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Theorem dchrvmasumlem2 26181

Description: Lemma for dchrvmasum 26208. (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 10680	. 2 ⊢ (𝜑 → 1 ∈ ℝ)
2		dchrvmasum.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
3		elrege0 12886	. . . . . . 7 ⊢ (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
4	2, 3	sylib 221	. . . . . 6 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
5	4	simpld 498	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ)
6	5	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
7		fzfid 13390	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
8		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
9		elfznn 12985	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
10	9	nnrpd 12470	. . . . . . . 8 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℝ⁺)
11		rpdivcl 12455	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺) → (𝑥 / 𝑑) ∈ ℝ⁺)
12	8, 10, 11	syl2an 598	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ⁺)
13	12	relogcld 25313	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑑)) ∈ ℝ)
14	8	adantr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
15	13, 14	rerpdivcld 12503	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑑)) / 𝑥) ∈ ℝ)
16	7, 15	fsumrecl 15139	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥) ∈ ℝ)
17	6, 16	remulcld 10709	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ∈ ℝ)
18		dchrvmasum.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ)
19		3nn 11753	. . . . . . 7 ⊢ 3 ∈ ℕ
20		nnrp 12441	. . . . . . 7 ⊢ (3 ∈ ℕ → 3 ∈ ℝ⁺)
21		relogcl 25266	. . . . . . 7 ⊢ (3 ∈ ℝ⁺ → (log‘3) ∈ ℝ)
22	19, 20, 21	mp2b 10	. . . . . 6 ⊢ (log‘3) ∈ ℝ
23		1re 10679	. . . . . 6 ⊢ 1 ∈ ℝ
24	22, 23	readdcli 10694	. . . . 5 ⊢ ((log‘3) + 1) ∈ ℝ
25		remulcl 10660	. . . . 5 ⊢ ((𝑅 ∈ ℝ ∧ ((log‘3) + 1) ∈ ℝ) → (𝑅 · ((log‘3) + 1)) ∈ ℝ)
26	18, 24, 25	sylancl 589	. . . 4 ⊢ (𝜑 → (𝑅 · ((log‘3) + 1)) ∈ ℝ)
27	26	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑅 · ((log‘3) + 1)) ∈ ℝ)
28		rpssre 12437	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
29	5	recnd 10707	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ)
30		o1const 15024	. . . . 5 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 𝐶 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 𝐶) ∈ 𝑂(1))
31	28, 29, 30	sylancr 590	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 𝐶) ∈ 𝑂(1))
32		logfacrlim2 25909	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ⇝_𝑟 1
33		rlimo1 15021	. . . . 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 13	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ∈ 𝑂(1))
35	6, 16, 31, 34	o1mul2 15029	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥))) ∈ 𝑂(1))
36	26	recnd 10707	. . . 4 ⊢ (𝜑 → (𝑅 · ((log‘3) + 1)) ∈ ℂ)
37		o1const 15024	. . . 4 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (𝑅 · ((log‘3) + 1)) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (𝑅 · ((log‘3) + 1))) ∈ 𝑂(1))
38	28, 36, 37	sylancr 590	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝑅 · ((log‘3) + 1))) ∈ 𝑂(1))
39	17, 27, 35, 38	o1add2 15028	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1)))) ∈ 𝑂(1))
40	17, 27	readdcld 10708	. 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 2836	. . . . . . . 8 ⊢ (𝑚 = (𝑥 / 𝑑) → (𝐹 ∈ ℂ ↔ 𝐾 ∈ ℂ))
43		dchrvmasum.f	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 𝐹 ∈ ℂ)
44	43	ralrimiva 3113	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ ℝ⁺ 𝐹 ∈ ℂ)
45	44	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ∀𝑚 ∈ ℝ⁺ 𝐹 ∈ ℂ)
46	42, 45, 12	rspcdva 3543	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐾 ∈ ℂ)
47		dchrvmasum.t	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℂ)
48	47	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑇 ∈ ℂ)
49	46, 48	subcld 11035	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐾 − 𝑇) ∈ ℂ)
