Step | Hyp | Ref
| Expression |
1 | | o1fsum.2 |
. . 3
⊢ (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴) ∈ 𝑂(1)) |
2 | | nnssre 10901 |
. . . . 5
⊢ ℕ
⊆ ℝ |
3 | 2 | a1i 11 |
. . . 4
⊢ (𝜑 → ℕ ⊆
ℝ) |
4 | | o1fsum.1 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ 𝑉) |
5 | 4, 1 | o1mptrcl 14201 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℂ) |
6 | | 1red 9934 |
. . . 4
⊢ (𝜑 → 1 ∈
ℝ) |
7 | 3, 5, 6 | elo1mpt2 14114 |
. . 3
⊢ (𝜑 → ((𝑘 ∈ ℕ ↦ 𝐴) ∈ 𝑂(1) ↔ ∃𝑐 ∈
(1[,)+∞)∃𝑚
∈ ℝ ∀𝑘
∈ ℕ (𝑐 ≤
𝑘 → (abs‘𝐴) ≤ 𝑚))) |
8 | 1, 7 | mpbid 221 |
. 2
⊢ (𝜑 → ∃𝑐 ∈ (1[,)+∞)∃𝑚 ∈ ℝ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) |
9 | | rpssre 11719 |
. . . . . 6
⊢
ℝ+ ⊆ ℝ |
10 | 9 | a1i 11 |
. . . . 5
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → ℝ+ ⊆
ℝ) |
11 | | nfcv 2751 |
. . . . . . . 8
⊢
Ⅎ𝑛𝐴 |
12 | | nfcsb1v 3515 |
. . . . . . . 8
⊢
Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐴 |
13 | | csbeq1a 3508 |
. . . . . . . 8
⊢ (𝑘 = 𝑛 → 𝐴 = ⦋𝑛 / 𝑘⦌𝐴) |
14 | 11, 12, 13 | cbvsumi 14275 |
. . . . . . 7
⊢
Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴 = Σ𝑛 ∈ (1...(⌊‘𝑥))⦋𝑛 / 𝑘⦌𝐴 |
15 | | fzfid 12634 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) →
(1...(⌊‘𝑥))
∈ Fin) |
16 | | o1f 14108 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ ℕ ↦ 𝐴) ∈ 𝑂(1) →
(𝑘 ∈ ℕ ↦
𝐴):dom (𝑘 ∈ ℕ ↦ 𝐴)⟶ℂ) |
17 | 1, 16 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴):dom (𝑘 ∈ ℕ ↦ 𝐴)⟶ℂ) |
18 | 4 | ralrimiva 2949 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑉) |
19 | | dmmptg 5549 |
. . . . . . . . . . . . . 14
⊢
(∀𝑘 ∈
ℕ 𝐴 ∈ 𝑉 → dom (𝑘 ∈ ℕ ↦ 𝐴) = ℕ) |
20 | 18, 19 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → dom (𝑘 ∈ ℕ ↦ 𝐴) = ℕ) |
21 | 20 | feq2d 5944 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑘 ∈ ℕ ↦ 𝐴):dom (𝑘 ∈ ℕ ↦ 𝐴)⟶ℂ ↔ (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ)) |
22 | 17, 21 | mpbid 221 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ) |
23 | | eqid 2610 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ ↦ 𝐴) = (𝑘 ∈ ℕ ↦ 𝐴) |
24 | 23 | fmpt 6289 |
. . . . . . . . . . 11
⊢
(∀𝑘 ∈
ℕ 𝐴 ∈ ℂ
↔ (𝑘 ∈ ℕ
↦ 𝐴):ℕ⟶ℂ) |
25 | 22, 24 | sylibr 223 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ) |
26 | 25 | ad3antrrr 762 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) →
∀𝑘 ∈ ℕ
𝐴 ∈
ℂ) |
27 | | elfznn 12241 |
. . . . . . . . 9
⊢ (𝑛 ∈
(1...(⌊‘𝑥))
→ 𝑛 ∈
ℕ) |
28 | 12 | nfel1 2765 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ |
29 | 13 | eleq1d 2672 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑛 → (𝐴 ∈ ℂ ↔ ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ)) |
30 | 28, 29 | rspc 3276 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ →
(∀𝑘 ∈ ℕ
𝐴 ∈ ℂ →
⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ)) |
31 | 30 | impcom 445 |
. . . . . . . . 9
⊢
((∀𝑘 ∈
ℕ 𝐴 ∈ ℂ
∧ 𝑛 ∈ ℕ)
→ ⦋𝑛 /
𝑘⦌𝐴 ∈
ℂ) |
32 | 26, 27, 31 | syl2an 493 |
. . . . . . . 8
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ⦋𝑛 /
𝑘⦌𝐴 ∈
ℂ) |
33 | 15, 32 | fsumcl 14311 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) →
Σ𝑛 ∈
(1...(⌊‘𝑥))⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ) |
34 | 14, 33 | syl5eqel 2692 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) →
Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴 ∈ ℂ) |
35 | | rpcn 11717 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℂ) |
36 | 35 | adantl 481 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈
ℂ) |
37 | | rpne0 11724 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ≠
0) |
38 | 37 | adantl 481 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0) |
39 | 34, 36, 38 | divcld 10680 |
. . . . 5
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) →
(Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴 / 𝑥) ∈ ℂ) |
40 | | simplrl 796 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑐 ∈ (1[,)+∞)) |
41 | | 1re 9918 |
. . . . . . . 8
⊢ 1 ∈
ℝ |
42 | | elicopnf 12140 |
. . . . . . . 8
⊢ (1 ∈
ℝ → (𝑐 ∈
(1[,)+∞) ↔ (𝑐
