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Theorem o1fsum 15795
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 12249 . . . . 5 ℕ ⊆ ℝ
32a1i 11 . . . 4 (𝜑 → ℕ ⊆ ℝ)
4 o1fsum.1 . . . . 5 ((𝜑𝑘 ∈ ℕ) → 𝐴𝑉)
54, 1o1mptrcl 15603 . . . 4 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
6 1red 11247 . . . 4 (𝜑 → 1 ∈ ℝ)
73, 5, 6elo1mpt2 15515 . . 3 (𝜑 → ((𝑘 ∈ ℕ ↦ 𝐴) ∈ 𝑂(1) ↔ ∃𝑐 ∈ (1[,)+∞)∃𝑚 ∈ ℝ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)))
81, 7mpbid 231 . 2 (𝜑 → ∃𝑐 ∈ (1[,)+∞)∃𝑚 ∈ ℝ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚))
9 rpssre 13016 . . . . . 6 + ⊆ ℝ
109a1i 11 . . . . 5 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → ℝ+ ⊆ ℝ)
11 nfcv 2891 . . . . . . . 8 𝑛𝐴
12 nfcsb1v 3914 . . . . . . . 8 𝑘𝑛 / 𝑘𝐴
13 csbeq1a 3903 . . . . . . . 8 (𝑘 = 𝑛𝐴 = 𝑛 / 𝑘𝐴)
1411, 12, 13cbvsumi 15679 . . . . . . 7 Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 = Σ𝑛 ∈ (1...(⌊‘𝑥))𝑛 / 𝑘𝐴
15 fzfid 13974 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → (1...(⌊‘𝑥)) ∈ Fin)
16 o1f 15509 . . . . . . . . . . . . 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 3135 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑘 ∈ ℕ 𝐴𝑉)
19 dmmptg 6248 . . . . . . . . . . . . . 14 (∀𝑘 ∈ ℕ 𝐴𝑉 → dom (𝑘 ∈ ℕ ↦ 𝐴) = ℕ)
2018, 19syl 17 . . . . . . . . . . . . 13 (𝜑 → dom (𝑘 ∈ ℕ ↦ 𝐴) = ℕ)
2120feq2d 6709 . . . . . . . . . . . 12 (𝜑 → ((𝑘 ∈ ℕ ↦ 𝐴):dom (𝑘 ∈ ℕ ↦ 𝐴)⟶ℂ ↔ (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ))
2217, 21mpbid 231 . . . . . . . . . . 11 (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ)
23 eqid 2725 . . . . . . . . . . . 12 (𝑘 ∈ ℕ ↦ 𝐴) = (𝑘 ∈ ℕ ↦ 𝐴)
2423fmpt 7119 . . . . . . . . . . 11 (∀𝑘 ∈ ℕ 𝐴 ∈ ℂ ↔ (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ)
2522, 24sylibr 233 . . . . . . . . . 10 (𝜑 → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ)
2625ad3antrrr 728 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ)
27 elfznn 13565 . . . . . . . . 9 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
2812nfel1 2908 . . . . . . . . . . 11 𝑘𝑛 / 𝑘𝐴 ∈ ℂ
2913eleq1d 2810 . . . . . . . . . . 11 (𝑘 = 𝑛 → (𝐴 ∈ ℂ ↔ 𝑛 / 𝑘𝐴 ∈ ℂ))
3028, 29rspc 3594 . . . . . . . . . 10 (𝑛 ∈ ℕ → (∀𝑘 ∈ ℕ 𝐴 ∈ ℂ → 𝑛 / 𝑘𝐴 ∈ ℂ))
3130impcom 406 . . . . . . . . 9 ((∀𝑘 ∈ ℕ 𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ) → 𝑛 / 𝑘𝐴 ∈ ℂ)
3226, 27, 31syl2an 594 . . . . . . . 8 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 / 𝑘𝐴 ∈ ℂ)
3315, 32fsumcl 15715 . . . . . . 7 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))𝑛 / 𝑘𝐴 ∈ ℂ)
3414, 33eqeltrid 2829 . . . . . 6 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 ∈ ℂ)
35 rpcn 13019 . . . . . . 7 (𝑥 ∈ ℝ+𝑥 ∈ ℂ)
3635adantl 480 . . . . . 6 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ)
37 rpne0 13025 . . . . . . 7 (𝑥 ∈ ℝ+𝑥 ≠ 0)
3837adantl 480 . . . . . 6 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0)
3934, 36, 38divcld 12023 . . . . 5 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥) ∈ ℂ)
40 simplrl 775 . . . . . . 7 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑐 ∈ (1[,)+∞))
41 1re 11246 . . . . . . . 8 1 ∈ ℝ
42 elicopnf 13457 . . . . . . . 8 (1 ∈ ℝ → (𝑐 ∈ (1[,)+∞) ↔ (𝑐 ∈ ℝ ∧ 1 ≤ 𝑐)))
4341, 42ax-mp 5 . . . . . . 7 (𝑐 ∈ (1[,)+∞) ↔ (𝑐 ∈ ℝ ∧ 1 ≤ 𝑐))
4440, 43sylib 217 . . . . . 6 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → (𝑐 ∈ ℝ ∧ 1 ≤ 𝑐))
4544simpld 493 . . . . 5 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑐 ∈ ℝ)
46 fzfid 13974 . . . . . . 7 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → (1...(⌊‘𝑐)) ∈ Fin)
4725ad2antrr 724 . . . . . . . . 9 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ)
48 elfznn 13565 . . . . . . . . 9 (𝑛 ∈ (1...(⌊‘𝑐)) → 𝑛 ∈ ℕ)
4947, 48, 31syl2an 594 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑐))) → 𝑛 / 𝑘𝐴 ∈ ℂ)
5049abscld 15419 . . . . . . 7 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑐))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
5146, 50fsumrecl 15716 . . . . . 6 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
52 simplrr 776 . . . . . 6 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑚 ∈ ℝ)
