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Theorem o1fsum 15158
Description: If 𝐴(𝑘) is O(1), then Σ𝑘𝑥, 𝐴(𝑘) is O(𝑥). (Contributed by Mario Carneiro, 23-May-2016.)
Hypotheses
Ref Expression
o1fsum.1 ((𝜑𝑘 ∈ ℕ) → 𝐴𝑉)
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.
StepHypRef Expression
1 o1fsum.2 . . 3 (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴) ∈ 𝑂(1))
2 nnssre 11631 . . . . 5 ℕ ⊆ ℝ
32a1i 11 . . . 4 (𝜑 → ℕ ⊆ ℝ)
4 o1fsum.1 . . . . 5 ((𝜑𝑘 ∈ ℕ) → 𝐴𝑉)
54, 1o1mptrcl 14969 . . . 4 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
6 1red 10631 . . . 4 (𝜑 → 1 ∈ ℝ)
73, 5, 6elo1mpt2 14882 . . 3 (𝜑 → ((𝑘 ∈ ℕ ↦ 𝐴) ∈ 𝑂(1) ↔ ∃𝑐 ∈ (1[,)+∞)∃𝑚 ∈ ℝ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)))
81, 7mpbid 233 . 2 (𝜑 → ∃𝑐 ∈ (1[,)+∞)∃𝑚 ∈ ℝ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚))
9 rpssre 12386 . . . . . 6 + ⊆ ℝ
109a1i 11 . . . . 5 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → ℝ+ ⊆ ℝ)
11 nfcv 2982 . . . . . . . 8 𝑛𝐴
12 nfcsb1v 3911 . . . . . . . 8 𝑘𝑛 / 𝑘𝐴
13 csbeq1a 3901 . . . . . . . 8 (𝑘 = 𝑛𝐴 = 𝑛 / 𝑘𝐴)
1411, 12, 13cbvsumi 15044 . . . . . . 7 Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 = Σ𝑛 ∈ (1...(⌊‘𝑥))𝑛 / 𝑘𝐴
15 fzfid 13331 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → (1...(⌊‘𝑥)) ∈ Fin)
16 o1f 14876 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ ↦ 𝐴) ∈ 𝑂(1) → (𝑘 ∈ ℕ ↦ 𝐴):dom (𝑘 ∈ ℕ ↦ 𝐴)⟶ℂ)
171, 16syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴):dom (𝑘 ∈ ℕ ↦ 𝐴)⟶ℂ)
184ralrimiva 3187 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑘 ∈ ℕ 𝐴𝑉)
19 dmmptg 6094 . . . . . . . . . . . . . 14 (∀𝑘 ∈ ℕ 𝐴𝑉 → dom (𝑘 ∈ ℕ ↦ 𝐴) = ℕ)
2018, 19syl 17 . . . . . . . . . . . . 13 (𝜑 → dom (𝑘 ∈ ℕ ↦ 𝐴) = ℕ)
2120feq2d 6497 . . . . . . . . . . . 12 (𝜑 → ((𝑘 ∈ ℕ ↦ 𝐴):dom (𝑘 ∈ ℕ ↦ 𝐴)⟶ℂ ↔ (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ))
2217, 21mpbid 233 . . . . . . . . . . 11 (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ)
23 eqid 2826 . . . . . . . . . . . 12 (𝑘 ∈ ℕ ↦ 𝐴) = (𝑘 ∈ ℕ ↦ 𝐴)
2423fmpt 6870 . . . . . . . . . . 11 (∀𝑘 ∈ ℕ 𝐴 ∈ ℂ ↔ (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ)
2522, 24sylibr 235 . . . . . . . . . 10 (𝜑 → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ)
2625ad3antrrr 726 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ)
27 elfznn 12926 . . . . . . . . 9 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
2812nfel1 2999 . . . . . . . . . . 11 𝑘𝑛 / 𝑘𝐴 ∈ ℂ
2913eleq1d 2902 . . . . . . . . . . 11 (𝑘 = 𝑛 → (𝐴 ∈ ℂ ↔ 𝑛 / 𝑘𝐴 ∈ ℂ))
3028, 29rspc 3615 . . . . . . . . . 10 (𝑛 ∈ ℕ → (∀𝑘 ∈ ℕ 𝐴 ∈ ℂ → 𝑛 / 𝑘𝐴 ∈ ℂ))
3130impcom 408 . . . . . . . . 9 ((∀𝑘 ∈ ℕ 𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ) → 𝑛 / 𝑘𝐴 ∈ ℂ)
3226, 27, 31syl2an 595 . . . . . . . 8 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 / 𝑘𝐴 ∈ ℂ)
3315, 32fsumcl 15080 . . . . . . 7 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))𝑛 / 𝑘𝐴 ∈ ℂ)
3414, 33eqeltrid 2922 . . . . . 6 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 ∈ ℂ)
35 rpcn 12389 . . . . . . 7 (𝑥 ∈ ℝ+𝑥 ∈ ℂ)
3635adantl 482 . . . . . 6 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ)
37 rpne0 12395 . . . . . . 7 (𝑥 ∈ ℝ+𝑥 ≠ 0)
3837adantl 482 . . . . . 6 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0)
3934, 36, 38divcld 11405 . . . . 5 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥) ∈ ℂ)
40 simplrl 773 . . . . . . 7 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑐 ∈ (1[,)+∞))
41 1re 10630 . . . . . . . 8 1 ∈ ℝ
42 elicopnf 12823 . . . . . . . 8 (1 ∈ ℝ → (𝑐 ∈ (1[,)+∞) ↔ (𝑐 ∈ ℝ ∧ 1 ≤ 𝑐)))
4341, 42ax-mp 5 . . . . . . 7 (𝑐 ∈ (1[,)+∞) ↔ (𝑐 ∈ ℝ ∧ 1 ≤ 𝑐))
4440, 43sylib 219 . . . . . 6 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → (𝑐 ∈ ℝ ∧ 1 ≤ 𝑐))
