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Theorem o1fsum 14589
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 11062 . . . . 5 ℕ ⊆ ℝ
32a1i 11 . . . 4 (𝜑 → ℕ ⊆ ℝ)
4 o1fsum.1 . . . . 5 ((𝜑𝑘 ∈ ℕ) → 𝐴𝑉)
54, 1o1mptrcl 14397 . . . 4 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
6 1red 10093 . . . 4 (𝜑 → 1 ∈ ℝ)
73, 5, 6elo1mpt2 14310 . . 3 (𝜑 → ((𝑘 ∈ ℕ ↦ 𝐴) ∈ 𝑂(1) ↔ ∃𝑐 ∈ (1[,)+∞)∃𝑚 ∈ ℝ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)))
81, 7mpbid 222 . 2 (𝜑 → ∃𝑐 ∈ (1[,)+∞)∃𝑚 ∈ ℝ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚))
9 rpssre 11881 . . . . . 6 + ⊆ ℝ
109a1i 11 . . . . 5 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → ℝ+ ⊆ ℝ)
11 nfcv 2793 . . . . . . . 8 𝑛𝐴
12 nfcsb1v 3582 . . . . . . . 8 𝑘𝑛 / 𝑘𝐴
13 csbeq1a 3575 . . . . . . . 8 (𝑘 = 𝑛𝐴 = 𝑛 / 𝑘𝐴)
1411, 12, 13cbvsumi 14471 . . . . . . 7 Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 = Σ𝑛 ∈ (1...(⌊‘𝑥))𝑛 / 𝑘𝐴
15 fzfid 12812 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → (1...(⌊‘𝑥)) ∈ Fin)
16 o1f 14304 . . . . . . . . . . . . 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 2995 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑘 ∈ ℕ 𝐴𝑉)
19 dmmptg 5670 . . . . . . . . . . . . . 14 (∀𝑘 ∈ ℕ 𝐴𝑉 → dom (𝑘 ∈ ℕ ↦ 𝐴) = ℕ)
2018, 19syl 17 . . . . . . . . . . . . 13 (𝜑 → dom (𝑘 ∈ ℕ ↦ 𝐴) = ℕ)
2120feq2d 6069 . . . . . . . . . . . 12 (𝜑 → ((𝑘 ∈ ℕ ↦ 𝐴):dom (𝑘 ∈ ℕ ↦ 𝐴)⟶ℂ ↔ (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ))
2217, 21mpbid 222 . . . . . . . . . . 11 (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ)
23 eqid 2651 . . . . . . . . . . . 12 (𝑘 ∈ ℕ ↦ 𝐴) = (𝑘 ∈ ℕ ↦ 𝐴)
2423fmpt 6421 . . . . . . . . . . 11 (∀𝑘 ∈ ℕ 𝐴 ∈ ℂ ↔ (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ)
2522, 24sylibr 224 . . . . . . . . . 10 (𝜑 → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ)
2625ad3antrrr 766 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ)
27 elfznn 12408 . . . . . . . . 9 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
2812nfel1 2808 . . . . . . . . . . 11 𝑘𝑛 / 𝑘𝐴 ∈ ℂ
2913eleq1d 2715 . . . . . . . . . . 11 (𝑘 = 𝑛 → (𝐴 ∈ ℂ ↔ 𝑛 / 𝑘𝐴 ∈ ℂ))
3028, 29rspc 3334 . . . . . . . . . 10 (𝑛 ∈ ℕ → (∀𝑘 ∈ ℕ 𝐴 ∈ ℂ → 𝑛 / 𝑘𝐴 ∈ ℂ))
3130impcom 445 . . . . . . . . 9 ((∀𝑘 ∈ ℕ 𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ) → 𝑛 / 𝑘𝐴 ∈ ℂ)
3226, 27, 31syl2an 493 . . . . . . . 8 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 / 𝑘𝐴 ∈ ℂ)
3315, 32fsumcl 14508 . . . . . . 7 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))𝑛 / 𝑘𝐴 ∈ ℂ)
3414, 33syl5eqel 2734 . . . . . 6 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 ∈ ℂ)
35 rpcn 11879 . . . . . . 7 (𝑥 ∈ ℝ+𝑥 ∈ ℂ)
3635adantl 481 . . . . . 6 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ)
37 rpne0 11886 . . . . . . 7 (𝑥 ∈ ℝ+𝑥 ≠ 0)
3837adantl 481 . . . . . 6 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0)
3934, 36, 38divcld 10839 . . . . 5 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥) ∈ ℂ)
40 simplrl 817 . . . . . . 7 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑐 ∈ (1[,)+∞))
41 1re 10077 . . . . . . . 8 1 ∈ ℝ
42 elicopnf 12307 . . . . . . . 8 (1 ∈ ℝ → (𝑐 ∈ (1[,)+∞) ↔ (𝑐 ∈ ℝ ∧ 1 ≤ 𝑐)))
4341, 42ax-mp 5 . . . . . . 7 (𝑐 ∈ (1[,)+∞) ↔ (𝑐 ∈ ℝ ∧ 1 ≤ 𝑐))
4440, 43sylib 208 . . . . . 6 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → (𝑐 ∈ ℝ ∧ 1 ≤ 𝑐))
4544simpld 474 . . . . 5 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑐 ∈ ℝ)
46 fzfid 12812 . . . . . . 7 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → (1...(⌊‘𝑐)) ∈ Fin)
4725ad2antrr 762 . . . . . . . . 9 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ)
48 elfznn 12408 . . . . . . . . 9 (𝑛 ∈ (1...(⌊‘𝑐)) → 𝑛 ∈ ℕ)
4947, 48, 31syl2an 493 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑐))) → 𝑛 / 𝑘𝐴 ∈ ℂ)
5049abscld 14219 . . . . . . 7 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑐))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
5146, 50fsumrecl 14509 . . . . . 6 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
52 simplrr 818 . . . . . 6 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑚 ∈ ℝ)
5351, 52readdcld 10107 . . . . 5 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) + 𝑚) ∈ ℝ)
5434, 36, 38absdivd 14238 . . . . . . . 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 753 . . . . . . 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 11885 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
