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Theorem o1fsum 15261

Description: If 𝐴(𝑘) is O(1), then Σ𝑘 ≤ 𝑥, 𝐴(𝑘) is O(𝑥). (Contributed by Mario Carneiro, 23-May-2016.)

**Hypotheses**
Ref	Expression
o1fsum.1	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ 𝑉)
o1fsum.2	⊢ (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴) ∈ 𝑂(1))

**Assertion**
Ref	Expression
o1fsum	⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1))

Distinct variable groups: 𝑥,𝐴 𝑥,𝑘,𝜑 Allowed substitution hints: 𝐴(𝑘) 𝑉(𝑥,𝑘)

Proof of Theorem o1fsum Dummy variables 𝑚 𝑐 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		o1fsum.2	. . 3 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴) ∈ 𝑂(1))
2		nnssre 11720	. . . . 5 ⊢ ℕ ⊆ ℝ
3	2	a1i 11	. . . 4 ⊢ (𝜑 → ℕ ⊆ ℝ)
4		o1fsum.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ 𝑉)
5	4, 1	o1mptrcl 15070	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
6		1red 10720	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
7	3, 5, 6	elo1mpt2 14982	. . 3 ⊢ (𝜑 → ((𝑘 ∈ ℕ ↦ 𝐴) ∈ 𝑂(1) ↔ ∃𝑐 ∈ (1[,)+∞)∃𝑚 ∈ ℝ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)))
8	1, 7	mpbid 235	. 2 ⊢ (𝜑 → ∃𝑐 ∈ (1[,)+∞)∃𝑚 ∈ ℝ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚))
9		rpssre 12479	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
10	9	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → ℝ⁺ ⊆ ℝ)
11		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑛𝐴
12		nfcsb1v 3814	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐴
13		csbeq1a 3804	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → 𝐴 = ⦋𝑛 / 𝑘⦌𝐴)
14	11, 12, 13	cbvsumi 15147	. . . . . . 7 ⊢ Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 = Σ𝑛 ∈ (1...(⌊‘𝑥))⦋𝑛 / 𝑘⦌𝐴
15		fzfid 13432	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
16		o1f 14976	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ↦ 𝐴) ∈ 𝑂(1) → (𝑘 ∈ ℕ ↦ 𝐴):dom (𝑘 ∈ ℕ ↦ 𝐴)⟶ℂ)
17	1, 16	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴):dom (𝑘 ∈ ℕ ↦ 𝐴)⟶ℂ)
18	4	ralrimiva 3096	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑉)
19		dmmptg 6074	. . . . . . . . . . . . . 14 ⊢ (∀𝑘 ∈ ℕ 𝐴 ∈ 𝑉 → dom (𝑘 ∈ ℕ ↦ 𝐴) = ℕ)
20	18, 19	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (𝑘 ∈ ℕ ↦ 𝐴) = ℕ)
21	20	feq2d 6490	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑘 ∈ ℕ ↦ 𝐴):dom (𝑘 ∈ ℕ ↦ 𝐴)⟶ℂ ↔ (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ))
22	17, 21	mpbid 235	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ)
23		eqid 2738	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ ↦ 𝐴) = (𝑘 ∈ ℕ ↦ 𝐴)
24	23	fmpt 6884	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ ℕ 𝐴 ∈ ℂ ↔ (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ)
25	22, 24	sylibr 237	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ)
26	25	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ⁺) → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ)
27		elfznn 13027	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
28	12	nfel1 2915	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ
29	13	eleq1d 2817	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → (𝐴 ∈ ℂ ↔ ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ))
30	28, 29	rspc 3514	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (∀𝑘 ∈ ℕ 𝐴 ∈ ℂ → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ))
31	30	impcom 411	. . . . . . . . 9 ⊢ ((∀𝑘 ∈ ℕ 𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ)
32	26, 27, 31	syl2an 599	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ)
33	15, 32	fsumcl 15183	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ)
34	14, 33	eqeltrid 2837	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ⁺) → Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 ∈ ℂ)
35		rpcn 12482	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
36	35	adantl 485	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
37		rpne0 12488	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
38	37	adantl 485	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
39	34, 36, 38	divcld 11494	. . . . 5 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥) ∈ ℂ)
40		simplrl 777	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑐 ∈ (1[,)+∞))
41		1re 10719	. . . . . . . 8 ⊢ 1 ∈ ℝ
42		elicopnf 12919	. . . . . . . 8 ⊢ (1 ∈ ℝ → (𝑐 ∈ (1[,)+∞) ↔ (𝑐 ∈ ℝ ∧ 1 ≤ 𝑐)))
43	41, 42	ax-mp 5	. . . . . . 7 ⊢ (𝑐 ∈ (1[,)+∞) ↔ (𝑐 ∈ ℝ ∧ 1 ≤ 𝑐))
44	40, 43	sylib 221	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → (𝑐 ∈ ℝ ∧ 1 ≤ 𝑐))
45	44	simpld 498	. . . . 5 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑐 ∈ ℝ)
46		fzfid 13432	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → (1...(⌊‘𝑐)) ∈ Fin)
