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Theorem dchrvmasumlem2 25539

Description: Lemma for dchrvmasum 25566. (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 10329	. 2 ⊢ (𝜑 → 1 ∈ ℝ)
2		dchrvmasum.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
3		elrege0 12529	. . . . . . 7 ⊢ (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
4	2, 3	sylib 210	. . . . . 6 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
5	4	simpld 489	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ)
6	5	adantr 473	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
7		fzfid 13027	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
8		simpr 478	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
9		elfznn 12624	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
10	9	nnrpd 12115	. . . . . . . 8 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℝ⁺)
11		rpdivcl 12101	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺) → (𝑥 / 𝑑) ∈ ℝ⁺)
12	8, 10, 11	syl2an 590	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ⁺)
13		relogcl 24663	. . . . . . 7 ⊢ ((𝑥 / 𝑑) ∈ ℝ⁺ → (log‘(𝑥 / 𝑑)) ∈ ℝ)
14	12, 13	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑑)) ∈ ℝ)
15	8	adantr 473	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
16	14, 15	rerpdivcld 12148	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑑)) / 𝑥) ∈ ℝ)
17	7, 16	fsumrecl 14806	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥) ∈ ℝ)
18	6, 17	remulcld 10359	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ∈ ℝ)
19		dchrvmasum.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ)
20		3nn 11392	. . . . . . 7 ⊢ 3 ∈ ℕ
21		nnrp 12087	. . . . . . 7 ⊢ (3 ∈ ℕ → 3 ∈ ℝ⁺)
22		relogcl 24663	. . . . . . 7 ⊢ (3 ∈ ℝ⁺ → (log‘3) ∈ ℝ)
23	20, 21, 22	mp2b 10	. . . . . 6 ⊢ (log‘3) ∈ ℝ
24		1re 10328	. . . . . 6 ⊢ 1 ∈ ℝ
25	23, 24	readdcli 10344	. . . . 5 ⊢ ((log‘3) + 1) ∈ ℝ
26		remulcl 10309	. . . . 5 ⊢ ((𝑅 ∈ ℝ ∧ ((log‘3) + 1) ∈ ℝ) → (𝑅 · ((log‘3) + 1)) ∈ ℝ)
27	19, 25, 26	sylancl 581	. . . 4 ⊢ (𝜑 → (𝑅 · ((log‘3) + 1)) ∈ ℝ)
28	27	adantr 473	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑅 · ((log‘3) + 1)) ∈ ℝ)
29		rpssre 12081	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
30	5	recnd 10357	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ)
31		o1const 14691	. . . . 5 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 𝐶 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 𝐶) ∈ 𝑂(1))
32	29, 30, 31	sylancr 582	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 𝐶) ∈ 𝑂(1))
33		logfacrlim2 25303	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ⇝_𝑟 1
34		rlimo1 14688	. . . . 5 ⊢ ((𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ⇝_𝑟 1 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ∈ 𝑂(1))
35	33, 34	mp1i 13	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ∈ 𝑂(1))
36	6, 17, 32, 35	o1mul2 14696	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥))) ∈ 𝑂(1))
37	27	recnd 10357	. . . 4 ⊢ (𝜑 → (𝑅 · ((log‘3) + 1)) ∈ ℂ)
38		o1const 14691	. . . 4 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (𝑅 · ((log‘3) + 1)) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (𝑅 · ((log‘3) + 1))) ∈ 𝑂(1))
39	29, 37, 38	sylancr 582	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝑅 · ((log‘3) + 1))) ∈ 𝑂(1))
40	18, 28, 36, 39	o1add2 14695	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1)))) ∈ 𝑂(1))
41	18, 28	readdcld 10358	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1))) ∈ ℝ)
42		dchrvmasum.g	. . . . . . . . 9 ⊢ (𝑚 = (𝑥 / 𝑑) → 𝐹 = 𝐾)
43	42	eleq1d 2863	. . . . . . . 8 ⊢ (𝑚 = (𝑥 / 𝑑) → (𝐹 ∈ ℂ ↔ 𝐾 ∈ ℂ))
44		dchrvmasum.f	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 𝐹 ∈ ℂ)
45	44	ralrimiva 3147	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ ℝ⁺ 𝐹 ∈ ℂ)
46	45	ad2antrr 718	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ∀𝑚 ∈ ℝ⁺ 𝐹 ∈ ℂ)
47	43, 46, 12	rspcdva 3503	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐾 ∈ ℂ)
48		dchrvmasum.t	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℂ)