5756ad2antrl 764 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
58 absid 14080 . . . . . . . . 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 6706 . . . . . . 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 2685 . . . . . 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 753 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 ∈ ℂ)
6362abscld 14219 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) ∈ ℝ)
64 fzfid 12812 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
6547, 27, 31syl2an 493 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 / 𝑘𝐴 ∈ ℂ)
6665adantlr 751 . . . . . . . . . 10 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 / 𝑘𝐴 ∈ ℂ)
6766abscld 14219 . . . . . . . . 9 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
6864, 67fsumrecl 14509 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
6957simpld 474 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑥 ∈ ℝ)
7051adantr 480 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
7152adantr 480 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑚 ∈ ℝ)
7270, 71readdcld 10107 . . . . . . . . 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 10108 . . . . . . . 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 6232 . . . . . . . . 9 (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) = (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))𝑛 / 𝑘𝐴)
7564, 66fsumabs 14577 . . . . . . . . 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, 75syl5eqbr 4720 . . . . . . . 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 12812 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin)
78 ssun2 3810 . . . . . . . . . . . . . 14 (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ⊆ ((1...(⌊‘𝑐)) ∪ (((⌊‘𝑐) + 1)...(⌊‘𝑥)))
79 flge1nn 12662 . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (⌊‘𝑐) ∈ ℕ)
8281nnred 11073 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (⌊‘𝑐) ∈ ℝ)
8345adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑐 ∈ ℝ)
84 flle 12640 . . . . . . . . . . . . . . . . . 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 811 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑐𝑥)
8782, 83, 69, 85, 86letrd 10232 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (⌊‘𝑐) ≤ 𝑥)
88 fznnfl 12701 . . . . . . . . . . . . . . . . 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 977 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (⌊‘𝑐) ∈ (1...(⌊‘𝑥)))
91 fzsplit 12405 . . . . . . . . . . . . . . 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, 92syl5sseqr 3687 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑥)))
9493sselda 3636 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) → 𝑛 ∈ (1...(⌊‘𝑥)))
9565abscld 14219 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
9695adantlr 751 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
9794, 96syldan 486 . . . . . . . . . . 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 14509 . . . . . . . . . 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 10108 . . . . . . . . . 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 10108 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (𝑥 · 𝑚) ∈ ℝ)
10170recnd 10106 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) ∈ ℂ)
102101mulid2d 10096 . . . . . . . . . . 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 10093 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 1 ∈ ℝ)
10449absge0d 14227 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑐))) → 0 ≤ (abs‘𝑛 / 𝑘𝐴))
10546, 50, 104fsumge0 14571 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴))
10651, 105jca 553 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 478 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → 1 ≤ 𝑐)
109108adantr 480 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 1 ≤ 𝑐)
110103, 83, 69, 109, 86letrd 10232 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 1 ≤ 𝑥)
111 lemul1a 10915 . . . . . . . . . . . 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 1369 . . . . . . . . . . 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 4709 . . . . . . . . . 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 13185 . . . . . . . . . . . . 13 ((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin → (#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ∈ ℕ0)
