Step | Hyp | Ref
| Expression |
1 | | rlimrel 14952 |
. . . . 5
⊢ Rel
⇝𝑟 |
2 | 1 | brrelex2i 5590 |
. . . 4
⊢ (𝐹 ⇝𝑟
𝐶 → 𝐶 ∈ V) |
3 | 2 | a1i 11 |
. . 3
⊢ (𝜑 → (𝐹 ⇝𝑟 𝐶 → 𝐶 ∈ V)) |
4 | | elex 3418 |
. . . . 5
⊢ (𝐶 ∈ ℂ → 𝐶 ∈ V) |
5 | 4 | ad2antrl 728 |
. . . 4
⊢ ((𝐹 ∈ (ℂ
↑pm ℝ) ∧ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑦 ∈ ℝ
∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))) → 𝐶 ∈ V) |
6 | 5 | a1i 11 |
. . 3
⊢ (𝜑 → ((𝐹 ∈ (ℂ ↑pm
ℝ) ∧ (𝐶 ∈
ℂ ∧ ∀𝑥
∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))) → 𝐶 ∈ V)) |
7 | | rlim.1 |
. . . . 5
⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
8 | | rlim.2 |
. . . . 5
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
9 | | cnex 10708 |
. . . . . 6
⊢ ℂ
∈ V |
10 | | reex 10718 |
. . . . . 6
⊢ ℝ
∈ V |
11 | | elpm2r 8467 |
. . . . . 6
⊢
(((ℂ ∈ V ∧ ℝ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ)) → 𝐹 ∈ (ℂ ↑pm
ℝ)) |
12 | 9, 10, 11 | mpanl12 702 |
. . . . 5
⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → 𝐹 ∈ (ℂ ↑pm
ℝ)) |
13 | 7, 8, 12 | syl2anc 587 |
. . . 4
⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm
ℝ)) |
14 | | eleq1 2821 |
. . . . . . . . 9
⊢ (𝑓 = 𝐹 → (𝑓 ∈ (ℂ ↑pm ℝ)
↔ 𝐹 ∈ (ℂ
↑pm ℝ))) |
15 | | eleq1 2821 |
. . . . . . . . 9
⊢ (𝑤 = 𝐶 → (𝑤 ∈ ℂ ↔ 𝐶 ∈ ℂ)) |
16 | 14, 15 | bi2anan9 639 |
. . . . . . . 8
⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → ((𝑓 ∈ (ℂ ↑pm ℝ)
∧ 𝑤 ∈ ℂ)
↔ (𝐹 ∈ (ℂ
↑pm ℝ) ∧ 𝐶 ∈ ℂ))) |
17 | | simpl 486 |
. . . . . . . . . . . 12
⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → 𝑓 = 𝐹) |
18 | 17 | dmeqd 5758 |
. . . . . . . . . . 11
⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → dom 𝑓 = dom 𝐹) |
19 | | fveq1 6685 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = 𝐹 → (𝑓‘𝑧) = (𝐹‘𝑧)) |
20 | | oveq12 7191 |
. . . . . . . . . . . . . . 15
⊢ (((𝑓‘𝑧) = (𝐹‘𝑧) ∧ 𝑤 = 𝐶) → ((𝑓‘𝑧) − 𝑤) = ((𝐹‘𝑧) − 𝐶)) |
21 | 19, 20 | sylan 583 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → ((𝑓‘𝑧) − 𝑤) = ((𝐹‘𝑧) − 𝐶)) |
22 | 21 | fveq2d 6690 |
. . . . . . . . . . . . 13
⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → (abs‘((𝑓‘𝑧) − 𝑤)) = (abs‘((𝐹‘𝑧) − 𝐶))) |
23 | 22 | breq1d 5050 |
. . . . . . . . . . . 12
⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → ((abs‘((𝑓‘𝑧) − 𝑤)) < 𝑥 ↔ (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)) |
24 | 23 | imbi2d 344 |
. . . . . . . . . . 11
⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → ((𝑦 ≤ 𝑧 → (abs‘((𝑓‘𝑧) − 𝑤)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))) |
25 | 18, 24 | raleqbidv 3305 |
. . . . . . . . . 10
⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → (∀𝑧 ∈ dom 𝑓(𝑦 ≤ 𝑧 → (abs‘((𝑓‘𝑧) − 𝑤)) < 𝑥) ↔ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))) |
26 | 25 | rexbidv 3208 |
. . . . . . . . 9
⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝑓(𝑦 ≤ 𝑧 → (abs‘((𝑓‘𝑧) − 𝑤)) < 𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))) |
27 | 26 | ralbidv 3110 |
. . . . . . . 8
⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝑓(𝑦 ≤ 𝑧 → (abs‘((𝑓‘𝑧) − 𝑤)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))) |
