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Theorem dchrvmasumlem2 26990

Description: Lemma for dchrvmasum 27017. (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 11211	. 2 ⊢ (𝜑 → 1 ∈ ℝ)
2		dchrvmasum.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
3		elrege0 13427	. . . . . . 7 ⊢ (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
4	2, 3	sylib 217	. . . . . 6 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
5	4	simpld 495	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ)
6	5	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
7		fzfid 13934	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
8		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
9		elfznn 13526	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
10	9	nnrpd 13010	. . . . . . . 8 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℝ⁺)
11		rpdivcl 12995	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺) → (𝑥 / 𝑑) ∈ ℝ⁺)
12	8, 10, 11	syl2an 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ⁺)
13	12	relogcld 26122	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑑)) ∈ ℝ)
14	8	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
15	13, 14	rerpdivcld 13043	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑑)) / 𝑥) ∈ ℝ)
16	7, 15	fsumrecl 15676	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥) ∈ ℝ)
17	6, 16	remulcld 11240	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ∈ ℝ)
18		dchrvmasum.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ)
19		3nn 12287	. . . . . . 7 ⊢ 3 ∈ ℕ
20		nnrp 12981	. . . . . . 7 ⊢ (3 ∈ ℕ → 3 ∈ ℝ⁺)
21		relogcl 26075	. . . . . . 7 ⊢ (3 ∈ ℝ⁺ → (log‘3) ∈ ℝ)
22	19, 20, 21	mp2b 10	. . . . . 6 ⊢ (log‘3) ∈ ℝ
23		1re 11210	. . . . . 6 ⊢ 1 ∈ ℝ
24	22, 23	readdcli 11225	. . . . 5 ⊢ ((log‘3) + 1) ∈ ℝ
25		remulcl 11191	. . . . 5 ⊢ ((𝑅 ∈ ℝ ∧ ((log‘3) + 1) ∈ ℝ) → (𝑅 · ((log‘3) + 1)) ∈ ℝ)
26	18, 24, 25	sylancl 586	. . . 4 ⊢ (𝜑 → (𝑅 · ((log‘3) + 1)) ∈ ℝ)
27	26	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑅 · ((log‘3) + 1)) ∈ ℝ)
28		rpssre 12977	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
29	5	recnd 11238	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ)
30		o1const 15560	. . . . 5 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 𝐶 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 𝐶) ∈ 𝑂(1))
31	28, 29, 30	sylancr 587	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 𝐶) ∈ 𝑂(1))
32		logfacrlim2 26718	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ⇝_𝑟 1
33		rlimo1 15557	. . . . 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 15565	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥))) ∈ 𝑂(1))
36	26	recnd 11238	. . . 4 ⊢ (𝜑 → (𝑅 · ((log‘3) + 1)) ∈ ℂ)
37		o1const 15560	. . . 4 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (𝑅 · ((log‘3) + 1)) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (𝑅 · ((log‘3) + 1))) ∈ 𝑂(1))
38	28, 36, 37	sylancr 587	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝑅 · ((log‘3) + 1))) ∈ 𝑂(1))
39	17, 27, 35, 38	o1add2 15564	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1)))) ∈ 𝑂(1))
40	17, 27	readdcld 11239	. 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 2818	. . . . . . . 8 ⊢ (𝑚 = (𝑥 / 𝑑) → (𝐹 ∈ ℂ ↔ 𝐾 ∈ ℂ))
43		dchrvmasum.f	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 𝐹 ∈ ℂ)
44	43	ralrimiva 3146	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ ℝ⁺ 𝐹 ∈ ℂ)
45	44	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ∀𝑚 ∈ ℝ⁺ 𝐹 ∈ ℂ)
46	42, 45, 12	rspcdva 3613	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐾 ∈ ℂ)
47		dchrvmasum.t	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℂ)
48	47	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑇 ∈ ℂ)
49	46, 48	subcld 11567	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐾 − 𝑇) ∈ ℂ)
50	49	abscld 15379	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝐾 − 𝑇)) ∈ ℝ)
51	9	adantl 482	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
52	50, 51	nndivred 12262	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝐾 − 𝑇)) / 𝑑) ∈ ℝ)
53	7, 52	fsumrecl 15676	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑) ∈ ℝ)
54	53	recnd 11238	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑) ∈ ℂ)
55	51	nnrpd 13010	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ⁺)
56	49	absge0d 15387	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(𝐾 − 𝑇)))
57	50, 55, 56	divge0d 13052	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((abs‘(𝐾 − 𝑇)) / 𝑑))
58	7, 52, 57	fsumge0 15737	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑))
59	53, 58	absidd 15365	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) = Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑))
60	59, 53	eqeltrd 2833	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ∈ ℝ)
61	40	recnd 11238	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1))) ∈ ℂ)
62	61	abscld 15379	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1)))) ∈ ℝ)
63		3re 12288	. . . . . . . 8 ⊢ 3 ∈ ℝ
64	63	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 3 ∈ ℝ)
65		1le3 12420	. . . . . . 7 ⊢ 1 ≤ 3
66	64, 65	jctir 521	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (3 ∈ ℝ ∧ 1 ≤ 3))
67	18	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑅 ∈ ℝ)
68	23	rexri 11268	. . . . . . . . . 10 ⊢ 1 ∈ ℝ^*
69	63	rexri 11268	. . . . . . . . . 10 ⊢ 3 ∈ ℝ^*
70		1lt3 12381	. . . . . . . . . 10 ⊢ 1 < 3