∈ ℝ ∧ 1 ≤ 𝑐))) |
43 | 41, 42 | ax-mp 5 |
. . . . . . 7
⊢ (𝑐 ∈ (1[,)+∞) ↔
(𝑐 ∈ ℝ ∧ 1
≤ 𝑐)) |
44 | 40, 43 | sylib 207 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → (𝑐 ∈ ℝ ∧ 1 ≤ 𝑐)) |
45 | 44 | simpld 474 |
. . . . 5
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑐 ∈ ℝ) |
46 | | fzfid 12634 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → (1...(⌊‘𝑐)) ∈ Fin) |
47 | 25 | ad2antrr 758 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ) |
48 | | elfznn 12241 |
. . . . . . . . 9
⊢ (𝑛 ∈
(1...(⌊‘𝑐))
→ 𝑛 ∈
ℕ) |
49 | 47, 48, 31 | syl2an 493 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑐))) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ) |
50 | 49 | abscld 14023 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑐))) →
(abs‘⦋𝑛
/ 𝑘⦌𝐴) ∈
ℝ) |
51 | 46, 50 | fsumrecl 14312 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ) |
52 | | simplrr 797 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑚 ∈ ℝ) |
53 | 51, 52 | readdcld 9948 |
. . . . 5
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚) ∈ ℝ) |
54 | 34, 36, 38 | absdivd 14042 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) →
(abs‘(Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴 / 𝑥)) = ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) / (abs‘𝑥))) |
55 | 54 | adantrr 749 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (abs‘(Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴 / 𝑥)) = ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) / (abs‘𝑥))) |
56 | | rprege0 11723 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℝ
∧ 0 ≤ 𝑥)) |
57 | 56 | ad2antrl 760 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) |
58 | | absid 13884 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ ∧ 0 ≤
𝑥) → (abs‘𝑥) = 𝑥) |
59 | 57, 58 | syl 17 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (abs‘𝑥) = 𝑥) |
60 | 59 | oveq2d 6565 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → ((abs‘Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴) / (abs‘𝑥)) = ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) / 𝑥)) |
61 | 55, 60 | eqtrd 2644 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (abs‘(Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴 / 𝑥)) = ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) / 𝑥)) |
62 | 34 | adantrr 749 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 ∈ ℂ) |
63 | 62 | abscld 14023 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (abs‘Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴) ∈ ℝ) |
64 | | fzfid 12634 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin) |
65 | 47, 27, 31 | syl2an 493 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ) |
66 | 65 | adantlr 747 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ) |
67 | 66 | abscld 14023 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(abs‘⦋𝑛
/ 𝑘⦌𝐴) ∈
ℝ) |
68 | 64, 67 | fsumrecl 14312 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ) |
69 | 57 | simpld 474 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → 𝑥 ∈ ℝ) |
70 | 51 | adantr 480 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ) |
71 | 52 | adantr 480 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → 𝑚 ∈ ℝ) |
72 | 70, 71 | readdcld 9948 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚) ∈ ℝ) |
73 | 69, 72 | remulcld 9949 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (𝑥 · (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚)) ∈ ℝ) |
74 | 14 | fveq2i 6106 |
. . . . . . . . 9
⊢
(abs‘Σ𝑘
∈ (1...(⌊‘𝑥))𝐴) = (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))⦋𝑛 / 𝑘⦌𝐴) |
75 | 64, 66 | fsumabs 14374 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (abs‘Σ𝑛 ∈
(1...(⌊‘𝑥))⦋𝑛 / 𝑘⦌𝐴) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴)) |
76 | 74, 75 | syl5eqbr 4618 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (abs‘Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴)) |
77 | | fzfid 12634 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin) |