49	48	ad2antrr 718	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑇 ∈ ℂ)
50	47, 49	subcld 10684	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐾 − 𝑇) ∈ ℂ)
51	50	abscld 14516	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝐾 − 𝑇)) ∈ ℝ)
52	9	adantl 474	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
53	51, 52	nndivred 11367	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝐾 − 𝑇)) / 𝑑) ∈ ℝ)
54	7, 53	fsumrecl 14806	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑) ∈ ℝ)
55	54	recnd 10357	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑) ∈ ℂ)
56	52	nnrpd 12115	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ⁺)
57	50	absge0d 14524	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(𝐾 − 𝑇)))
58	51, 56, 57	divge0d 12157	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((abs‘(𝐾 − 𝑇)) / 𝑑))
59	7, 53, 58	fsumge0 14865	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑))
60	54, 59	absidd 14502	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) = Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑))
61	60, 54	eqeltrd 2878	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ∈ ℝ)
62	41	recnd 10357	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1))) ∈ ℂ)
63	62	abscld 14516	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1)))) ∈ ℝ)
64		3re 11393	. . . . . . . 8 ⊢ 3 ∈ ℝ
65	64	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 3 ∈ ℝ)
66		1le3 11532	. . . . . . 7 ⊢ 1 ≤ 3
67	65, 66	jctir 517	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (3 ∈ ℝ ∧ 1 ≤ 3))
68	19	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑅 ∈ ℝ)
69	24	rexri 10387	. . . . . . . . . 10 ⊢ 1 ∈ ℝ^*
70	64	rexri 10387	. . . . . . . . . 10 ⊢ 3 ∈ ℝ^*
71		1lt3 11493	. . . . . . . . . 10 ⊢ 1 < 3
72		lbico1 12477	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ^* ∧ 3 ∈ ℝ^* ∧ 1 < 3) → 1 ∈ (1[,)3))
73	69, 70, 71, 72	mp3an 1586	. . . . . . . . 9 ⊢ 1 ∈ (1[,)3)
74		0red 10332	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 0 ∈ ℝ)
75		elico2 12486	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℝ^*) → (𝑚 ∈ (1[,)3) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚 ∧ 𝑚 < 3)))
76	24, 70, 75	mp2an 684	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚 ∧ 𝑚 < 3))
77	76	simp1bi 1176	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (1[,)3) → 𝑚 ∈ ℝ)
78		0red 10332	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) → 0 ∈ ℝ)
79		1red 10329	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) → 1 ∈ ℝ)
80		0lt1 10842	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
81	80	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) → 0 < 1)
82	76	simp2bi 1177	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) → 1 ≤ 𝑚)
83	78, 79, 77, 81, 82	ltletrd 10487	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (1[,)3) → 0 < 𝑚)
84	77, 83	elrpd 12114	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (1[,)3) → 𝑚 ∈ ℝ⁺)
85	48	adantr 473	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 𝑇 ∈ ℂ)
86	44, 85	subcld 10684	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (𝐹 − 𝑇) ∈ ℂ)
87	86	abscld 14516	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (abs‘(𝐹 − 𝑇)) ∈ ℝ)
88	84, 87	sylan2 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (abs‘(𝐹 − 𝑇)) ∈ ℝ)
89	19	adantr 473	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 𝑅 ∈ ℝ)
90	86	absge0d 14524	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 0 ≤ (abs‘(𝐹 − 𝑇)))
91	84, 90	sylan2 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 0 ≤ (abs‘(𝐹 − 𝑇)))
92		dchrvmasum.2	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑚 ∈ (1[,)3)(abs‘(𝐹 − 𝑇)) ≤ 𝑅)
93	92	r19.21bi 3113	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (abs‘(𝐹 − 𝑇)) ≤ 𝑅)
94	74, 88, 89, 91, 93	letrd 10484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 0 ≤ 𝑅)
95	94	ralrimiva 3147	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ (1[,)3)0 ≤ 𝑅)
96		biidd 254	. . . . . . . . . 10 ⊢ (𝑚 = 1 → (0 ≤ 𝑅 ↔ 0 ≤ 𝑅))
97	96	rspcv 3493	. . . . . . . . 9 ⊢ (1 ∈ (1[,)3) → (∀𝑚 ∈ (1[,)3)0 ≤ 𝑅 → 0 ≤ 𝑅))
