Theorem: orderTopology_of_nhds_abs
File: Mathlib/Topology/Order/LeftRightNhds.lean

Proof steps:
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : TopologicalSpace α✝
inst✝³ : LinearOrderedAddCommGroup α✝
inst✝² : OrderTopology α✝
l : Filter β
f g : β → α✝
α : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : LinearOrderedAddCommGroup α
h_nhds : ∀ (a : α), 𝓝 a = ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |a - b| < r}
⊢ OrderTopology α
state_after: α✝ : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : TopologicalSpace α✝
inst✝³ : LinearOrderedAddCommGroup α✝
inst✝² : OrderTopology α✝
l : Filter β
f g : β → α✝
α : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : LinearOrderedAddCommGroup α
h_nhds : ∀ (a : α), 𝓝 a = ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |a - b| < r}
a : α
⊢ 𝓝 a = 𝓝 a
tactic: refine ⟨TopologicalSpace.ext_nhds fun a => ?_⟩
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : TopologicalSpace α✝
inst✝³ : LinearOrderedAddCommGroup α✝
inst✝² : OrderTopology α✝
l : Filter β
f g : β → α✝
α : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : LinearOrderedAddCommGroup α
h_nhds : ∀ (a : α), 𝓝 a = ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |a - b| < r}
a : α
⊢ 𝓝 a = 𝓝 a
state_after: α✝ : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : TopologicalSpace α✝
inst✝³ : LinearOrderedAddCommGroup α✝
inst✝² : OrderTopology α✝
l : Filter β
f g : β → α✝
α : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : LinearOrderedAddCommGroup α
h_nhds : ∀ (a : α), 𝓝 a = ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |a - b| < r}
a : α
⊢ ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |a - b| < r} = 𝓝 a
tactic: rw [h_nhds]
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : TopologicalSpace α✝
inst✝³ : LinearOrderedAddCommGroup α✝
inst✝² : OrderTopology α✝
l : Filter β
f g : β → α✝
α : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : LinearOrderedAddCommGroup α
h_nhds : ∀ (a : α), 𝓝 a = ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |a - b| < r}
a : α
⊢ ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |a - b| < r} = 𝓝 a
state_after: α✝ : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : TopologicalSpace α✝
inst✝³ : LinearOrderedAddCommGroup α✝
inst✝² : OrderTopology α✝
l : Filter β
f g : β → α✝
α : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : LinearOrderedAddCommGroup α
h_nhds : ∀ (a : α), 𝓝 a = ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |a - b| < r}
a : α
this : TopologicalSpace α := Preorder.topology α
⊢ ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |a - b| < r} = 𝓝 a
tactic: letI := Preorder.topology α
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : TopologicalSpace α✝
inst✝³ : LinearOrderedAddCommGroup α✝
inst✝² : OrderTopology α✝
l : Filter β
f g : β → α✝
α : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : LinearOrderedAddCommGroup α
h_nhds : ∀ (a : α), 𝓝 a = ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |a - b| < r}
a : α
this : TopologicalSpace α := Preorder.topology α
⊢ ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |a - b| < r} = 𝓝 a
state_after: α✝ : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : TopologicalSpace α✝
inst✝³ : LinearOrderedAddCommGroup α✝
inst✝² : OrderTopology α✝
l : Filter β
f g : β → α✝
α : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : LinearOrderedAddCommGroup α
h_nhds : ∀ (a : α), 𝓝 a = ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |a - b| < r}
a : α
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α := { topology_eq_generate_intervals := rfl }
⊢ ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |a - b| < r} = 𝓝 a
tactic: letI : OrderTopology α := ⟨rfl⟩
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : TopologicalSpace α✝
inst✝³ : LinearOrderedAddCommGroup α✝
inst✝² : OrderTopology α✝
l : Filter β
f g : β → α✝
α : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : LinearOrderedAddCommGroup α
h_nhds : ∀ (a : α), 𝓝 a = ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |a - b| < r}
a : α
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α := { topology_eq_generate_intervals := rfl }
⊢ ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |a - b| < r} = 𝓝 a
state_after: no goals
tactic: exact (nhds_eq_iInf_abs_sub a).symm