78 | | ssun2 3739 |
. . . . . . . . . . . . . 14
⊢
(((⌊‘𝑐)
+ 1)...(⌊‘𝑥))
⊆ ((1...(⌊‘𝑐)) ∪ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) |
79 | | flge1nn 12484 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑐 ∈ ℝ ∧ 1 ≤
𝑐) →
(⌊‘𝑐) ∈
ℕ) |
80 | 44, 79 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → (⌊‘𝑐) ∈ ℕ) |
81 | 80 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑐) ∈ ℕ) |
82 | 81 | nnred 10912 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑐) ∈ ℝ) |
83 | 45 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → 𝑐 ∈ ℝ) |
84 | | flle 12462 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑐 ∈ ℝ →
(⌊‘𝑐) ≤
𝑐) |
85 | 83, 84 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑐) ≤ 𝑐) |
86 | | simprr 792 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → 𝑐 ≤ 𝑥) |
87 | 82, 83, 69, 85, 86 | letrd 10073 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑐) ≤ 𝑥) |
88 | | fznnfl 12523 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ℝ →
((⌊‘𝑐) ∈
(1...(⌊‘𝑥))
↔ ((⌊‘𝑐)
∈ ℕ ∧ (⌊‘𝑐) ≤ 𝑥))) |
89 | 69, 88 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → ((⌊‘𝑐) ∈ (1...(⌊‘𝑥)) ↔ ((⌊‘𝑐) ∈ ℕ ∧
(⌊‘𝑐) ≤
𝑥))) |
90 | 81, 87, 89 | mpbir2and 959 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑐) ∈ (1...(⌊‘𝑥))) |
91 | | fzsplit 12238 |
. . . . . . . . . . . . . . 15
⊢
((⌊‘𝑐)
∈ (1...(⌊‘𝑥)) → (1...(⌊‘𝑥)) = ((1...(⌊‘𝑐)) ∪ (((⌊‘𝑐) + 1)...(⌊‘𝑥)))) |
92 | 90, 91 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (1...(⌊‘𝑥)) = ((1...(⌊‘𝑐)) ∪ (((⌊‘𝑐) + 1)...(⌊‘𝑥)))) |
93 | 78, 92 | syl5sseqr 3617 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑥))) |
94 | 93 | sselda 3568 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) → 𝑛 ∈ (1...(⌊‘𝑥))) |
95 | 65 | abscld 14023 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(abs‘⦋𝑛
/ 𝑘⦌𝐴) ∈
ℝ) |
96 | 95 | adantlr 747 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(abs‘⦋𝑛
/ 𝑘⦌𝐴) ∈
ℝ) |
97 | 94, 96 | syldan 486 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) → (abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ) |
98 | 77, 97 | fsumrecl 14312 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ) |
99 | 69, 70 | remulcld 9949 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴)) ∈ ℝ) |
100 | 69, 71 | remulcld 9949 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (𝑥 · 𝑚) ∈ ℝ) |
101 | 70 | recnd 9947 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℂ) |
102 | 101 | mulid2d 9937 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (1 · Σ𝑛 ∈
(1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴)) = Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴)) |
103 | | 1red 9934 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → 1 ∈ ℝ) |
104 | 49 | absge0d 14031 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑐))) → 0 ≤
(abs‘⦋𝑛
/ 𝑘⦌𝐴)) |
105 | 46, 50, 104 | fsumge0 14368 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴)) |
106 | 51, 105 | jca 553 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ ∧ 0 ≤ Σ𝑛 ∈
(1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴))) |
107 | 106 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ ∧ 0 ≤ Σ𝑛 ∈
(1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴))) |
108 | 44 | simprd 478 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → 1 ≤ 𝑐) |
109 | 108 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → 1 ≤ 𝑐) |
110 | 103, 83, 69, 109, 86 | letrd 10073 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → 1 ≤ 𝑥) |
111 | | lemul1a 10756 |
. . . . . . . . . . . 12
⊢ (((1
∈ ℝ ∧ 𝑥
∈ ℝ ∧ (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ ∧ 0 ≤ Σ𝑛 ∈
(1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴))) ∧ 1 ≤ 𝑥) → (1 · Σ𝑛 ∈
(1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴)) ≤ (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴))) |
112 | 103, 69, 107, 110, 111 | syl31anc 1321 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (1 · Σ𝑛 ∈