115 nn0re 11339 . . . . . . . . . . . . 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 10108 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ((#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚) ∈ ℝ)
11871adantr 480 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) → 𝑚 ∈ ℝ)
119 elfzuz 12376 . . . . . . . . . . . . . 14 (𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥)) → 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1)))
12081peano2nnd 11075 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ((⌊‘𝑐) + 1) ∈ ℕ)
121 eluznn 11796 . . . . . . . . . . . . . . . 16 ((((⌊‘𝑐) + 1) ∈ ℕ ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑛 ∈ ℕ)
122120, 121sylan 487 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑛 ∈ ℕ)
123 simpllr 815 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑐 ∈ ℝ)
125 reflcl 12637 . . . . . . . . . . . . . . . . 17 (𝑐 ∈ ℝ → (⌊‘𝑐) ∈ ℝ)
126 peano2re 10247 . . . . . . . . . . . . . . . . 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 11073 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑛 ∈ ℝ)
129 fllep1 12642 . . . . . . . . . . . . . . . . 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 11738 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1)) → ((⌊‘𝑐) + 1) ≤ 𝑛)
132131adantl 481 . . . . . . . . . . . . . . . 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 10232 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑐𝑛)
134 nfv 1883 . . . . . . . . . . . . . . . . 17 𝑘 𝑐𝑛
135 nfcv 2793 . . . . . . . . . . . . . . . . . . 19 𝑘abs
136135, 12nffv 6236 . . . . . . . . . . . . . . . . . 18 𝑘(abs‘𝑛 / 𝑘𝐴)
137 nfcv 2793 . . . . . . . . . . . . . . . . . 18 𝑘
138 nfcv 2793 . . . . . . . . . . . . . . . . . 18 𝑘𝑚
139136, 137, 138nfbr 4732 . . . . . . . . . . . . . . . . 17 𝑘(abs‘𝑛 / 𝑘𝐴) ≤ 𝑚
140134, 139nfim 1865 . . . . . . . . . . . . . . . 16 𝑘(𝑐𝑛 → (abs‘𝑛 / 𝑘𝐴) ≤ 𝑚)
141 breq2 4689 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝑐𝑘𝑐𝑛))
14213fveq2d 6233 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (abs‘𝐴) = (abs‘𝑛 / 𝑘𝐴))
143142breq1d 4695 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → ((abs‘𝐴) ≤ 𝑚 ↔ (abs‘𝑛 / 𝑘𝐴) ≤ 𝑚))
144141, 143imbi12d 333 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ((𝑐𝑘 → (abs‘𝐴) ≤ 𝑚) ↔ (𝑐𝑛 → (abs‘𝑛 / 𝑘𝐴) ≤ 𝑚)))
145140, 144rspc 3334 . . . . . . . . . . . . . . 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 490 . . . . . . . . . . . . 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 14575 . . . . . . . . . . . 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 10106 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑚 ∈ ℂ)
150 fsumconst 14566 . . . . . . . . . . . . 13 (((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin ∧ 𝑚 ∈ ℂ) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))𝑚 = ((#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚))
15177, 149, 150syl2anc 694 . . . . . . . . . . . 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 4711 . . . . . . . . . . 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))
153120nnzd 11519 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ((⌊‘𝑐) + 1) ∈ ℤ)
154 uzid 11740 . . . . . . . . . . . . . 14 (((⌊‘𝑐) + 1) ∈ ℤ → ((⌊‘𝑐) + 1) ∈ (ℤ‘((⌊‘𝑐) + 1)))
155153, 154syl 17 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ((⌊‘𝑐) + 1) ∈ (ℤ‘((⌊‘𝑐) + 1)))
156 0red 10079 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 0 ∈ ℝ)
15747, 30mpan9 485 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 / 𝑘𝐴 ∈ ℂ)
158157adantlr 751 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ ℕ) → 𝑛 / 𝑘𝐴 ∈ ℂ)
159122, 158syldan 486 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑛 / 𝑘𝐴 ∈ ℂ)
160159abscld 14219 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
16171adantr 480 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑚 ∈ ℝ)
162159absge0d 14227 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 0 ≤ (abs‘𝑛 / 𝑘𝐴))
163156, 160, 161, 162, 146letrd 10232 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 0 ≤ 𝑚)
164163ralrimiva 2995 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ∀𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))0 ≤ 𝑚)
165 biidd 252 . . . . . . . . . . . . . 14 (𝑛 = ((⌊‘𝑐) + 1) → (0 ≤ 𝑚 ↔ 0 ≤ 𝑚))
166165rspcv 3336 . . . . . . . . . . . . 13 (((⌊‘𝑐) + 1) ∈ (ℤ‘((⌊‘𝑐) + 1)) → (∀𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))0 ≤ 𝑚 → 0 ≤ 𝑚))
167155, 164, 166sylc 65 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 0 ≤ 𝑚)
