| Step | Hyp | Ref
| Expression |
| 1 | | o1fsum.2 |
. . 3
⊢ (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴) ∈ 𝑂(1)) |
| 2 | | nnssre 12270 |
. . . . 5
⊢ ℕ
⊆ ℝ |
| 3 | 2 | a1i 11 |
. . . 4
⊢ (𝜑 → ℕ ⊆
ℝ) |
| 4 | | o1fsum.1 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ 𝑉) |
| 5 | 4, 1 | o1mptrcl 15659 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℂ) |
| 6 | | 1red 11262 |
. . . 4
⊢ (𝜑 → 1 ∈
ℝ) |
| 7 | 3, 5, 6 | elo1mpt2 15571 |
. . 3
⊢ (𝜑 → ((𝑘 ∈ ℕ ↦ 𝐴) ∈ 𝑂(1) ↔ ∃𝑐 ∈
(1[,)+∞)∃𝑚
∈ ℝ ∀𝑘
∈ ℕ (𝑐 ≤
𝑘 → (abs‘𝐴) ≤ 𝑚))) |
| 8 | 1, 7 | mpbid 232 |
. 2
⊢ (𝜑 → ∃𝑐 ∈ (1[,)+∞)∃𝑚 ∈ ℝ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) |
| 9 | | rpssre 13042 |
. . . . . 6
⊢
ℝ+ ⊆ ℝ |
| 10 | 9 | a1i 11 |
. . . . 5
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → ℝ+ ⊆
ℝ) |
| 11 | | csbeq1a 3913 |
. . . . . . . 8
⊢ (𝑘 = 𝑛 → 𝐴 = ⦋𝑛 / 𝑘⦌𝐴) |
| 12 | | nfcv 2905 |
. . . . . . . 8
⊢
Ⅎ𝑛𝐴 |
| 13 | | nfcsb1v 3923 |
. . . . . . . 8
⊢
Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐴 |
| 14 | 11, 12, 13 | cbvsum 15731 |
. . . . . . 7
⊢
Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴 = Σ𝑛 ∈ (1...(⌊‘𝑥))⦋𝑛 / 𝑘⦌𝐴 |
| 15 | | fzfid 14014 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) →
(1...(⌊‘𝑥))
∈ Fin) |
| 16 | | o1f 15565 |
. . . . . . . . . . . . 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 3146 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑉) |
| 19 | | dmmptg 6262 |
. . . . . . . . . . . . . 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 6722 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑘 ∈ ℕ ↦ 𝐴):dom (𝑘 ∈ ℕ ↦ 𝐴)⟶ℂ ↔ (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ)) |
| 22 | 17, 21 | mpbid 232 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶ℂ) |
| 23 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ ↦ 𝐴) = (𝑘 ∈ ℕ ↦ 𝐴) |
| 24 | 23 | fmpt 7130 |
. . . . . . . . . . 11
⊢
(∀𝑘 ∈
ℕ 𝐴 ∈ ℂ
↔ (𝑘 ∈ ℕ
↦ 𝐴):ℕ⟶ℂ) |
| 25 | 22, 24 | sylibr 234 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ) |
| 26 | 25 | ad3antrrr 730 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) →
∀𝑘 ∈ ℕ
𝐴 ∈
ℂ) |
| 27 | | elfznn 13593 |
. . . . . . . . 9
⊢ (𝑛 ∈
(1...(⌊‘𝑥))
→ 𝑛 ∈
ℕ) |
| 28 | 13 | nfel1 2922 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ |
| 29 | 11 | eleq1d 2826 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑛 → (𝐴 ∈ ℂ ↔ ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ)) |
| 30 | 28, 29 | rspc 3610 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ →
(∀𝑘 ∈ ℕ
𝐴 ∈ ℂ →
⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ)) |
| 31 | 30 | impcom 407 |
. . . . . . . . 9
⊢
((∀𝑘 ∈
ℕ 𝐴 ∈ ℂ
∧ 𝑛 ∈ ℕ)
→ ⦋𝑛 /
𝑘⦌𝐴 ∈
ℂ) |
| 32 | 26, 27, 31 | syl2an 596 |
. . . . . . . 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 15769 |
. . . . . . 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 | eqeltrid 2845 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) →
Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴 ∈ ℂ) |
| 35 | | rpcn 13045 |
. . . . . . 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 13051 |
. . . . . . 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 12043 |
. . . . 5
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑥 ∈ ℝ+) →
(Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴 / 𝑥) ∈ ℂ) |
| 40 | | simplrl 777 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑐 ∈ (1[,)+∞)) |
| 41 | | 1re 11261 |
. . . . . . . 8
⊢ 1 ∈
ℝ |
| 42 | | elicopnf 13485 |
. . . . . . . 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 218 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → (𝑐 ∈ ℝ ∧ 1 ≤ 𝑐)) |
| 45 | 44 | simpld 494 |
. . . . 5
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑐 ∈ ℝ) |