5351, 52readdcld 11275 . . . . 5 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) + 𝑚) ∈ ℝ)
5434, 36, 38absdivd 15438 . . . . . . . 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 715 . . . . . . 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 13024 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
5756ad2antrl 726 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
58 absid 15279 . . . . . . . . 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 7435 . . . . . . 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 2765 . . . . . 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 715 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 ∈ ℂ)
6362abscld 15419 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) ∈ ℝ)
64 fzfid 13974 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
6547, 27, 31syl2an 594 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 / 𝑘𝐴 ∈ ℂ)
6665adantlr 713 . . . . . . . . . 10 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 / 𝑘𝐴 ∈ ℂ)
6766abscld 15419 . . . . . . . . 9 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
6864, 67fsumrecl 15716 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
6957simpld 493 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑥 ∈ ℝ)
7051adantr 479 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
7152adantr 479 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑚 ∈ ℝ)
7270, 71readdcld 11275 . . . . . . . . 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 11276 . . . . . . . 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 6899 . . . . . . . . 9 (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) = (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))𝑛 / 𝑘𝐴)
7564, 66fsumabs 15783 . . . . . . . . 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 5184 . . . . . . . 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 13974 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin)
78 ssun2 4171 . . . . . . . . . . . . . 14 (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ⊆ ((1...(⌊‘𝑐)) ∪ (((⌊‘𝑐) + 1)...(⌊‘𝑥)))
79 flge1nn 13822 . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (⌊‘𝑐) ∈ ℕ)
8281nnred 12260 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (⌊‘𝑐) ∈ ℝ)
8345adantr 479 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑐 ∈ ℝ)
84 flle 13800 . . . . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑐𝑥)
8782, 83, 69, 85, 86letrd 11403 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (⌊‘𝑐) ≤ 𝑥)
88 fznnfl 13863 . . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (⌊‘𝑐) ∈ (1...(⌊‘𝑥)))
91 fzsplit 13562 . . . . . . . . . . . . . . 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 4030 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑥)))
9493sselda 3976 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) → 𝑛 ∈ (1...(⌊‘𝑥)))
9565abscld 15419 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
9695adantlr 713 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
9794, 96syldan 589 . . . . . . . . . . 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 15716 . . . . . . . . . 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 11276 . . . . . . . . . 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 11276 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (𝑥 · 𝑚) ∈ ℝ)
10170recnd 11274 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) ∈ ℂ)
102101mullidd 11264 . . . . . . . . . . 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 11247 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 1 ∈ ℝ)
10449absge0d 15427 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑐))) → 0 ≤ (abs‘𝑛 / 𝑘𝐴))
10546, 50, 104fsumge0 15777 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴))
10651, 105jca 510 . . . . . . . . . . . . 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 479 . . . . . . . . . . . 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 494 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → 1 ≤ 𝑐)
