MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  o1fsum Structured version   Visualization version   GIF version

Theorem o1fsum 15846
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 12268 . . . . 5 ℕ ⊆ ℝ
32a1i 11 . . . 4 (𝜑 → ℕ ⊆ ℝ)
4 o1fsum.1 . . . . 5 ((𝜑𝑘 ∈ ℕ) → 𝐴𝑉)
54, 1o1mptrcl 15656 . . . 4 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
6 1red 11260 . . . 4 (𝜑 → 1 ∈ ℝ)
73, 5, 6elo1mpt2 15568 . . 3 (𝜑 → ((𝑘 ∈ ℕ ↦ 𝐴) ∈ 𝑂(1) ↔ ∃𝑐 ∈ (1[,)+∞)∃𝑚 ∈ ℝ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)))
81, 7mpbid 232 . 2 (𝜑 → ∃𝑐 ∈ (1[,)+∞)∃𝑚 ∈ ℝ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚))
9 rpssre 13040 . . . . . 6 + ⊆ ℝ
109a1i 11 . . . . 5 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → ℝ+ ⊆ ℝ)
11 csbeq1a 3922 . . . . . . . 8 (𝑘 = 𝑛𝐴 = 𝑛 / 𝑘𝐴)
12 nfcv 2903 . . . . . . . 8 𝑛𝐴
13 nfcsb1v 3933 . . . . . . . 8 𝑘𝑛 / 𝑘𝐴
1411, 12, 13cbvsum 15728 . . . . . . 7 Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 = Σ𝑛 ∈ (1...(⌊‘𝑥))𝑛 / 𝑘𝐴
15 fzfid 14011 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → (1...(⌊‘𝑥)) ∈ Fin)
16 o1f 15562 . . . . . . . . . . . . 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 3144 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑘 ∈ ℕ 𝐴𝑉)
19 dmmptg 6264 . . . . . . . . . . . . . 14 (∀𝑘 ∈ ℕ 𝐴𝑉 → dom (𝑘 ∈ ℕ ↦ 𝐴) = ℕ)
2018, 19syl 17 . . . . . . . . . . . . 13 (𝜑 → dom (𝑘 ∈ ℕ ↦ 𝐴) = ℕ)
2120feq2d 6723 . . . . . . . . . . . 12 (𝜑 → ((𝑘 ∈ ℕ ↦ 𝐴):dom (𝑘 ∈ ℕ ↦ 𝐴)⟶ℂ ↔ (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ))
2217, 21mpbid 232 . . . . . . . . . . 11 (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ)
23 eqid 2735 . . . . . . . . . . . 12 (𝑘 ∈ ℕ ↦ 𝐴) = (𝑘 ∈ ℕ ↦ 𝐴)
2423fmpt 7130 . . . . . . . . . . 11 (∀𝑘 ∈ ℕ 𝐴 ∈ ℂ ↔ (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ)
2522, 24sylibr 234 . . . . . . . . . 10 (𝜑 → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ)
2625ad3antrrr 730 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ)
27 elfznn 13590 . . . . . . . . 9 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
2813nfel1 2920 . . . . . . . . . . 11 𝑘𝑛 / 𝑘𝐴 ∈ ℂ
2911eleq1d 2824 . . . . . . . . . . 11 (𝑘 = 𝑛 → (𝐴 ∈ ℂ ↔ 𝑛 / 𝑘𝐴 ∈ ℂ))
3028, 29rspc 3610 . . . . . . . . . 10 (𝑛 ∈ ℕ → (∀𝑘 ∈ ℕ 𝐴 ∈ ℂ → 𝑛 / 𝑘𝐴 ∈ ℂ))
3130impcom 407 . . . . . . . . 9 ((∀𝑘 ∈ ℕ 𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ) → 𝑛 / 𝑘𝐴 ∈ ℂ)
3226, 27, 31syl2an 596 . . . . . . . 8 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 / 𝑘𝐴 ∈ ℂ)
3315, 32fsumcl 15766 . . . . . . 7 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))𝑛 / 𝑘𝐴 ∈ ℂ)
3414, 33eqeltrid 2843 . . . . . 6 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 ∈ ℂ)
35 rpcn 13043 . . . . . . 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 13049 . . . . . . 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 12041 . . . . 5 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥) ∈ ℂ)
40 simplrl 777 . . . . . . 7 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑐 ∈ (1[,)+∞))
41 1re 11259 . . . . . . . 8 1 ∈ ℝ
42 elicopnf 13482 . . . . . . . 8 (1 ∈ ℝ → (𝑐 ∈ (1[,)+∞) ↔ (𝑐 ∈ ℝ ∧ 1 ≤ 𝑐)))
4341, 42ax-mp 5 . . . . . . 7 (𝑐 ∈ (1[,)+∞) ↔ (𝑐 ∈ ℝ ∧ 1 ≤ 𝑐))
4440, 43sylib 218 . . . . . 6 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → (𝑐 ∈ ℝ ∧ 1 ≤ 𝑐))
4544simpld 494 . . . . 5 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑐 ∈ ℝ)
46 fzfid 14011 . . . . . . 7 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → (1...(⌊‘𝑐)) ∈ Fin)
4725ad2antrr 726 . . . . . . . . 9 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ)
48 elfznn 13590 . . . . . . . . 9 (𝑛 ∈ (1...(⌊‘𝑐)) → 𝑛 ∈ ℕ)
4947, 48, 31syl2an 596 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑐))) → 𝑛 / 𝑘𝐴 ∈ ℂ)
5049abscld 15472 . . . . . . 7 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑐))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
5146, 50fsumrecl 15767 . . . . . 6 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
52 simplrr 778 . . . . . 6 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑚 ∈ ℝ)
5351, 52readdcld 11288 . . . . 5 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) + 𝑚) ∈ ℝ)
5434, 36, 38absdivd 15491 . . . . . . . 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 717 . . . . . . 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 13048 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
5756ad2antrl 728 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
58 absid 15332 . . . . . . . . 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 7447 . . . . . . 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 2775 . . . . . 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 717 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 ∈ ℂ)
6362abscld 15472 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) ∈ ℝ)
64 fzfid 14011 . . . . . . . . 9 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
6547, 27, 31syl2an 596 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 / 𝑘𝐴 ∈ ℂ)