71		lbico1 13374	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ^* ∧ 3 ∈ ℝ^* ∧ 1 < 3) → 1 ∈ (1[,)3))
72	68, 69, 70, 71	mp3an 1461	. . . . . . . . 9 ⊢ 1 ∈ (1[,)3)
73		0red 11213	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 0 ∈ ℝ)
74		elico2 13384	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℝ^*) → (𝑚 ∈ (1[,)3) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚 ∧ 𝑚 < 3)))
75	23, 69, 74	mp2an 690	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚 ∧ 𝑚 < 3))
76	75	simp1bi 1145	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (1[,)3) → 𝑚 ∈ ℝ)
77		0red 11213	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) → 0 ∈ ℝ)
78		1red 11211	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) → 1 ∈ ℝ)
79		0lt1 11732	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
80	79	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) → 0 < 1)
81	75	simp2bi 1146	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) → 1 ≤ 𝑚)
82	77, 78, 76, 80, 81	ltletrd 11370	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (1[,)3) → 0 < 𝑚)
83	76, 82	elrpd 13009	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (1[,)3) → 𝑚 ∈ ℝ⁺)
84	47	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 𝑇 ∈ ℂ)
85	43, 84	subcld 11567	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (𝐹 − 𝑇) ∈ ℂ)
86	85	abscld 15379	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (abs‘(𝐹 − 𝑇)) ∈ ℝ)
87	83, 86	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (abs‘(𝐹 − 𝑇)) ∈ ℝ)
88	18	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 𝑅 ∈ ℝ)
89	85	absge0d 15387	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 0 ≤ (abs‘(𝐹 − 𝑇)))
90	83, 89	sylan2 593	. . . . . . . . . . 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 3248	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (abs‘(𝐹 − 𝑇)) ≤ 𝑅)
93	73, 87, 88, 90, 92	letrd 11367	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 0 ≤ 𝑅)
94	93	ralrimiva 3146	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ (1[,)3)0 ≤ 𝑅)
95		biidd 261	. . . . . . . . . 10 ⊢ (𝑚 = 1 → (0 ≤ 𝑅 ↔ 0 ≤ 𝑅))
96	95	rspcv 3608	. . . . . . . . 9 ⊢ (1 ∈ (1[,)3) → (∀𝑚 ∈ (1[,)3)0 ≤ 𝑅 → 0 ≤ 𝑅))
97	72, 94, 96	mpsyl 68	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝑅)
98	97	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ 𝑅)
99	67, 98	jca 512	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
100	50	recnd 11238	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝐾 − 𝑇)) ∈ ℂ)
101	5	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐶 ∈ ℝ)
102	101, 15	remulcld 11240	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) ∈ ℝ)
103	4	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
104		log1 26085	. . . . . . . . 9 ⊢ (log‘1) = 0
105	51	nncnd 12224	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℂ)
106	105	mullidd 11228	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 · 𝑑) = 𝑑)
107		rpre 12978	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
108	107	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
109		fznnfl 13823	. . . . . . . . . . . . . 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 500	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ≤ 𝑥)
112	106, 111	eqbrtrd 5169	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 · 𝑑) ≤ 𝑥)
113		1red 11211	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
114	107	ad2antlr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
115	113, 114, 55	lemuldivd 13061	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((1 · 𝑑) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑑)))
116	112, 115	mpbid 231	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ≤ (𝑥 / 𝑑))
117		1rp 12974	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ⁺
118	117	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ⁺)
119	118, 12	logled 26126	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 ≤ (𝑥 / 𝑑) ↔ (log‘1) ≤ (log‘(𝑥 / 𝑑))))
120	116, 119	mpbid 231	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (log‘1) ≤ (log‘(𝑥 / 𝑑)))
121	104, 120	eqbrtrrid 5183	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (log‘(𝑥 / 𝑑)))
122		rpregt0 12984	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
123	122	ad2antlr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
124		divge0 12079	. . . . . . . 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 11728	. . . . . . 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 15266	. . . . . . . . 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 481	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (abs‘(abs‘(𝐾 − 𝑇))) = (abs‘(𝐾 − 𝑇)))
131	41	fvoveq1d 7427	. . . . . . . . . 10 ⊢ (𝑚 = (𝑥 / 𝑑) → (abs‘(𝐹 − 𝑇)) = (abs‘(𝐾 − 𝑇)))
132		fveq2 6888	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑥 / 𝑑) → (log‘𝑚) = (log‘(𝑥 / 𝑑)))
133		id 22	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑥 / 𝑑) → 𝑚 = (𝑥 / 𝑑))
134	132, 133	oveq12d 7423	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑥 / 𝑑) → ((log‘𝑚) / 𝑚) = ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑)))
135	134	oveq2d 7421	. . . . . . . . . 10 ⊢ (𝑚 = (𝑥 / 𝑑) → (𝐶 · ((log‘𝑚) / 𝑚)) = (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))))
136	131, 135	breq12d 5160	. . . . . . . . 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 3146	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ (3[,)+∞)(abs‘(𝐹 − 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚)))