50	49	abscld 14844	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝐾 − 𝑇)) ∈ ℝ)
51	9	adantl 485	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
52	50, 51	nndivred 11728	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝐾 − 𝑇)) / 𝑑) ∈ ℝ)
53	7, 52	fsumrecl 15139	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑) ∈ ℝ)
54	53	recnd 10707	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑) ∈ ℂ)
55	51	nnrpd 12470	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ⁺)
56	49	absge0d 14852	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(𝐾 − 𝑇)))
57	50, 55, 56	divge0d 12512	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((abs‘(𝐾 − 𝑇)) / 𝑑))
58	7, 52, 57	fsumge0 15198	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑))
59	53, 58	absidd 14830	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) = Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑))
60	59, 53	eqeltrd 2852	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ∈ ℝ)
61	40	recnd 10707	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1))) ∈ ℂ)
62	61	abscld 14844	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1)))) ∈ ℝ)
63		3re 11754	. . . . . . . 8 ⊢ 3 ∈ ℝ
64	63	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 3 ∈ ℝ)
65		1le3 11886	. . . . . . 7 ⊢ 1 ≤ 3
66	64, 65	jctir 524	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (3 ∈ ℝ ∧ 1 ≤ 3))
67	18	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑅 ∈ ℝ)
68	23	rexri 10737	. . . . . . . . . 10 ⊢ 1 ∈ ℝ^*
69	63	rexri 10737	. . . . . . . . . 10 ⊢ 3 ∈ ℝ^*
70		1lt3 11847	. . . . . . . . . 10 ⊢ 1 < 3
71		lbico1 12833	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ^* ∧ 3 ∈ ℝ^* ∧ 1 < 3) → 1 ∈ (1[,)3))
72	68, 69, 70, 71	mp3an 1458	. . . . . . . . 9 ⊢ 1 ∈ (1[,)3)
73		0red 10682	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 0 ∈ ℝ)
74		elico2 12843	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℝ^*) → (𝑚 ∈ (1[,)3) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚 ∧ 𝑚 < 3)))
75	23, 69, 74	mp2an 691	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚 ∧ 𝑚 < 3))
76	75	simp1bi 1142	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (1[,)3) → 𝑚 ∈ ℝ)
77		0red 10682	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) → 0 ∈ ℝ)
78		1red 10680	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) → 1 ∈ ℝ)
79		0lt1 11200	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
80	79	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) → 0 < 1)
81	75	simp2bi 1143	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) → 1 ≤ 𝑚)
82	77, 78, 76, 80, 81	ltletrd 10838	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (1[,)3) → 0 < 𝑚)
83	76, 82	elrpd 12469	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (1[,)3) → 𝑚 ∈ ℝ⁺)
84	47	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 𝑇 ∈ ℂ)
85	43, 84	subcld 11035	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (𝐹 − 𝑇) ∈ ℂ)
86	85	abscld 14844	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (abs‘(𝐹 − 𝑇)) ∈ ℝ)
87	83, 86	sylan2 595	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (abs‘(𝐹 − 𝑇)) ∈ ℝ)
88	18	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 𝑅 ∈ ℝ)
89	85	absge0d 14852	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 0 ≤ (abs‘(𝐹 − 𝑇)))
90	83, 89	sylan2 595	. . . . . . . . . . 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 3137	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (abs‘(𝐹 − 𝑇)) ≤ 𝑅)
93	73, 87, 88, 90, 92	letrd 10835	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 0 ≤ 𝑅)
94	93	ralrimiva 3113	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ (1[,)3)0 ≤ 𝑅)
95		biidd 265	. . . . . . . . . 10 ⊢ (𝑚 = 1 → (0 ≤ 𝑅 ↔ 0 ≤ 𝑅))
96	95	rspcv 3536	. . . . . . . . 9 ⊢ (1 ∈ (1[,)3) → (∀𝑚 ∈ (1[,)3)0 ≤ 𝑅 → 0 ≤ 𝑅))
97	72, 94, 96	mpsyl 68	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝑅)
98	97	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ 𝑅)
