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

tactic: simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrderedAddCommGroup α
inst✝ : OrderTopology α
l : Filter β
f g : β → α
a : α
⊢ 𝓝 a = ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |a - b| < r}
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrderedAddCommGroup α
inst✝ : OrderTopology α
l : Filter β
f g : β → α
a : α
⊢ (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) =
    (⨅ x, ⨅ (_ : x > 0), 𝓟 {a_1 | -x < a - a_1}) ⊓ ⨅ x, ⨅ (_ : x > 0), 𝓟 {a_1 | a - a_1 < x}

tactic: refine (congr_arg₂ _ ?_ ?_).trans (inf_comm ..)
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrderedAddCommGroup α
inst✝ : OrderTopology α
l : Filter β
f g : β → α
a : α
⊢ (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) =
    (⨅ x, ⨅ (_ : x > 0), 𝓟 {a_1 | -x < a - a_1}) ⊓ ⨅ x, ⨅ (_ : x > 0), 𝓟 {a_1 | a - a_1 < x}
state_after: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrderedAddCommGroup α
inst✝ : OrderTopology α
l : Filter β
f g : β → α
a : α
⊢ ⨅ b ∈ Iio a, 𝓟 (Ioi b) = ⨅ x, ⨅ (_ : x > 0), 𝓟 {a_1 | a - a_1 < x}

case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrderedAddCommGroup α
inst✝ : OrderTopology α
l : Filter β
f g : β → α
a : α
⊢ ⨅ b ∈ Ioi a, 𝓟 (Iio b) = ⨅ x, ⨅ (_ : x > 0), 𝓟 {a_1 | -x < a - a_1}

tactic: refine (Equiv.subLeft a).iInf_congr fun x => ?_
state_before: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrderedAddCommGroup α
inst✝ : OrderTopology α
l : Filter β
f g : β → α
a : α
⊢ ⨅ b ∈ Iio a, 𝓟 (Ioi b) = ⨅ x, ⨅ (_ : x > 0), 𝓟 {a_1 | a - a_1 < x}
state_after: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrderedAddCommGroup α
inst✝ : OrderTopology α
l : Filter β
f g : β → α
a x : α
⊢ ⨅ (_ : (Equiv.subLeft a) x > 0), 𝓟 {a_1 | a - a_1 < (Equiv.subLeft a) x} = ⨅ (_ : x ∈ Iio a), 𝓟 (Ioi x)

tactic: simp [Ioi]
state_before: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrderedAddCommGroup α
inst✝ : OrderTopology α
l : Filter β
f g : β → α
a x : α
⊢ ⨅ (_ : (Equiv.subLeft a) x > 0), 𝓟 {a_1 | a - a_1 < (Equiv.subLeft a) x} = ⨅ (_ : x ∈ Iio a), 𝓟 (Ioi x)
state_after: no goals

tactic: refine (Equiv.subRight a).iInf_congr fun x => ?_
state_before: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrderedAddCommGroup α
inst✝ : OrderTopology α
l : Filter β
f g : β → α
a : α
⊢ ⨅ b ∈ Ioi a, 𝓟 (Iio b) = ⨅ x, ⨅ (_ : x > 0), 𝓟 {a_1 | -x < a - a_1}
state_after: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrderedAddCommGroup α
inst✝ : OrderTopology α
l : Filter β
f g : β → α
a x : α
⊢ ⨅ (_ : (Equiv.subRight a) x > 0), 𝓟 {a_1 | -(Equiv.subRight a) x < a - a_1} = ⨅ (_ : x ∈ Ioi a), 𝓟 (Iio x)

tactic: simp [Iio]
state_before: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrderedAddCommGroup α
inst✝ : OrderTopology α
l : Filter β
f g : β → α
a x : α
⊢ ⨅ (_ : (Equiv.subRight a) x > 0), 𝓟 {a_1 | -(Equiv.subRight a) x < a - a_1} = ⨅ (_ : x ∈ Ioi a), 𝓟 (Iio x)
state_after: no goals