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Theorem o1fsum 15861

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 12297	. . . . 5 ⊢ ℕ ⊆ ℝ
3	2	a1i 11	. . . 4 ⊢ (𝜑 → ℕ ⊆ ℝ)
4		o1fsum.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ 𝑉)
5	4, 1	o1mptrcl 15669	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
6		1red 11291	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
7	3, 5, 6	elo1mpt2 15581	. . 3 ⊢ (𝜑 → ((𝑘 ∈ ℕ ↦ 𝐴) ∈ 𝑂(1) ↔ ∃𝑐 ∈ (1[,)+∞)∃𝑚 ∈ ℝ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)))
8	1, 7	mpbid 232	. 2 ⊢ (𝜑 → ∃𝑐 ∈ (1[,)+∞)∃𝑚 ∈ ℝ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚))
9		rpssre 13064	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
10	9	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → ℝ⁺ ⊆ ℝ)
11		csbeq1a 3935	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → 𝐴 = ⦋𝑛 / 𝑘⦌𝐴)
12		nfcv 2908	. . . . . . . 8 ⊢ Ⅎ𝑛𝐴
13		nfcsb1v 3946	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐴
14	11, 12, 13	cbvsum 15743	. . . . . . 7 ⊢ Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 = Σ𝑛 ∈ (1...(⌊‘𝑥))⦋𝑛 / 𝑘⦌𝐴
15		fzfid 14024	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
16		o1f 15575	. . . . . . . . . . . . 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 3152	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑉)
19		dmmptg 6273	. . . . . . . . . . . . . 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 6733	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑘 ∈ ℕ ↦ 𝐴):dom (𝑘 ∈ ℕ ↦ 𝐴)⟶ℂ ↔ (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ))
22	17, 21	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ)
23		eqid 2740	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ ↦ 𝐴) = (𝑘 ∈ ℕ ↦ 𝐴)
24	23	fmpt 7144	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ ℕ 𝐴 ∈ ℂ ↔ (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ)
25	22, 24	sylibr 234	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ)
26	25	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ⁺) → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ)
27		elfznn 13613	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
28	13	nfel1 2925	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ
29	11	eleq1d 2829	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → (𝐴 ∈ ℂ ↔ ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ))
30	28, 29	rspc 3623	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (∀𝑘 ∈ ℕ 𝐴 ∈ ℂ → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ))
31	30	impcom 407	. . . . . . . . 9 ⊢ ((∀𝑘 ∈ ℕ 𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ)
32	26, 27, 31	syl2an 595	. . . . . . . 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 15781	. . . . . . 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 2848	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ⁺) → Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 ∈ ℂ)
35		rpcn 13067	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
36	35	adantl 481	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
37		rpne0 13073	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
38	37	adantl 481	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
39	34, 36, 38	divcld 12070	. . . . 5 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥) ∈ ℂ)
40		simplrl 776	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑐 ∈ (1[,)+∞))
41		1re 11290	. . . . . . . 8 ⊢ 1 ∈ ℝ
42		elicopnf 13505	. . . . . . . 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 218	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → (𝑐 ∈ ℝ ∧ 1 ≤ 𝑐))
45	44	simpld 494	. . . . 5 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑐 ∈ ℝ)
46		fzfid 14024	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → (1...(⌊‘𝑐)) ∈ Fin)
47	25	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ)
48		elfznn 13613	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝑐)) → 𝑛 ∈ ℕ)
49	47, 48, 31	syl2an 595	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑐))) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ)
50	49	abscld 15485	. . . . . . 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 15782	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ)
52		simplrr 777	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑚 ∈ ℝ)
53	51, 52	readdcld 11319	. . . . 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 15504	. . . . . . . 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 716	. . . . . . 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 13072	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
57	56	ad2antrl 727	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
58		absid 15345	. . . . . . . . 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 7464	. . . . . . 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 2780	. . . . . 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 716	. . . . . . . . 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 15485	. . . . . . . 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 14024	. . . . . . . . 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 595	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ)
66	65	adantlr 714	. . . . . . . . . 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 15485	. . . . . . . . 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 15782	. . . . . . . 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 494	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → 𝑥 ∈ ℝ)
70	51	adantr 480	. . . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → 𝑚 ∈ ℝ)
72	70, 71	readdcld 11319	. . . . . . . . 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 11320	. . . . . . . 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 6923	. . . . . . . . 9 ⊢ (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) = (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))⦋𝑛 / 𝑘⦌𝐴)
75	64, 66	fsumabs 15849	. . . . . . . . 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 5201	. . . . . . . 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 14024	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin)
78		ssun2 4202	. . . . . . . . . . . . . 14 ⊢ (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ⊆ ((1...(⌊‘𝑐)) ∪ (((⌊‘𝑐) + 1)...(⌊‘𝑥)))
79		flge1nn 13872	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑐) ∈ ℕ)
82	81	nnred 12308	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑐) ∈ ℝ)
83	45	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → 𝑐 ∈ ℝ)
84		flle 13850	. . . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . 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 11447	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑐) ≤ 𝑥)
88		fznnfl 13913	. . . . . . . . . . . . . . . . 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 712	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑐) ∈ (1...(⌊‘𝑥)))
91		fzsplit 13610	. . . . . . . . . . . . . . 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 4062	. . . . . . . . . . . . 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 4008	. . . . . . . . . . . 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 15485	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ)
96	95	adantlr 714	. . . . . . . . . . . 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 590	. . . . . . . . . . 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 15782	. . . . . . . . . 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 11320	. . . . . . . . . 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 11320	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (𝑥 · 𝑚) ∈ ℝ)
101	70	recnd 11318	. . . . . . . . . . . 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	mullidd 11308	. . . . . . . . . . 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 11291	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → 1 ∈ ℝ)
