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Theorem dchrvmasumlem3 25188

Description: Lemma for dchrvmasum 25214. (Contributed by Mario Carneiro, 3-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
dchrvmasumlem3	⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (𝐾 − 𝑇))) ∈ 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 dchrvmasumlem3**
Step	Hyp	Ref	Expression
1		1red 10055	. 2 ⊢ (𝜑 → 1 ∈ ℝ)
2		rpvmasum.z	. . 3 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
3		rpvmasum.l	. . 3 ⊢ 𝐿 = (ℤRHom‘𝑍)
4		rpvmasum.a	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
5		rpvmasum.g	. . 3 ⊢ 𝐺 = (DChr‘𝑁)
6		rpvmasum.d	. . 3 ⊢ 𝐷 = (Base‘𝐺)
7		rpvmasum.1	. . 3 ⊢ 1 = (0_g‘𝐺)
8		dchrisum.b	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
9		dchrisum.n1	. . 3 ⊢ (𝜑 → 𝑋 ≠ 1 )
10		dchrvmasum.f	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 𝐹 ∈ ℂ)
11		dchrvmasum.g	. . 3 ⊢ (𝑚 = (𝑥 / 𝑑) → 𝐹 = 𝐾)
12		dchrvmasum.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
13		dchrvmasum.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℂ)
14		dchrvmasum.1	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → (abs‘(𝐹 − 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚)))
15		dchrvmasum.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ)
16		dchrvmasum.2	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ (1[,)3)(abs‘(𝐹 − 𝑇)) ≤ 𝑅)
17	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16	dchrvmasumlem2 25187	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ∈ 𝑂(1))
18		fzfid 12772	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
19		simpr 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
20		elfznn 12370	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
21	20	nnrpd 11870	. . . . . . . 8 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℝ⁺)
22		rpdivcl 11856	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺) → (𝑥 / 𝑑) ∈ ℝ⁺)
23	19, 21, 22	syl2an 494	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ⁺)
24	10	ralrimiva 2966	. . . . . . . 8 ⊢ (𝜑 → ∀𝑚 ∈ ℝ⁺ 𝐹 ∈ ℂ)
25	24	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ∀𝑚 ∈ ℝ⁺ 𝐹 ∈ ℂ)
26	11	eleq1d 2686	. . . . . . . 8 ⊢ (𝑚 = (𝑥 / 𝑑) → (𝐹 ∈ ℂ ↔ 𝐾 ∈ ℂ))
27	26	rspcv 3305	. . . . . . 7 ⊢ ((𝑥 / 𝑑) ∈ ℝ⁺ → (∀𝑚 ∈ ℝ⁺ 𝐹 ∈ ℂ → 𝐾 ∈ ℂ))
28	23, 25, 27	sylc 65	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐾 ∈ ℂ)
29	13	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑇 ∈ ℂ)
30	28, 29	subcld 10392	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐾 − 𝑇) ∈ ℂ)
31	30	abscld 14175	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝐾 − 𝑇)) ∈ ℝ)
32	20	adantl 482	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
33	31, 32	nndivred 11069	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝐾 − 𝑇)) / 𝑑) ∈ ℝ)
34	18, 33	fsumrecl 14465	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑) ∈ ℝ)
35	8	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
36		elfzelz 12342	. . . . . . 7 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℤ)
37	36	adantl 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℤ)
38	5, 2, 6, 3, 35, 37	dchrzrhcl 24970	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑑)) ∈ ℂ)
39		mucl 24867	. . . . . . . . 9 ⊢ (𝑑 ∈ ℕ → (μ‘𝑑) ∈ ℤ)
40	32, 39	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (μ‘𝑑) ∈ ℤ)
41	40	zred 11482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (μ‘𝑑) ∈ ℝ)
42	41, 32	nndivred 11069	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑑) / 𝑑) ∈ ℝ)
43	42	recnd 10068	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑑) / 𝑑) ∈ ℂ)
44	38, 43	mulcld 10060	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ)
45	44, 30	mulcld 10060	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (𝐾 − 𝑇)) ∈ ℂ)
46	18, 45	fsumcl 14464	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (𝐾 − 𝑇)) ∈ ℂ)
47	46	abscld 14175	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (𝐾 − 𝑇))) ∈ ℝ)
48	34	recnd 10068	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑) ∈ ℂ)
49	48	abscld 14175	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ∈ ℝ)
50	45	abscld 14175	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (𝐾 − 𝑇))) ∈ ℝ)
51	18, 50	fsumrecl 14465	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (𝐾 − 𝑇))) ∈ ℝ)
52	18, 45	fsumabs 14533	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (𝐾 − 𝑇))) ≤ Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (𝐾 − 𝑇))))
53	44	abscld 14175	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) ∈ ℝ)
54	32	nnrecred 11066	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 / 𝑑) ∈ ℝ)
55	30	absge0d 14183	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(𝐾 − 𝑇)))
56	38, 43	absmuld 14193	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) = ((abs‘(𝑋‘(𝐿‘𝑑))) · (abs‘((μ‘𝑑) / 𝑑))))
57	38	abscld 14175	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑋‘(𝐿‘𝑑))) ∈ ℝ)
58		1red 10055	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
59	43	abscld 14175	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑑) / 𝑑)) ∈ ℝ)
60	38	absge0d 14183	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(𝑋‘(𝐿‘𝑑))))
61	43	absge0d 14183	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((μ‘𝑑) / 𝑑)))
62		eqid 2622	. . . . . . . . . . . 12 ⊢ (Base‘𝑍) = (Base‘𝑍)
63	4	nnnn0d 11351	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