47	25	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ)
48		elfznn 13027	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝑐)) → 𝑛 ∈ ℕ)
49	47, 48, 31	syl2an 599	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑐))) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ)
50	49	abscld 14886	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑐))) → (abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ)
51	46, 50	fsumrecl 15184	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ)
52		simplrr 778	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑚 ∈ ℝ)
53	51, 52	readdcld 10748	. . . . 5 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚) ∈ ℝ)
54	34, 36, 38	absdivd 14905	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ⁺) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) = ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) / (abs‘𝑥)))
55	54	adantrr 717	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) = ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) / (abs‘𝑥)))
56		rprege0 12487	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
57	56	ad2antrl 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
58		absid 14746	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (abs‘𝑥) = 𝑥)
59	57, 58	syl 17	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (abs‘𝑥) = 𝑥)
60	59	oveq2d 7186	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) / (abs‘𝑥)) = ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) / 𝑥))
61	55, 60	eqtrd 2773	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) = ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) / 𝑥))
62	34	adantrr 717	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 ∈ ℂ)
63	62	abscld 14886	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) ∈ ℝ)
64		fzfid 13432	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
65	47, 27, 31	syl2an 599	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ)
66	65	adantlr 715	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ)
67	66	abscld 14886	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ)
68	64, 67	fsumrecl 15184	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ)
69	57	simpld 498	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → 𝑥 ∈ ℝ)
70	51	adantr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ)
71	52	adantr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → 𝑚 ∈ ℝ)
72	70, 71	readdcld 10748	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚) ∈ ℝ)
73	69, 72	remulcld 10749	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (𝑥 · (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚)) ∈ ℝ)
74	14	fveq2i 6677	. . . . . . . . 9 ⊢ (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) = (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))⦋𝑛 / 𝑘⦌𝐴)
75	64, 66	fsumabs 15249	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))⦋𝑛 / 𝑘⦌𝐴) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴))
76	74, 75	eqbrtrid 5065	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴))
77		fzfid 13432	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin)
78		ssun2 4063	. . . . . . . . . . . . . 14 ⊢ (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ⊆ ((1...(⌊‘𝑐)) ∪ (((⌊‘𝑐) + 1)...(⌊‘𝑥)))
79		flge1nn 13282	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 ∈ ℝ ∧ 1 ≤ 𝑐) → (⌊‘𝑐) ∈ ℕ)
80	44, 79	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → (⌊‘𝑐) ∈ ℕ)
81	80	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑐) ∈ ℕ)
82	81	nnred 11731	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑐) ∈ ℝ)
83	45	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → 𝑐 ∈ ℝ)
84		flle 13260	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 ∈ ℝ → (⌊‘𝑐) ≤ 𝑐)
85	83, 84	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑐) ≤ 𝑐)
86		simprr 773	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → 𝑐 ≤ 𝑥)
87	82, 83, 69, 85, 86	letrd 10875	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑐) ≤ 𝑥)
88		fznnfl 13321	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ → ((⌊‘𝑐) ∈ (1...(⌊‘𝑥)) ↔ ((⌊‘𝑐) ∈ ℕ ∧ (⌊‘𝑐) ≤ 𝑥)))
89	69, 88	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → ((⌊‘𝑐) ∈ (1...(⌊‘𝑥)) ↔ ((⌊‘𝑐) ∈ ℕ ∧ (⌊‘𝑐) ≤ 𝑥)))
90	81, 87, 89	mpbir2and 713	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑐) ∈ (1...(⌊‘𝑥)))
91		fzsplit 13024	. . . . . . . . . . . . . . 15 ⊢ ((⌊‘𝑐) ∈ (1...(⌊‘𝑥)) → (1...(⌊‘𝑥)) = ((1...(⌊‘𝑐)) ∪ (((⌊‘𝑐) + 1)...(⌊‘𝑥))))
92	90, 91	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (1...(⌊‘𝑥)) = ((1...(⌊‘𝑐)) ∪ (((⌊‘𝑐) + 1)...(⌊‘𝑥))))