| 46 | | fzfid 14014 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → (1...(⌊‘𝑐)) ∈ Fin) |
| 47 | 25 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → ∀𝑘 ∈ ℕ 𝐴 ∈ ℂ) |
| 48 | | elfznn 13593 |
. . . . . . . . 9
⊢ (𝑛 ∈
(1...(⌊‘𝑐))
→ 𝑛 ∈
ℕ) |
| 49 | 47, 48, 31 | syl2an 596 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑐))) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ) |
| 50 | 49 | abscld 15475 |
. . . . . . 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 15770 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → Σ𝑛 ∈ (1...(⌊‘𝑐))(abs‘⦋𝑛 / 𝑘⦌𝐴) ∈ ℝ) |
| 52 | | simplrr 778 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → 𝑚 ∈ ℝ) |
| 53 | 51, 52 | readdcld 11290 |
. . . . 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 15494 |
. . . . . . . 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 717 |
. . . . . . 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 13050 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℝ
∧ 0 ≤ 𝑥)) |
| 57 | 56 | ad2antrl 728 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) |
| 58 | | absid 15335 |
. . . . . . . . 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 7447 |
. . . . . . 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 2777 |
. . . . . 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 717 |
. . . . . . . . 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 15475 |
. . . . . . . 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 14014 |
. . . . . . . . 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 596 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ) |
| 66 | 65 | adantlr 715 |
. . . . . . . . . 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 15475 |
. . . . . . . . 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 15770 |
. . . . . . . 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 494 |
. . . . . . . . 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 11290 |
. . . . . . . . 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 11291 |
. . . . . . . 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 6909 |
. . . . . . . . 9
⊢
(abs‘Σ𝑘
∈ (1...(⌊‘𝑥))𝐴) = (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))⦋𝑛 / 𝑘⦌𝐴) |
| 75 | 64, 66 | fsumabs 15837 |
. . . . . . . . 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 | eqbrtrid 5178 |
. . . . . . . 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 14014 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin) |
| 78 | | ssun2 4179 |
. . . . . . . . . . . . . 14
⊢
(((⌊‘𝑐)
+ 1)...(⌊‘𝑥))
⊆ ((1...(⌊‘𝑐)) ∪ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) |
| 79 | | flge1nn 13861 |
. . . . . . . . . . . . . . . . . 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 12281 |
. . . . . . . . . . . . . . . . 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 13839 |
. . . . . . . . . . . . . . . . . 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 773 |
. . . . . . . . . . . . . . . . 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 11418 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑐) ≤ 𝑥) |
| 88 | | fznnfl 13902 |
. . . . . . . . . . . . . . . . 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 713 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑐) ∈ (1...(⌊‘𝑥))) |
| 91 | | fzsplit 13590 |
. . . . . . . . . . . . . . 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 | sseqtrrid 4027 |
. . . . . . . . . . . . 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 3983 |
. . . . . . . . . . . 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 15475 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(abs‘⦋𝑛
/ 𝑘⦌𝐴) ∈
ℝ) |
| 96 | 95 | adantlr 715 |
. . . . . . . . . . . 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 591 |