64	2, 62, 3	znzrhfo 19896	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑍))
65	63, 64	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐿:ℤ–onto→(Base‘𝑍))
66		fof 6115	. . . . . . . . . . . . . . 15 ⊢ (𝐿:ℤ–onto→(Base‘𝑍) → 𝐿:ℤ⟶(Base‘𝑍))
67	65, 66	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍))
68	67	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐿:ℤ⟶(Base‘𝑍))
69	68, 37	ffvelrnd 6360	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐿‘𝑑) ∈ (Base‘𝑍))
70	5, 6, 2, 62, 35, 69	dchrabs2 24987	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑋‘(𝐿‘𝑑))) ≤ 1)
71	41	recnd 10068	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (μ‘𝑑) ∈ ℂ)
72	32	nncnd 11036	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℂ)
73	32	nnne0d 11065	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ≠ 0)
74	71, 72, 73	absdivd 14194	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑑) / 𝑑)) = ((abs‘(μ‘𝑑)) / (abs‘𝑑)))
75	32	nnrpd 11870	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ⁺)
76	75	rprege0d 11879	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑑 ∈ ℝ ∧ 0 ≤ 𝑑))
77		absid 14036	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 ∈ ℝ ∧ 0 ≤ 𝑑) → (abs‘𝑑) = 𝑑)
78	76, 77	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘𝑑) = 𝑑)
79	78	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘(μ‘𝑑)) / (abs‘𝑑)) = ((abs‘(μ‘𝑑)) / 𝑑))
80	74, 79	eqtrd 2656	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑑) / 𝑑)) = ((abs‘(μ‘𝑑)) / 𝑑))
81	71	abscld 14175	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑑)) ∈ ℝ)
82		mule1 24874	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ ℕ → (abs‘(μ‘𝑑)) ≤ 1)
83	32, 82	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑑)) ≤ 1)
84	81, 58, 75, 83	lediv1dd 11930	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘(μ‘𝑑)) / 𝑑) ≤ (1 / 𝑑))
85	80, 84	eqbrtrd 4675	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑑) / 𝑑)) ≤ (1 / 𝑑))
86	57, 58, 59, 54, 60, 61, 70, 85	lemul12ad 10966	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑋‘(𝐿‘𝑑))) · (abs‘((μ‘𝑑) / 𝑑))) ≤ (1 · (1 / 𝑑)))
87	54	recnd 10068	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 / 𝑑) ∈ ℂ)
88	87	mulid2d 10058	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 · (1 / 𝑑)) = (1 / 𝑑))
89	86, 88	breqtrd 4679	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑋‘(𝐿‘𝑑))) · (abs‘((μ‘𝑑) / 𝑑))) ≤ (1 / 𝑑))
90	56, 89	eqbrtrd 4675	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) ≤ (1 / 𝑑))
91	53, 54, 31, 55, 90	lemul1ad 10963	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) · (abs‘(𝐾 − 𝑇))) ≤ ((1 / 𝑑) · (abs‘(𝐾 − 𝑇))))
92	44, 30	absmuld 14193	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (𝐾 − 𝑇))) = ((abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) · (abs‘(𝐾 − 𝑇))))
93	31	recnd 10068	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝐾 − 𝑇)) ∈ ℂ)
94	93, 72, 73	divrec2d 10805	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝐾 − 𝑇)) / 𝑑) = ((1 / 𝑑) · (abs‘(𝐾 − 𝑇))))
95	91, 92, 94	3brtr4d 4685	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (𝐾 − 𝑇))) ≤ ((abs‘(𝐾 − 𝑇)) / 𝑑))
96	18, 50, 33, 95	fsumle 14531	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (𝐾 − 𝑇))) ≤ Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑))
97	47, 51, 34, 52, 96	letrd 10194	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (𝐾 − 𝑇))) ≤ Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑))
98	34	leabsd 14153	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑) ≤ (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)))
99	47, 34, 49, 97, 98	letrd 10194	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (𝐾 − 𝑇))) ≤ (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)))
100	99	adantrr 753	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (𝐾 − 𝑇))) ≤ (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)))
101	1, 17, 34, 46, 100	o1le 14383	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (𝐾 − 𝑇))) ∈ 𝑂(1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 class class class wbr 4653 ↦ cmpt 4729 ⟶wf 5884 –onto→wfo 5886 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 · cmul 9941 +∞cpnf 10071 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 3c3 11071 ℕ₀cn0 11292 ℤcz 11377 ℝ⁺crp 11832 [,)cico 12177 ...cfz 12326 ⌊cfl 12591 abscabs 13974 𝑂(1)co1 14217 Σcsu 14416 Basecbs 15857 0_gc0g 16100 ℤRHomczrh 19848 ℤ/nℤczn 19851 logclog 24301 μcmu 24821 DChrcdchr 24957

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-o1 14221 df-lo1 14222 df-sum 14417 df-ef 14798 df-e 14799 df-sin 14800 df-cos 14801 df-pi 14803 df-dvds 14984 df-prm 15386 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-qus 16169 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-nsg 17592 df-eqg 17593 df-ghm 17658 df-cntz 17750 df-od 17948 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-rnghom 18715 df-drng 18749 df-subrg 18778 df-lmod 18865 df-lss 18933 df-lsp 18972 df-sra 19172 df-rgmod 19173 df-lidl 19174 df-rsp 19175 df-2idl 19232 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-zring 19819 df-zrh 19852 df-zn 19855 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 df-cxp 24304 df-em 24719 df-mu 24827 df-dchr 24958

This theorem is referenced by: dchrvmasumiflem1 25190

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