93	78, 92	sseqtrrid 3930	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑥)))
94	93	sselda 3877	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) → 𝑛 ∈ (1...(⌊‘𝑥)))
95	65	abscld 14886	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ)
96	95	adantlr 715	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ)
97	94, 96	syldan 594	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) → (abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ)
98	77, 97	fsumrecl 15184	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ)
99	69, 70	remulcld 10749	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴)) ∈ ℝ)
100	69, 71	remulcld 10749	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (𝑥 · 𝑚) ∈ ℝ)
101	70	recnd 10747	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℂ)
102	101	mulid2d 10737	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (1 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴)) = Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴))
103		1red 10720	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → 1 ∈ ℝ)
104	49	absge0d 14894	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑐))) → 0 ≤ (abs‘⦋𝑛 / 𝑘⦌𝐴))
105	46, 50, 104	fsumge0 15243	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴))
106	51, 105	jca 515	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ ∧ 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴)))
107	106	adantr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ ∧ 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴)))
108	44	simprd 499	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → 1 ≤ 𝑐)
109	108	adantr 484	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → 1 ≤ 𝑐)
110	103, 83, 69, 109, 86	letrd 10875	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → 1 ≤ 𝑥)
111		lemul1a 11572	. . . . . . . . . . . 12 ⊢ (((1 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ ∧ 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴))) ∧ 1 ≤ 𝑥) → (1 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴)) ≤ (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴)))
112	103, 69, 107, 110, 111	syl31anc 1374	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (1 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴)) ≤ (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴)))
113	102, 112	eqbrtrrd 5054	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴)))
114		hashcl 13809	. . . . . . . . . . . . 13 ⊢ ((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ∈ ℕ₀)
115		nn0re 11985	. . . . . . . . . . . . 13 ⊢ ((♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ∈ ℕ₀ → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ∈ ℝ)
116	77, 114, 115	3syl 18	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ∈ ℝ)
117	116, 71	remulcld 10749	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → ((♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚) ∈ ℝ)
118	71	adantr 484	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) → 𝑚 ∈ ℝ)
119		elfzuz 12994	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥)) → 𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1)))
120	81	peano2nnd 11733	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → ((⌊‘𝑐) + 1) ∈ ℕ)
121		eluznn 12400	. . . . . . . . . . . . . . . 16 ⊢ ((((⌊‘𝑐) + 1) ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1))) → 𝑛 ∈ ℕ)
122	120, 121	sylan 583	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1))) → 𝑛 ∈ ℕ)
123		simpllr 776	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1))) → ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚))
124	83	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1))) → 𝑐 ∈ ℝ)
125		reflcl 13257	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ ℝ → (⌊‘𝑐) ∈ ℝ)
126		peano2re 10891	. . . . . . . . . . . . . . . . 17 ⊢ ((⌊‘𝑐) ∈ ℝ → ((⌊‘𝑐) + 1) ∈ ℝ)
127	124, 125, 126	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1))) → ((⌊‘𝑐) + 1) ∈ ℝ)
128	122	nnred 11731	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1))) → 𝑛 ∈ ℝ)
129		fllep1 13262	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ ℝ → 𝑐 ≤ ((⌊‘𝑐) + 1))
130	124, 129	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1))) → 𝑐 ≤ ((⌊‘𝑐) + 1))
131		eluzle 12337	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1)) → ((⌊‘𝑐) + 1) ≤ 𝑛)
132	131	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1))) → ((⌊‘𝑐) + 1) ≤ 𝑛)
133	124, 127, 128, 130, 132	letrd 10875	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1))) → 𝑐 ≤ 𝑛)
134		nfv 1921	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘 𝑐 ≤ 𝑛
135		nfcv 2899	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘abs
136	135, 12	nffv 6684	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(abs‘⦋𝑛 / 𝑘⦌𝐴)
137		nfcv 2899	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘 ≤
138		nfcv 2899	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘𝑚
139	136, 137, 138	nfbr 5077	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚
140	134, 139	nfim 1903	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(𝑐 ≤ 𝑛 → (abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚)
141		breq2 5034	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → (𝑐 ≤ 𝑘 ↔ 𝑐 ≤ 𝑛))
142	13	fveq2d 6678	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑛 → (abs‘𝐴) = (abs‘⦋𝑛 / 𝑘⦌𝐴))
143	142	breq1d 5040	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → ((abs‘𝐴) ≤ 𝑚 ↔ (abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚))
144	141, 143	imbi12d 348	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → ((𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚) ↔ (𝑐 ≤ 𝑛 → (abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚)))
145	140, 144	rspc 3514	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚) → (𝑐 ≤ 𝑛 → (abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚)))
146	122, 123, 133, 145	syl3c 66	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1))) → (abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚)
147	119, 146	sylan2 596	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) → (abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚)
148	77, 97, 118, 147	fsumle 15247	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))𝑚)
149	71	recnd 10747	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → 𝑚 ∈ ℂ)
150		fsumconst 15238	. . . . . . . . . . . . 13 ⊢ (((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin ∧ 𝑚 ∈ ℂ) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))𝑚 = ((♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚))
151	77, 149, 150	syl2anc 587	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))𝑚 = ((♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚))
152	148, 151	breqtrd 5056	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ ((♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚))
153		biidd 265	. . . . . . . . . . . . 13 ⊢ (𝑛 = ((⌊‘𝑐) + 1) → (0 ≤ 𝑚 ↔ 0 ≤ 𝑚))
154		0red 10722	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1))) → 0 ∈ ℝ)
155	47, 30	mpan9 510	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ ℕ) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ)
156	155	adantlr 715	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ ℕ) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ)
157	122, 156	syldan 594	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1))) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ)
158	157	abscld 14886	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1))) → (abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ)
159	71	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1))) → 𝑚 ∈ ℝ)
160	157	absge0d 14894	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1))) → 0 ≤ (abs‘⦋𝑛 / 𝑘⦌𝐴))
161	154, 158, 159, 160, 146	letrd 10875	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1))) → 0 ≤ 𝑚)
162	161	ralrimiva 3096	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → ∀𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1))0 ≤ 𝑚)
163	120	nnzd 12167	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → ((⌊‘𝑐) + 1) ∈ ℤ)
164		uzid 12339	. . . . . . . . . . . . . 14 ⊢ (((⌊‘𝑐) + 1) ∈ ℤ → ((⌊‘𝑐) + 1) ∈ (ℤ_≥‘((⌊‘𝑐) + 1)))
165	163, 164	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → ((⌊‘𝑐) + 1) ∈ (ℤ_≥‘((⌊‘𝑐) + 1)))
166	153, 162, 165	rspcdva 3528	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → 0 ≤ 𝑚)
167		reflcl 13257	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
168	69, 167	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑥) ∈ ℝ)
169		ssdomg 8601	. . . . . . . . . . . . . . . 16 ⊢ ((1...(⌊‘𝑥)) ∈ Fin → ((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ≼ (1...(⌊‘𝑥))))
170	64, 93, 169	sylc 65	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ≼ (1...(⌊‘𝑥)))
171		hashdomi 13833	. . . . . . . . . . . . . . 15 ⊢ ((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ≼ (1...(⌊‘𝑥)) → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ (♯‘(1...(⌊‘𝑥))))
172	170, 171	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ (♯‘(1...(⌊‘𝑥))))
173		flge0nn0 13281	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
174		hashfz1 13798	. . . . . . . . . . . . . . 15 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
175	57, 173, 174	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
