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Theorem dchrvmasumlem2 27825

Description: Lemma for dchrvmasum 27852. (Contributed by Mario Carneiro, 4-May-2016.)

**Hypotheses**
Ref	Expression
rpvmasum.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
rpvmasum.l	⊢ 𝐿 = (ℤRHom‘𝑍)
rpvmasum.a	⊢ (𝜑 → 𝑁 ∈ ℕ)
rpvmasum.g	⊢ 𝐺 = (DChr‘𝑁)
rpvmasum.d	⊢ 𝐷 = (Base‘𝐺)
rpvmasum.1	⊢ 1 = (0_g‘𝐺)
dchrisum.b	⊢ (𝜑 → 𝑋 ∈ 𝐷)
dchrisum.n1	⊢ (𝜑 → 𝑋 ≠ 1 )
dchrvmasum.f	⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 𝐹 ∈ ℂ)
dchrvmasum.g	⊢ (𝑚 = (𝑥 / 𝑑) → 𝐹 = 𝐾)
dchrvmasum.c	⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
dchrvmasum.t	⊢ (𝜑 → 𝑇 ∈ ℂ)
dchrvmasum.1	⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → (abs‘(𝐹 − 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚)))
dchrvmasum.r	⊢ (𝜑 → 𝑅 ∈ ℝ)
dchrvmasum.2	⊢ (𝜑 → ∀𝑚 ∈ (1[,)3)(abs‘(𝐹 − 𝑇)) ≤ 𝑅)

**Assertion**
Ref	Expression
dchrvmasumlem2	⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ∈ 𝑂(1))

Distinct variable groups: 𝑥,𝑚, 1 𝑚,𝑑,𝑥,𝐶 𝐹,𝑑,𝑥 𝑚,𝐾 𝑚,𝑁,𝑥 𝜑,𝑑,𝑚,𝑥 𝑇,𝑑,𝑚,𝑥 𝑅,𝑑,𝑚,𝑥 𝑚,𝑍,𝑥 𝐷,𝑚,𝑥 𝐿,𝑑,𝑚,𝑥 𝑋,𝑑,𝑚,𝑥 Allowed substitution hints: 𝐷(𝑑) 1 (𝑑) 𝐹(𝑚) 𝐺(𝑥, 𝑚, 𝑑) 𝐾(𝑥, 𝑑) 𝑁(𝑑) 𝑍(𝑑)

**Proof of Theorem dchrvmasumlem2**
Step	Hyp	Ref	Expression
1		1red 11309	. 2 ⊢ (𝜑 → 1 ∈ ℝ)
2		dchrvmasum.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
3		elrege0 13585	. . . . . . 7 ⊢ (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
4	2, 3	sylib 221	. . . . . 6 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
5	4	simpld 500	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ)
6	5	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
7		fzfid 14116	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
8		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
9		elfznn 13687	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
10	9	nnrpd 13162	. . . . . . . 8 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℝ⁺)
11		rpdivcl 13147	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺) → (𝑥 / 𝑑) ∈ ℝ⁺)
12	8, 10, 11	syl2an 608	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ⁺)
13	12	relogcld 26951	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑑)) ∈ ℝ)
14	8	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
15	13, 14	rerpdivcld 13195	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑑)) / 𝑥) ∈ ℝ)
16	7, 15	fsumrecl 15900	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥) ∈ ℝ)
17	6, 16	remulcld 11339	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ∈ ℝ)
18		dchrvmasum.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ)
19		3nn 12422	. . . . . . 7 ⊢ 3 ∈ ℕ
20		nnrp 13132	. . . . . . 7 ⊢ (3 ∈ ℕ → 3 ∈ ℝ⁺)
21		relogcl 26903	. . . . . . 7 ⊢ (3 ∈ ℝ⁺ → (log‘3) ∈ ℝ)
22	19, 20, 21	mp2b 10	. . . . . 6 ⊢ (log‘3) ∈ ℝ
23		1re 11308	. . . . . 6 ⊢ 1 ∈ ℝ
24	22, 23	readdcli 11324	. . . . 5 ⊢ ((log‘3) + 1) ∈ ℝ
25		remulcl 11285	. . . . 5 ⊢ ((𝑅 ∈ ℝ ∧ ((log‘3) + 1) ∈ ℝ) → (𝑅 · ((log‘3) + 1)) ∈ ℝ)
26	18, 24, 25	sylancl 598	. . . 4 ⊢ (𝜑 → (𝑅 · ((log‘3) + 1)) ∈ ℝ)
27	26	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑅 · ((log‘3) + 1)) ∈ ℝ)
28		rpssre 13128	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
29	5	recnd 11337	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ)
30		o1const 15787	. . . . 5 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 𝐶 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 𝐶) ∈ 𝑂(1))
31	28, 29, 30	sylancr 599	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 𝐶) ∈ 𝑂(1))
32		logfacrlim2 27553	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ⇝_𝑟 1
33		rlimo1 15784	. . . . 5 ⊢ ((𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ⇝_𝑟 1 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ∈ 𝑂(1))
34	32, 33	mp1i 14	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ∈ 𝑂(1))
35	6, 16, 31, 34	o1mul2 15792	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥))) ∈ 𝑂(1))
36	26	recnd 11337	. . . 4 ⊢ (𝜑 → (𝑅 · ((log‘3) + 1)) ∈ ℂ)
37		o1const 15787	. . . 4 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (𝑅 · ((log‘3) + 1)) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (𝑅 · ((log‘3) + 1))) ∈ 𝑂(1))
