Theorem selberg2lem 27399

Description: Lemma for selberg2 27400. Equation 10.4.12 of [Shapiro], p. 420. (Contributed by Mario Carneiro, 23-May-2016.)

Assertion

Ref	Expression
selberg2lem	⊢ (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1)

Distinct variable group:   𝑥,𝑛

Proof of Theorem selberg2lem

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpre 12979	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ)
2		chpcl 26972	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
3	1, 2	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℝ)
4	3	recnd 11239	. . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℂ)
5		rprege0 12986	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
6		flge0nn0 13782	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ0)
7	5, 6	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℕ0)
8		nn0p1nn 12508	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑥) ∈ ℕ0 → ((⌊‘𝑥) + 1) ∈ ℕ)
9	7, 8	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ∈ ℕ)
10	9	nnrpd 13011	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ∈ ℝ+)
11	10	relogcld 26473	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → (log‘((⌊‘𝑥) + 1)) ∈ ℝ)
12	11	recnd 11239	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ+ → (log‘((⌊‘𝑥) + 1)) ∈ ℂ)
13		relogcl 26426	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
14	13	recnd 11239	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℂ)
15	12, 14	subcld 11568	. . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ)
16	4, 15	mulcld 11231	. . . . . 6 ⊢ (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ ℂ)
17		fzfid 13935	. . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → (1...(⌊‘𝑥)) ∈ Fin)
18		elfznn 13527	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
19	18	adantl 481	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
20	19	nnrpd 13011	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
21		1rp 12975	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ+
22		rpaddcl 12993	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝑛 + 1) ∈ ℝ+)
23	21, 22	mpan2 688	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ+ → (𝑛 + 1) ∈ ℝ+)
24	23	relogcld 26473	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ+ → (log‘(𝑛 + 1)) ∈ ℝ)
25		relogcl 26426	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ+ → (log‘𝑛) ∈ ℝ)
26	24, 25	resubcld 11639	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ+ → ((log‘(𝑛 + 1)) − (log‘𝑛)) ∈ ℝ)
27		rpre 12979	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ+ → 𝑛 ∈ ℝ)
28		chpcl 26972	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ → (ψ‘𝑛) ∈ ℝ)
29	27, 28	syl 17	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ+ → (ψ‘𝑛) ∈ ℝ)
30	26, 29	remulcld 11241	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ+ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℝ)
31	30	recnd 11239	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ+ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
32	20, 31	syl 17	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
33	17, 32	fsumcl 15676	. . . . . 6 ⊢ (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
34		rpcnne0 12989	. . . . . 6 ⊢ (𝑥 ∈ ℝ+ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
35		divsubdir 11905	. . . . . 6 ⊢ ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ ℂ ∧ Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) / 𝑥) = ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)))
36	16, 33, 34, 35	syl3anc 1368	. . . . 5 ⊢ (𝑥 ∈ ℝ+ → ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) / 𝑥) = ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)))
37	4, 12	mulcld 11231	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) ∈ ℂ)
38	4, 14	mulcld 11231	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
39	37, 38, 33	sub32d 11600	. . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → ((((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − ((ψ‘𝑥) · (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = ((((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) − ((ψ‘𝑥) · (log‘𝑥))))
40	4, 12, 14	subdid 11667	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − ((ψ‘𝑥) · (log‘𝑥))))
41	40	oveq1d 7416	. . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = ((((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − ((ψ‘𝑥) · (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))))
