2788	. . . . . . . . . . . 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 4519	. . . . . . . . . 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 2788	. . . . . . . . 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 7373	. . . . . . . 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 2776	. . . . . . 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 7375	. . . . . 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 2776	. . . . 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 15588	. . . 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 13878	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
116		inss1 4188	. . . . . . . . 9 ⊢ ((1...(⌊‘𝑥)) ∩ 𝑇) ⊆ (1...(⌊‘𝑥))
117		ssfi 9117	. . . . . . . . 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 16644	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (ϕ‘𝑁) ∈ ℕ)
120	119	nncnd 12169	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (ϕ‘𝑁) ∈ ℂ)
121	116	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1...(⌊‘𝑥)) ∩ 𝑇) ⊆ (1...(⌊‘𝑥)))
122	121	sselda 3944	. . . . . . . . 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 15669	. . . . . . 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 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ϕ‘𝑁) ∈ ℂ)
126	125, 39	mulcld 11175	. . . . . . . . . 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 3143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∀𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((ϕ‘𝑁) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
129	115	olcd 872	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1...(⌊‘𝑥)) ⊆ (ℤ_≥‘1) ∨ (1...(⌊‘𝑥)) ∈ Fin))
130		sumss2 15611	. . . . . . . 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 836	. . . . . . 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 3926	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇) ↔ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ 𝑇))
133	132	baib 536	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → (𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇) ↔ 𝑛 ∈ 𝑇))
134	133	adantl 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇) ↔ 𝑛 ∈ 𝑇))
135		rpvmasum2.t	. . . . . . . . . . . . 13 ⊢ 𝑇 = (^◡𝐿 “ {𝐴})
136	135	eleq2i 2829	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑇 ↔ 𝑛 ∈ (^◡𝐿 “ {𝐴}))
137	27	ffnd 6669	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐿 Fn ℤ)
138		fniniseg 7010	. . . . . . . . . . . . . 14 ⊢ (𝐿 Fn ℤ → (𝑛 ∈ (^◡𝐿 “ {𝐴}) ↔ (𝑛 ∈ ℤ ∧ (𝐿‘𝑛) = 𝐴)))
139	138	baibd 540	. . . . . . . . . . . . 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 282	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ 𝑇 ↔ (𝐿‘𝑛) = 𝐴))
142	134, 141	bitr2d 279	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝐿‘𝑛) = 𝐴 ↔ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)))
143	39	mul02d 11353	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (0 · ((Λ‘𝑛) / 𝑛)) = 0)
144	142, 143	ifbieq2d 4512	. . . . . . . . 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 7454	. . . . . . . . . 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 724	. . . . . . . . . . . . 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 750	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑓 ∈ 𝐷) → (∗‘(𝑓‘𝐴)) ∈ ℂ)
149	32, 148	mulcld 11175	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑓 ∈ 𝐷) → ((𝑓‘(𝐿‘𝑛)) · (∗‘(𝑓‘𝐴))) ∈ ℂ)
150	147, 39, 149	fsummulc1 15670	. . . . . . . . . . 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 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐴 ∈ 𝑈)
152	3, 4, 8, 9, 12, 146, 30, 151	sum2dchr 26622	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑓 ∈ 𝐷 ((𝑓‘(𝐿‘𝑛)) · (∗‘(𝑓‘𝐴))) = if((𝐿‘𝑛) = 𝐴, (ϕ‘𝑁), 0))
153	152	oveq1d 7372	. . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑓 ∈ 𝐷) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