. . . . . . . . . . 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 15770 |
. . . . . . . . . 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 11291 |
. . . . . . . . . 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 11291 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (𝑥 · 𝑚) ∈ ℝ) |
| 101 | 70 | recnd 11289 |
. . . . . . . . . . . 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 | mullidd 11279 |
. . . . . . . . . . 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 11262 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → 1 ∈ ℝ) |
| 104 | 49 | absge0d 15483 |
. . . . . . . . . . . . . . 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 15831 |
. . . . . . . . . . . . . 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 511 |
. . . . . . . . . . . . 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 495 |
. . . . . . . . . . . . . 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 11418 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → 1 ≤ 𝑥) |
| 111 | | lemul1a 12121 |
. . . . . . . . . . . 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 1375 |
. . . . . . . . . . 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 5167 |
. . . . . . . . . 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 14395 |
. . . . . . . . . . . . 13
⊢
((((⌊‘𝑐)
+ 1)...(⌊‘𝑥))
∈ Fin → (♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ∈
ℕ0) |
| 115 | | nn0re 12535 |
. . . . . . . . . . . . 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 11291 |
. . . . . . . . . . 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 13560 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥)) → 𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))) |
| 120 | 81 | peano2nnd 12283 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → ((⌊‘𝑐) + 1) ∈ ℕ) |
| 121 | | eluznn 12960 |
. . . . . . . . . . . . . . . 16
⊢
((((⌊‘𝑐)
+ 1) ∈ ℕ ∧ 𝑛
∈ (ℤ≥‘((⌊‘𝑐) + 1))) → 𝑛 ∈ ℕ) |
| 122 | 120, 121 | sylan 580 |
. . . . . . . . . . . . . . 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 776 |
. . . . . . . . . . . . . . 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 13836 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑐 ∈ ℝ →
(⌊‘𝑐) ∈
ℝ) |
| 126 | | peano2re 11434 |
. . . . . . . . . . . . . . . . 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 12281 |
. . . . . . . . . . . . . . . 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 13841 |
. . . . . . . . . . . . . . . . 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 12891 |
. . . . . . . . . . . . . . . . 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 11418 |
. . . . . . . . . . . . . . 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 1914 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘 𝑐 ≤ 𝑛 |
| 135 | | nfcv 2905 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘abs |
| 136 | 135, 13 | nffv 6916 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘(abs‘⦋𝑛 / 𝑘⦌𝐴) |
| 137 | | nfcv 2905 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘
≤ |
| 138 | | nfcv 2905 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘𝑚 |
| 139 | 136, 137,
138 | nfbr 5190 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘(abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚 |
| 140 | 134, 139 | nfim 1896 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑘(𝑐 ≤ 𝑛 → (abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚) |
| 141 | | breq2 5147 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑛 → (𝑐 ≤ 𝑘 ↔ 𝑐 ≤ 𝑛)) |
| 142 | 11 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑛 → (abs‘𝐴) = (abs‘⦋𝑛 / 𝑘⦌𝐴)) |