4544simpld 495 . . . . 5 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑐 ∈ ℝ)
46 fzfid 13331 . . . . . . 7 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → (1...(⌊‘𝑐)) ∈ Fin)
4725ad2antrr 722 . . . . . . . . 9 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ)
48 elfznn 12926 . . . . . . . . 9 (𝑛 ∈ (1...(⌊‘𝑐)) → 𝑛 ∈ ℕ)
4947, 48, 31syl2an 595 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑐))) → 𝑛 / 𝑘𝐴 ∈ ℂ)
5049abscld 14786 . . . . . . 7 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑐))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
5146, 50fsumrecl 15081 . . . . . 6 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
52 simplrr 774 . . . . . 6 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑚 ∈ ℝ)
5351, 52readdcld 10659 . . . . 5 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) + 𝑚) ∈ ℝ)
5434, 36, 38absdivd 14805 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) = ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) / (abs‘𝑥)))
5554adantrr 713 . . . . . . 7 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) = ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) / (abs‘𝑥)))
56 rprege0 12394 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
5756ad2antrl 724 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
58 absid 14646 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (abs‘𝑥) = 𝑥)
5957, 58syl 17 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (abs‘𝑥) = 𝑥)
6059oveq2d 7164 . . . . . . 7 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) / (abs‘𝑥)) = ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) / 𝑥))
6155, 60eqtrd 2861 . . . . . 6 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) = ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) / 𝑥))
6234adantrr 713 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 ∈ ℂ)
6362abscld 14786 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) ∈ ℝ)
64 fzfid 13331 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
6547, 27, 31syl2an 595 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 / 𝑘𝐴 ∈ ℂ)
6665adantlr 711 . . . . . . . . . 10 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 / 𝑘𝐴 ∈ ℂ)
6766abscld 14786 . . . . . . . . 9 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
6864, 67fsumrecl 15081 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
6957simpld 495 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑥 ∈ ℝ)
7051adantr 481 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
7152adantr 481 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑚 ∈ ℝ)
7270, 71readdcld 10659 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) + 𝑚) ∈ ℝ)
7369, 72remulcld 10660 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (𝑥 · (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) + 𝑚)) ∈ ℝ)
7414fveq2i 6670 . . . . . . . . 9 (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) = (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))𝑛 / 𝑘𝐴)
7564, 66fsumabs 15146 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))𝑛 / 𝑘𝐴) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘𝑛 / 𝑘𝐴))
7674, 75eqbrtrid 5098 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘𝑛 / 𝑘𝐴))
77 fzfid 13331 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin)
78 ssun2 4153 . . . . . . . . . . . . . 14 (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ⊆ ((1...(⌊‘𝑐)) ∪ (((⌊‘𝑐) + 1)...(⌊‘𝑥)))
79 flge1nn 13181 . . . . . . . . . . . . . . . . . 18 ((𝑐 ∈ ℝ ∧ 1 ≤ 𝑐) → (⌊‘𝑐) ∈ ℕ)
8044, 79syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → (⌊‘𝑐) ∈ ℕ)
8180adantr 481 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (⌊‘𝑐) ∈ ℕ)
