| Step | Hyp | Ref
| Expression |
| 1 | | df-cau 25290 |
. 2
⊢ 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 | | dmeq 5914 |
. . . . . 6
⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷) |
| 3 | 2 | dmeqd 5916 |
. . . . 5
⊢ (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷) |
| 4 | | xmetf 24339 |
. . . . . . . 8
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) |
| 5 | 4 | fdmd 6746 |
. . . . . . 7
⊢ (𝐷 ∈ (∞Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋)) |
| 6 | 5 | dmeqd 5916 |
. . . . . 6
⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = dom (𝑋 × 𝑋)) |
| 7 | | dmxpid 5941 |
. . . . . 6
⊢ dom
(𝑋 × 𝑋) = 𝑋 |
| 8 | 6, 7 | eqtrdi 2793 |
. . . . 5
⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = 𝑋) |
| 9 | 3, 8 | sylan9eqr 2799 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋) |
| 10 | 9 | oveq1d 7446 |
. . 3
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (dom dom 𝑑 ↑pm ℂ) = (𝑋 ↑pm
ℂ)) |
| 11 | | simpr 484 |
. . . . . . . 8
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷) |
| 12 | 11 | fveq2d 6910 |
. . . . . . 7
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (ball‘𝑑) = (ball‘𝐷)) |
| 13 | 12 | oveqd 7448 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ((𝑓‘𝑘)(ball‘𝑑)𝑥) = ((𝑓‘𝑘)(ball‘𝐷)𝑥)) |
| 14 | 13 | feq3d 6723 |
. . . . 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))) |
| 15 | 14 | rexbidv 3179 |
. . . 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))) |
| 16 | 15 | ralbidv 3178 |
. . 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))) |
| 17 | 10, 16 | rabeqbidv 3455 |
. 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)}) |
| 18 | | fvssunirn 6939 |
. . 3
⊢
(∞Met‘𝑋)
⊆ ∪ ran ∞Met |
| 19 | 18 | sseli 3979 |
. 2
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran
∞Met) |
| 20 | | ovex 7464 |
. . . 4
⊢ (𝑋 ↑pm ℂ)
∈ V |
| 21 | 20 | rabex 5339 |
. . 3
⊢ {𝑓 ∈ (𝑋 ↑pm ℂ) ∣
∀𝑥 ∈
ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑥)} ∈ V |
| 22 | 21 | 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) |
| 23 | 1, 17, 19, 22 | fvmptd2 7024 |
1
⊢ (𝐷 ∈ (∞Met‘𝑋) → (Cau‘𝐷) = {𝑓 ∈ (𝑋 ↑pm ℂ) ∣
∀𝑥 ∈
ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑥)}) |