176	172, 175	breqtrd 5056	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ (⌊‘𝑥))
177		flle 13260	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
178	69, 177	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑥) ≤ 𝑥)
179	116, 168, 69, 176, 178	letrd 10875	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ 𝑥)
180	116, 69, 71, 166, 179	lemul1ad 11657	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → ((♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚) ≤ (𝑥 · 𝑚))
181	98, 117, 100, 152, 180	letrd 10875	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ (𝑥 · 𝑚))
182	70, 98, 99, 100, 113, 181	le2addd 11337	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴)) ≤ ((𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴)) + (𝑥 · 𝑚)))
183		ltp1 11558	. . . . . . . . . . 11 ⊢ ((⌊‘𝑐) ∈ ℝ → (⌊‘𝑐) < ((⌊‘𝑐) + 1))
184		fzdisj 13025	. . . . . . . . . . 11 ⊢ ((⌊‘𝑐) < ((⌊‘𝑐) + 1) → ((1...(⌊‘𝑐)) ∩ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) = ∅)
185	82, 183, 184	3syl 18	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → ((1...(⌊‘𝑐)) ∩ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) = ∅)
186	96	recnd 10747	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℂ)
187	185, 92, 64, 186	fsumsplit 15190	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴) = (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴)))
188	36	adantrr 717	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → 𝑥 ∈ ℂ)
189	188, 101, 149	adddid 10743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (𝑥 · (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚)) = ((𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴)) + (𝑥 · 𝑚)))
190	182, 187, 189	3brtr4d 5062	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ (𝑥 · (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚)))
191	63, 68, 73, 76, 190	letrd 10875	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) ≤ (𝑥 · (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚)))
192		rpregt0 12486	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
193	192	ad2antrl 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
194		ledivmul 11594	. . . . . . . 8 ⊢ (((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) ∈ ℝ ∧ (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚) ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) / 𝑥) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚) ↔ (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) ≤ (𝑥 · (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚))))
195	63, 72, 193, 194	syl3anc 1372	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) / 𝑥) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚) ↔ (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) ≤ (𝑥 · (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚))))
196	191, 195	mpbird 260	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) / 𝑥) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚))
197	61, 196	eqbrtrd 5052	. . . . 5 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚))
198	10, 39, 45, 53, 197	elo1d 14983	. . . 4 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1))
199	198	ex 416	. . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) → (∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1)))
200	199	rexlimdvva 3204	. 2 ⊢ (𝜑 → (∃𝑐 ∈ (1[,)+∞)∃𝑚 ∈ ℝ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1)))
201	8, 200	mpd 15	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∀wral 3053 ∃wrex 3054 ⦋csb 3790 ∪ cun 3841 ∩ cin 3842 ⊆ wss 3843 ∅c0 4211 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5525 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 ≼ cdom 8553 Fincfn 8555 ℂcc 10613 ℝcr 10614 0cc0 10615 1c1 10616 + caddc 10618 · cmul 10620 +∞cpnf 10750 < clt 10753 ≤ cle 10754 / cdiv 11375 ℕcn 11716 ℕ₀cn0 11976 ℤcz 12062 ℤ_≥cuz 12324 ℝ⁺crp 12472 [,)cico 12823 ...cfz 12981 ⌊cfl 13251 ♯chash 13782 abscabs 14683 𝑂(1)co1 14933 Σcsu 15135

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-inf2 9177 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-oadd 8135 df-er 8320 df-pm 8440 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-sup 8979 df-inf 8980 df-oi 9047 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-n0 11977 df-xnn0 12049 df-z 12063 df-uz 12325 df-rp 12473 df-ico 12827 df-fz 12982 df-fzo 13125 df-fl 13253 df-seq 13461 df-exp 13522 df-hash 13783 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-clim 14935 df-o1 14937 df-lo1 14938 df-sum 15136

This theorem is referenced by: selberg2lem 26286

W3C validator