139	138	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → ∀𝑚 ∈ (3[,)+∞)(abs‘(𝐹 − 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚)))
140		nndivre 12249	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑑 ∈ ℕ) → (𝑥 / 𝑑) ∈ ℝ)
141	108, 9, 140	syl2an 596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ)
142	141	adantr 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (𝑥 / 𝑑) ∈ ℝ)
143		simpr 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → 3 ≤ (𝑥 / 𝑑))
144		elicopnf 13418	. . . . . . . . . . 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 583	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (𝑥 / 𝑑) ∈ (3[,)+∞))
147	136, 139, 146	rspcdva 3613	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (abs‘(𝐾 − 𝑇)) ≤ (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))))
148	13	recnd 11238	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑑)) ∈ ℂ)
149		rpcnne0 12988	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
150	149	ad2antlr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
151	55	rpcnne0d 13021	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0))
152		divdiv2 11922	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑)) = (((log‘(𝑥 / 𝑑)) · 𝑑) / 𝑥))
154		div23 11887	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑑)) · 𝑑) / 𝑥) = (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑))
156	153, 155	eqtrd 2772	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑)) = (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑))
157	156	oveq2d 7421	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))) = (𝐶 · (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑)))
158	29	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐶 ∈ ℂ)
159	15	recnd 11238	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑑)) / 𝑥) ∈ ℂ)
160	158, 159, 105	mulassd 11233	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑) = (𝐶 · (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑)))
161	157, 160	eqtr4d 2775	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))) = ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑))
162	161	adantr 481	. . . . . . . 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 5173	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (abs‘(𝐾 − 𝑇)) ≤ ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑))
164	130, 163	eqbrtrd 5169	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 3 ≤ (𝑥 / 𝑑)) → (abs‘(abs‘(𝐾 − 𝑇))) ≤ ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑))
165	129	adantr 481	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (abs‘(abs‘(𝐾 − 𝑇))) = (abs‘(𝐾 − 𝑇)))
166	131	breq1d 5157	. . . . . . . 8 ⊢ (𝑚 = (𝑥 / 𝑑) → ((abs‘(𝐹 − 𝑇)) ≤ 𝑅 ↔ (abs‘(𝐾 − 𝑇)) ≤ 𝑅))
167	91	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → ∀𝑚 ∈ (1[,)3)(abs‘(𝐹 − 𝑇)) ≤ 𝑅)
168	141	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (𝑥 / 𝑑) ∈ ℝ)
169	116	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → 1 ≤ (𝑥 / 𝑑))
170		simpr 485	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (𝑥 / 𝑑) < 3)
171		elico2 13384	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℝ^*) → ((𝑥 / 𝑑) ∈ (1[,)3) ↔ ((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑) ∧ (𝑥 / 𝑑) < 3)))
172	23, 69, 171	mp2an 690	. . . . . . . . 9 ⊢ ((𝑥 / 𝑑) ∈ (1[,)3) ↔ ((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑) ∧ (𝑥 / 𝑑) < 3))
173	168, 169, 170, 172	syl3anbrc 1343	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (𝑥 / 𝑑) ∈ (1[,)3))
174	166, 167, 173	rspcdva 3613	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑑) < 3) → (abs‘(𝐾 − 𝑇)) ≤ 𝑅)
175	165, 174	eqbrtrd 5169	. . . . . 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 26505	. . . . 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 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
178	7, 177, 159	fsummulc2 15726	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) = Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)))
179	178	oveq1d 7420	. . . . 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 5175	. . . 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 15357	. . . 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 11367	. . 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 715	. 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 15595	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ∈ 𝑂(1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ⊆ wss 3947 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 +∞cpnf 11241 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 3c3 12264 ℝ⁺crp 12970 [,)cico 13322 ...cfz 13480 ⌊cfl 13751 abscabs 15177 ⇝_𝑟 crli 15425 𝑂(1)co1 15426 Σcsu 15628 Basecbs 17140 0_gc0g 17381 ℤRHomczrh 21040 ℤ/nℤczn 21043 logclog 26054 DChrcdchr 26724

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 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 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 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-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-o1 15430 df-lo1 15431 df-sum 15629 df-ef 16007 df-e 16008 df-sin 16009 df-cos 16010 df-tan 16011 df-pi 16012 df-dvds 16194 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-cmp 22882 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375 df-ulm 25880 df-log 26056 df-cxp 26057 df-atan 26361 df-em 26486

This theorem is referenced by: dchrvmasumlem3 26991

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