Theorem: Monotone.tendsto_nhdsWithin_Iio
File: Mathlib/Topology/Order/Basic.lean

tactic: rcases eq_empty_or_nonempty (Iio x) with (h | h)
state_before: α✝ : Type u
β✝ : Type v
γ : Type w
inst✝¹² : ConditionallyCompleteLinearOrder α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : OrderTopology α✝
inst✝⁹ : ConditionallyCompleteLinearOrder β✝
inst✝⁸ : TopologicalSpace β✝
inst✝⁷ : OrderClosedTopology β✝
inst✝⁶ : Nonempty γ
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
Mf : Monotone f
x : α
⊢ Tendsto f (𝓝[Iio x] x) (𝓝 (sSup (f '' Iio x)))

tactic: refine' tendsto_order.2 ⟨fun l hl => _, fun m hm => _⟩
state_before: case inr
α✝ : Type u
β✝ : Type v
γ : Type w
inst✝¹² : ConditionallyCompleteLinearOrder α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : OrderTopology α✝
inst✝⁹ : ConditionallyCompleteLinearOrder β✝
inst✝⁸ : TopologicalSpace β✝
inst✝⁷ : OrderClosedTopology β✝
inst✝⁶ : Nonempty γ
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
Mf : Monotone f
x : α
h : Set.Nonempty (Iio x)
⊢ Tendsto f (𝓝[Iio x] x) (𝓝 (sSup (f '' Iio x)))

tactic: simp [h]
state_before: case inl
α✝ : Type u
β✝ : Type v
γ : Type w
inst✝¹² : ConditionallyCompleteLinearOrder α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : OrderTopology α✝
inst✝⁹ : ConditionallyCompleteLinearOrder β✝
inst✝⁸ : TopologicalSpace β✝
inst✝⁷ : OrderClosedTopology β✝
inst✝⁶ : Nonempty γ
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
Mf : Monotone f
x : α
h : Iio x = ∅
⊢ Tendsto f (𝓝[Iio x] x) (𝓝 (sSup (f '' Iio x)))

tactic: obtain ⟨z, zx, lz⟩ : ∃ a : α, a < x ∧ l < f a := by
  simpa only [mem_image, exists_prop, exists_exists_and_eq_and] using
    exists_lt_of_lt_csSup (nonempty_image_iff.2 h) hl
state_before: case inr.refine'_1
α✝ : Type u
β✝ : Type v
γ : Type w
inst✝¹² : ConditionallyCompleteLinearOrder α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : OrderTopology α✝
inst✝⁹ : ConditionallyCompleteLinearOrder β✝
inst✝⁸ : TopologicalSpace β✝
inst✝⁷ : OrderClosedTopology β✝
inst✝⁶ : Nonempty γ
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
Mf : Monotone f
x : α
h : Set.Nonempty (Iio x)
l : β
hl : l < sSup (f '' Iio x)
⊢ ∀ᶠ (b : α) in 𝓝[Iio x] x, l < f b

tactic: exact mem_of_superset (Ioo_mem_nhdsWithin_Iio' zx) fun y hy => lz.trans_le (Mf hy.1.le)
state_before: case inr.refine'_1.intro.intro
α✝ : Type u
β✝ : Type v
γ : Type w
inst✝¹² : ConditionallyCompleteLinearOrder α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : OrderTopology α✝
inst✝⁹ : ConditionallyCompleteLinearOrder β✝
inst✝⁸ : TopologicalSpace β✝
inst✝⁷ : OrderClosedTopology β✝
inst✝⁶ : Nonempty γ
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
Mf : Monotone f
x : α
h : Set.Nonempty (Iio x)
l : β
hl : l < sSup (f '' Iio x)
z : α
zx : z < x
lz : l < f z
⊢ ∀ᶠ (b : α) in 𝓝[Iio x] x, l < f b

tactic: simpa only [mem_image, exists_prop, exists_exists_and_eq_and] using
  exists_lt_of_lt_csSup (nonempty_image_iff.2 h) hl
state_before: α✝ : Type u
β✝ : Type v
γ : Type w
inst✝¹² : ConditionallyCompleteLinearOrder α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : OrderTopology α✝
inst✝⁹ : ConditionallyCompleteLinearOrder β✝
inst✝⁸ : TopologicalSpace β✝
inst✝⁷ : OrderClosedTopology β✝
inst✝⁶ : Nonempty γ
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
Mf : Monotone f
x : α
h : Set.Nonempty (Iio x)
l : β
hl : l < sSup (f '' Iio x)
⊢ ∃ a, a < x ∧ l < f a

tactic: refine mem_of_superset self_mem_nhdsWithin fun _ hy => lt_of_le_of_lt ?_ hm
state_before: case inr.refine'_2
α✝ : Type u
β✝ : Type v
γ : Type w
inst✝¹² : ConditionallyCompleteLinearOrder α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : OrderTopology α✝
inst✝⁹ : ConditionallyCompleteLinearOrder β✝
inst✝⁸ : TopologicalSpace β✝
inst✝⁷ : OrderClosedTopology β✝
inst✝⁶ : Nonempty γ
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
Mf : Monotone f
x : α
h : Set.Nonempty (Iio x)
m : β
hm : m > sSup (f '' Iio x)
⊢ ∀ᶠ (b : α) in 𝓝[Iio x] x, f b < m

tactic: exact le_csSup (Mf.map_bddAbove bddAbove_Iio) (mem_image_of_mem _ hy)
state_before: case inr.refine'_2
α✝ : Type u
β✝ : Type v
γ : Type w
inst✝¹² : ConditionallyCompleteLinearOrder α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : OrderTopology α✝
inst✝⁹ : ConditionallyCompleteLinearOrder β✝
inst✝⁸ : TopologicalSpace β✝
inst✝⁷ : OrderClosedTopology β✝
inst✝⁶ : Nonempty γ
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
Mf : Monotone f
x : α
h : Set.Nonempty (Iio x)
m : β
hm : m > sSup (f '' Iio x)
x✝ : α
hy : x✝ ∈ Iio x
⊢ f x✝ ≤ sSup (f '' Iio x)