Theorem: CantorScheme.VanishingDiam.map_continuous
File: Mathlib/Topology/MetricSpace/CantorScheme.lean

Proof steps:
state_before: β : Type u_1
α : Type u_2
A : List β → Set α
inst✝² : PseudoMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : DiscreteTopology β
hA : VanishingDiam A
⊢ Continuous (inducedMap A).snd
state_after: β : Type u_1
α : Type u_2
A : List β → Set α
inst✝² : PseudoMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : DiscreteTopology β
hA : VanishingDiam A
⊢ ∀ (a : ↑(inducedMap A).fst),
    ∀ ε > 0, ∀ᶠ (x : ↑(inducedMap A).fst) in 𝓝 a, dist ((inducedMap A).snd x) ((inducedMap A).snd a) < ε
tactic: rw [Metric.continuous_iff']
state_before: β : Type u_1
α : Type u_2
A : List β → Set α
inst✝² : PseudoMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : DiscreteTopology β
hA : VanishingDiam A
⊢ ∀ (a : ↑(inducedMap A).fst),
    ∀ ε > 0, ∀ᶠ (x : ↑(inducedMap A).fst) in 𝓝 a, dist ((inducedMap A).snd x) ((inducedMap A).snd a) < ε
state_after: case mk
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝² : PseudoMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : DiscreteTopology β
hA : VanishingDiam A
x : ℕ → β
hx : x ∈ (inducedMap A).fst
ε : ℝ
ε_pos : ε > 0
⊢ ∀ᶠ (x_1 : ↑(inducedMap A).fst) in 𝓝 ⟨x, hx⟩, dist ((inducedMap A).snd x_1) ((inducedMap A).snd ⟨x, hx⟩) < ε
tactic: rintro ⟨x, hx⟩ ε ε_pos
state_before: case mk
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝² : PseudoMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : DiscreteTopology β
hA : VanishingDiam A
x : ℕ → β
hx : x ∈ (inducedMap A).fst
ε : ℝ
ε_pos : ε > 0
⊢ ∀ᶠ (x_1 : ↑(inducedMap A).fst) in 𝓝 ⟨x, hx⟩, dist ((inducedMap A).snd x_1) ((inducedMap A).snd ⟨x, hx⟩) < ε
state_after: case mk.intro
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝² : PseudoMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : DiscreteTopology β
hA : VanishingDiam A
x : ℕ → β
hx : x ∈ (inducedMap A).fst
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
⊢ ∀ᶠ (x_1 : ↑(inducedMap A).fst) in 𝓝 ⟨x, hx⟩, dist ((inducedMap A).snd x_1) ((inducedMap A).snd ⟨x, hx⟩) < ε
tactic: cases' hA.dist_lt _ ε_pos x with n hn
state_before: case mk.intro
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝² : PseudoMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : DiscreteTopology β
hA : VanishingDiam A
x : ℕ → β
hx : x ∈ (inducedMap A).fst
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
⊢ ∀ᶠ (x_1 : ↑(inducedMap A).fst) in 𝓝 ⟨x, hx⟩, dist ((inducedMap A).snd x_1) ((inducedMap A).snd ⟨x, hx⟩) < ε
state_after: case mk.intro
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝² : PseudoMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : DiscreteTopology β
hA : VanishingDiam A
x : ℕ → β
hx : x ∈ (inducedMap A).fst
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
⊢ ∃ t, (∀ y ∈ t, dist ((inducedMap A).snd y) ((inducedMap A).snd ⟨x, hx⟩) < ε) ∧ IsOpen t ∧ ⟨x, hx⟩ ∈ t
tactic: rw [_root_.eventually_nhds_iff]
state_before: case mk.intro
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝² : PseudoMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : DiscreteTopology β
hA : VanishingDiam A
x : ℕ → β
hx : x ∈ (inducedMap A).fst
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
⊢ ∃ t, (∀ y ∈ t, dist ((inducedMap A).snd y) ((inducedMap A).snd ⟨x, hx⟩) < ε) ∧ IsOpen t ∧ ⟨x, hx⟩ ∈ t
state_after: case mk.intro.refine_1
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝² : PseudoMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : DiscreteTopology β
hA : VanishingDiam A
x : ℕ → β
hx : x ∈ (inducedMap A).fst
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
⊢ ∀ y ∈ Subtype.val ⁻¹' cylinder x n, dist ((inducedMap A).snd y) ((inducedMap A).snd ⟨x, hx⟩) < ε

case mk.intro.refine_2
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝² : PseudoMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : DiscreteTopology β
hA : VanishingDiam A
x : ℕ → β
hx : x ∈ (inducedMap A).fst
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
⊢ IsOpen (Subtype.val ⁻¹' cylinder x n)
tactic: refine ⟨(↑)⁻¹' cylinder x n, ?_, ?_, by simp⟩
state_before: case mk.intro.refine_2
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝² : PseudoMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : DiscreteTopology β
hA : VanishingDiam A
x : ℕ → β
hx : x ∈ (inducedMap A).fst
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
⊢ IsOpen (Subtype.val ⁻¹' cylinder x n)
state_after: case mk.intro.refine_2.a
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝² : PseudoMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : DiscreteTopology β
hA : VanishingDiam A
x : ℕ → β
hx : x ∈ (inducedMap A).fst
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
⊢ IsOpen (cylinder x n)
tactic: apply continuous_subtype_val.isOpen_preimage
state_before: case mk.intro.refine_2.a
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝² : PseudoMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : DiscreteTopology β
hA : VanishingDiam A
x : ℕ → β
hx : x ∈ (inducedMap A).fst
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
⊢ IsOpen (cylinder x n)
state_after: no goals
tactic: apply isOpen_cylinder