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Theorem dchrvmasumlem2 27788

Description: Lemma for dchrvmasum 27815. (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 11280	. 2 ⊢ (𝜑 → 1 ∈ ℝ)
2		dchrvmasum.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
3		elrege0 13554	. . . . . . 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 14084	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
8		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
9		elfznn 13655	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
10	9	nnrpd 13131	. . . . . . . 8 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℝ⁺)
11		rpdivcl 13116	. . . . . . . 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 26914	. . . . . 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 13164	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑑)) / 𝑥) ∈ ℝ)
16	7, 15	fsumrecl 15867	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥) ∈ ℝ)
17	6, 16	remulcld 11310	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ∈ ℝ)
18		dchrvmasum.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ)
19		3nn 12391	. . . . . . 7 ⊢ 3 ∈ ℕ
20		nnrp 13101	. . . . . . 7 ⊢ (3 ∈ ℕ → 3 ∈ ℝ⁺)
21		relogcl 26866	. . . . . . 7 ⊢ (3 ∈ ℝ⁺ → (log‘3) ∈ ℝ)
22	19, 20, 21	mp2b 10	. . . . . 6 ⊢ (log‘3) ∈ ℝ
23		1re 11279	. . . . . 6 ⊢ 1 ∈ ℝ
24	22, 23	readdcli 11295	. . . . 5 ⊢ ((log‘3) + 1) ∈ ℝ
25		remulcl 11256	. . . . 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 13097	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
29	5	recnd 11308	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ)
30		o1const 15754	. . . . 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 27516	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ⇝_𝑟 1
33		rlimo1 15751	. . . . 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 15759	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥))) ∈ 𝑂(1))
36	26	recnd 11308	. . . 4 ⊢ (𝜑 → (𝑅 · ((log‘3) + 1)) ∈ ℂ)
37		o1const 15754	. . . 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 15758	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1)))) ∈ 𝑂(1))
40	17, 27	readdcld 11309	. 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 11640	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐾 − 𝑇) ∈ ℂ)
50	49	abscld 15573	. . . . 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 12361	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝐾 − 𝑇)) / 𝑑) ∈ ℝ)
53	7, 52	fsumrecl 15867	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑) ∈ ℝ)
54	53	recnd 11308	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑) ∈ ℂ)
55	51	nnrpd 13131	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ⁺)
56	49	absge0d 15581	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(𝐾 − 𝑇)))
57	50, 55, 56	divge0d 13173	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((abs‘(𝐾 − 𝑇)) / 𝑑))
58	7, 52, 57	fsumge0 15929	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑))
59	53, 58	absidd 15557	. . . . 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 11308	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1))) ∈ ℂ)
62	61	abscld 15573	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1)))) ∈ ℝ)
63		3re 12392	. . . . . . . 8 ⊢ 3 ∈ ℝ
64	63	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 3 ∈ ℝ)
65		1le3 12526	. . . . . . 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 11338	. . . . . . . . . 10 ⊢ 1 ∈ ℝ^*
69	63	rexri 11338	. . . . . . . . . 10 ⊢ 3 ∈ ℝ^*
70		1lt3 12487	. . . . . . . . . 10 ⊢ 1 < 3
71		lbico1 13500	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ^* ∧ 3 ∈ ℝ^* ∧ 1 < 3) → 1 ∈ (1[,)3))
72	68, 69, 70, 71	mp3an 1490	. . . . . . . . 9 ⊢ 1 ∈ (1[,)3)
73		0red 11282	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 0 ∈ ℝ)
74		elico2 13510	. . . . . . . . . . . . . . 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 11282	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) → 0 ∈ ℝ)
78		1red 11280	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) → 1 ∈ ℝ)
79		0lt1 11807	. . . . . . . . . . . . . . 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 11441	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (1[,)3) → 0 < 𝑚)
83	76, 82	elrpd 13130	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (1[,)3) → 𝑚 ∈ ℝ⁺)
84	47	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 𝑇 ∈ ℂ)
85	43, 84	subcld 11640	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (𝐹 − 𝑇) ∈ ℂ)
86	85	abscld 15573	. . . . . . . . . . . 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 15581	. . . . . . . . . . . 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 11438	. . . . . . . . . 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 11308	. . . . . 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 11310	. . . . . 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 26876	. . . . . . . . 9 ⊢ (log‘1) = 0
105	51	nncnd 12320	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℂ)
106	105	mullidd 11298	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 · 𝑑) = 𝑑)
107		rpre 13098	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
108	107	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
109		fznnfl 13970	. . . . . . . . . . . . . 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 11280	. . . . . . . . . . . 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 13182	. . . . . . . . . . 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 13093	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ⁺
118	117	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ⁺)
119	118, 12	logled 26918	. . . . . . . . . 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 13104	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
123	122	ad2antlr 740	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
124		divge0 12155	. . . . . . . 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 11803	. . . . . . 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 15458	. . . . . . . . 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 7430	. . . . . . . . . 10 ⊢ (𝑚 = (𝑥 / 𝑑) → (abs‘(𝐹 − 𝑇)) = (abs‘(𝐾 − 𝑇)))
132		fveq2 6873	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑥 / 𝑑) → (log‘𝑚) = (log‘(𝑥 / 𝑑)))
133		id 23	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑥 / 𝑑) → 𝑚 = (𝑥 / 𝑑))
134	132, 133	oveq12d 7426	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑥 / 𝑑) → ((log‘𝑚) / 𝑚) = ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑)))
135	134	oveq2d 7424	. . . . . . . . . 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 12348	. . . . . . . . . . . 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 13545	. . . . . . . . . . 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 11308	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑑)) ∈ ℂ)
149		rpcnne0 13108	. . . . . . . . . . . . . 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 13142	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0))
152		divdiv2 11998	. . . . . . . . . . . . 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 11962	. . . . . . . . . . . . 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 7424	. . . . . . . . . 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 11308	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑑)) / 𝑥) ∈ ℂ)
160	158, 159, 105	mulassd 11303	. . . . . . . . . 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 13510	. . . . . . . . . 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 27302	. . . . 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 15917	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) = Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)))
179	178	oveq1d 7423	. . . . 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 15549	. . . 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 11438	. . 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 15787	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 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ⊆ wss 3898 class class class wbr 5102 ↦ cmpt 5185 ‘cfv 6527 (class class class)co 7408 ℂcc 11169 ℝcr 11170 0cc0 11171 1c1 11172 + caddc 11174 · cmul 11176 +∞cpnf 11311 ℝ^*cxr 11313 < clt 11314 ≤ cle 11315 − cmin 11512 / cdiv 11942 ℕcn 12304 3c3 12367 ℝ⁺crp 13089 [,)cico 13447 ...cfz 13608 ⌊cfl 13898 abscabs 15368 ⇝_𝑟 crli 15619 𝑂(1)co1 15620 Σcsu 15820 Basecbs 17348 0_gc0g 17571 ℤRHomczrh 21766 ℤ/nℤczn 21769 logclog 26845 DChrcdchr 27522

