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 12620 |
. . . . . 6
⊢ 1 ∈
ℝ+ |
6 | 5 | a1i 11 |
. . . . 5
⊢ (𝜑 → 1 ∈
ℝ+) |
7 | 1, 2, 3, 4, 6 | caucvgrlem 15269 |
. . . 4
⊢ (𝜑 → ∃𝑗 ∈ 𝐴 ((lim sup‘𝐹) ∈ ℝ ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · 1)))) |
8 | | simpl 486 |
. . . . 5
⊢ (((lim
sup‘𝐹) ∈ ℝ
∧ ∀𝑘 ∈
𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · 1))) → (lim
sup‘𝐹) ∈
ℝ) |
9 | 8 | rexlimivw 3211 |
. . . 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 10891 |
. 2
⊢ (𝜑 → (lim sup‘𝐹) ∈
ℂ) |
12 | 1 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝐴 ⊆
ℝ) |
13 | 2 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝐹:𝐴⟶ℝ) |
14 | 3 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → sup(𝐴, ℝ*, < ) =
+∞) |
15 | 4 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) →
∀𝑥 ∈
ℝ+ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) |
16 | | simpr 488 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈
ℝ+) |
17 | | 3rp 12622 |
. . . . . . . 8
⊢ 3 ∈
ℝ+ |
18 | | rpdivcl 12641 |
. . . . . . . 8
⊢ ((𝑦 ∈ ℝ+
∧ 3 ∈ ℝ+) → (𝑦 / 3) ∈
ℝ+) |
19 | 16, 17, 18 | sylancl 589 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (𝑦 / 3) ∈
ℝ+) |
20 | 12, 13, 14, 15, 19 | caucvgrlem 15269 |
. . . . . 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 488 |
. . . . . . 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 3175 |
. . . . . 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 3985 |
. . . . 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 12626 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℝ+
→ 𝑦 ∈
ℂ) |
27 | 26 | adantl 485 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈
ℂ) |
28 | | 3cn 11941 |
. . . . . . . . 9
⊢ 3 ∈
ℂ |
29 | 28 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 3 ∈
ℂ) |
30 | | 3ne0 11966 |
. . . . . . . . 9
⊢ 3 ≠
0 |
31 | 30 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 3 ≠
0) |
32 | 27, 29, 31 | divcan2d 11640 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (3
· (𝑦 / 3)) = 𝑦) |
33 | 32 | breq2d 5082 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) →
((abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · (𝑦 / 3)) ↔ (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑦)) |
34 | 33 | imbi2d 344 |
. . . . 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 3230 |
. . . 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 235 |
. . 3
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) →
∃𝑗 ∈ ℝ
∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑦)) |
37 | 36 | ralrimiva 3108 |
. 2
⊢ (𝜑 → ∀𝑦 ∈ ℝ+ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑦)) |
38 | | ax-resscn 10816 |
. . . 4
⊢ ℝ
⊆ ℂ |
39 | | fss 6584 |
. . . 4
⊢ ((𝐹:𝐴⟶ℝ ∧ ℝ ⊆
ℂ) → 𝐹:𝐴⟶ℂ) |
40 | 2, 38, 39 | sylancl 589 |
. . 3
⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
41 | | eqidd 2740 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) = (𝐹‘𝑘)) |
42 | 40, 1, 41 | rlim 15089 |
. 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)) |