104	49	absge0d 15493	. . . . . . . . . . . . . . 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 15843	. . . . . . . . . . . . . 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 511	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 495	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → 1 ≤ 𝑐)
109	108	adantr 480	. . . . . . . . . . . . 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 11447	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → 1 ≤ 𝑥)
111		lemul1a 12148	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 5190	. . . . . . . . . 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 14405	. . . . . . . . . . . . 13 ⊢ ((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ∈ ℕ₀)
115		nn0re 12562	. . . . . . . . . . . . 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 11320	. . . . . . . . . . 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 480	. . . . . . . . . . . . 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 13580	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥)) → 𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1)))
120	81	peano2nnd 12310	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → ((⌊‘𝑐) + 1) ∈ ℕ)
121		eluznn 12983	. . . . . . . . . . . . . . . 16 ⊢ ((((⌊‘𝑐) + 1) ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1))) → 𝑛 ∈ ℕ)
122	120, 121	sylan 579	. . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 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 13847	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ ℝ → (⌊‘𝑐) ∈ ℝ)
126		peano2re 11463	. . . . . . . . . . . . . . . . 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 12308	. . . . . . . . . . . . . . . 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 13852	. . . . . . . . . . . . . . . . 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 12916	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (ℤ_≥‘((⌊‘𝑐) + 1)) → ((⌊‘𝑐) + 1) ≤ 𝑛)
132	131	adantl 481	. . . . . . . . . . . . . . . 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 11447	. . . . . . . . . . . . . . 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 1913	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘 𝑐 ≤ 𝑛
135		nfcv 2908	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘abs
136	135, 13	nffv 6930	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(abs‘⦋𝑛 / 𝑘⦌𝐴)
137		nfcv 2908	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘 ≤
138		nfcv 2908	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘𝑚
139	136, 137, 138	nfbr 5213	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚
140	134, 139	nfim 1895	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(𝑐 ≤ 𝑛 → (abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚)
141		breq2 5170	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → (𝑐 ≤ 𝑘 ↔ 𝑐 ≤ 𝑛))
142	11	fveq2d 6924	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑛 → (abs‘𝐴) = (abs‘⦋𝑛 / 𝑘⦌𝐴))
143	142	breq1d 5176	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → ((abs‘𝐴) ≤ 𝑚 ↔ (abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚))
144	141, 143	imbi12d 344	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → ((𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚) ↔ (𝑐 ≤ 𝑛 → (abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚)))
145	140, 144	rspc 3623	. . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . 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 15847	. . . . . . . . . . . 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 11318	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → 𝑚 ∈ ℂ)
150		fsumconst 15838	. . . . . . . . . . . . 13 ⊢ (((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin ∧ 𝑚 ∈ ℂ) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))𝑚 = ((♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚))
151	77, 149, 150	syl2anc 583	. . . . . . . . . . . 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 5192	. . . . . . . . . . 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 262	. . . . . . . . . . . . 13 ⊢ (𝑛 = ((⌊‘𝑐) + 1) → (0 ≤ 𝑚 ↔ 0 ≤ 𝑚))
154		0red 11293	. . . . . . . . . . . . . . 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 506	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ ℕ) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ)
156	155	adantlr 714	. . . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . . 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 15485	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 15493	. . . . . . . . . . . . . . 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 11447	. . . . . . . . . . . . . 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 3152	. . . . . . . . . . . . 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 12666	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → ((⌊‘𝑐) + 1) ∈ ℤ)
164		uzid 12918	. . . . . . . . . . . . . 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 3636	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → 0 ≤ 𝑚)
167		reflcl 13847	. . . . . . . . . . . . . 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 9060	. . . . . . . . . . . . . . . 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 14429	. . . . . . . . . . . . . . 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 13871	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
174		hashfz1 14395	. . . . . . . . . . . . . . 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 5192	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ (⌊‘𝑥))
177		flle 13850	. . . . . . . . . . . . . 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 11447	. . . . . . . . . . . 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 12234	. . . . . . . . . . 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 11447	. . . . . . . . . 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 11909	. . . . . . . . 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 12134	. . . . . . . . . . 11 ⊢ ((⌊‘𝑐) ∈ ℝ → (⌊‘𝑐) < ((⌊‘𝑐) + 1))
184		fzdisj 13611	. . . . . . . . . . 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 11318	. . . . . . . . . 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 15789	. . . . . . . . 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 716	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → 𝑥 ∈ ℂ)
189	188, 101, 149	adddid 11314	. . . . . . . . 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 5198	. . . . . . . 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 11447	. . . . . . 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 13071	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
193	192	ad2antrl 727	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑐 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
194		ledivmul 12171	. . . . . . . 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 1371	. . . . . . 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 257	. . . . . 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 5188	. . . . 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 15582	. . . 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 412	. . 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 3219	. 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 ⦋csb 3921 ∪ cun 3974 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 class class class wbr 5166 ↦ cmpt 5249 dom cdm 5700 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ≼ cdom 9001 Fincfn 9003 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 +∞cpnf 11321 < clt 11324 ≤ cle 11325 / cdiv 11947 ℕcn 12293 ℕ₀cn0 12553 ℤcz 12639 ℤ_≥cuz 12903 ℝ⁺crp 13057 [,)cico 13409 ...cfz 13567 ⌊cfl 13841 ♯chash 14379 abscabs 15283 𝑂(1)co1 15532 Σcsu 15734

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-oadd 8526 df-er 8763 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-xnn0 12626 df-z 12640 df-uz 12904 df-rp 13058 df-ico 13413 df-fz 13568 df-fzo 13712 df-fl 13843 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-o1 15536 df-lo1 15537 df-sum 15735

This theorem is referenced by: selberg2lem 27612

W3C validator