109108adantr 479 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 1 ≤ 𝑐)
110103, 83, 69, 109, 86letrd 11403 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 1 ≤ 𝑥)
111 lemul1a 12101 . . . . . . . . . . . 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 1370 . . . . . . . . . . 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 5173 . . . . . . . . . 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 14351 . . . . . . . . . . . . 13 ((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ∈ ℕ0)
115 nn0re 12514 . . . . . . . . . . . . 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 11276 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ((♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚) ∈ ℝ)
11871adantr 479 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) → 𝑚 ∈ ℝ)
119 elfzuz 13532 . . . . . . . . . . . . . 14 (𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥)) → 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1)))
12081peano2nnd 12262 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ((⌊‘𝑐) + 1) ∈ ℕ)
121 eluznn 12935 . . . . . . . . . . . . . . . 16 ((((⌊‘𝑐) + 1) ∈ ℕ ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑛 ∈ ℕ)
122120, 121sylan 578 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑛 ∈ ℕ)
123 simpllr 774 . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑐 ∈ ℝ)
125 reflcl 13797 . . . . . . . . . . . . . . . . 17 (𝑐 ∈ ℝ → (⌊‘𝑐) ∈ ℝ)
126 peano2re 11419 . . . . . . . . . . . . . . . . 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 12260 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑛 ∈ ℝ)
129 fllep1 13802 . . . . . . . . . . . . . . . . 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 12868 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1)) → ((⌊‘𝑐) + 1) ≤ 𝑛)
132131adantl 480 . . . . . . . . . . . . . . . 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 11403 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑐𝑛)
134 nfv 1909 . . . . . . . . . . . . . . . . 17 𝑘 𝑐𝑛
135 nfcv 2891 . . . . . . . . . . . . . . . . . . 19 𝑘abs
136135, 12nffv 6906 . . . . . . . . . . . . . . . . . 18 𝑘(abs‘𝑛 / 𝑘𝐴)
137 nfcv 2891 . . . . . . . . . . . . . . . . . 18 𝑘
138 nfcv 2891 . . . . . . . . . . . . . . . . . 18 𝑘𝑚
139136, 137, 138nfbr 5196 . . . . . . . . . . . . . . . . 17 𝑘(abs‘𝑛 / 𝑘𝐴) ≤ 𝑚
140134, 139nfim 1891 . . . . . . . . . . . . . . . 16 𝑘(𝑐𝑛 → (abs‘𝑛 / 𝑘𝐴) ≤ 𝑚)
141 breq2 5153 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝑐𝑘𝑐𝑛))
14213fveq2d 6900 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (abs‘𝐴) = (abs‘𝑛 / 𝑘𝐴))
143142breq1d 5159 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → ((abs‘𝐴) ≤ 𝑚 ↔ (abs‘𝑛 / 𝑘𝐴) ≤ 𝑚))
144141, 143imbi12d 343 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ((𝑐𝑘 → (abs‘𝐴) ≤ 𝑚) ↔ (𝑐𝑛 → (abs‘𝑛 / 𝑘𝐴) ≤ 𝑚)))
145140, 144rspc 3594 . . . . . . . . . . . . . . 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 591 . . . . . . . . . . . . 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 15781 . . . . . . . . . . . 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 11274 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑚 ∈ ℂ)
150 fsumconst 15772 . . . . . . . . . . . . 13 (((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin ∧ 𝑚 ∈ ℂ) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))𝑚 = ((♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚))
15177, 149, 150syl2anc 582 . . . . . . . . . . . 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 5175 . . . . . . . . . . 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 261 . . . . . . . . . . . . 13 (𝑛 = ((⌊‘𝑐) + 1) → (0 ≤ 𝑚 ↔ 0 ≤ 𝑚))
154 0red 11249 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 0 ∈ ℝ)
15547, 30mpan9 505 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 / 𝑘𝐴 ∈ ℂ)
156155adantlr 713 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ ℕ) → 𝑛 / 𝑘𝐴 ∈ ℂ)
157122, 156syldan 589 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑛 / 𝑘𝐴 ∈ ℂ)
158157abscld 15419 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