42		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (log‘𝑚) = (log‘𝑛))
43		fvoveq1 7424	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1)))
44	42, 43	jca 511	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((log‘𝑚) = (log‘𝑛) ∧ (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1))))
45		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (log‘𝑚) = (log‘(𝑛 + 1)))
46		fvoveq1 7424	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1)))
47	45, 46	jca 511	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → ((log‘𝑚) = (log‘(𝑛 + 1)) ∧ (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1))))
48		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (log‘𝑚) = (log‘1))
49		log1 26436	. . . . . . . . . . . 12 ⊢ (log‘1) = 0
50	48, 49	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (log‘𝑚) = 0)
51		oveq1 7408	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
52		1m1e0 12281	. . . . . . . . . . . . . 14 ⊢ (1 − 1) = 0
53	51, 52	eqtrdi 2780	. . . . . . . . . . . . 13 ⊢ (𝑚 = 1 → (𝑚 − 1) = 0)
54	53	fveq2d 6885	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (ψ‘(𝑚 − 1)) = (ψ‘0))
55		2pos 12312	. . . . . . . . . . . . 13 ⊢ 0 < 2
56		0re 11213	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
57		chpeq0 27057	. . . . . . . . . . . . . 14 ⊢ (0 ∈ ℝ → ((ψ‘0) = 0 ↔ 0 < 2))
58	56, 57	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((ψ‘0) = 0 ↔ 0 < 2)
59	55, 58	mpbir 230	. . . . . . . . . . . 12 ⊢ (ψ‘0) = 0
60	54, 59	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (ψ‘(𝑚 − 1)) = 0)
61	50, 60	jca 511	. . . . . . . . . 10 ⊢ (𝑚 = 1 → ((log‘𝑚) = 0 ∧ (ψ‘(𝑚 − 1)) = 0))
62		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (log‘𝑚) = (log‘((⌊‘𝑥) + 1)))
63		fvoveq1 7424	. . . . . . . . . . 11 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1)))
64	62, 63	jca 511	. . . . . . . . . 10 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → ((log‘𝑚) = (log‘((⌊‘𝑥) + 1)) ∧ (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1))))
65		nnuz 12862	. . . . . . . . . . 11 ⊢ ℕ = (ℤ≥‘1)
66	9, 65	eleqtrdi 2835	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ∈ (ℤ≥‘1))
67		elfznn 13527	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1...((⌊‘𝑥) + 1)) → 𝑚 ∈ ℕ)
68	67	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
69	68	nnrpd 13011	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ+)
70	69	relogcld 26473	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (log‘𝑚) ∈ ℝ)
71	70	recnd 11239	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (log‘𝑚) ∈ ℂ)
72	68	nnred 12224	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ)
73		peano2rem 11524	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℝ → (𝑚 − 1) ∈ ℝ)
74	72, 73	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (𝑚 − 1) ∈ ℝ)
75		chpcl 26972	. . . . . . . . . . . 12 ⊢ ((𝑚 − 1) ∈ ℝ → (ψ‘(𝑚 − 1)) ∈ ℝ)
76	74, 75	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℝ)
77	76	recnd 11239	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℂ)
78	44, 47, 61, 64, 66, 71, 77	fsumparts 15749	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) − Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘((𝑛 + 1) − 1)))))
79	7	nn0zd 12581	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℤ)
80		fzval3 13698	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑥) ∈ ℤ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
81	79, 80	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ+ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
82	81	eqcomd 2730	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → (1..^((⌊‘𝑥) + 1)) = (1...(⌊‘𝑥)))
83		nnm1nn0 12510	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
84	19, 83	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℕ0)
85	84	nn0red 12530	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
86		chpcl 26972	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 − 1) ∈ ℝ → (ψ‘(𝑛 − 1)) ∈ ℝ)
87	85, 86	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℝ)
88	87	recnd 11239	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℂ)
89		vmacl 26966	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
90	19, 89	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
91	90	recnd 11239	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
92	19	nncnd 12225	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
93		ax-1cn 11164	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