98	73, 95, 97	mpsyl 68	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝑅)
99	98	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ 𝑅)
100	68, 99	jca 508	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
101	51	recnd 10357	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝐾 − 𝑇)) ∈ ℂ)
102	5	ad2antrr 718	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐶 ∈ ℝ)
103	102, 16	remulcld 10359	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) ∈ ℝ)
104	4	ad2antrr 718	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
105		log1 24673	. . . . . . . . 9 ⊢ (log‘1) = 0
106	52	nncnd 11330	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℂ)
107	106	mulid2d 10347	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 · 𝑑) = 𝑑)
108		rpre 12082	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
109	108	adantl 474	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
110		fznnfl 12916	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝑥)))
111	109, 110	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝑥)))
112	111	simplbda 494	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ≤ 𝑥)
113	107, 112	eqbrtrd 4865	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 · 𝑑) ≤ 𝑥)
114		1red 10329	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
115	108	ad2antlr 719	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
116	114, 115, 56	lemuldivd 12166	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((1 · 𝑑) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑑)))
117	113, 116	mpbid 224	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ≤ (𝑥 / 𝑑))
118		1rp 12078	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ⁺
119	118	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ⁺)
120	119, 12	logled 24714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 ≤ (𝑥 / 𝑑) ↔ (log‘1) ≤ (log‘(𝑥 / 𝑑))))
121	117, 120	mpbid 224	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (log‘1) ≤ (log‘(𝑥 / 𝑑)))
122	105, 121	syl5eqbrr 4879	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (log‘(𝑥 / 𝑑)))
123		rpregt0 12090	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
124	123	ad2antlr 719	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
125		divge0 11184	. . . . . . . 8 ⊢ ((((log‘(𝑥 / 𝑑)) ∈ ℝ ∧ 0 ≤ (log‘(𝑥 / 𝑑))) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → 0 ≤ ((log‘(𝑥 / 𝑑)) / 𝑥))
126	14, 122, 124, 125	syl21anc 867	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((log‘(𝑥 / 𝑑)) / 𝑥))
127		mulge0 10838	. . . . . . 7 ⊢ (((𝐶 ∈ ℝ ∧ 0 ≤ 𝐶) ∧ (((log‘(𝑥 / 𝑑)) / 𝑥) ∈ ℝ ∧ 0 ≤ ((log‘(𝑥 / 𝑑)) / 𝑥))) → 0 ≤ (𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)))
128	104, 16, 126, 127	syl12anc 866	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)))
129		absidm 14404	. . . . . . . . 9 ⊢ ((𝐾 − 𝑇) ∈ ℂ → (abs‘(abs‘(𝐾 − 𝑇))) = (abs‘(𝐾 − 𝑇)))
130	50, 129	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(abs‘(𝐾 − 𝑇))) = (abs‘(𝐾 − 𝑇)))
131	130	adantr 473	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (abs‘(abs‘(𝐾 − 𝑇))) = (abs‘(𝐾 − 𝑇)))
132	42	fvoveq1d 6900	. . . . . . . . . 10 ⊢ (𝑚 = (𝑥 / 𝑑) → (abs‘(𝐹 − 𝑇)) = (abs‘(𝐾 − 𝑇)))
133		fveq2 6411	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑥 / 𝑑) → (log‘𝑚) = (log‘(𝑥 / 𝑑)))
134		id 22	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑥 / 𝑑) → 𝑚 = (𝑥 / 𝑑))
135	133, 134	oveq12d 6896	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑥 / 𝑑) → ((log‘𝑚) / 𝑚) = ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑)))
136	135	oveq2d 6894	. . . . . . . . . 10 ⊢ (𝑚 = (𝑥 / 𝑑) → (𝐶 · ((log‘𝑚) / 𝑚)) = (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))))
137	132, 136	breq12d 4856	. . . . . . . . 9 ⊢ (𝑚 = (𝑥 / 𝑑) → ((abs‘(𝐹 − 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚)) ↔ (abs‘(𝐾 − 𝑇)) ≤ (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑)))))
138		dchrvmasum.1	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → (abs‘(𝐹 − 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚)))