38	28, 36, 37	sylancr 599	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝑅 · ((log‘3) + 1))) ∈ 𝑂(1))
39	17, 27, 35, 38	o1add2 15791	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1)))) ∈ 𝑂(1))
40	17, 27	readdcld 11338	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1))) ∈ ℝ)
41		dchrvmasum.g	. . . . . . . . 9 ⊢ (𝑚 = (𝑥 / 𝑑) → 𝐹 = 𝐾)
42	41	eleq1d 2845	. . . . . . . 8 ⊢ (𝑚 = (𝑥 / 𝑑) → (𝐹 ∈ ℂ ↔ 𝐾 ∈ ℂ))
43		dchrvmasum.f	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 𝐹 ∈ ℂ)
44	43	ralrimiva 3154	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ ℝ⁺ 𝐹 ∈ ℂ)
45	44	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ∀𝑚 ∈ ℝ⁺ 𝐹 ∈ ℂ)
46	42, 45, 12	rspcdva 3577	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐾 ∈ ℂ)
47		dchrvmasum.t	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℂ)
48	47	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑇 ∈ ℂ)
49	46, 48	subcld 11669	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐾 − 𝑇) ∈ ℂ)
50	49	abscld 15606	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝐾 − 𝑇)) ∈ ℝ)
51	9	adantl 487	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
52	50, 51	nndivred 12392	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝐾 − 𝑇)) / 𝑑) ∈ ℝ)
53	7, 52	fsumrecl 15900	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑) ∈ ℝ)
54	53	recnd 11337	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑) ∈ ℂ)
55	51	nnrpd 13162	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ⁺)
56	49	absge0d 15614	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(𝐾 − 𝑇)))
57	50, 55, 56	divge0d 13204	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((abs‘(𝐾 − 𝑇)) / 𝑑))
58	7, 52, 57	fsumge0 15962	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑))
59	53, 58	absidd 15590	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) = Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑))
60	59, 53	eqeltrd 2860	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ∈ ℝ)
61	40	recnd 11337	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1))) ∈ ℂ)
62	61	abscld 15606	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1)))) ∈ ℝ)
63		3re 12423	. . . . . . . 8 ⊢ 3 ∈ ℝ
64	63	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 3 ∈ ℝ)
65		1le3 12557	. . . . . . 7 ⊢ 1 ≤ 3
66	64, 65	jctir 530	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (3 ∈ ℝ ∧ 1 ≤ 3))
67	18	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑅 ∈ ℝ)
68	23	rexri 11367	. . . . . . . . . 10 ⊢ 1 ∈ ℝ^*
69	63	rexri 11367	. . . . . . . . . 10 ⊢ 3 ∈ ℝ^*
70		1lt3 12518	. . . . . . . . . 10 ⊢ 1 < 3
71		lbico1 13531	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ^* ∧ 3 ∈ ℝ^* ∧ 1 < 3) → 1 ∈ (1[,)3))
72	68, 69, 70, 71	mp3an 1490	. . . . . . . . 9 ⊢ 1 ∈ (1[,)3)
73		0red 11311	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 0 ∈ ℝ)
74		elico2 13541	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℝ^*) → (𝑚 ∈ (1[,)3) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚 ∧ 𝑚 < 3)))
75	23, 69, 74	mp2an 705	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚 ∧ 𝑚 < 3))
76	75	simp1bi 1163	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (1[,)3) → 𝑚 ∈ ℝ)
77		0red 11311	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) → 0 ∈ ℝ)
78		1red 11309	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) → 1 ∈ ℝ)
79		0lt1 11838	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
80	79	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) → 0 < 1)
81	75	simp2bi 1164	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) → 1 ≤ 𝑚)
82	77, 78, 76, 80, 81	ltletrd 11470	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (1[,)3) → 0 < 𝑚)
83	76, 82	elrpd 13161	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (1[,)3) → 𝑚 ∈ ℝ⁺)
84	47	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 𝑇 ∈ ℂ)
85	43, 84	subcld 11669	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (𝐹 − 𝑇) ∈ ℂ)