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-inf2 9620 ax-cnex 11227 ax-resscn 11228 ax-1cn 11229 ax-icn 11230 ax-addcl 11231 ax-addrcl 11232 ax-mulcl 11233 ax-mulrcl 11234 ax-mulcom 11235 ax-addass 11236 ax-mulass 11237 ax-distr 11238 ax-i2m1 11239 ax-1ne0 11240 ax-1rid 11241 ax-rnegex 11242 ax-rrecex 11243 ax-cnre 11244 ax-pre-lttri 11245 ax-pre-lttrn 11246 ax-pre-ltadd 11247 ax-pre-mulgt0 11248 ax-pre-sup 11249 ax-addf 11250

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 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-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 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 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-isom 6536 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8156 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8454 df-2o 8455 df-oadd 8458 df-er 8695 df-map 8827 df-pm 8828 df-ixp 8904 df-en 8952 df-dom 8953 df-sdom 8954 df-fin 8955 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9991 df-pnf 11316 df-mnf 11317 df-xr 11318 df-ltxr 11319 df-le 11320 df-sub 11514 df-neg 11515 df-div 11943 df-nn 12305 df-2 12374 df-3 12375 df-4 12376 df-5 12377 df-6 12378 df-7 12379 df-8 12380 df-9 12381 df-n0 12576 df-xnn0 12649 df-z 12663 df-dec 12784 df-uz 12935 df-q 13045 df-rp 13090 df-xneg 13210 df-xadd 13211 df-xmul 13212 df-ioo 13449 df-ioc 13450 df-ico 13451 df-icc 13452 df-fz 13609 df-fzo 13757 df-fl 13900 df-mod 13978 df-seq 14113 df-exp 14173 df-fac 14385 df-bc 14414 df-hash 14442 df-shft 15187 df-cj 15233 df-re 15234 df-im 15235 df-sqrt 15369 df-abs 15370 df-limsup 15605 df-clim 15622 df-rlim 15623 df-o1 15624 df-lo1 15625 df-sum 15821 df-ef 16200 df-e 16201 df-sin 16202 df-cos 16203 df-tan 16204 df-pi 16205 df-dvds 16390 df-struct 17286 df-sets 17303 df-slot 17321 df-ndx 17333 df-base 17349 df-ress 17370 df-plusg 17402 df-mulr 17403 df-starv 17404 df-sca 17405 df-vsca 17406 df-ip 17407 df-tset 17408 df-ple 17409 df-ds 17411 df-unif 17412 df-hom 17413 df-cco 17414 df-rest 17554 df-topn 17555 df-0g 17573 df-gsum 17574 df-topgen 17575 df-pt 17576 df-prds 17579 df-xrs 17635 df-qtop 17640 df-imas 17641 df-xps 17643 df-mre 17717 df-mrc 17718 df-acs 17720 df-mgm 18777 df-sgrp 18869 df-mnd 18885 df-submnd 18940 df-mulg 19239 df-cntz 19492 df-cmn 19957 df-psmet 21631 df-xmet 21632 df-met 21633 df-bl 21634 df-mopn 21635 df-fbas 21636 df-fg 21637 df-cnfld 21640 df-top 23173 df-topon 23190 df-topsp 23212 df-bases 23225 df-cld 23298 df-ntr 23299 df-cls 23300 df-nei 23377 df-lp 23415 df-perf 23416 df-cn 23506 df-cnp 23507 df-haus 23594 df-cmp 23666 df-tx 23842 df-hmeo 24035 df-fil 24126 df-fm 24218 df-flim 24219 df-flf 24220 df-xms 24600 df-ms 24601 df-tms 24602 df-cncf 25160 df-limc 26147 df-dv 26148 df-ulm 26667 df-log 26847 df-cxp 26848 df-atan 27158 df-em 27283

This theorem is used by: dchrvmasumlem3 27789

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