155		mulass 11139	. . . . . . . . . . . . . 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 11320	. . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 15588	. . . . . . . . . . 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 2784	. . . . . . . . . 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 2790	. . . . . . . . 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 2778	. . . . . . . 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 15588	. . . . . . 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 2780	. . . . . 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 15660	. . . . . 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 2776	. . . . 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 26602	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel)
168		ablgrp 19567	. . . . . . . . . 10 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
169	4, 65	grpidcl 18778	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 1 ∈ 𝐷)
170	2, 167, 168, 169	4syl 19	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ 𝐷)
171	47	mulid1d 11172	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · 1) = (log‘𝑥))
172	171, 47	eqeltrd 2838	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · 1) ∈ ℂ)
173		iftrue 4492	. . . . . . . . . . 11 ⊢ (𝑓 = 1 → if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)) = 1)
174	173	oveq2d 7373	. . . . . . . . . 10 ⊢ (𝑓 = 1 → ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))) = ((log‘𝑥) · 1))
175	174	sumsn 15631	. . . . . . . . 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 4747	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝐷 ∖ { 1 }) ↔ (𝑓 ∈ 𝐷 ∧ 𝑓 ≠ 1 ))
178		ifnefalse 4498	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ≠ 1 → if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)) = if(𝑓 ∈ 𝑊, -1, 0))
179	178	ad2antll 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑓 ∈ 𝐷 ∧ 𝑓 ≠ 1 )) → if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0)) = if(𝑓 ∈ 𝑊, -1, 0))
180		negeq 11393	. . . . . . . . . . . . . . 15 ⊢ (if(𝑓 ∈ 𝑊, 1, 0) = 1 → -if(𝑓 ∈ 𝑊, 1, 0) = -1)
181		negeq 11393	. . . . . . . . . . . . . . . 16 ⊢ (if(𝑓 ∈ 𝑊, 1, 0) = 0 → -if(𝑓 ∈ 𝑊, 1, 0) = -0)
182		neg0 11447	. . . . . . . . . . . . . . . 16 ⊢ -0 = 0
183	181, 182	eqtrdi 2792	. . . . . . . . . . . . . . 15 ⊢ (if(𝑓 ∈ 𝑊, 1, 0) = 0 → -if(𝑓 ∈ 𝑊, 1, 0) = 0)
184	180, 183	ifsb 4499	. . . . . . . . . . . . . 14 ⊢ -if(𝑓 ∈ 𝑊, 1, 0) = if(𝑓 ∈ 𝑊, -1, 0)
185	179, 184	eqtr4di 2794	. . . . . . . . . . . . 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 7373	. . . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑓 ∈ 𝐷 ∧ 𝑓 ≠ 1 )) → (log‘𝑥) ∈ ℂ)
188	49, 51	ifcli 4533	. . . . . . . . . . . . 13 ⊢ if(𝑓 ∈ 𝑊, 1, 0) ∈ ℂ
189		mulneg2 11592	. . . . . . . . . . . . 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 2776	. . . . . . . . . . 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 15588	. . . . . . . . 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 9123	. . . . . . . . . . 11 ⊢ (𝐷 ∈ Fin → (𝐷 ∖ { 1 }) ∈ Fin)
195	6, 194	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝐷 ∖ { 1 }) ∈ Fin)
196	47	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ (𝐷 ∖ { 1 })) → (log‘𝑥) ∈ ℂ)
197		mulcl 11135	. . . . . . . . . . 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 15672	. . . . . . . . 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 15669	. . . . . . . . . . 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 4040	. . . . . . . . . . . . . . . 16 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
204		difss 4091	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∖ { 1 }) ⊆ 𝐷
205	203, 204	sstri 3953	. . . . . . . . . . . . . . 15 ⊢ 𝑊 ⊆ 𝐷
206		ssfi 9117	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑊 ⊆ 𝐷) → 𝑊 ∈ Fin)
207	6, 205, 206	sylancl 586	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑊 ∈ Fin)
208		fsumconst 15675	. . . . . . . . . . . . . 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 3147	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∀𝑓 ∈ 𝑊 1 ∈ ℂ)
213	195	olcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝐷 ∖ { 1 }) ⊆ (ℤ_≥‘1) ∨ (𝐷 ∖ { 1 }) ∈ Fin))
214		sumss2 15611	. . . . . . . . . . . . . 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 836	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑓 ∈ 𝑊 1 = Σ𝑓 ∈ (𝐷 ∖ { 1 })if(𝑓 ∈ 𝑊, 1, 0))
216		hashcl 14256	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ Fin → (♯‘𝑊) ∈ ℕ₀)
217	207, 216	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (♯‘𝑊) ∈ ℕ₀)
218	217	nn0cnd 12475	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (♯‘𝑊) ∈ ℂ)
219	218	mulid1d 11172	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((♯‘𝑊) · 1) = (♯‘𝑊))
220	209, 215, 219	3eqtr3d 2784	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑓 ∈ (𝐷 ∖ { 1 })if(𝑓 ∈ 𝑊, 1, 0) = (♯‘𝑊))
221	220	oveq2d 7373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · Σ𝑓 ∈ (𝐷 ∖ { 1 })if(𝑓 ∈ 𝑊, 1, 0)) = ((log‘𝑥) · (♯‘𝑊)))
222	201, 221	eqtr3d 2778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑓 ∈ (𝐷 ∖ { 1 })((log‘𝑥) · if(𝑓 ∈ 𝑊, 1, 0)) = ((log‘𝑥) · (♯‘𝑊)))
223	222	negeqd 11395	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → -Σ𝑓 ∈ (𝐷 ∖ { 1 })((log‘𝑥) · if(𝑓 ∈ 𝑊, 1, 0)) = -((log‘𝑥) · (♯‘𝑊)))
224	193, 199, 223	3eqtrd 2780	. . . . . . . 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 7375	. . . . . . 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 11175	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · (♯‘𝑊)) ∈ ℂ)
227	172, 226	negsubd 11518	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((log‘𝑥) · 1) + -((log‘𝑥) · (♯‘𝑊))) = (((log‘𝑥) · 1) − ((log‘𝑥) · (♯‘𝑊))))
228	225, 227	eqtrd 2776	. . . . . 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 4431	. . . . . . . 8 ⊢ ({ 1 } ∩ (𝐷 ∖ { 1 })) = ∅
230	229	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ({ 1 } ∩ (𝐷 ∖ { 1 })) = ∅)
231		undif2 4436	. . . . . . . 8 ⊢ ({ 1 } ∪ (𝐷 ∖ { 1 })) = ({ 1 } ∪ 𝐷)
232	170	snssd 4769	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → { 1 } ⊆ 𝐷)
233		ssequn1 4140	. . . . . . . . 9 ⊢ ({ 1 } ⊆ 𝐷 ↔ ({ 1 } ∪ 𝐷) = 𝐷)
234	232, 233	sylib 217	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ({ 1 } ∪ 𝐷) = 𝐷)
235	231, 234	eqtr2id 2789	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐷 = ({ 1 } ∪ (𝐷 ∖ { 1 })))
236	230, 235, 6, 55	fsumsplit 15626	. . . . . 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 11611	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · (1 − (♯‘𝑊))) = (((log‘𝑥) · 1) − ((log‘𝑥) · (♯‘𝑊))))
238	228, 236, 237	3eqtr4rd 2787	. . . . 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 7375	. . . 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 2786	. . 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 5205	. 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 12922	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
243	242	a1i 11	. . 3 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
244	1, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐷 ∈ Fin)
245	17	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → (𝑓‘𝐴) ∈ ℂ)
246	245	cjcld 15081	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑓 ∈ 𝐷) → (∗‘(𝑓‘𝐴)) ∈ ℂ)
247	58, 55	subcld 11512	. . . . 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 11175	. . . 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 467	. . 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 481	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → (∗‘(𝑓‘𝐴)) ∈ ℂ)
251	247	an32s 650	. . . 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 15502	. . . . 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 6841	. . . . . . . . . . . 12 ⊢ (𝑓 = 1 → (𝑓‘(𝐿‘𝑛)) = ( 1 ‘(𝐿‘𝑛)))