(1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴)) ≤ (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴))) |
113 | 102, 112 | eqbrtrrd 4607 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴))) |
114 | | hashcl 13009 |
. . . . . . . . . . . . 13
⊢
((((⌊‘𝑐)
+ 1)...(⌊‘𝑥))
∈ Fin → (#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ∈
ℕ0) |
115 | | nn0re 11178 |
. . . . . . . . . . . . 13
⊢
((#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ∈ ℕ0 →
(#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ∈ ℝ) |
116 | 77, 114, 115 | 3syl 18 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ∈
ℝ) |
117 | 116, 71 | remulcld 9949 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → ((#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚) ∈ ℝ) |
118 | 71 | adantr 480 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) → 𝑚 ∈ ℝ) |
119 | | elfzuz 12209 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥)) → 𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))) |
120 | 81 | peano2nnd 10914 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → ((⌊‘𝑐) + 1) ∈ ℕ) |
121 | | eluznn 11634 |
. . . . . . . . . . . . . . . 16
⊢
((((⌊‘𝑐)
+ 1) ∈ ℕ ∧ 𝑛
∈ (ℤ≥‘((⌊‘𝑐) + 1))) → 𝑛 ∈ ℕ) |
122 | 120, 121 | sylan 487 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))) → 𝑛 ∈ ℕ) |
123 | | simpllr 795 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))) → ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) |
124 | 83 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))) → 𝑐 ∈ ℝ) |
125 | | reflcl 12459 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑐 ∈ ℝ →
(⌊‘𝑐) ∈
ℝ) |
126 | | peano2re 10088 |
. . . . . . . . . . . . . . . . 17
⊢
((⌊‘𝑐)
∈ ℝ → ((⌊‘𝑐) + 1) ∈ ℝ) |
127 | 124, 125,
126 | 3syl 18 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))) → ((⌊‘𝑐) + 1) ∈
ℝ) |
128 | 122 | nnred 10912 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))) → 𝑛 ∈ ℝ) |
129 | | fllep1 12464 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑐 ∈ ℝ → 𝑐 ≤ ((⌊‘𝑐) + 1)) |
130 | 124, 129 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))) → 𝑐 ≤ ((⌊‘𝑐) + 1)) |
131 | | eluzle 11576 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1)) → ((⌊‘𝑐) + 1) ≤ 𝑛) |
132 | 131 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))) → ((⌊‘𝑐) + 1) ≤ 𝑛) |
133 | 124, 127,
128, 130, 132 | letrd 10073 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))) → 𝑐 ≤ 𝑛) |
134 | | nfv 1830 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘 𝑐 ≤ 𝑛 |
135 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘abs |
136 | 135, 12 | nffv 6110 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘(abs‘⦋𝑛 / 𝑘⦌𝐴) |
137 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘
≤ |
138 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘𝑚 |
139 | 136, 137,
138 | nfbr 4629 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘(abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚 |
140 | 134, 139 | nfim 1813 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑘(𝑐 ≤ 𝑛 → (abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚) |
141 | | breq2 4587 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑛 → (𝑐 ≤ 𝑘 ↔ 𝑐 ≤ 𝑛)) |
142 | 13 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑛 → (abs‘𝐴) = (abs‘⦋𝑛 / 𝑘⦌𝐴)) |
143 | 142 | breq1d 4593 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑛 → ((abs‘𝐴) ≤ 𝑚 ↔ (abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚)) |
144 | 141, 143 | imbi12d 333 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑛 → ((𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚) ↔ (𝑐 ≤ 𝑛 → (abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚))) |
145 | 140, 144 | rspc 3276 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ →
(∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚) → (𝑐 ≤ 𝑛 → (abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚))) |
146 | 122, 123,
133, 145 | syl3c 64 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))) →