8281nnred 11642 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (⌊‘𝑐) ∈ ℝ)
8345adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑐 ∈ ℝ)
84 flle 13159 . . . . . . . . . . . . . . . . . 18 (𝑐 ∈ ℝ → (⌊‘𝑐) ≤ 𝑐)
8583, 84syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (⌊‘𝑐) ≤ 𝑐)
86 simprr 769 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑐𝑥)
8782, 83, 69, 85, 86letrd 10786 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (⌊‘𝑐) ≤ 𝑥)
88 fznnfl 13220 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ → ((⌊‘𝑐) ∈ (1...(⌊‘𝑥)) ↔ ((⌊‘𝑐) ∈ ℕ ∧ (⌊‘𝑐) ≤ 𝑥)))
8969, 88syl 17 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ((⌊‘𝑐) ∈ (1...(⌊‘𝑥)) ↔ ((⌊‘𝑐) ∈ ℕ ∧ (⌊‘𝑐) ≤ 𝑥)))
9081, 87, 89mpbir2and 709 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (⌊‘𝑐) ∈ (1...(⌊‘𝑥)))
91 fzsplit 12923 . . . . . . . . . . . . . . 15 ((⌊‘𝑐) ∈ (1...(⌊‘𝑥)) → (1...(⌊‘𝑥)) = ((1...(⌊‘𝑐)) ∪ (((⌊‘𝑐) + 1)...(⌊‘𝑥))))
9290, 91syl 17 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (1...(⌊‘𝑥)) = ((1...(⌊‘𝑐)) ∪ (((⌊‘𝑐) + 1)...(⌊‘𝑥))))
9378, 92sseqtrrid 4024 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑥)))
9493sselda 3971 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) → 𝑛 ∈ (1...(⌊‘𝑥)))
9565abscld 14786 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
9695adantlr 711 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
9794, 96syldan 591 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
9877, 97fsumrecl 15081 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))(abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
9969, 70remulcld 10660 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴)) ∈ ℝ)
10069, 71remulcld 10660 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (𝑥 · 𝑚) ∈ ℝ)
10170recnd 10658 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) ∈ ℂ)
102101mulid2d 10648 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (1 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴)) = Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴))
103 1red 10631 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 1 ∈ ℝ)
10449absge0d 14794 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑐))) → 0 ≤ (abs‘𝑛 / 𝑘𝐴))
10546, 50, 104fsumge0 15140 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴))
10651, 105jca 512 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) ∈ ℝ ∧ 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴)))
107106adantr 481 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) ∈ ℝ ∧ 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴)))
10844simprd 496 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → 1 ≤ 𝑐)
109108adantr 481 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 1 ≤ 𝑐)
110103, 83, 69, 109, 86letrd 10786 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 1 ≤ 𝑥)
111 lemul1a 11483 . . . . . . . . . . . 12 (((1 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) ∈ ℝ ∧ 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴))) ∧ 1 ≤ 𝑥) → (1 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴)) ≤ (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴)))
112103, 69, 107, 110, 111syl31anc 1367 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (1 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴)) ≤ (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴)))
113102, 112eqbrtrrd 5087 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) ≤ (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴)))
114 hashcl 13707 . . . . . . . . . . . . 13 ((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ∈ ℕ0)
