Step | Hyp | Ref
| Expression |
1 | | ulmrel 23936 |
. . . 4
⊢ Rel
(⇝𝑢‘𝑆) |
2 | | brrelex12 5079 |
. . . 4
⊢ ((Rel
(⇝𝑢‘𝑆) ∧ 𝐹(⇝𝑢‘𝑆)𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V)) |
3 | 1, 2 | mpan 702 |
. . 3
⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V)) |
4 | 3 | a1i 11 |
. 2
⊢ (𝑆 ∈ 𝑉 → (𝐹(⇝𝑢‘𝑆)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))) |
5 | | 3simpa 1051 |
. . . 4
⊢ ((𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) → (𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ)) |
6 | | fvex 6113 |
. . . . . . 7
⊢
(ℤ≥‘𝑛) ∈ V |
7 | | fex 6394 |
. . . . . . 7
⊢ ((𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ (ℤ≥‘𝑛) ∈ V) → 𝐹 ∈ V) |
8 | 6, 7 | mpan2 703 |
. . . . . 6
⊢ (𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) → 𝐹 ∈ V) |
9 | 8 | a1i 11 |
. . . . 5
⊢ (𝑆 ∈ 𝑉 → (𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) → 𝐹 ∈ V)) |
10 | | fex 6394 |
. . . . . 6
⊢ ((𝐺:𝑆⟶ℂ ∧ 𝑆 ∈ 𝑉) → 𝐺 ∈ V) |
11 | 10 | expcom 450 |
. . . . 5
⊢ (𝑆 ∈ 𝑉 → (𝐺:𝑆⟶ℂ → 𝐺 ∈ V)) |
12 | 9, 11 | anim12d 584 |
. . . 4
⊢ (𝑆 ∈ 𝑉 → ((𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ) → (𝐹 ∈ V ∧ 𝐺 ∈ V))) |
13 | 5, 12 | syl5 33 |
. . 3
⊢ (𝑆 ∈ 𝑉 → ((𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) → (𝐹 ∈ V ∧ 𝐺 ∈ V))) |
14 | 13 | rexlimdvw 3016 |
. 2
⊢ (𝑆 ∈ 𝑉 → (∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) → (𝐹 ∈ V ∧ 𝐺 ∈ V))) |
15 | | elex 3185 |
. . . . . 6
⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) |
16 | | simpr1 1060 |
. . . . . . . . . . . . 13
⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → 𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆)) |
17 | | uzssz 11583 |
. . . . . . . . . . . . 13
⊢
(ℤ≥‘𝑛) ⊆ ℤ |
18 | | ovex 6577 |
. . . . . . . . . . . . . 14
⊢ (ℂ
↑𝑚 𝑆) ∈ V |
19 | | zex 11263 |
. . . . . . . . . . . . . 14
⊢ ℤ
∈ V |
20 | | elpm2r 7761 |
. . . . . . . . . . . . . 14
⊢
((((ℂ ↑𝑚 𝑆) ∈ V ∧ ℤ ∈ V) ∧
(𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ (ℤ≥‘𝑛) ⊆ ℤ)) → 𝑓 ∈ ((ℂ
↑𝑚 𝑆) ↑pm
ℤ)) |
21 | 18, 19, 20 | mpanl12 714 |
. . . . . . . . . . . . 13
⊢ ((𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ (ℤ≥‘𝑛) ⊆ ℤ) → 𝑓 ∈ ((ℂ
↑𝑚 𝑆) ↑pm
ℤ)) |
22 | 16, 17, 21 | sylancl 693 |
. . . . . . . . . . . 12
⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → 𝑓 ∈ ((ℂ ↑𝑚
𝑆)
↑pm ℤ)) |
23 | | simpr2 1061 |
. . . . . . . . . . . . 13
⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → 𝑦:𝑆⟶ℂ) |
24 | | cnex 9896 |
. . . . . . . . . . . . . 14
⊢ ℂ
∈ V |
25 | | simpl 472 |
. . . . . . . . . . . . . 14
⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → 𝑆 ∈ 𝑉) |
26 | | elmapg 7757 |
. . . . . . . . . . . . . 14
⊢ ((ℂ
∈ V ∧ 𝑆 ∈
𝑉) → (𝑦 ∈ (ℂ
↑𝑚 𝑆) ↔ 𝑦:𝑆⟶ℂ)) |
27 | 24, 25, 26 | sylancr 694 |
. . . . . . . . . . . . 13
⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → (𝑦 ∈ (ℂ ↑𝑚
𝑆) ↔ 𝑦:𝑆⟶ℂ)) |
28 | 23, 27 | mpbird 246 |
. . . . . . . . . . . 12
⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → 𝑦 ∈ (ℂ ↑𝑚
𝑆)) |
29 | 22, 28 | jca 553 |
. . . . . . . . . . 11
⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → (𝑓 ∈ ((ℂ ↑𝑚
𝑆)
↑pm ℤ) ∧ 𝑦 ∈ (ℂ ↑𝑚
𝑆))) |
30 | 29 | ex 449 |
. . . . . . . . . 10
⊢ (𝑆 ∈ 𝑉 → ((𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥) → (𝑓 ∈ ((ℂ ↑𝑚
𝑆)
↑pm ℤ) ∧ 𝑦 ∈ (ℂ ↑𝑚
𝑆)))) |
31 | 30 | rexlimdvw 3016 |
. . . . . . . . 9
⊢ (𝑆 ∈ 𝑉 → (∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥) → (𝑓 ∈ ((ℂ ↑𝑚
𝑆)
↑pm ℤ) ∧ 𝑦 ∈ (ℂ ↑𝑚
𝑆)))) |
32 | 31 | ssopab2dv 4929 |
. . . . . . . 8
⊢ (𝑆 ∈ 𝑉 → {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} ⊆ {〈𝑓, 𝑦〉 ∣ (𝑓 ∈ ((ℂ ↑𝑚
𝑆)
↑pm ℤ) ∧ 𝑦 ∈ (ℂ ↑𝑚
𝑆))}) |
33 | | df-xp 5044 |
. . . . . . . 8
⊢
(((ℂ ↑𝑚 𝑆) ↑pm ℤ)
× (ℂ ↑𝑚 𝑆)) = {〈𝑓, 𝑦〉 ∣ (𝑓 ∈ ((ℂ ↑𝑚
𝑆)
↑pm ℤ) ∧ 𝑦 ∈ (ℂ ↑𝑚
𝑆))} |
34 | 32, 33 | syl6sseqr 3615 |
. . . . . . 7
⊢ (𝑆 ∈ 𝑉 → {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} ⊆ (((ℂ
↑𝑚 𝑆) ↑pm ℤ)
× (ℂ ↑𝑚 𝑆))) |
35 | | ovex 6577 |
. . . . . . . . 9
⊢ ((ℂ
↑𝑚 𝑆) ↑pm ℤ) ∈
V |
36 | 35, 18 | xpex 6860 |
. . . . . . . 8
⊢
(((ℂ ↑𝑚 𝑆) ↑pm ℤ)
× (ℂ ↑𝑚 𝑆)) ∈ V |
37 | 36 | ssex 4730 |
. . . . . . 7
⊢
({〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} ⊆ (((ℂ
↑𝑚 𝑆) ↑pm ℤ)
× (ℂ ↑𝑚 𝑆)) → {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} ∈ V) |
38 | 34, 37 | syl 17 |
. . . . . 6
⊢ (𝑆 ∈ 𝑉 → {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} ∈ V) |
39 | | oveq2 6557 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑆 → (ℂ ↑𝑚
𝑠) = (ℂ
↑𝑚 𝑆)) |
