| Step | Hyp | Ref
| Expression |
| 1 | | caurcvgr.1 |
. . . . 5
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
| 2 | | caurcvgr.2 |
. . . . 5
⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
| 3 | | caurcvgr.3 |
. . . . 5
⊢ (𝜑 → sup(𝐴, ℝ*, < ) =
+∞) |
| 4 | | caurcvgr.4 |
. . . . 5
⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) |
| 5 | | 1rp 13038 |
. . . . . 6
⊢ 1 ∈
ℝ+ |
| 6 | 5 | a1i 11 |
. . . . 5
⊢ (𝜑 → 1 ∈
ℝ+) |
| 7 | 1, 2, 3, 4, 6 | caucvgrlem 15709 |
. . . 4
⊢ (𝜑 → ∃𝑗 ∈ 𝐴 ((lim sup‘𝐹) ∈ ℝ ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · 1)))) |
| 8 | | simpl 482 |
. . . . 5
⊢ (((lim
sup‘𝐹) ∈ ℝ
∧ ∀𝑘 ∈
𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · 1))) → (lim
sup‘𝐹) ∈
ℝ) |
| 9 | 8 | rexlimivw 3151 |
. . . 4
⊢
(∃𝑗 ∈
𝐴 ((lim sup‘𝐹) ∈ ℝ ∧
∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · 1))) → (lim
sup‘𝐹) ∈
ℝ) |
| 10 | 7, 9 | syl 17 |
. . 3
⊢ (𝜑 → (lim sup‘𝐹) ∈
ℝ) |
| 11 | 10 | recnd 11289 |
. 2
⊢ (𝜑 → (lim sup‘𝐹) ∈
ℂ) |
| 12 | 1 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝐴 ⊆
ℝ) |
| 13 | 2 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝐹:𝐴⟶ℝ) |
| 14 | 3 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → sup(𝐴, ℝ*, < ) =
+∞) |
| 15 | 4 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) →
∀𝑥 ∈
ℝ+ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) |
| 16 | | simpr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈
ℝ+) |
| 17 | | 3rp 13040 |
. . . . . . . 8
⊢ 3 ∈
ℝ+ |
| 18 | | rpdivcl 13060 |
. . . . . . . 8
⊢ ((𝑦 ∈ ℝ+
∧ 3 ∈ ℝ+) → (𝑦 / 3) ∈
ℝ+) |
| 19 | 16, 17, 18 | sylancl 586 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (𝑦 / 3) ∈
ℝ+) |
| 20 | 12, 13, 14, 15, 19 | caucvgrlem 15709 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) →
∃𝑗 ∈ 𝐴 ((lim sup‘𝐹) ∈ ℝ ∧
∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · (𝑦 / 3))))) |
| 21 | | simpr 484 |
. . . . . . 7
⊢ (((lim
sup‘𝐹) ∈ ℝ
∧ ∀𝑘 ∈
𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · (𝑦 / 3)))) → ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · (𝑦 / 3)))) |
| 22 | 21 | reximi 3084 |
. . . . . 6
⊢
(∃𝑗 ∈
𝐴 ((lim sup‘𝐹) ∈ ℝ ∧
∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · (𝑦 / 3)))) → ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · (𝑦 / 3)))) |
| 23 | 20, 22 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) →
∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · (𝑦 / 3)))) |
| 24 | | ssrexv 4053 |
. . . . 5
⊢ (𝐴 ⊆ ℝ →
(∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · (𝑦 / 3))) → ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · (𝑦 / 3))))) |
| 25 | 12, 23, 24 | sylc 65 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) →
∃𝑗 ∈ ℝ
∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · (𝑦 / 3)))) |
| 26 | | rpcn 13045 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℝ+
→ 𝑦 ∈
ℂ) |
| 27 | 26 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈
ℂ) |
| 28 | | 3cn 12347 |
. . . . . . . . 9
⊢ 3 ∈
ℂ |
| 29 | 28 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 3 ∈
ℂ) |
| 30 | | 3ne0 12372 |
. . . . . . . . 9
⊢ 3 ≠
0 |
| 31 | 30 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 3 ≠
0) |
| 32 | 27, 29, 31 | divcan2d 12045 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (3
· (𝑦 / 3)) = 𝑦) |
| 33 | 32 | breq2d 5155 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) →
((abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · (𝑦 / 3)) ↔ (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑦)) |
| 34 | 33 | imbi2d 340 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ((𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · (𝑦 / 3))) ↔ (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑦))) |
| 35 | 34 | rexralbidv 3223 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) →
(∃𝑗 ∈ ℝ
∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · (𝑦 / 3))) ↔ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑦))) |
| 36 | 25, 35 | mpbid 232 |
. . 3
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) →
∃𝑗 ∈ ℝ
∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑦)) |
| 37 | 36 | ralrimiva 3146 |
. 2
⊢ (𝜑 → ∀𝑦 ∈ ℝ+ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑦)) |
| 38 | | ax-resscn 11212 |
. . . 4
⊢ ℝ
⊆ ℂ |
| 39 | | fss 6752 |
. . . 4
⊢ ((𝐹:𝐴⟶ℝ ∧ ℝ ⊆
ℂ) → 𝐹:𝐴⟶ℂ) |
| 40 | 2, 38, 39 | sylancl 586 |
. . 3
⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
| 41 | | eqidd 2738 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) = (𝐹‘𝑘)) |
| 42 | 40, 1, 41 | rlim 15531 |
. 2
⊢ (𝜑 → (𝐹 ⇝𝑟 (lim
sup‘𝐹) ↔ ((lim
sup‘𝐹) ∈ ℂ
∧ ∀𝑦 ∈
ℝ+ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑦)))) |
| 43 | 11, 37, 42 | mpbir2and 713 |
1
⊢ (𝜑 → 𝐹 ⇝𝑟 (lim
sup‘𝐹)) |