94		pncan 11463	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
95	92, 93, 94	sylancl 585	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = 𝑛)
96		npcan 11466	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
97	92, 93, 96	sylancl 585	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 − 1) + 1) = 𝑛)
98	95, 97	eqtr4d 2767	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = ((𝑛 − 1) + 1))
99	98	fveq2d 6885	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = (ψ‘((𝑛 − 1) + 1)))
100		chpp1 27003	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 − 1) ∈ ℕ0 → (ψ‘((𝑛 − 1) + 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))))
101	84, 100	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 − 1) + 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))))
102	97	fveq2d 6885	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘((𝑛 − 1) + 1)) = (Λ‘𝑛))
103	102	oveq2d 7417	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
104	99, 101, 103	3eqtrd 2768	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
105	88, 91, 104	mvrladdd 11624	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) = (Λ‘𝑛))
106	105	oveq2d 7417	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((log‘𝑛) · (Λ‘𝑛)))
107	20	relogcld 26473	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
108	107	recnd 11239	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
109	91, 108	mulcomd 11232	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) = ((log‘𝑛) · (Λ‘𝑛)))
110	106, 109	eqtr4d 2767	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((Λ‘𝑛) · (log‘𝑛)))
111	82, 110	sumeq12rdv 15650	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)))
112	7	nn0cnd 12531	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℂ)
113		pncan 11463	. . . . . . . . . . . . . . . . 17 ⊢ (((⌊‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
114	112, 93, 113	sylancl 585	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ+ → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
115	114	fveq2d 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ+ → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘(⌊‘𝑥)))
116		chpfl 26998	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
117	1, 116	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ+ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
118	115, 117	eqtrd 2764	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ+ → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘𝑥))
119	118	oveq2d 7417	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) = ((log‘((⌊‘𝑥) + 1)) · (ψ‘𝑥)))
120	12, 4	mulcomd 11232	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘𝑥)) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
121	119, 120	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
122		0cn 11203	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℂ
123	122	mul01i 11401	. . . . . . . . . . . . 13 ⊢ (0 · 0) = 0
124	123	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ → (0 · 0) = 0)
125	121, 124	oveq12d 7419	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ+ → (((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − 0))
126	37	subid1d 11557	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ+ → (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − 0) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
127	125, 126	eqtrd 2764	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → (((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
128	95	fveq2d 6885	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = (ψ‘𝑛))
129	128	oveq2d 7417	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘((𝑛 + 1) − 1))) = (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
130	82, 129	sumeq12rdv 15650	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘((𝑛 + 1) − 1))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
131	127, 130	oveq12d 7419	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → ((((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) − Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘((𝑛 + 1) − 1)))) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))))
132	78, 111, 131	3eqtr3d 2772	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))))
133	132	oveq1d 7416	. . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) = ((((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) − ((ψ‘𝑥) · (log‘𝑥))))