40 | 39 | feq3d 5945 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑆 → (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑠) ↔ 𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆))) |
41 | | feq2 5940 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑆 → (𝑦:𝑠⟶ℂ ↔ 𝑦:𝑆⟶ℂ)) |
42 | | raleq 3115 |
. . . . . . . . . . . 12
⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) |
43 | 42 | rexralbidv 3040 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑆 → (∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) |
44 | 43 | ralbidv 2969 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) |
45 | 40, 41, 44 | 3anbi123d 1391 |
. . . . . . . . 9
⊢ (𝑠 = 𝑆 → ((𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥) ↔ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥))) |
46 | 45 | rexbidv 3034 |
. . . . . . . 8
⊢ (𝑠 = 𝑆 → (∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥) ↔ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥))) |
47 | 46 | opabbidv 4648 |
. . . . . . 7
⊢ (𝑠 = 𝑆 → {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} = {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)}) |
48 | | df-ulm 23935 |
. . . . . . 7
⊢
⇝𝑢 = (𝑠 ∈ V ↦ {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)}) |
49 | 47, 48 | fvmptg 6189 |
. . . . . 6
⊢ ((𝑆 ∈ V ∧ {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} ∈ V) →
(⇝𝑢‘𝑆) = {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)}) |
50 | 15, 38, 49 | syl2anc 691 |
. . . . 5
⊢ (𝑆 ∈ 𝑉 →
(⇝𝑢‘𝑆) = {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)}) |
51 | 50 | breqd 4594 |
. . . 4
⊢ (𝑆 ∈ 𝑉 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ 𝐹{〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)}𝐺)) |
52 | | simpl 472 |
. . . . . . . 8
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → 𝑓 = 𝐹) |
53 | 52 | feq1d 5943 |
. . . . . . 7
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ↔ 𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆))) |
54 | | simpr 476 |
. . . . . . . 8
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → 𝑦 = 𝐺) |
55 | 54 | feq1d 5943 |
. . . . . . 7
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (𝑦:𝑆⟶ℂ ↔ 𝐺:𝑆⟶ℂ)) |
56 | 52 | fveq1d 6105 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (𝑓‘𝑘) = (𝐹‘𝑘)) |
57 | 56 | fveq1d 6105 |
. . . . . . . . . . . . 13
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → ((𝑓‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑧)) |
58 | 54 | fveq1d 6105 |
. . . . . . . . . . . . 13
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (𝑦‘𝑧) = (𝐺‘𝑧)) |
59 | 57, 58 | oveq12d 6567 |
. . . . . . . . . . . 12
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧)) = (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) |
60 | 59 | fveq2d 6107 |
. . . . . . . . . . 11
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) = (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))) |
61 | 60 | breq1d 4593 |
. . . . . . . . . 10
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → ((abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) |
62 | 61 | ralbidv 2969 |
. . . . . . . . 9
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) |
63 | 62 | rexralbidv 3040 |
. . . . . . . 8
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) |
64 | 63 | ralbidv 2969 |
. . . . . . 7
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) |
65 | 53, 55, 64 | 3anbi123d 1391 |
. . . . . 6
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → ((𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥) ↔ (𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))) |
66 | 65 | rexbidv 3034 |
. . . . 5
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥) ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))) |
67 | | eqid 2610 |
. . . . 5
⊢
{〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} = {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} |
68 | 66, 67 | brabga 4914 |
. . . 4
⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹{〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)}𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))) |
69 | 51, 68 | sylan9bb 732 |
. . 3
⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ V ∧ 𝐺 ∈ V)) → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))) |
70 | 69 | ex 449 |
. 2
⊢ (𝑆 ∈ 𝑉 → ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)))) |
71 | 4, 14, 70 | pm5.21ndd 368 |
1
⊢ (𝑆 ∈ 𝑉 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))) |