| 143 | 142 | breq1d 5153 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑛 → ((abs‘𝐴) ≤ 𝑚 ↔ (abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚)) |
| 144 | 141, 143 | imbi12d 344 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑛 → ((𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚) ↔ (𝑐 ≤ 𝑛 → (abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚))) |
| 145 | 140, 144 | rspc 3610 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ →
(∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚) → (𝑐 ≤ 𝑛 → (abs‘⦋𝑛 / 𝑘⦌𝐴) ≤ 𝑚))) |
| 146 | 122, 123,
133, 145 | syl3c 66 |
. . . . . . . . . . . . . 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 593 |
. . . . . . . . . . . . 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 15835 |
. . . . . . . . . . . 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 11289 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → 𝑚 ∈ ℂ) |
| 150 | | fsumconst 15826 |
. . . . . . . . . . . . 13
⊢
(((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ∈ Fin ∧ 𝑚 ∈ ℂ) → Σ𝑛 ∈ (((⌊‘𝑐) + 1)...(⌊‘𝑥))𝑚 = ((♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚)) |
| 151 | 77, 149, 150 | syl2anc 584 |
. . . . . . . . . . . 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 5169 |
. . . . . . . . . . 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 | | biidd 262 |
. . . . . . . . . . . . 13
⊢ (𝑛 = ((⌊‘𝑐) + 1) → (0 ≤ 𝑚 ↔ 0 ≤ 𝑚)) |
| 154 | | 0red 11264 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))) → 0 ∈
ℝ) |
| 155 | 47, 30 | mpan9 506 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ 𝑛 ∈ ℕ) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ) |
| 156 | 155 | adantlr 715 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈ ℕ) → ⦋𝑛 / 𝑘⦌𝐴 ∈ ℂ) |
| 157 | 122, 156 | syldan 591 |
. . . . . . . . . . . . . . . 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) |
| 158 | 157 | abscld 15475 |
. . . . . . . . . . . . . . 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) |
| 159 | 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) |
| 160 | 157 | absge0d 15483 |
. . . . . . . . . . . . . . 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)) |
| 161 | 154, 158,
159, 160, 146 | letrd 11418 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧
𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) ∧ 𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))) → 0 ≤ 𝑚) |
| 162 | 161 | ralrimiva 3146 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → ∀𝑛 ∈
(ℤ≥‘((⌊‘𝑐) + 1))0 ≤ 𝑚) |
| 163 | 120 | nnzd 12640 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → ((⌊‘𝑐) + 1) ∈ ℤ) |
| 164 | | uzid 12893 |
. . . . . . . . . . . . . 14
⊢
(((⌊‘𝑐)
+ 1) ∈ ℤ → ((⌊‘𝑐) + 1) ∈
(ℤ≥‘((⌊‘𝑐) + 1))) |
| 165 | 163, 164 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → ((⌊‘𝑐) + 1) ∈
(ℤ≥‘((⌊‘𝑐) + 1))) |
| 166 | 153, 162,
165 | rspcdva 3623 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → 0 ≤ 𝑚) |
| 167 | | reflcl 13836 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℝ →
(⌊‘𝑥) ∈
ℝ) |
| 168 | 69, 167 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑥) ∈ ℝ) |
| 169 | | ssdomg 9040 |
. . . . . . . . . . . . . . . 16
⊢
((1...(⌊‘𝑥)) ∈ Fin → ((((⌊‘𝑐) + 1)...(⌊‘𝑥)) ⊆
(1...(⌊‘𝑥))
→ (((⌊‘𝑐)
+ 1)...(⌊‘𝑥))
≼ (1...(⌊‘𝑥)))) |
| 170 | 64, 93, 169 | sylc 65 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (((⌊‘𝑐) + 1)...(⌊‘𝑥)) ≼ (1...(⌊‘𝑥))) |
| 171 | | hashdomi 14419 |
. . . . . . . . . . . . . . 15
⊢
((((⌊‘𝑐)
+ 1)...(⌊‘𝑥))
≼ (1...(⌊‘𝑥)) →
(♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤
(♯‘(1...(⌊‘𝑥)))) |
| 172 | 170, 171 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) →
(♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤
(♯‘(1...(⌊‘𝑥)))) |