(abs‘⦋𝑛
/ 𝑘⦌𝐴) ≤ 𝑚) |
147 | 119, 146 | sylan2 490 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) → (abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚) |
148 | 77, 97, 118, 147 | fsumle 14372 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))𝑚) |
149 | 71 | recnd 9947 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → 𝑚 ∈ ℂ) |
150 | | fsumconst 14364 |
. . . . . . . . . . . . 13
⊢
(((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin ∧ 𝑚 ∈ ℂ) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))𝑚 = ((#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚)) |
151 | 77, 149, 150 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))𝑚 = ((#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚)) |
152 | 148, 151 | breqtrd 4609 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ ((#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚)) |
153 | 120 | nnzd 11357 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → ((⌊‘𝑐) + 1) ∈ ℤ) |
154 | | uzid 11578 |
. . . . . . . . . . . . . 14
⊢
(((⌊‘𝑐)
+ 1) ∈ ℤ → ((⌊‘𝑐) + 1) ∈
(ℤ≥‘((⌊‘𝑐) + 1))) |
155 | 153, 154 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → ((⌊‘𝑐) + 1) ∈
(ℤ≥‘((⌊‘𝑐) + 1))) |
156 | | 0red 9920 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))) → 0 ∈
ℝ) |
157 | 47, 30 | mpan9 485 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ ℕ) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ) |
158 | 157 | adantlr 747 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ ℕ) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ) |
159 | 122, 158 | syldan 486 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ) |
160 | 159 | abscld 14023 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))) →
(abs‘⦋𝑛
/ 𝑘⦌𝐴) ∈
ℝ) |
161 | 71 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))) → 𝑚 ∈ ℝ) |
162 | 159 | absge0d 14031 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))) → 0 ≤
(abs‘⦋𝑛
/ 𝑘⦌𝐴)) |
163 | 156, 160,
161, 162, 146 | letrd 10073 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))) → 0 ≤ 𝑚) |
164 | 163 | ralrimiva 2949 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → ∀𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))0 ≤ 𝑚) |
165 | | biidd 251 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = ((⌊‘𝑐) + 1) → (0 ≤ 𝑚 ↔ 0 ≤ 𝑚)) |
166 | 165 | rspcv 3278 |
. . . . . . . . . . . . 13
⊢
(((⌊‘𝑐)
+ 1) ∈ (ℤ≥‘((⌊‘𝑐) + 1)) → (∀𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))0 ≤ 𝑚 → 0 ≤ 𝑚)) |
167 | 155, 164,
166 | sylc 63 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → 0 ≤ 𝑚) |
168 | | reflcl 12459 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℝ →
(⌊‘𝑥) ∈
ℝ) |
169 | 69, 168 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑥) ∈ ℝ) |
170 | | ssdomg 7887 |
. . . . . . . . . . . . . . . 16
⊢
((1...(⌊‘𝑥)) ∈ Fin → ((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ⊆
(1...(⌊‘𝑥))
→ (((⌊‘𝑐)
+ 1)...(⌊‘𝑥))
≼ (1...(⌊‘𝑥)))) |
171 | 64, 93, 170 | sylc 63 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ≼ (1...(⌊‘𝑥))) |
172 | | hashdomi 13030 |
. . . . . . . . . . . . . . 15
⊢
((((⌊‘𝑐)
+ 1)...(⌊‘𝑥))
≼ (1...(⌊‘𝑥)) → (#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤
(#‘(1...(⌊‘𝑥)))) |
173 | 171, 172 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤
(#‘(1...(⌊‘𝑥)))) |
174 | | flge0nn0 12483 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 0 ≤
𝑥) →
(⌊‘𝑥) ∈
ℕ0) |
175 | | hashfz1 12996 |
. . . . . . . . . . . . . . 15
⊢
((⌊‘𝑥)
∈ ℕ0 → (#‘(1...(⌊‘𝑥))) = (⌊‘𝑥)) |
176 | 57, 174, 175 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (#‘(1...(⌊‘𝑥))) = (⌊‘𝑥)) |
177 | 173, 176 | breqtrd 4609 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ (⌊‘𝑥)) |
178 | | flle 12462 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℝ →
(⌊‘𝑥) ≤
𝑥) |
179 | 69, 178 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑥) ≤ 𝑥) |
180 | 116, 169,
69, 177, 179 | letrd 10073 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ 𝑥) |