6665adantlr 715 . . . . . . . . . 10 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 / 𝑘𝐴 ∈ ℂ)
6766abscld 15472 . . . . . . . . 9 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
6864, 67fsumrecl 15767 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
6957simpld 494 . . . . . . . . 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 11288 . . . . . . . . 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 11289 . . . . . . . 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 6910 . . . . . . . . 9 (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) = (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))𝑛 / 𝑘𝐴)
7564, 66fsumabs 15834 . . . . . . . . 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 5183 . . . . . . . 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 14011 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin)
78 ssun2 4189 . . . . . . . . . . . . . 14 (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ⊆ ((1...(⌊‘𝑐)) ∪ (((⌊‘𝑐) + 1)...(⌊‘𝑥)))
79 flge1nn 13858 . . . . . . . . . . . . . . . . . 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 12279 . . . . . . . . . . . . . . . . 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 13836 . . . . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑐𝑥)
8782, 83, 69, 85, 86letrd 11416 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (⌊‘𝑐) ≤ 𝑥)
88 fznnfl 13899 . . . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (⌊‘𝑐) ∈ (1...(⌊‘𝑥)))
91 fzsplit 13587 . . . . . . . . . . . . . . 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 4049 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑥)))
9493sselda 3995 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) → 𝑛 ∈ (1...(⌊‘𝑥)))
9565abscld 15472 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
9695adantlr 715 . . . . . . . . . . . 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 15767 . . . . . . . . . 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 11289 . . . . . . . . . 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 11289 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (𝑥 · 𝑚) ∈ ℝ)
10170recnd 11287 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴) ∈ ℂ)
102101mullidd 11277 . . . . . . . . . . 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 11260 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 1 ∈ ℝ)
10449absge0d 15480 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑐))) → 0 ≤ (abs‘𝑛 / 𝑘𝐴))
10546, 50, 104fsumge0 15828 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘𝑛 / 𝑘𝐴))
10651, 105jca 511 . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . 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 11416 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 1 ≤ 𝑥)
111 lemul1a 12119 . . . . . . . . . . . 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 1372 . . . . . . . . . . 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 5172 . . . . . . . . . 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 14392 . . . . . . . . . . . . 13 ((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ∈ ℕ0)
115 nn0re 12533 . . . . . . . . . . . . 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 11289 . . . . . . . . . . 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 13557 . . . . . . . . . . . . . 14 (𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥)) → 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1)))
12081peano2nnd 12281 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ((⌊‘𝑐) + 1) ∈ ℕ)
121 eluznn 12958 . . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . 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 13833 . . . . . . . . . . . . . . . . 17 (𝑐 ∈ ℝ → (⌊‘𝑐) ∈ ℝ)
126 peano2re 11432 . . . . . . . . . . . . . . . . 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 12279 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑛 ∈ ℝ)
129 fllep1 13838 . . . . . . . . . . . . . . . . 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 12889 . . . . . . . . . . . . . . . . 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 11416 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑐𝑛)
134 nfv 1912 . . . . . . . . . . . . . . . . 17 𝑘 𝑐𝑛
135 nfcv 2903 . . . . . . . . . . . . . . . . . . 19 𝑘abs
136135, 13nffv 6917 . . . . . . . . . . . . . . . . . 18 𝑘(abs‘𝑛 / 𝑘𝐴)
137 nfcv 2903 . . . . . . . . . . . . . . . . . 18 𝑘
138 nfcv 2903 . . . . . . . . . . . . . . . . . 18 𝑘𝑚
139136, 137, 138nfbr 5195 . . . . . . . . . . . . . . . . 17 𝑘(abs‘𝑛 / 𝑘𝐴) ≤ 𝑚
140134, 139nfim 1894 . . . . . . . . . . . . . . . 16 𝑘(𝑐𝑛 → (abs‘𝑛 / 𝑘𝐴) ≤ 𝑚)
141 breq2 5152 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝑐𝑘𝑐𝑛))
14211fveq2d 6911 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (abs‘𝐴) = (abs‘𝑛 / 𝑘𝐴))
143142breq1d 5158 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → ((abs‘𝐴) ≤ 𝑚 ↔ (abs‘𝑛 / 𝑘𝐴) ≤ 𝑚))