28 | 16, 27 | anbi12d 634 |
. . . . . . 7
⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → (((𝑓 ∈ (ℂ ↑pm ℝ)
∧ 𝑤 ∈ ℂ)
∧ ∀𝑥 ∈
ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝑓(𝑦 ≤ 𝑧 → (abs‘((𝑓‘𝑧) − 𝑤)) < 𝑥)) ↔ ((𝐹 ∈ (ℂ ↑pm
ℝ) ∧ 𝐶 ∈
ℂ) ∧ ∀𝑥
∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))) |
29 | | df-rlim 14948 |
. . . . . . 7
⊢
⇝𝑟 = {〈𝑓, 𝑤〉 ∣ ((𝑓 ∈ (ℂ ↑pm ℝ)
∧ 𝑤 ∈ ℂ)
∧ ∀𝑥 ∈
ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝑓(𝑦 ≤ 𝑧 → (abs‘((𝑓‘𝑧) − 𝑤)) < 𝑥))} |
30 | 28, 29 | brabga 5399 |
. . . . . 6
⊢ ((𝐹 ∈ (ℂ
↑pm ℝ) ∧ 𝐶 ∈ V) → (𝐹 ⇝𝑟 𝐶 ↔ ((𝐹 ∈ (ℂ ↑pm
ℝ) ∧ 𝐶 ∈
ℂ) ∧ ∀𝑥
∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))) |
31 | | anass 472 |
. . . . . 6
⊢ (((𝐹 ∈ (ℂ
↑pm ℝ) ∧ 𝐶 ∈ ℂ) ∧ ∀𝑥 ∈ ℝ+
∃𝑦 ∈ ℝ
∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)) ↔ (𝐹 ∈ (ℂ ↑pm
ℝ) ∧ (𝐶 ∈
ℂ ∧ ∀𝑥
∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))) |
32 | 30, 31 | bitrdi 290 |
. . . . 5
⊢ ((𝐹 ∈ (ℂ
↑pm ℝ) ∧ 𝐶 ∈ V) → (𝐹 ⇝𝑟 𝐶 ↔ (𝐹 ∈ (ℂ ↑pm
ℝ) ∧ (𝐶 ∈
ℂ ∧ ∀𝑥
∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))) |
33 | 32 | ex 416 |
. . . 4
⊢ (𝐹 ∈ (ℂ
↑pm ℝ) → (𝐶 ∈ V → (𝐹 ⇝𝑟 𝐶 ↔ (𝐹 ∈ (ℂ ↑pm
ℝ) ∧ (𝐶 ∈
ℂ ∧ ∀𝑥
∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))))) |
34 | 13, 33 | syl 17 |
. . 3
⊢ (𝜑 → (𝐶 ∈ V → (𝐹 ⇝𝑟 𝐶 ↔ (𝐹 ∈ (ℂ ↑pm
ℝ) ∧ (𝐶 ∈
ℂ ∧ ∀𝑥
∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))))) |
35 | 3, 6, 34 | pm5.21ndd 384 |
. 2
⊢ (𝜑 → (𝐹 ⇝𝑟 𝐶 ↔ (𝐹 ∈ (ℂ ↑pm
ℝ) ∧ (𝐶 ∈
ℂ ∧ ∀𝑥
∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))) |
36 | 13 | biantrurd 536 |
. 2
⊢ (𝜑 → ((𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑦 ∈ ℝ
∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)) ↔ (𝐹 ∈ (ℂ ↑pm
ℝ) ∧ (𝐶 ∈
ℂ ∧ ∀𝑥
∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))) |
37 | 7 | fdmd 6525 |
. . . . . . 7
⊢ (𝜑 → dom 𝐹 = 𝐴) |
38 | 37 | raleqdv 3317 |
. . . . . 6
⊢ (𝜑 → (∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))) |
39 | | rlim.4 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = 𝐵) |
40 | 39 | fvoveq1d 7204 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (abs‘((𝐹‘𝑧) − 𝐶)) = (abs‘(𝐵 − 𝐶))) |
41 | 40 | breq1d 5050 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥 ↔ (abs‘(𝐵 − 𝐶)) < 𝑥)) |
42 | 41 | imbi2d 344 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥))) |
43 | 42 | ralbidva 3109 |
. . . . . 6
⊢ (𝜑 → (∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥))) |
44 | 38, 43 | bitrd 282 |
. . . . 5
⊢ (𝜑 → (∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥))) |
45 | 44 | rexbidv 3208 |
. . . 4
⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥))) |
46 | 45 | ralbidv 3110 |
. . 3
⊢ (𝜑 → (∀𝑥 ∈ ℝ+
∃𝑦 ∈ ℝ
∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥))) |
47 | 46 | anbi2d 632 |
. 2
⊢ (𝜑 → ((𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑦 ∈ ℝ
∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑦 ∈ ℝ
∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))) |
48 | 35, 36, 47 | 3bitr2d 310 |
1
⊢ (𝜑 → (𝐹 ⇝𝑟 𝐶 ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑦 ∈ ℝ
∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))) |