86	85	abscld 15606	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (abs‘(𝐹 − 𝑇)) ∈ ℝ)
87	83, 86	sylan2 605	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (abs‘(𝐹 − 𝑇)) ∈ ℝ)
88	18	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 𝑅 ∈ ℝ)
89	85	absge0d 15614	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 0 ≤ (abs‘(𝐹 − 𝑇)))
90	83, 89	sylan2 605	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 0 ≤ (abs‘(𝐹 − 𝑇)))
91		dchrvmasum.2	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑚 ∈ (1[,)3)(abs‘(𝐹 − 𝑇)) ≤ 𝑅)
92	91	r19.21bi 3254	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (abs‘(𝐹 − 𝑇)) ≤ 𝑅)
93	73, 87, 88, 90, 92	letrd 11467	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 0 ≤ 𝑅)
94	93	ralrimiva 3154	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ (1[,)3)0 ≤ 𝑅)
95		biidd 265	. . . . . . . . . 10 ⊢ (𝑚 = 1 → (0 ≤ 𝑅 ↔ 0 ≤ 𝑅))
96	95	rspcv 3572	. . . . . . . . 9 ⊢ (1 ∈ (1[,)3) → (∀𝑚 ∈ (1[,)3)0 ≤ 𝑅 → 0 ≤ 𝑅))
97	72, 94, 96	mpsyl 69	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝑅)
98	97	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ 𝑅)
99	67, 98	jca 521	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
100	50	recnd 11337	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝐾 − 𝑇)) ∈ ℂ)
101	5	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐶 ∈ ℝ)
102	101, 15	remulcld 11339	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) ∈ ℝ)
103	4	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
104		log1 26913	. . . . . . . . 9 ⊢ (log‘1) = 0
105	51	nncnd 12351	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℂ)
106	105	mullidd 11327	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 · 𝑑) = 𝑑)
107		rpre 13129	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
108	107	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
109		fznnfl 14002	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝑥)))
110	108, 109	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝑥)))
111	110	simplbda 505	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ≤ 𝑥)
112	106, 111	eqbrtrd 5126	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 · 𝑑) ≤ 𝑥)
113		1red 11309	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
114	107	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
115	113, 114, 55	lemuldivd 13213	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((1 · 𝑑) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑑)))
116	112, 115	mpbid 235	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ≤ (𝑥 / 𝑑))
117		1rp 13124	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ⁺
118	117	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ⁺)
119	118, 12	logled 26955	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 ≤ (𝑥 / 𝑑) ↔ (log‘1) ≤ (log‘(𝑥 / 𝑑))))
120	116, 119	mpbid 235	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (log‘1) ≤ (log‘(𝑥 / 𝑑)))
121	104, 120	eqbrtrrid 5140	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (log‘(𝑥 / 𝑑)))
122		rpregt0 13135	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
123	122	ad2antlr 740	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
124		divge0 12186	. . . . . . . 8 ⊢ ((((log‘(𝑥 / 𝑑)) ∈ ℝ ∧ 0 ≤ (log‘(𝑥 / 𝑑))) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → 0 ≤ ((log‘(𝑥 / 𝑑)) / 𝑥))
125	13, 121, 123, 124	syl21anc 851	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((log‘(𝑥 / 𝑑)) / 𝑥))
126		mulge0 11834	. . . . . . 7 ⊢ (((𝐶 ∈ ℝ ∧ 0 ≤ 𝐶) ∧ (((log‘(𝑥 / 𝑑)) / 𝑥) ∈ ℝ ∧ 0 ≤ ((log‘(𝑥 / 𝑑)) / 𝑥))) → 0 ≤ (𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)))
127	103, 15, 125, 126	syl12anc 850	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)))
128		absidm 15491	. . . . . . . . 9 ⊢ ((𝐾 − 𝑇) ∈ ℂ → (abs‘(abs‘(𝐾 − 𝑇))) = (abs‘(𝐾 − 𝑇)))
129	49, 128	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(abs‘(𝐾 − 𝑇))) = (abs‘(𝐾 − 𝑇)))
130	129	adantr 486	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (abs‘(abs‘(𝐾 − 𝑇))) = (abs‘(𝐾 − 𝑇)))
131	41	fvoveq1d 7442	. . . . . . . . . 10 ⊢ (𝑚 = (𝑥 / 𝑑) → (abs‘(𝐹 − 𝑇)) = (abs‘(𝐾 − 𝑇)))