99	67, 98	jca 515	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
100	50	recnd 10707	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝐾 − 𝑇)) ∈ ℂ)
101	5	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐶 ∈ ℝ)
102	101, 15	remulcld 10709	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) ∈ ℝ)
103	4	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
104		log1 25276	. . . . . . . . 9 ⊢ (log‘1) = 0
105	51	nncnd 11690	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℂ)
106	105	mulid2d 10697	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 · 𝑑) = 𝑑)
107		rpre 12438	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
108	107	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
109		fznnfl 13279	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝑥)))
110	108, 109	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝑥)))
111	110	simplbda 503	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ≤ 𝑥)
112	106, 111	eqbrtrd 5054	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 · 𝑑) ≤ 𝑥)
113		1red 10680	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
114	107	ad2antlr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
115	113, 114, 55	lemuldivd 12521	. . . . . . . . . . 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 12434	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ⁺
118	117	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ⁺)
119	118, 12	logled 25317	. . . . . . . . . 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 5068	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (log‘(𝑥 / 𝑑)))
122		rpregt0 12444	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
123	122	ad2antlr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
124		divge0 11547	. . . . . . . 8 ⊢ ((((log‘(𝑥 / 𝑑)) ∈ ℝ ∧ 0 ≤ (log‘(𝑥 / 𝑑))) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → 0 ≤ ((log‘(𝑥 / 𝑑)) / 𝑥))
125	13, 121, 123, 124	syl21anc 836	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((log‘(𝑥 / 𝑑)) / 𝑥))
126		mulge0 11196	. . . . . . 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 835	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)))
128		absidm 14731	. . . . . . . . 9 ⊢ ((𝐾 − 𝑇) ∈ ℂ → (abs‘(abs‘(𝐾 − 𝑇))) = (abs‘(𝐾 − 𝑇)))
129	49, 128	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(abs‘(𝐾 − 𝑇))) = (abs‘(𝐾 − 𝑇)))
130	129	adantr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (abs‘(abs‘(𝐾 − 𝑇))) = (abs‘(𝐾 − 𝑇)))
131	41	fvoveq1d 7172	. . . . . . . . . 10 ⊢ (𝑚 = (𝑥 / 𝑑) → (abs‘(𝐹 − 𝑇)) = (abs‘(𝐾 − 𝑇)))
132		fveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑥 / 𝑑) → (log‘𝑚) = (log‘(𝑥 / 𝑑)))
133		id 22	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑥 / 𝑑) → 𝑚 = (𝑥 / 𝑑))
134	132, 133	oveq12d 7168	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑥 / 𝑑) → ((log‘𝑚) / 𝑚) = ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑)))
135	134	oveq2d 7166	. . . . . . . . . 10 ⊢ (𝑚 = (𝑥 / 𝑑) → (𝐶 · ((log‘𝑚) / 𝑚)) = (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))))
136	131, 135	breq12d 5045	. . . . . . . . 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 3113	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ (3[,)+∞)(abs‘(𝐹 − 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚)))
139	138	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → ∀𝑚 ∈ (3[,)+∞)(abs‘(𝐹 − 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚)))
140		nndivre 11715	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑑 ∈ ℕ) → (𝑥 / 𝑑) ∈ ℝ)
141	108, 9, 140	syl2an 598	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ)
142	141	adantr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (𝑥 / 𝑑) ∈ ℝ)
143		simpr 488	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → 3 ≤ (𝑥 / 𝑑))
144		elicopnf 12877	. . . . . . . . . . 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 586	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (𝑥 / 𝑑) ∈ (3[,)+∞))