115 nn0re 11895 . . . . . . . . . . . . 13 ((♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ∈ ℕ0 → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ∈ ℝ)
11677, 114, 1153syl 18 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ∈ ℝ)
117116, 71remulcld 10660 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ((♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚) ∈ ℝ)
11871adantr 481 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) → 𝑚 ∈ ℝ)
119 elfzuz 12894 . . . . . . . . . . . . . 14 (𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥)) → 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1)))
12081peano2nnd 11644 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ((⌊‘𝑐) + 1) ∈ ℕ)
121 eluznn 12307 . . . . . . . . . . . . . . . 16 ((((⌊‘𝑐) + 1) ∈ ℕ ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑛 ∈ ℕ)
122120, 121sylan 580 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑛 ∈ ℕ)
123 simpllr 772 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚))
12483adantr 481 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑐 ∈ ℝ)
125 reflcl 13156 . . . . . . . . . . . . . . . . 17 (𝑐 ∈ ℝ → (⌊‘𝑐) ∈ ℝ)
126 peano2re 10802 . . . . . . . . . . . . . . . . 17 ((⌊‘𝑐) ∈ ℝ → ((⌊‘𝑐) + 1) ∈ ℝ)
127124, 125, 1263syl 18 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → ((⌊‘𝑐) + 1) ∈ ℝ)
128122nnred 11642 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑛 ∈ ℝ)
129 fllep1 13161 . . . . . . . . . . . . . . . . 17 (𝑐 ∈ ℝ → 𝑐 ≤ ((⌊‘𝑐) + 1))
130124, 129syl 17 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑐 ≤ ((⌊‘𝑐) + 1))
131 eluzle 12245 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1)) → ((⌊‘𝑐) + 1) ≤ 𝑛)
132131adantl 482 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → ((⌊‘𝑐) + 1) ≤ 𝑛)
133124, 127, 128, 130, 132letrd 10786 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑐𝑛)
134 nfv 1908 . . . . . . . . . . . . . . . . 17 𝑘 𝑐𝑛
135 nfcv 2982 . . . . . . . . . . . . . . . . . . 19 𝑘abs
136135, 12nffv 6677 . . . . . . . . . . . . . . . . . 18 𝑘(abs‘𝑛 / 𝑘𝐴)
137 nfcv 2982 . . . . . . . . . . . . . . . . . 18 𝑘
138 nfcv 2982 . . . . . . . . . . . . . . . . . 18 𝑘𝑚
139136, 137, 138nfbr 5110 . . . . . . . . . . . . . . . . 17 𝑘(abs‘𝑛 / 𝑘𝐴) ≤ 𝑚
140134, 139nfim 1890 . . . . . . . . . . . . . . . 16 𝑘(𝑐𝑛 → (abs‘𝑛 / 𝑘𝐴) ≤ 𝑚)
141 breq2 5067 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝑐𝑘𝑐𝑛))
14213fveq2d 6671 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (abs‘𝐴) = (abs‘𝑛 / 𝑘𝐴))
143142breq1d 5073 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → ((abs‘𝐴) ≤ 𝑚 ↔ (abs‘𝑛 / 𝑘𝐴) ≤ 𝑚))
144141, 143imbi12d 346 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ((𝑐𝑘 → (abs‘𝐴) ≤ 𝑚) ↔ (𝑐𝑛 → (abs‘𝑛 / 𝑘𝐴) ≤ 𝑚)))
145140, 144rspc 3615 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚) → (𝑐𝑛 → (abs‘𝑛 / 𝑘𝐴) ≤ 𝑚)))
146122, 123, 133, 145syl3c 66 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → (abs‘𝑛 / 𝑘𝐴) ≤ 𝑚)
147119, 146sylan2 592 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) → (abs‘𝑛 / 𝑘𝐴) ≤ 𝑚)
14877, 97, 118, 147fsumle 15144 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))(abs‘𝑛 / 𝑘𝐴) ≤ Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))𝑚)
14971recnd 10658 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑚 ∈ ℂ)
150 fsumconst 15135 . . . . . . . . . . . . 13 (((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin ∧ 𝑚 ∈ ℂ) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))𝑚 = ((♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚))
15177, 149, 150syl2anc 584 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))𝑚 = ((♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚))