132		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑥 / 𝑑) → (log‘𝑚) = (log‘(𝑥 / 𝑑)))
133		id 23	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑥 / 𝑑) → 𝑚 = (𝑥 / 𝑑))
134	132, 133	oveq12d 7438	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑥 / 𝑑) → ((log‘𝑚) / 𝑚) = ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑)))
135	134	oveq2d 7436	. . . . . . . . . 10 ⊢ (𝑚 = (𝑥 / 𝑑) → (𝐶 · ((log‘𝑚) / 𝑚)) = (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))))
136	131, 135	breq12d 5115	. . . . . . . . 9 ⊢ (𝑚 = (𝑥 / 𝑑) → ((abs‘(𝐹 − 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚)) ↔ (abs‘(𝐾 − 𝑇)) ≤ (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑)))))
137		dchrvmasum.1	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → (abs‘(𝐹 − 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚)))
138	137	ralrimiva 3154	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ (3[,)+∞)(abs‘(𝐹 − 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚)))
139	138	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → ∀𝑚 ∈ (3[,)+∞)(abs‘(𝐹 − 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚)))
140		nndivre 12379	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑑 ∈ ℕ) → (𝑥 / 𝑑) ∈ ℝ)
141	108, 9, 140	syl2an 608	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ)
142	141	adantr 486	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (𝑥 / 𝑑) ∈ ℝ)
143		simpr 490	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → 3 ≤ (𝑥 / 𝑑))
144		elicopnf 13576	. . . . . . . . . . 11 ⊢ (3 ∈ ℝ → ((𝑥 / 𝑑) ∈ (3[,)+∞) ↔ ((𝑥 / 𝑑) ∈ ℝ ∧ 3 ≤ (𝑥 / 𝑑))))
145	63, 144	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝑥 / 𝑑) ∈ (3[,)+∞) ↔ ((𝑥 / 𝑑) ∈ ℝ ∧ 3 ≤ (𝑥 / 𝑑)))
146	142, 143, 145	sylanbrc 595	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (𝑥 / 𝑑) ∈ (3[,)+∞))
147	136, 139, 146	rspcdva 3577	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (abs‘(𝐾 − 𝑇)) ≤ (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))))
148	13	recnd 11337	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑑)) ∈ ℂ)
149		rpcnne0 13139	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
150	149	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
151	55	rpcnne0d 13173	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0))
152		divdiv2 12029	. . . . . . . . . . . . 13 ⊢ (((log‘(𝑥 / 𝑑)) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0)) → ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑)) = (((log‘(𝑥 / 𝑑)) · 𝑑) / 𝑥))
153	148, 150, 151, 152	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑)) = (((log‘(𝑥 / 𝑑)) · 𝑑) / 𝑥))
154		div23 11993	. . . . . . . . . . . . 13 ⊢ (((log‘(𝑥 / 𝑑)) ∈ ℂ ∧ 𝑑 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((log‘(𝑥 / 𝑑)) · 𝑑) / 𝑥) = (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑))
155	148, 105, 150, 154	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑑)) · 𝑑) / 𝑥) = (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑))
156	153, 155	eqtrd 2795	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑)) = (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑))
157	156	oveq2d 7436	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))) = (𝐶 · (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑)))
158	29	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐶 ∈ ℂ)
159	15	recnd 11337	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑑)) / 𝑥) ∈ ℂ)
160	158, 159, 105	mulassd 11332	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑) = (𝐶 · (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑)))
161	157, 160	eqtr4d 2798	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))) = ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑))
162	161	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))) = ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑))
163	147, 162	breqtrd 5130	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (abs‘(𝐾 − 𝑇)) ≤ ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑))