| 173 | | flge0nn0 13860 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 0 ≤
𝑥) →
(⌊‘𝑥) ∈
ℕ0) |
| 174 | | hashfz1 14385 |
. . . . . . . . . . . . . . 15
⊢
((⌊‘𝑥)
∈ ℕ0 → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥)) |
| 175 | 57, 173, 174 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) →
(♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥)) |
| 176 | 172, 175 | breqtrd 5169 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) →
(♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ (⌊‘𝑥)) |
| 177 | | flle 13839 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℝ →
(⌊‘𝑥) ≤
𝑥) |
| 178 | 69, 177 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (⌊‘𝑥) ≤ 𝑥) |
| 179 | 116, 168,
69, 176, 178 | letrd 11418 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) →
(♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) ≤ 𝑥) |
| 180 | 116, 69, 71, 166, 179 | lemul1ad 12207 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) →
((♯‘(((⌊‘𝑐) + 1)...(⌊‘𝑥))) · 𝑚) ≤ (𝑥 · 𝑚)) |
| 181 | 98, 117, 100, 152, 180 | letrd 11418 |
. . . . . . . . . 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)) |
| 182 | 70, 98, 99, 100, 113, 181 | le2addd 11882 |
. . . . . . . . 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))) |
| 183 | | ltp1 12107 |
. . . . . . . . . . 11
⊢
((⌊‘𝑐)
∈ ℝ → (⌊‘𝑐) < ((⌊‘𝑐) + 1)) |
| 184 | | fzdisj 13591 |
. . . . . . . . . . 11
⊢
((⌊‘𝑐)
< ((⌊‘𝑐) +
1) → ((1...(⌊‘𝑐)) ∩ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) = ∅) |
| 185 | 82, 183, 184 | 3syl 18 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → ((1...(⌊‘𝑐)) ∩ (((⌊‘𝑐) + 1)...(⌊‘𝑥))) = ∅) |
| 186 | 96 | recnd 11289 |
. . . . . . . . . 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) |
| 187 | 185, 92, 64, 186 | fsumsplit 15777 |
. . . . . . . . 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))) |
| 188 | 36 | adantrr 717 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → 𝑥 ∈ ℂ) |
| 189 | 188, 101,
149 | adddid 11285 |
. . . . . . . . 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))) |
| 190 | 182, 187,
189 | 3brtr4d 5175 |
. . . . . . . 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))) |
| 191 | 63, 68, 73, 76, 190 | letrd 11418 |
. . . . . . 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))) |
| 192 | | rpregt0 13049 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℝ
∧ 0 < 𝑥)) |
| 193 | 192 | ad2antrl 728 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) ∧ (𝑥 ∈ ℝ+ ∧ 𝑐 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 < 𝑥)) |
| 194 | | ledivmul 12144 |
. . . . . . . 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)))) |
| 195 | 63, 72, 193, 194 | syl3anc 1373 |
. . . . . . 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)))) |
| 196 | 191, 195 | mpbird 257 |
. . . . . 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)) |
| 197 | 61, 196 | eqbrtrd 5165 |
. . . . 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)) |
| 198 | 10, 39, 45, 53, 197 | elo1d 15572 |
. . . 4
⊢ (((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) ∧
∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚)) → (𝑥 ∈ ℝ+ ↦
(Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1)) |
| 199 | 198 | ex 412 |
. . 3
⊢ ((𝜑 ∧ (𝑐 ∈ (1[,)+∞) ∧ 𝑚 ∈ ℝ)) →
(∀𝑘 ∈ ℕ
(𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚) → (𝑥 ∈ ℝ+ ↦
(Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1))) |
| 200 | 199 | rexlimdvva 3213 |
. 2
⊢ (𝜑 → (∃𝑐 ∈ (1[,)+∞)∃𝑚 ∈ ℝ ∀𝑘 ∈ ℕ (𝑐 ≤ 𝑘 → (abs‘𝐴) ≤ 𝑚) → (𝑥 ∈ ℝ+ ↦
(Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1))) |
| 201 | 8, 200 | mpd 15 |
1
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦
(Σ𝑘 ∈
(1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1)) |