255	254	oveq1d 7372	. . . . . . . . . . 11 ⊢ (𝑓 = 1 → ((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) = (( 1 ‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)))
256	255	sumeq2sdv 15589	. . . . . . . . . 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 7375	. . . . . . . . 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 482	. . . . . . . 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 11183	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℂ)
260	259	mulid1d 11172	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘𝑥) · 1) = (log‘𝑥))
261	260	oveq2d 7373	. . . . . . . 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 2796	. . . . . . 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 5205	. . . . . 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 26835	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) − (log‘𝑥))) ∈ 𝑂(1))
265	264	ad2antrr 724	. . . . . 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 2838	. . . . 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 7373	. . . . . . . . . 10 ⊢ (𝑓 ≠ 1 → ((log‘𝑥) · if(𝑓 = 1 , 1, if(𝑓 ∈ 𝑊, -1, 0))) = ((log‘𝑥) · if(𝑓 ∈ 𝑊, -1, 0)))
268	267	oveq2d 7373	. . . . . . . . 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 713	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
270		mulcom 11137	. . . . . . . . . . . . . . 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 11607	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → (-1 · (log‘𝑥)) = -(log‘𝑥))
273	271, 272	eqtrd 2776	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · -1) = -(log‘𝑥))
274	269	mul01d 11354	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · 0) = 0)
275	273, 274	ifeq12d 4507	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → if(𝑓 ∈ 𝑊, ((log‘𝑥) · -1), ((log‘𝑥) · 0)) = if(𝑓 ∈ 𝑊, -(log‘𝑥), 0))
276		ovif2 7455	. . . . . . . . . . . 12 ⊢ ((log‘𝑥) · if(𝑓 ∈ 𝑊, -1, 0)) = if(𝑓 ∈ 𝑊, ((log‘𝑥) · -1), ((log‘𝑥) · 0))
277		negeq 11393	. . . . . . . . . . . . 13 ⊢ (if(𝑓 ∈ 𝑊, (log‘𝑥), 0) = (log‘𝑥) → -if(𝑓 ∈ 𝑊, (log‘𝑥), 0) = -(log‘𝑥))
278		negeq 11393	. . . . . . . . . . . . . 14 ⊢ (if(𝑓 ∈ 𝑊, (log‘𝑥), 0) = 0 → -if(𝑓 ∈ 𝑊, (log‘𝑥), 0) = -0)
279	278, 182	eqtrdi 2792	. . . . . . . . . . . . 13 ⊢ (if(𝑓 ∈ 𝑊, (log‘𝑥), 0) = 0 → -if(𝑓 ∈ 𝑊, (log‘𝑥), 0) = 0)
280	277, 279	ifsb 4499	. . . . . . . . . . . 12 ⊢ -if(𝑓 ∈ 𝑊, (log‘𝑥), 0) = if(𝑓 ∈ 𝑊, -(log‘𝑥), 0)
281	275, 276, 280	3eqtr4g 2801	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · if(𝑓 ∈ 𝑊, -1, 0)) = -if(𝑓 ∈ 𝑊, (log‘𝑥), 0))
282	281	oveq2d 7373	. . . . . . . . . 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 650	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑓‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
284		ifcl 4531	. . . . . . . . . . . 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 11519	. . . . . . . . . 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 2776	. . . . . . . . 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 2798	. . . . . . . 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 650	. . . . . . 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 5205	. . . . . 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 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → 𝑁 ∈ ℕ)
292		simplr 767	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → 𝑓 ∈ 𝐷)
293		simpr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ≠ 1 ) → 𝑓 ≠ 1 )
294		eqid 2736	. . . . . . . 8 ⊢ (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎)) = (𝑎 ∈ ℕ ↦ ((𝑓‘(𝐿‘𝑎)) / 𝑎))
295	8, 23, 291, 3, 4, 65, 292, 293, 294	dchrmusumlema 26841	. . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝑁 ∈ ℕ)