134	39, 41, 133	3eqtr4d 2774	. . . . . 6 ⊢ (𝑥 ∈ ℝ+ → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))))
135	134	oveq1d 7416	. . . . 5 ⊢ (𝑥 ∈ ℝ+ → ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))
136		div23 11888	. . . . . . 7 ⊢ (((ψ‘𝑥) ∈ ℂ ∧ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) = (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
137	4, 15, 34, 136	syl3anc 1368	. . . . . 6 ⊢ (𝑥 ∈ ℝ+ → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) = (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
138	137	oveq1d 7416	. . . . 5 ⊢ (𝑥 ∈ ℝ+ → ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)) = ((((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)))
139	36, 135, 138	3eqtr3rd 2773	. . . 4 ⊢ (𝑥 ∈ ℝ+ → ((((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))
140	139	mpteq2ia 5241	. . 3 ⊢ (𝑥 ∈ ℝ+ ↦ ((((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥))) = (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))
141		ovexd 7436	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ V)
142		ovexd 7436	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥) ∈ V)
143		reex 11197	. . . . . . . 8 ⊢ ℝ ∈ V
144		rpssre 12978	. . . . . . . 8 ⊢ ℝ+ ⊆ ℝ
145	143, 144	ssexi 5312	. . . . . . 7 ⊢ ℝ+ ∈ V
146	145	a1i 11	. . . . . 6 ⊢ (⊤ → ℝ+ ∈ V)
147		ovexd 7436	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((ψ‘𝑥) / 𝑥) ∈ V)
148	15	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ)
149		eqidd 2725	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) = (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)))
150		eqidd 2725	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) = (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
151	146, 147, 148, 149, 150	offval2 7683	. . . . 5 ⊢ (⊤ → ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∘f · (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))))
152		chpo1ub 27329	. . . . . 6 ⊢ (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
153		0red 11214	. . . . . . . 8 ⊢ (⊤ → 0 ∈ ℝ)
154		1red 11212	. . . . . . . 8 ⊢ (⊤ → 1 ∈ ℝ)
155		divrcnv 15795	. . . . . . . . 9 ⊢ (1 ∈ ℂ → (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) ⇝𝑟 0)
156	93, 155	mp1i 13	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) ⇝𝑟 0)
157		rpreccl 12997	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ+)
158	157	rpred 13013	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ)
159	158	adantl 481	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ)
160	11, 13	resubcld 11639	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℝ)
161	160	adantl 481	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℝ)
162		rpaddcl 12993	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝑥 + 1) ∈ ℝ+)
163	21, 162	mpan2 688	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ → (𝑥 + 1) ∈ ℝ+)
164	163	relogcld 26473	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ+ → (log‘(𝑥 + 1)) ∈ ℝ)
165	164, 13	resubcld 11639	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → ((log‘(𝑥 + 1)) − (log‘𝑥)) ∈ ℝ)
166	7	nn0red 12530	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℝ)
167		1red 11212	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ+ → 1 ∈ ℝ)
168		flle 13761	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
169	1, 168	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ+ → (⌊‘𝑥) ≤ 𝑥)
170	166, 1, 167, 169	leadd1dd 11825	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ≤ (𝑥 + 1))
171	10, 163	logled 26477	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ → (((⌊‘𝑥) + 1) ≤ (𝑥 + 1) ↔ (log‘((⌊‘𝑥) + 1)) ≤ (log‘(𝑥 + 1))))
172	170, 171	mpbid 231	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ+ → (log‘((⌊‘𝑥) + 1)) ≤ (log‘(𝑥 + 1)))
173	11, 164, 13, 172	lesub1dd 11827	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ ((log‘(𝑥 + 1)) − (log‘𝑥)))
174		logdifbnd 26842	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → ((log‘(𝑥 + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
175	160, 165, 158, 173, 174	letrd 11368	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
176	175	ad2antrl 725	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
177		fllep1 13763	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
178	1, 177	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ≤ ((⌊‘𝑥) + 1))
179		logleb 26453	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ ((⌊‘𝑥) + 1) ∈ ℝ+) → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