152148, 151breqtrd 5089 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))(abs‘𝑛 / 𝑘𝐴) ≤ ((♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚))
153 biidd 263 . . . . . . . . . . . . 13 (𝑛 = ((⌊‘𝑐) + 1) → (0 ≤ 𝑚 ↔ 0 ≤ 𝑚))
154 0red 10633 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 0 ∈ ℝ)
15547, 30mpan9 507 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 / 𝑘𝐴 ∈ ℂ)
156155adantlr 711 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ ℕ) → 𝑛 / 𝑘𝐴 ∈ ℂ)
157122, 156syldan 591 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑛 / 𝑘𝐴 ∈ ℂ)
158157abscld 14786 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
15971adantr 481 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑚 ∈ ℝ)
160157absge0d 14794 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 0 ≤ (abs‘𝑛 / 𝑘𝐴))
161154, 158, 159, 160, 146letrd 10786 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 0 ≤ 𝑚)
162161ralrimiva 3187 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ∀𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))0 ≤ 𝑚)
163120nnzd 12075 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ((⌊‘𝑐) + 1) ∈ ℤ)
164 uzid 12247 . . . . . . . . . . . . . 14 (((⌊‘𝑐) + 1) ∈ ℤ → ((⌊‘𝑐) + 1) ∈ (ℤ‘((⌊‘𝑐) + 1)))
165163, 164syl 17 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ((⌊‘𝑐) + 1) ∈ (ℤ‘((⌊‘𝑐) + 1)))
166153, 162, 165rspcdva 3629 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 0 ≤ 𝑚)
167 reflcl 13156 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
16869, 167syl 17 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (⌊‘𝑥) ∈ ℝ)
169 ssdomg 8544 . . . . . . . . . . . . . . . 16 ((1...(⌊‘𝑥)) ∈ Fin → ((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ≼ (1...(⌊‘𝑥))))
17064, 93, 169sylc 65 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ≼ (1...(⌊‘𝑥)))
171 hashdomi 13731 . . . . . . . . . . . . . . 15 ((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ≼ (1...(⌊‘𝑥)) → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ (♯‘(1...(⌊‘𝑥))))
172170, 171syl 17 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ (♯‘(1...(⌊‘𝑥))))
173 flge0nn0 13180 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ0)
174 hashfz1 13696 . . . . . . . . . . . . . . 15 ((⌊‘𝑥) ∈ ℕ0 → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
17557, 173, 1743syl 18 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
176172, 175breqtrd 5089 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ (⌊‘𝑥))
177 flle 13159 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
17869, 177syl 17 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (⌊‘𝑥) ≤ 𝑥)
179116, 168, 69, 176, 178letrd 10786 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ 𝑥)
180116, 69, 71, 166, 179lemul1ad 11568 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ((♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚) ≤ (𝑥 · 𝑚))
18198, 117, 100, 152, 180letrd 10786 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))(abs‘𝑛 / 𝑘𝐴) ≤ (𝑥 · 𝑚))
18270, 98, 99, 100, 113, 181le2addd 11248 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) + Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))(abs‘𝑛 / 𝑘𝐴)) ≤ ((𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴)) + (𝑥 · 𝑚)))
183 ltp1 11469 . . . . . . . . . . 11 ((⌊‘𝑐) ∈ ℝ → (⌊‘𝑐) < ((⌊‘𝑐) + 1))
184 fzdisj 12924 . . . . . . . . . . 11 ((⌊‘𝑐) < ((⌊‘𝑐) + 1) → ((1...(⌊‘𝑐)) ∩ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) = ∅)
18582, 183, 1843syl 18 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ((1...(⌊‘𝑐)) ∩ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) = ∅)
18696recnd 10658 . . . . . . . . . 10 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℂ)