297	296	ad2antrr 724	. . . . . . . . . . . 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 481	. . . . . . . . . . . 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 767	. . . . . . . . . . . 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 769	. . . . . . . . . . . 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 779	. . . . . . . . . . . 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 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)))) → ∀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 26852	. . . . . . . . . . 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 4508	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ 𝑊 ↔ 𝑡 = 0) → if(𝑓 ∈ 𝑊, (log‘𝑥), 0) = if(𝑡 = 0, (log‘𝑥), 0))
305	304	oveq2d 7373	. . . . . . . . . . . 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 5207	. . . . . . . . . . 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 26851	. . . . . . . . . 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 2838	. . . . . . . . 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 3153	. . . . . . . 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 1936	. . . . . . 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 2838	. . . . 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 3032	. . . 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 15507	. . 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 15697	. 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 2839	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 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 {crab 3407 ∖ cdif 3907 ∪ cun 3908 ∩ cin 3909 ⊆ wss 3910 ∅c0 4282 ifcif 4486 {csn 4586 class class class wbr 5105 ↦ cmpt 5188 ^◡ccnv 5632 “ cima 5636 Fn wfn 6491 ⟶wf 6492 –onto→wfo 6494 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 +∞cpnf 11186 ≤ cle 11190 − cmin 11385 -cneg 11386 / cdiv 11812 ℕcn 12153 ℕ₀cn0 12413 ℤcz 12499 ℤ_≥cuz 12763 ℝ⁺crp 12915 [,)cico 13266 ...cfz 13424 ⌊cfl 13695 seqcseq 13906 ♯chash 14230 ∗ccj 14981 abscabs 15119 ⇝ cli 15366 𝑂(1)co1 15368 Σcsu 15570 ϕcphi 16636 Basecbs 17083 0_gc0g 17321 MndHom cmhm 18599 Grpcgrp 18748 Abelcabl 19563 mulGrpcmgp 19896 1_rcur 19913 Unitcui 20068 ℂ_fldccnfld 20796 ℤRHomczrh 20900 ℤ/nℤczn 20903 logclog 25910 Λcvma 26441 DChrcdchr 26580

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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-disj 5071 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-rpss 7660 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-omul 8417 df-er 8648 df-ec 8650 df-qs 8654 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-acn 9878 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-xnn0 12486 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-word 14403 df-concat 14459 df-s1 14484 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-o1 15372 df-lo1 15373 df-sum 15571 df-ef 15950 df-e 15951 df-sin 15952 df-cos 15953 df-tan 15954 df-pi 15955 df-dvds 16137 df-gcd 16375 df-prm 16548 df-phi 16638 df-pc 16709 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-qus 17391 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-nsg 18926 df-eqg 18927 df-ghm 19006 df-gim 19049 df-ga 19070 df-cntz 19097 df-oppg 19124 df-od 19310 df-gex 19311 df-pgp 19312 df-lsm 19418 df-pj1 19419 df-cmn 19564 df-abl 19565 df-cyg 19655 df-dprd 19774 df-dpj 19775 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-rnghom 20146 df-drng 20187 df-subrg 20220 df-lmod 20324 df-lss 20393 df-lsp 20433 df-sra 20633 df-rgmod 20634 df-lidl 20635 df-rsp 20636 df-2idl 20702 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-zring 20870 df-zrh 20904 df-zn 20907 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-0p 25034 df-limc 25230 df-dv 25231 df-ply 25549 df-idp 25550 df-coe 25551 df-dgr 25552 df-quot 25651 df-ulm 25736 df-log 25912 df-cxp 25913 df-atan 26217 df-em 26342 df-cht 26446 df-vma 26447 df-chp 26448 df-ppi 26449 df-mu 26450 df-dchr 26581

This theorem is referenced by: dchrisum0re 26861 rpvmasum 26874

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