139	138	ralrimiva 3147	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ (3[,)+∞)(abs‘(𝐹 − 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚)))
140	139	ad3antrrr 722	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → ∀𝑚 ∈ (3[,)+∞)(abs‘(𝐹 − 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚)))
141		nndivre 11354	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑑 ∈ ℕ) → (𝑥 / 𝑑) ∈ ℝ)
142	109, 9, 141	syl2an 590	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ)
143	142	adantr 473	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (𝑥 / 𝑑) ∈ ℝ)
144		simpr 478	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → 3 ≤ (𝑥 / 𝑑))
145		elicopnf 12519	. . . . . . . . . . 11 ⊢ (3 ∈ ℝ → ((𝑥 / 𝑑) ∈ (3[,)+∞) ↔ ((𝑥 / 𝑑) ∈ ℝ ∧ 3 ≤ (𝑥 / 𝑑))))
146	64, 145	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝑥 / 𝑑) ∈ (3[,)+∞) ↔ ((𝑥 / 𝑑) ∈ ℝ ∧ 3 ≤ (𝑥 / 𝑑)))
147	143, 144, 146	sylanbrc 579	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (𝑥 / 𝑑) ∈ (3[,)+∞))
148	137, 140, 147	rspcdva 3503	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (abs‘(𝐾 − 𝑇)) ≤ (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))))
149	14	recnd 10357	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑑)) ∈ ℂ)
150		rpcnne0 12094	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
151	150	ad2antlr 719	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
152	56	rpcnne0d 12126	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0))
153		divdiv2 11029	. . . . . . . . . . . . 13 ⊢ (((log‘(𝑥 / 𝑑)) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0)) → ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑)) = (((log‘(𝑥 / 𝑑)) · 𝑑) / 𝑥))
154	149, 151, 152, 153	syl3anc 1491	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑)) = (((log‘(𝑥 / 𝑑)) · 𝑑) / 𝑥))
155		div23 10996	. . . . . . . . . . . . 13 ⊢ (((log‘(𝑥 / 𝑑)) ∈ ℂ ∧ 𝑑 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((log‘(𝑥 / 𝑑)) · 𝑑) / 𝑥) = (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑))
156	149, 106, 151, 155	syl3anc 1491	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑑)) · 𝑑) / 𝑥) = (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑))
157	154, 156	eqtrd 2833	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑)) = (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑))
158	157	oveq2d 6894	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))) = (𝐶 · (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑)))
159	30	ad2antrr 718	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐶 ∈ ℂ)
160	16	recnd 10357	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑑)) / 𝑥) ∈ ℂ)
161	159, 160, 106	mulassd 10352	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑) = (𝐶 · (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑)))
162	158, 161	eqtr4d 2836	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))) = ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑))
163	162	adantr 473	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))) = ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑))
164	148, 163	breqtrd 4869	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (abs‘(𝐾 − 𝑇)) ≤ ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑))
165	131, 164	eqbrtrd 4865	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (abs‘(abs‘(𝐾 − 𝑇))) ≤ ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑))
166	130	adantr 473	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (abs‘(abs‘(𝐾 − 𝑇))) = (abs‘(𝐾 − 𝑇)))
167	132	breq1d 4853	. . . . . . . 8 ⊢ (𝑚 = (𝑥 / 𝑑) → ((abs‘(𝐹 − 𝑇)) ≤ 𝑅 ↔ (abs‘(𝐾 − 𝑇)) ≤ 𝑅))
168	92	ad3antrrr 722	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → ∀𝑚 ∈ (1[,)3)(abs‘(𝐹 − 𝑇)) ≤ 𝑅)
169	142	adantr 473	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (𝑥 / 𝑑) ∈ ℝ)
170	117	adantr 473	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → 1 ≤ (𝑥 / 𝑑))
171		simpr 478	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (𝑥 / 𝑑) < 3)
172		elico2 12486	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℝ^*) → ((𝑥 / 𝑑) ∈ (1[,)3) ↔ ((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑) ∧ (𝑥 / 𝑑) < 3)))