168 reflcl 12637 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
16969, 168syl 17 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (⌊‘𝑥) ∈ ℝ)
170 ssdomg 8043 . . . . . . . . . . . . . . . 16 ((1...(⌊‘𝑥)) ∈ Fin → ((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ≼ (1...(⌊‘𝑥))))
17164, 93, 170sylc 65 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ≼ (1...(⌊‘𝑥)))
172 hashdomi 13207 . . . . . . . . . . . . . . 15 ((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ≼ (1...(⌊‘𝑥)) → (#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ (#‘(1...(⌊‘𝑥))))
173171, 172syl 17 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ (#‘(1...(⌊‘𝑥))))
174 flge0nn0 12661 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ0)
175 hashfz1 13174 . . . . . . . . . . . . . . 15 ((⌊‘𝑥) ∈ ℕ0 → (#‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
17657, 174, 1753syl 18 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (#‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
177173, 176breqtrd 4711 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ (⌊‘𝑥))
178 flle 12640 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
17969, 178syl 17 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (⌊‘𝑥) ≤ 𝑥)
180116, 169, 69, 177, 179letrd 10232 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ 𝑥)
181116, 69, 71, 167, 180lemul1ad 11001 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ((#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚) ≤ (𝑥 · 𝑚))
18298, 117, 100, 152, 181letrd 10232 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))(abs‘𝑛 / 𝑘𝐴) ≤ (𝑥 · 𝑚))
18370, 98, 99, 100, 113, 182le2addd 10684 . . . . . . . . 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)))
184 ltp1 10899 . . . . . . . . . . 11 ((⌊‘𝑐) ∈ ℝ → (⌊‘𝑐) < ((⌊‘𝑐) + 1))
185 fzdisj 12406 . . . . . . . . . . 11 ((⌊‘𝑐) < ((⌊‘𝑐) + 1) → ((1...(⌊‘𝑐)) ∩ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) = ∅)
18682, 184, 1853syl 18 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ((1...(⌊‘𝑐)) ∩ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) = ∅)
18796recnd 10106 . . . . . . . . . 10 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℂ)
188186, 92, 64, 187fsumsplit 14515 . . . . . . . . 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)))
18936adantrr 753 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑥 ∈ ℂ)
190189, 101, 149adddid 10102 . . . . . . . . 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)))
191183, 188, 1903brtr4d 4717 . . . . . . . 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)))
19263, 68, 73, 76, 191letrd 10232 . . . . . . 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)))
193 rpregt0 11884 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
194193ad2antrl 764 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
195 ledivmul 10937 . . . . . . . 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))))
19663, 72, 194, 195syl3anc 1366 . . . . . . 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))))
197192, 196mpbird 247 . . . . . 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))
19861, 197eqbrtrd 4707 . . . . 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))
19910, 39, 45, 53, 198elo1d 14311 . . . 4 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1))
200199ex 449 . . 3 ((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) → (∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚) → (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1)))
201200rexlimdvva 3067 . 2 (𝜑 → (∃𝑐 ∈ (1[,)+∞)∃𝑚 ∈ ℝ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚) → (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1)))
2028, 201mpd 15 1 (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  csb 3566  cun 3605  cin 3606  wss 3607  c0 3948   class class class wbr 4685  cmpt 4762  dom cdm 5143  wf 5922  cfv 5926  (class class class)co 6690  cdom 7995  Fincfn 7997  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  +∞cpnf 10109   < clt 10112  cle 10113   / cdiv 10722  cn 11058  0cn0 11330  cz 11415  cuz 11725  +crp 11870  [,)cico 12215  ...cfz 12364  cfl 12631  #chash 13157  abscabs 14018  𝑂(1)co1 14261  Σcsu 14460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-rp 11871  df-ico 12219  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-o1 14265  df-lo1 14266  df-sum 14461
This theorem is referenced by:  selberg2lem  25284
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