Theorem: Metric.equicontinuousAt_iff_pair
File: Mathlib/Topology/MetricSpace/Equicontinuity.lean

tactic: rw [equicontinuousAt_iff_pair]
state_before: α : Type u_1
β : Type u_2
ι✝ : Type u_3
inst✝¹ : PseudoMetricSpace α
ι : Type u_4
inst✝ : TopologicalSpace β
F : ι → β → α
x₀ : β
⊢ EquicontinuousAt F x₀ ↔ ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ι), dist (F i x) (F i x') < ε
state_after: α : Type u_1
β : Type u_2
ι✝ : Type u_3
inst✝¹ : PseudoMetricSpace α
ι : Type u_4
inst✝ : TopologicalSpace β
F : ι → β → α
x₀ : β
⊢ (∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝 x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι), (F i x, F i y) ∈ U) ↔
    ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ι), dist (F i x) (F i x') < ε

tactic: constructor <;> intro H
state_before: α : Type u_1
β : Type u_2
ι✝ : Type u_3
inst✝¹ : PseudoMetricSpace α
ι : Type u_4
inst✝ : TopologicalSpace β
F : ι → β → α
x₀ : β
⊢ (∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝 x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι), (F i x, F i y) ∈ U) ↔
    ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ι), dist (F i x) (F i x') < ε
state_after: case mp
α : Type u_1
β : Type u_2
ι✝ : Type u_3
inst✝¹ : PseudoMetricSpace α
ι : Type u_4
inst✝ : TopologicalSpace β
F : ι → β → α
x₀ : β
H : ∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝 x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι), (F i x, F i y) ∈ U
⊢ ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ι), dist (F i x) (F i x') < ε

case mpr
α : Type u_1
β : Type u_2
ι✝ : Type u_3
inst✝¹ : PseudoMetricSpace α
ι : Type u_4
inst✝ : TopologicalSpace β
F : ι → β → α
x₀ : β
H : ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ι), dist (F i x) (F i x') < ε
⊢ ∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝 x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι), (F i x, F i y) ∈ U

tactic: intro ε hε
state_before: case mp
α : Type u_1
β : Type u_2
ι✝ : Type u_3
inst✝¹ : PseudoMetricSpace α
ι : Type u_4
inst✝ : TopologicalSpace β
F : ι → β → α
x₀ : β
H : ∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝 x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι), (F i x, F i y) ∈ U
⊢ ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ι), dist (F i x) (F i x') < ε
state_after: case mp
α : Type u_1
β : Type u_2
ι✝ : Type u_3
inst✝¹ : PseudoMetricSpace α
ι : Type u_4
inst✝ : TopologicalSpace β
F : ι → β → α
x₀ : β
H : ∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝 x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι), (F i x, F i y) ∈ U
ε : ℝ
hε : ε > 0
⊢ ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ι), dist (F i x) (F i x') < ε

tactic: exact H _ (dist_mem_uniformity hε)
state_before: case mp
α : Type u_1
β : Type u_2
ι✝ : Type u_3
inst✝¹ : PseudoMetricSpace α
ι : Type u_4
inst✝ : TopologicalSpace β
F : ι → β → α
x₀ : β
H : ∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝 x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι), (F i x, F i y) ∈ U
ε : ℝ
hε : ε > 0
⊢ ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ι), dist (F i x) (F i x') < ε
state_after: no goals

tactic: intro U hU
state_before: case mpr
α : Type u_1
β : Type u_2
ι✝ : Type u_3
inst✝¹ : PseudoMetricSpace α
ι : Type u_4
inst✝ : TopologicalSpace β
F : ι → β → α
x₀ : β
H : ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ι), dist (F i x) (F i x') < ε
⊢ ∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝 x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι), (F i x, F i y) ∈ U
state_after: case mpr
α : Type u_1
β : Type u_2
ι✝ : Type u_3
inst✝¹ : PseudoMetricSpace α
ι : Type u_4
inst✝ : TopologicalSpace β
F : ι → β → α
x₀ : β
H : ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ι), dist (F i x) (F i x') < ε
U : Set (α × α)
hU : U ∈ 𝓤 α
⊢ ∃ V ∈ 𝓝 x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι), (F i x, F i y) ∈ U

tactic: rcases mem_uniformity_dist.mp hU with ⟨ε, hε, hεU⟩
state_before: case mpr
α : Type u_1
β : Type u_2
ι✝ : Type u_3
inst✝¹ : PseudoMetricSpace α
ι : Type u_4
inst✝ : TopologicalSpace β
F : ι → β → α
x₀ : β
H : ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ι), dist (F i x) (F i x') < ε
U : Set (α × α)
hU : U ∈ 𝓤 α
⊢ ∃ V ∈ 𝓝 x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι), (F i x, F i y) ∈ U
state_after: case mpr.intro.intro
α : Type u_1
β : Type u_2
ι✝ : Type u_3
inst✝¹ : PseudoMetricSpace α
ι : Type u_4
inst✝ : TopologicalSpace β
F : ι → β → α
x₀ : β
H : ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ι), dist (F i x) (F i x') < ε
U : Set (α × α)
hU : U ∈ 𝓤 α
ε : ℝ
hε : ε > 0
hεU : ∀ {a b : α}, dist a b < ε → (a, b) ∈ U
⊢ ∃ V ∈ 𝓝 x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι), (F i x, F i y) ∈ U

tactic: refine Exists.imp (fun V => And.imp_right fun h => ?_) (H _ hε)
state_before: case mpr.intro.intro
α : Type u_1
β : Type u_2
ι✝ : Type u_3
inst✝¹ : PseudoMetricSpace α
ι : Type u_4
inst✝ : TopologicalSpace β
F : ι → β → α
x₀ : β
H : ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ι), dist (F i x) (F i x') < ε
U : Set (α × α)
hU : U ∈ 𝓤 α
ε : ℝ
hε : ε > 0
hεU : ∀ {a b : α}, dist a b < ε → (a, b) ∈ U
⊢ ∃ V ∈ 𝓝 x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι), (F i x, F i y) ∈ U
state_after: case mpr.intro.intro
α : Type u_1
β : Type u_2
ι✝ : Type u_3
inst✝¹ : PseudoMetricSpace α
ι : Type u_4
inst✝ : TopologicalSpace β
F : ι → β → α
x₀ : β
H : ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ι), dist (F i x) (F i x') < ε
U : Set (α × α)
hU : U ∈ 𝓤 α
ε : ℝ
hε : ε > 0
hεU : ∀ {a b : α}, dist a b < ε → (a, b) ∈ U
V : Set β
h : ∀ x ∈ V, ∀ x' ∈ V, ∀ (i : ι), dist (F i x) (F i x') < ε
⊢ ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι), (F i x, F i y) ∈ U

tactic: exact fun x hx x' hx' i => hεU (h _ hx _ hx' i)
state_before: case mpr.intro.intro
α : Type u_1
β : Type u_2
ι✝ : Type u_3
inst✝¹ : PseudoMetricSpace α
ι : Type u_4
inst✝ : TopologicalSpace β
F : ι → β → α
x₀ : β
H : ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ι), dist (F i x) (F i x') < ε
U : Set (α × α)
hU : U ∈ 𝓤 α
ε : ℝ
hε : ε > 0
hεU : ∀ {a b : α}, dist a b < ε → (a, b) ∈ U
V : Set β
h : ∀ x ∈ V, ∀ x' ∈ V, ∀ (i : ι), dist (F i x) (F i x') < ε
⊢ ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι), (F i x, F i y) ∈ U
state_after: no goals