164	130, 163	eqbrtrd 5126	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (abs‘(abs‘(𝐾 − 𝑇))) ≤ ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑))
165	129	adantr 486	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (abs‘(abs‘(𝐾 − 𝑇))) = (abs‘(𝐾 − 𝑇)))
166	131	breq1d 5112	. . . . . . . 8 ⊢ (𝑚 = (𝑥 / 𝑑) → ((abs‘(𝐹 − 𝑇)) ≤ 𝑅 ↔ (abs‘(𝐾 − 𝑇)) ≤ 𝑅))
167	91	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → ∀𝑚 ∈ (1[,)3)(abs‘(𝐹 − 𝑇)) ≤ 𝑅)
168	141	adantr 486	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (𝑥 / 𝑑) ∈ ℝ)
169	116	adantr 486	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → 1 ≤ (𝑥 / 𝑑))
170		simpr 490	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (𝑥 / 𝑑) < 3)
171		elico2 13541	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℝ^*) → ((𝑥 / 𝑑) ∈ (1[,)3) ↔ ((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑) ∧ (𝑥 / 𝑑) < 3)))
172	23, 69, 171	mp2an 705	. . . . . . . . 9 ⊢ ((𝑥 / 𝑑) ∈ (1[,)3) ↔ ((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑) ∧ (𝑥 / 𝑑) < 3))
173	168, 169, 170, 172	syl3anbrc 1362	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (𝑥 / 𝑑) ∈ (1[,)3))
174	166, 167, 173	rspcdva 3577	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (abs‘(𝐾 − 𝑇)) ≤ 𝑅)
175	165, 174	eqbrtrd 5126	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (abs‘(abs‘(𝐾 − 𝑇))) ≤ 𝑅)
176	8, 66, 99, 100, 102, 127, 164, 175	fsumharmonic 27339	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ≤ (Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1))))
177	29	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
178	7, 177, 159	fsummulc2 15950	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) = Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)))
179	178	oveq1d 7435	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1))) = (Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1))))
180	176, 179	breqtrrd 5132	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ≤ ((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1))))
181	40	leabsd 15582	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1))) ≤ (abs‘((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1)))))
182	60, 40, 62, 180, 181	letrd 11467	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ≤ (abs‘((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1)))))
183	182	adantrr 730	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ≤ (abs‘((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1)))))
184	1, 39, 40, 54, 183	o1le 15820	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ∈ 𝑂(1))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ⊆ wss 3895 class class class wbr 5102 ↦ cmpt 5185 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 ℝcr 11199 0cc0 11200 1c1 11201 + caddc 11203 · cmul 11205 +∞cpnf 11340 ℝ^*cxr 11342 < clt 11343 ≤ cle 11344 − cmin 11541 / cdiv 11973 ℕcn 12335 3c3 12398 ℝ⁺crp 13120 [,)cico 13478 ...cfz 13639 ⌊cfl 13930 abscabs 15401 ⇝_𝑟 crli 15652 𝑂(1)co1 15653 Σcsu 15853 Basecbs 17387 0_gc0g 17610 ℤRHomczrh 21805 ℤ/nℤczn 21808 logclog 26882 DChrcdchr 27559

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-oadd 8480 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-xnn0 12680 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ioc 13481 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-fac 14418 df-bc 14447 df-hash 14475 df-shft 15220 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-limsup 15638 df-clim 15655 df-rlim 15656 df-o1 15657 df-lo1 15658 df-sum 15854 df-ef 16233 df-e 16234 df-sin 16235 df-cos 16236 df-tan 16237 df-pi 16238 df-dvds 16423 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-mulg 19278 df-cntz 19531 df-cmn 19996 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cn 23545 df-cnp 23546 df-haus 23633 df-cmp 23705 df-tx 23881 df-hmeo 24074 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-xms 24639 df-ms 24640 df-tms 24641 df-cncf 25199 df-limc 26186 df-dv 26187 df-ulm 26704 df-log 26884 df-cxp 26885 df-atan 27195 df-em 27320

This theorem is used by: dchrvmasumlem3 27826

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