181 | 116, 69, 71, 167, 180 | lemul1ad 10842 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → ((#‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚) ≤ (𝑥 · 𝑚)) |
182 | 98, 117, 100, 152, 181 | letrd 10073 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ (𝑥 · 𝑚)) |
183 | 70, 98, 99, 100, 113, 182 | le2addd 10525 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴)) ≤ ((𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴)) + (𝑥 · 𝑚))) |
184 | | ltp1 10740 |
. . . . . . . . . . 11
⊢
((⌊‘𝑐)
∈ ℝ → (⌊‘𝑐) < ((⌊‘𝑐) + 1)) |
185 | | fzdisj 12239 |
. . . . . . . . . . 11
⊢
((⌊‘𝑐)
< ((⌊‘𝑐) +
1) → ((1...(⌊‘𝑐)) ∩ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) = ∅) |
186 | 82, 184, 185 | 3syl 18 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → ((1...(⌊‘𝑐)) ∩ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) = ∅) |
187 | 96 | recnd 9947 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(abs‘⦋𝑛
/ 𝑘⦌𝐴) ∈
ℂ) |
188 | 186, 92, 64, 187 | fsumsplit 14318 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴) = (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴))) |
189 | 36 | adantrr 749 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → 𝑥 ∈ ℂ) |
190 | 189, 101,
149 | adddid 9943 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (𝑥 · (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚)) = ((𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴)) + (𝑥 · 𝑚))) |
191 | 183, 188,
190 | 3brtr4d 4615 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ (𝑥 · (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚))) |
192 | 63, 68, 73, 76, 191 | letrd 10073 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (abs‘Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴) ≤ (𝑥 · (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚))) |
193 | | rpregt0 11722 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℝ
∧ 0 < 𝑥)) |
194 | 193 | ad2antrl 760 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 < 𝑥)) |
195 | | ledivmul 10778 |
. . . . . . . 8
⊢
(((abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) ∈ ℝ ∧ (Σ𝑛 ∈
(1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚) ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) →
(((abs‘Σ𝑘
∈ (1...(⌊‘𝑥))𝐴) / 𝑥) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚) ↔ (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) ≤ (𝑥 · (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚)))) |
196 | 63, 72, 194, 195 | syl3anc 1318 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (((abs‘Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴) / 𝑥) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚) ↔ (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴) ≤ (𝑥 · (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚)))) |
197 | 192, 196 | mpbird 246 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → ((abs‘Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴) / 𝑥) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚)) |
198 | 61, 197 | eqbrtrd 4605 |
. . . . 5
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (abs‘(Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴 / 𝑥)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) + 𝑚)) |
199 | 10, 39, 45, 53, 198 | elo1d 14115 |
. . . 4
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → (𝑥 ∈ ℝ+ ↦
(Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1)) |
200 | 199 | ex 449 |
. . 3
⊢ ((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) →
(∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚) → (𝑥 ∈ ℝ+ ↦
(Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1))) |
201 | 200 | rexlimdvva 3020 |
. 2
⊢ (𝜑 → (∃𝑐 ∈ (1[,)+∞)∃𝑚 ∈ ℝ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚) → (𝑥 ∈ ℝ+ ↦
(Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1))) |
202 | 8, 201 | mpd 15 |
1
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦
(Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1)) |