147	136, 139, 146	rspcdva 3543	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (abs‘(𝐾 − 𝑇)) ≤ (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))))
148	13	recnd 10707	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑑)) ∈ ℂ)
149		rpcnne0 12448	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
150	149	ad2antlr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
151	55	rpcnne0d 12481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0))
152		divdiv2 11390	. . . . . . . . . . . . 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 1368	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑)) = (((log‘(𝑥 / 𝑑)) · 𝑑) / 𝑥))
154		div23 11355	. . . . . . . . . . . . 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 1368	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑑)) · 𝑑) / 𝑥) = (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑))
156	153, 155	eqtrd 2793	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑)) = (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑))
157	156	oveq2d 7166	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))) = (𝐶 · (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑)))
158	29	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐶 ∈ ℂ)
159	15	recnd 10707	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑑)) / 𝑥) ∈ ℂ)
160	158, 159, 105	mulassd 10702	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑) = (𝐶 · (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑)))
161	157, 160	eqtr4d 2796	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))) = ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑))
162	161	adantr 484	. . . . . . . 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 5058	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (abs‘(𝐾 − 𝑇)) ≤ ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑))
164	130, 163	eqbrtrd 5054	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (abs‘(abs‘(𝐾 − 𝑇))) ≤ ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑))
165	129	adantr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (abs‘(abs‘(𝐾 − 𝑇))) = (abs‘(𝐾 − 𝑇)))
166	131	breq1d 5042	. . . . . . . 8 ⊢ (𝑚 = (𝑥 / 𝑑) → ((abs‘(𝐹 − 𝑇)) ≤ 𝑅 ↔ (abs‘(𝐾 − 𝑇)) ≤ 𝑅))
167	91	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → ∀𝑚 ∈ (1[,)3)(abs‘(𝐹 − 𝑇)) ≤ 𝑅)
168	141	adantr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (𝑥 / 𝑑) ∈ ℝ)
169	116	adantr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → 1 ≤ (𝑥 / 𝑑))
170		simpr 488	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (𝑥 / 𝑑) < 3)
171		elico2 12843	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℝ^*) → ((𝑥 / 𝑑) ∈ (1[,)3) ↔ ((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑) ∧ (𝑥 / 𝑑) < 3)))
172	23, 69, 171	mp2an 691	. . . . . . . . 9 ⊢ ((𝑥 / 𝑑) ∈ (1[,)3) ↔ ((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑) ∧ (𝑥 / 𝑑) < 3))
173	168, 169, 170, 172	syl3anbrc 1340	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (𝑥 / 𝑑) ∈ (1[,)3))
174	166, 167, 173	rspcdva 3543	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (abs‘(𝐾 − 𝑇)) ≤ 𝑅)
175	165, 174	eqbrtrd 5054	. . . . . 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 25696	. . . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
178	7, 177, 159	fsummulc2 15187	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) = Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)))
179	178	oveq1d 7165	. . . . 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 5060	. . . 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 14822	. . . 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 10835	. . 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 716	. 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 15057	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ∈ 𝑂(1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2951 ∀wral 3070 ⊆ wss 3858 class class class wbr 5032 ↦ cmpt 5112 ‘cfv 6335 (class class class)co 7150 ℂcc 10573 ℝcr 10574 0cc0 10575 1c1 10576 + caddc 10578 · cmul 10580 +∞cpnf 10710 ℝ^*cxr 10712 < clt 10713 ≤ cle 10714 − cmin 10908 / cdiv 11335 ℕcn 11674 3c3 11730 ℝ⁺crp 12430 [,)cico 12781 ...cfz 12939 ⌊cfl 13209 abscabs 14641 ⇝_𝑟 crli 14890 𝑂(1)co1 14891 Σcsu 15090 Basecbs 16541 0_gc0g 16771 ℤRHomczrh 20269 ℤ/nℤczn 20272 logclog 25245 DChrcdchr 25915

