Step | Hyp | Ref
| Expression |
1 | | df-cau 22862 |
. . 3
⊢ Cau =
(𝑑 ∈ ∪ ran ∞Met ↦ {𝑓 ∈ (dom dom 𝑑 ↑pm ℂ) ∣
∀𝑥 ∈
ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝑑)𝑥)}) |
2 | 1 | a1i 11 |
. 2
⊢ (𝐷 ∈ (∞Met‘𝑋) → Cau = (𝑑 ∈ ∪ ran ∞Met ↦ {𝑓 ∈ (dom dom 𝑑 ↑pm ℂ) ∣
∀𝑥 ∈
ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝑑)𝑥)})) |
3 | | dmeq 5246 |
. . . . . 6
⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷) |
4 | 3 | dmeqd 5248 |
. . . . 5
⊢ (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷) |
5 | | xmetf 21944 |
. . . . . . . 8
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) |
6 | | fdm 5964 |
. . . . . . . 8
⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom
𝐷 = (𝑋 × 𝑋)) |
7 | 5, 6 | syl 17 |
. . . . . . 7
⊢ (𝐷 ∈ (∞Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋)) |
8 | 7 | dmeqd 5248 |
. . . . . 6
⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = dom (𝑋 × 𝑋)) |
9 | | dmxpid 5266 |
. . . . . 6
⊢ dom
(𝑋 × 𝑋) = 𝑋 |
10 | 8, 9 | syl6eq 2660 |
. . . . 5
⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = 𝑋) |
11 | 4, 10 | sylan9eqr 2666 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋) |
12 | 11 | oveq1d 6564 |
. . 3
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (dom dom 𝑑 ↑pm ℂ) = (𝑋 ↑pm
ℂ)) |
13 | | simpr 476 |
. . . . . . . 8
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷) |
14 | 13 | fveq2d 6107 |
. . . . . . 7
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (ball‘𝑑) = (ball‘𝐷)) |
15 | 14 | oveqd 6566 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ((𝑓‘𝑘)(ball‘𝑑)𝑥) = ((𝑓‘𝑘)(ball‘𝐷)𝑥)) |
16 | 15 | feq3d 5945 |
. . . . 5
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ((𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝑑)𝑥) ↔ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑥))) |
17 | 16 | rexbidv 3034 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝑑)𝑥) ↔ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑥))) |
18 | 17 | ralbidv 2969 |
. . 3
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (∀𝑥 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾
(ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝑑)𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾
(ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑥))) |
19 | 12, 18 | rabeqbidv 3168 |
. 2
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → {𝑓 ∈ (dom dom 𝑑 ↑pm ℂ) ∣
∀𝑥 ∈
ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝑑)𝑥)} = {𝑓 ∈ (𝑋 ↑pm ℂ) ∣
∀𝑥 ∈
ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑥)}) |
20 | | fvssunirn 6127 |
. . 3
⊢
(∞Met‘𝑋)
⊆ ∪ ran ∞Met |
21 | 20 | sseli 3564 |
. 2
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran
∞Met) |
22 | | ovex 6577 |
. . . 4
⊢ (𝑋 ↑pm
ℂ) ∈ V |
23 | 22 | rabex 4740 |
. . 3
⊢ {𝑓 ∈ (𝑋 ↑pm ℂ) ∣
∀𝑥 ∈
ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑥)} ∈ V |
24 | 23 | a1i 11 |
. 2
⊢ (𝐷 ∈ (∞Met‘𝑋) → {𝑓 ∈ (𝑋 ↑pm ℂ) ∣
∀𝑥 ∈
ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑥)} ∈ V) |
25 | 2, 19, 21, 24 | fvmptd 6197 |
1
⊢ (𝐷 ∈ (∞Met‘𝑋) → (Cau‘𝐷) = {𝑓 ∈ (𝑋 ↑pm ℂ) ∣
∀𝑥 ∈
ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑥)}) |