15971adantr 479 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑚 ∈ ℝ)
160157absge0d 15427 . . . . . . . . . . . . . . 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 11403 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 0 ≤ 𝑚)
162161ralrimiva 3135 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ∀𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))0 ≤ 𝑚)
163120nnzd 12618 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ((⌊‘𝑐) + 1) ∈ ℤ)
164 uzid 12870 . . . . . . . . . . . . . 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 3607 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 0 ≤ 𝑚)
167 reflcl 13797 . . . . . . . . . . . . . 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 9021 . . . . . . . . . . . . . . . 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 14375 . . . . . . . . . . . . . . 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 13821 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ0)
174 hashfz1 14341 . . . . . . . . . . . . . . 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 5175 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ (⌊‘𝑥))
177 flle 13800 . . . . . . . . . . . . . 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 11403 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ 𝑥)
180116, 69, 71, 166, 179lemul1ad 12186 . . . . . . . . . . 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 11403 . . . . . . . . . 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 11865 . . . . . . . . 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 12087 . . . . . . . . . . 11 ((⌊‘𝑐) ∈ ℝ → (⌊‘𝑐) < ((⌊‘𝑐) + 1))
184 fzdisj 13563 . . . . . . . . . . 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 11274 . . . . . . . . . 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 15723 . . . . . . . . 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 715 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑥 ∈ ℂ)
189188, 101, 149adddid 11270 . . . . . . . . 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 5181 . . . . . . . 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 11403 . . . . . . 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 13023 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
193192ad2antrl 726 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
194 ledivmul 12123 . . . . . . . 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 1368 . . . . . . 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 256 . . . . . 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 5171 . . . . 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 15516 . . . 4 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1))
199198ex 411 . . 3 ((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) → (∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚) → (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1)))
200199rexlimdvva 3201 . 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 205  wa 394   = wceq 1533  wcel 2098  wne 2929  wral 3050  wrex 3059  csb 3889  cun 3942  cin 3943  wss 3944  c0 4322   class class class wbr 5149  cmpt 5232  dom cdm 5678  wf 6545  cfv 6549  (class class class)co 7419  cdom 8962  Fincfn 8964  cc 11138  cr 11139  0cc0 11140  1c1 11141   + caddc 11143   · cmul 11145  +∞cpnf 11277   < clt 11280  cle 11281   / cdiv 11903  cn 12245  0cn0 12505  cz 12591  cuz 12855  +crp 13009  [,)cico 13361  ...cfz 13519  cfl 13791  chash 14325  abscabs 15217  𝑂(1)co1 15466  Σcsu 15668
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-er 8725  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9467  df-inf 9468  df-oi 9535  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-n0 12506  df-xnn0 12578  df-z 12592  df-uz 12856  df-rp 13010  df-ico 13365  df-fz 13520  df-fzo 13663  df-fl 13793  df-seq 14003  df-exp 14063  df-hash 14326  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-clim 15468  df-o1 15470  df-lo1 15471  df-sum 15669
This theorem is referenced by:  selberg2lem  27528
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