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5156 ax-sep 5169 ax-nul 5176 ax-pow 5234 ax-pr 5298 ax-un 7459 ax-inf2 9137 ax-cnex 10631 ax-resscn 10632 ax-1cn 10633 ax-icn 10634 ax-addcl 10635 ax-addrcl 10636 ax-mulcl 10637 ax-mulrcl 10638 ax-mulcom 10639 ax-addass 10640 ax-mulass 10641 ax-distr 10642 ax-i2m1 10643 ax-1ne0 10644 ax-1rid 10645 ax-rnegex 10646 ax-rrecex 10647 ax-cnre 10648 ax-pre-lttri 10649 ax-pre-lttrn 10650 ax-pre-ltadd 10651 ax-pre-mulgt0 10652 ax-pre-sup 10653 ax-addf 10654 ax-mulf 10655

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3697 df-csb 3806 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-pss 3877 df-nul 4226 df-if 4421 df-pw 4496 df-sn 4523 df-pr 4525 df-tp 4527 df-op 4529 df-uni 4799 df-int 4839 df-iun 4885 df-iin 4886 df-br 5033 df-opab 5095 df-mpt 5113 df-tr 5139 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5483 df-se 5484 df-we 5485 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-rn 5535 df-res 5536 df-ima 5537 df-pred 6126 df-ord 6172 df-on 6173 df-lim 6174 df-suc 6175 df-iota 6294 df-fun 6337 df-fn 6338 df-f 6339 df-f1 6340 df-fo 6341 df-f1o 6342 df-fv 6343 df-isom 6344 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7405 df-om 7580 df-1st 7693 df-2nd 7694 df-supp 7836 df-wrecs 7957 df-recs 8018 df-rdg 8056 df-1o 8112 df-2o 8113 df-oadd 8116 df-er 8299 df-map 8418 df-pm 8419 df-ixp 8480 df-en 8528 df-dom 8529 df-sdom 8530 df-fin 8531 df-fsupp 8867 df-fi 8908 df-sup 8939 df-inf 8940 df-oi 9007 df-card 9401 df-pnf 10715 df-mnf 10716 df-xr 10717 df-ltxr 10718 df-le 10719 df-sub 10910 df-neg 10911 df-div 11336 df-nn 11675 df-2 11737 df-3 11738 df-4 11739 df-5 11740 df-6 11741 df-7 11742 df-8 11743 df-9 11744 df-n0 11935 df-xnn0 12007 df-z 12021 df-dec 12138 df-uz 12283 df-q 12389 df-rp 12431 df-xneg 12548 df-xadd 12549 df-xmul 12550 df-ioo 12783 df-ioc 12784 df-ico 12785 df-icc 12786 df-fz 12940 df-fzo 13083 df-fl 13211 df-mod 13287 df-seq 13419 df-exp 13480 df-fac 13684 df-bc 13713 df-hash 13741 df-shft 14474 df-cj 14506 df-re 14507 df-im 14508 df-sqrt 14642 df-abs 14643 df-limsup 14876 df-clim 14893 df-rlim 14894 df-o1 14895 df-lo1 14896 df-sum 15091 df-ef 15469 df-e 15470 df-sin 15471 df-cos 15472 df-tan 15473 df-pi 15474 df-dvds 15656 df-struct 16543 df-ndx 16544 df-slot 16545 df-base 16547 df-sets 16548 df-ress 16549 df-plusg 16636 df-mulr 16637 df-starv 16638 df-sca 16639 df-vsca 16640 df-ip 16641 df-tset 16642 df-ple 16643 df-ds 16645 df-unif 16646 df-hom 16647 df-cco 16648 df-rest 16754 df-topn 16755 df-0g 16773 df-gsum 16774 df-topgen 16775 df-pt 16776 df-prds 16779 df-xrs 16833 df-qtop 16838 df-imas 16839 df-xps 16841 df-mre 16915 df-mrc 16916 df-acs 16918 df-mgm 17918 df-sgrp 17967 df-mnd 17978 df-submnd 18023 df-mulg 18292 df-cntz 18514 df-cmn 18975 df-psmet 20158 df-xmet 20159 df-met 20160 df-bl 20161 df-mopn 20162 df-fbas 20163 df-fg 20164 df-cnfld 20167 df-top 21594 df-topon 21611 df-topsp 21633 df-bases 21646 df-cld 21719 df-ntr 21720 df-cls 21721 df-nei 21798 df-lp 21836 df-perf 21837 df-cn 21927 df-cnp 21928 df-haus 22015 df-cmp 22087 df-tx 22262 df-hmeo 22455 df-fil 22546 df-fm 22638 df-flim 22639 df-flf 22640 df-xms 23022 df-ms 23023 df-tms 23024 df-cncf 23579 df-limc 24565 df-dv 24566 df-ulm 25071 df-log 25247 df-cxp 25248 df-atan 25552 df-em 25677

This theorem is referenced by: dchrvmasumlem3 26182

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