187185, 92, 64, 186fsumsplit 15087 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘𝑛 / 𝑘𝐴) = (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) + Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))(abs‘𝑛 / 𝑘𝐴)))
18836adantrr 713 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑥 ∈ ℂ)
189188, 101, 149adddid 10654 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (𝑥 · (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) + 𝑚)) = ((𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴)) + (𝑥 · 𝑚)))
190182, 187, 1893brtr4d 5095 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘𝑛 / 𝑘𝐴) ≤ (𝑥 · (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) + 𝑚)))
19163, 68, 73, 76, 190letrd 10786 . . . . . . 7 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) ≤ (𝑥 · (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) + 𝑚)))
192 rpregt0 12393 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
193192ad2antrl 724 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
194 ledivmul 11505 . . . . . . . 8 (((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) ∈ ℝ ∧ (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) + 𝑚) ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) / 𝑥) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) + 𝑚) ↔ (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) ≤ (𝑥 · (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) + 𝑚))))
19563, 72, 193, 194syl3anc 1365 . . . . . . 7 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) / 𝑥) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) + 𝑚) ↔ (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) ≤ (𝑥 · (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) + 𝑚))))
196191, 195mpbird 258 . . . . . 6 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) / 𝑥) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) + 𝑚))
19761, 196eqbrtrd 5085 . . . . 5 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) + 𝑚))
19810, 39, 45, 53, 197elo1d 14883 . . . 4 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1))
199198ex 413 . . 3 ((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) → (∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚) → (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1)))
200199rexlimdvva 3299 . 2 (𝜑 → (∃𝑐 ∈ (1[,)+∞)∃𝑚 ∈ ℝ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚) → (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1)))
2018, 200mpd 15 1 (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wne 3021  wral 3143  wrex 3144  csb 3887  cun 3938  cin 3939  wss 3940  c0 4295   class class class wbr 5063  cmpt 5143  dom cdm 5554  wf 6348  cfv 6352  (class class class)co 7148  cdom 8496  Fincfn 8498  cc 10524  cr 10525  0cc0 10526  1c1 10527   + caddc 10529   · cmul 10531  +∞cpnf 10661   < clt 10664  cle 10665   / cdiv 11286  cn 11627  0cn0 11886  cz 11970  cuz 12232  +crp 12379  [,)cico 12730  ...cfz 12882  cfl 13150  chash 13680  abscabs 14583  𝑂(1)co1 14833  Σcsu 15032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-inf2 9093  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-isom 6361  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-pm 8399  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-sup 8895  df-inf 8896  df-oi 8963  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-3 11690  df-n0 11887  df-xnn0 11957  df-z 11971  df-uz 12233  df-rp 12380  df-ico 12734  df-fz 12883  df-fzo 13024  df-fl 13152  df-seq 13360  df-exp 13420  df-hash 13681  df-cj 14448  df-re 14449  df-im 14450  df-sqrt 14584  df-abs 14585  df-clim 14835  df-o1 14837  df-lo1 14838  df-sum 15033
This theorem is referenced by:  selberg2lem  26040
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