180	10, 179	mpdan 684	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ+ → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
181	178, 180	mpbid 231	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1)))
182	11, 13	subge0d 11801	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → (0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
183	181, 182	mpbird 257	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → 0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))
184	183	ad2antrl 725	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))
185	153, 154, 156, 159, 161, 176, 184	rlimsqz2 15594	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ⇝𝑟 0)
186		rlimo1 15558	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ⇝𝑟 0 → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ 𝑂(1))
187	185, 186	syl 17	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ 𝑂(1))
188		o1mul 15556	. . . . . 6 ⊢ (((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1) ∧ (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ 𝑂(1)) → ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∘f · (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
189	152, 187, 188	sylancr 586	. . . . 5 ⊢ (⊤ → ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∘f · (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
190	151, 189	eqeltrrd 2826	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
191		nnrp 12982	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ+)
192	191	ssriv 3978	. . . . . . . 8 ⊢ ℕ ⊆ ℝ+
193	192	a1i 11	. . . . . . 7 ⊢ (⊤ → ℕ ⊆ ℝ+)
194	193	sselda 3974	. . . . . 6 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ+)
195	194, 31	syl 17	. . . . 5 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
196		chpo1ub 27329	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ+ ↦ ((ψ‘𝑛) / 𝑛)) ∈ 𝑂(1)
197	196	a1i 11	. . . . . . 7 ⊢ (⊤ → (𝑛 ∈ ℝ+ ↦ ((ψ‘𝑛) / 𝑛)) ∈ 𝑂(1))
198		rerpdivcl 13001	. . . . . . . . 9 ⊢ (((ψ‘𝑛) ∈ ℝ ∧ 𝑛 ∈ ℝ+) → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
199	29, 198	mpancom 685	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ+ → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
200	199	adantl 481	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℝ+) → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
201	31	adantl 481	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℝ+) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
202		rpreccl 12997	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ+ → (1 / 𝑛) ∈ ℝ+)
203	202	rpred 13013	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ+ → (1 / 𝑛) ∈ ℝ)
204		chpge0 26974	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ → 0 ≤ (ψ‘𝑛))
205	27, 204	syl 17	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ+ → 0 ≤ (ψ‘𝑛))
206		logdifbnd 26842	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ+ → ((log‘(𝑛 + 1)) − (log‘𝑛)) ≤ (1 / 𝑛))
207	26, 203, 29, 205, 206	lemul1ad 12150	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ+ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ≤ ((1 / 𝑛) · (ψ‘𝑛)))
208	27	lep1d 12142	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℝ+ → 𝑛 ≤ (𝑛 + 1))
209		logleb 26453	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℝ+ ∧ (𝑛 + 1) ∈ ℝ+) → (𝑛 ≤ (𝑛 + 1) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
210	23, 209	mpdan 684	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℝ+ → (𝑛 ≤ (𝑛 + 1) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
211	208, 210	mpbid 231	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ+ → (log‘𝑛) ≤ (log‘(𝑛 + 1)))
212	24, 25	subge0d 11801	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ+ → (0 ≤ ((log‘(𝑛 + 1)) − (log‘𝑛)) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
213	211, 212	mpbird 257	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ+ → 0 ≤ ((log‘(𝑛 + 1)) − (log‘𝑛)))
214	26, 29, 213, 205	mulge0d 11788	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ+ → 0 ≤ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
215	30, 214	absidd 15366	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ+ → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
216		rpregt0 12985	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ+ → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
217		divge0 12080	. . . . . . . . . . . 12 ⊢ ((((ψ‘𝑛) ∈ ℝ ∧ 0 ≤ (ψ‘𝑛)) ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → 0 ≤ ((ψ‘𝑛) / 𝑛))