173	24, 70, 172	mp2an 684	. . . . . . . . 9 ⊢ ((𝑥 / 𝑑) ∈ (1[,)3) ↔ ((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑) ∧ (𝑥 / 𝑑) < 3))
174	169, 170, 171, 173	syl3anbrc 1444	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (𝑥 / 𝑑) ∈ (1[,)3))
175	167, 168, 174	rspcdva 3503	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (abs‘(𝐾 − 𝑇)) ≤ 𝑅)
176	166, 175	eqbrtrd 4865	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (abs‘(abs‘(𝐾 − 𝑇))) ≤ 𝑅)
177	8, 67, 100, 101, 103, 128, 165, 176	fsumharmonic 25090	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ≤ (Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1))))
178	30	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
179	7, 178, 160	fsummulc2 14854	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) = Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)))
180	179	oveq1d 6893	. . . . 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))))
181	177, 180	breqtrrd 4871	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ≤ ((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1))))
182	41	leabsd 14494	. . . 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)))))
183	61, 41, 63, 181, 182	letrd 10484	. . 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)))))
184	183	adantrr 709	. 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)))))
185	1, 40, 41, 55, 184	o1le 14724	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ∈ 𝑂(1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 ∀wral 3089 ⊆ wss 3769 class class class wbr 4843 ↦ cmpt 4922 ‘cfv 6101 (class class class)co 6878 ℂcc 10222 ℝcr 10223 0cc0 10224 1c1 10225 + caddc 10227 · cmul 10229 +∞cpnf 10360 ℝ^*cxr 10362 < clt 10363 ≤ cle 10364 − cmin 10556 / cdiv 10976 ℕcn 11312 3c3 11369 ℝ⁺crp 12074 [,)cico 12426 ...cfz 12580 ⌊cfl 12846 abscabs 14315 ⇝_𝑟 crli 14557 𝑂(1)co1 14558 Σcsu 14757 Basecbs 16184 0_gc0g 16415 ℤRHomczrh 20170 ℤ/nℤczn 20173 logclog 24642 DChrcdchr 25309

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 ax-addf 10303 ax-mulf 10304

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-om 7300 df-1st 7401 df-2nd 7402 df-supp 7533 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-2o 7800 df-oadd 7803 df-er 7982 df-map 8097 df-pm 8098 df-ixp 8149 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fsupp 8518 df-fi 8559 df-sup 8590 df-inf 8591 df-oi 8657 df-card 9051 df-cda 9278 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-q 12034 df-rp 12075 df-xneg 12193 df-xadd 12194 df-xmul 12195 df-ioo 12428 df-ioc 12429 df-ico 12430 df-icc 12431 df-fz 12581 df-fzo 12721 df-fl 12848 df-mod 12924 df-seq 13056 df-exp 13115 df-fac 13314 df-bc 13343 df-hash 13371 df-shft 14148 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-limsup 14543 df-clim 14560 df-rlim 14561 df-o1 14562 df-lo1 14563 df-sum 14758 df-ef 15134 df-e 15135 df-sin 15136 df-cos 15137 df-pi 15139 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-mulr 16281 df-starv 16282 df-sca 16283 df-vsca 16284 df-ip 16285 df-tset 16286 df-ple 16287 df-ds 16289 df-unif 16290 df-hom 16291 df-cco 16292 df-rest 16398 df-topn 16399 df-0g 16417 df-gsum 16418 df-topgen 16419 df-pt 16420 df-prds 16423 df-xrs 16477 df-qtop 16482 df-imas 16483 df-xps 16485 df-mre 16561 df-mrc 16562 df-acs 16564 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-submnd 17651 df-mulg 17857 df-cntz 18062 df-cmn 18510 df-psmet 20060 df-xmet 20061 df-met 20062 df-bl 20063 df-mopn 20064 df-fbas 20065 df-fg 20066 df-cnfld 20069 df-top 21027 df-topon 21044 df-topsp 21066 df-bases 21079 df-cld 21152 df-ntr 21153 df-cls 21154 df-nei 21231 df-lp 21269 df-perf 21270 df-cn 21360 df-cnp 21361 df-haus 21448 df-cmp 21519 df-tx 21694 df-hmeo 21887 df-fil 21978 df-fm 22070 df-flim 22071 df-flf 22072 df-xms 22453 df-ms 22454 df-tms 22455 df-cncf 23009 df-limc 23971 df-dv 23972 df-log 24644 df-cxp 24645 df-em 25071

This theorem is referenced by: dchrvmasumlem3 25540

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