144141, 143imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ((𝑐𝑘 → (abs‘𝐴) ≤ 𝑚) ↔ (𝑐𝑛 → (abs‘𝑛 / 𝑘𝐴) ≤ 𝑚)))
145140, 144rspc 3610 . . . . . . . . . . . . . . 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 593 . . . . . . . . . . . . 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 15832 . . . . . . . . . . . 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 11287 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑚 ∈ ℂ)
150 fsumconst 15823 . . . . . . . . . . . . 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 5174 . . . . . . . . . . 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 262 . . . . . . . . . . . . 13 (𝑛 = ((⌊‘𝑐) + 1) → (0 ≤ 𝑚 ↔ 0 ≤ 𝑚))
154 0red 11262 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 0 ∈ ℝ)
15547, 30mpan9 506 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 / 𝑘𝐴 ∈ ℂ)
156155adantlr 715 . . . . . . . . . . . . . . . . 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 15472 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → (abs‘𝑛 / 𝑘𝐴) ∈ ℝ)
15971adantr 480 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 𝑚 ∈ ℝ)
160157absge0d 15480 . . . . . . . . . . . . . . 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 11416 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) ∧ 𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))) → 0 ≤ 𝑚)
162161ralrimiva 3144 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ∀𝑛 ∈ (ℤ‘((⌊‘𝑐) + 1))0 ≤ 𝑚)
163120nnzd 12638 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → ((⌊‘𝑐) + 1) ∈ ℤ)
164 uzid 12891 . . . . . . . . . . . . . 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 3623 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 0 ≤ 𝑚)
167 reflcl 13833 . . . . . . . . . . . . . 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 9039 . . . . . . . . . . . . . . . 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 14416 . . . . . . . . . . . . . . 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 13857 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ0)
174 hashfz1 14382 . . . . . . . . . . . . . . 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 5174 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ (⌊‘𝑥))
177 flle 13836 . . . . . . . . . . . . . 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 11416 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ 𝑥)
180116, 69, 71, 166, 179lemul1ad 12205 . . . . . . . . . . 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 11416 . . . . . . . . . 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 11880 . . . . . . . . 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 12105 . . . . . . . . . . 11 ((⌊‘𝑐) ∈ ℝ → (⌊‘𝑐) < ((⌊‘𝑐) + 1))
184 fzdisj 13588 . . . . . . . . . . 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 11287 . . . . . . . . . 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 15774 . . . . . . . . 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 717 . . . . . . . . . 10 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → 𝑥 ∈ ℂ)
189188, 101, 149adddid 11283 . . . . . . . . 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 5180 . . . . . . . 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 11416 . . . . . . 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 13047 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
193192ad2antrl 728 . . . . . . . 8 ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+𝑐𝑥)) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
194 ledivmul 12142 . . . . . . . 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 1370 . . . . . . 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 257 . . . . . 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 5170 . . . . 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 15569 . . . 4 (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧ ∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚)) → (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1))
199198ex 412 . . 3 ((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) → (∀𝑘 ∈ ℕ (𝑐𝑘 → (abs‘𝐴) ≤ 𝑚) → (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1)))
200199rexlimdvva 3211 . 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 206  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  csb 3908  cun 3961  cin 3962  wss 3963  c0 4339   class class class wbr 5148  cmpt 5231  dom cdm 5689  wf 6559  cfv 6563  (class class class)co 7431  cdom 8982  Fincfn 8984  cc 11151  cr 11152  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158  +∞cpnf 11290   < clt 11293  cle 11294   / cdiv 11918  cn 12264  0cn0 12524  cz 12611  cuz 12876  +crp 13032  [,)cico 13386  ...cfz 13544  cfl 13827  chash 14366  abscabs 15270  𝑂(1)co1 15519  Σcsu 15719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-rp 13033  df-ico 13390  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-o1 15523  df-lo1 15524  df-sum 15720
This theorem is referenced by:  selberg2lem  27609
  Copyright terms: Public domain W3C validator