218	29, 205, 216, 217	syl21anc 835	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ+ → 0 ≤ ((ψ‘𝑛) / 𝑛))
219	199, 218	absidd 15366	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ+ → (abs‘((ψ‘𝑛) / 𝑛)) = ((ψ‘𝑛) / 𝑛))
220	29	recnd 11239	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ+ → (ψ‘𝑛) ∈ ℂ)
221		rpcn 12981	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ+ → 𝑛 ∈ ℂ)
222		rpne0 12987	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ+ → 𝑛 ≠ 0)
223	220, 221, 222	divrec2d 11991	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ+ → ((ψ‘𝑛) / 𝑛) = ((1 / 𝑛) · (ψ‘𝑛)))
224	219, 223	eqtrd 2764	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ+ → (abs‘((ψ‘𝑛) / 𝑛)) = ((1 / 𝑛) · (ψ‘𝑛)))
225	207, 215, 224	3brtr4d 5170	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ+ → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ≤ (abs‘((ψ‘𝑛) / 𝑛)))
226	225	ad2antrl 725	. . . . . . 7 ⊢ ((⊤ ∧ (𝑛 ∈ ℝ+ ∧ 1 ≤ 𝑛)) → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ≤ (abs‘((ψ‘𝑛) / 𝑛)))
227	154, 197, 200, 201, 226	o1le 15596	. . . . . 6 ⊢ (⊤ → (𝑛 ∈ ℝ+ ↦ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ∈ 𝑂(1))
228	193, 227	o1res2 15504	. . . . 5 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ∈ 𝑂(1))
229	195, 228	o1fsum 15756	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)) ∈ 𝑂(1))
230	141, 142, 190, 229	o1sub2 15567	. . 3 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥))) ∈ 𝑂(1))
231	140, 230	eqeltrrid 2830	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1))
232	231	mptru 1540	1 ⊢ (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1)

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533  ⊤wtru 1534   ∈ wcel 2098   ≠ wne 2932  Vcvv 3466   ⊆ wss 3940   class class class wbr 5138   ↦ cmpt 5221  ‘cfv 6533  (class class class)co 7401   ∘_f cof 7661  ℂcc 11104  ℝcr 11105  0cc0 11106  1c1 11107   + caddc 11109   · cmul 11111   < clt 11245   ≤ cle 11246   − cmin 11441   / cdiv 11868  ℕcn 12209  2c2 12264  ℕ₀cn0 12469  ℤcz 12555  ℤ_≥cuz 12819  ℝ⁺crp 12971  ...cfz 13481  ..^cfzo 13624  ⌊cfl 13752  abscabs 15178   ⇝_𝑟 crli 15426  𝑂(1)co1 15427  Σcsu 15629  logclog 26405  Λcvma 26940  ψcchp 26941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184  ax-addf 11185

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-of 7663  df-om 7849  df-1st 7968  df-2nd 7969  df-supp 8141  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-2o 8462  df-oadd 8465  df-er 8699  df-map 8818  df-pm 8819  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-fi 9402  df-sup 9433  df-inf 9434  df-oi 9501  df-dju 9892  df-card 9930  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-xnn0 12542  df-z 12556  df-dec 12675  df-uz 12820  df-q 12930  df-rp 12972  df-xneg 13089  df-xadd 13090  df-xmul 13091  df-ioo 13325  df-ioc 13326  df-ico 13327  df-icc 13328  df-fz 13482  df-fzo 13625  df-fl 13754  df-mod 13832  df-seq 13964  df-exp 14025  df-fac 14231  df-bc 14260  df-hash 14288  df-shft 15011  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-limsup 15412  df-clim 15429  df-rlim 15430  df-o1 15431  df-lo1 15432  df-sum 15630  df-ef 16008  df-e 16009  df-sin 16010  df-cos 16011  df-pi 16013  df-dvds 16195  df-gcd 16433  df-prm 16606  df-pc 16769  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-mulr 17210  df-starv 17211  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-unif 17219  df-hom 17220  df-cco 17221  df-rest 17367  df-topn 17368  df-0g 17386  df-gsum 17387  df-topgen 17388  df-pt 17389  df-prds 17392  df-xrs 17447  df-qtop 17452  df-imas 17453  df-xps 17455  df-mre 17529  df-mrc 17530  df-acs 17532  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-submnd 18704  df-mulg 18986  df-cntz 19223  df-cmn 19692  df-psmet 21220  df-xmet 21221  df-met 21222  df-bl 21223  df-mopn 21224  df-fbas 21225  df-fg 21226  df-cnfld 21229  df-top 22718  df-topon 22735  df-topsp 22757  df-bases 22771  df-cld 22845  df-ntr 22846  df-cls 22847  df-nei 22924  df-lp 22962  df-perf 22963  df-cn 23053  df-cnp 23054  df-haus 23141  df-tx 23388  df-hmeo 23581  df-fil 23672  df-fm 23764  df-flim 23765  df-flf 23766  df-xms 24148  df-ms 24149  df-tms 24150  df-cncf 24720  df-limc 25717  df-dv 25718  df-log 26407  df-cxp 26408  df-cht 26945  df-vma 26946  df-chp 26947  df-ppi 26948

This theorem is referenced by:  selberg2  27400  selberg3lem2  27407