Theorem: continuousWithinAt_right_of_monotoneOn_of_exists_between
File: Mathlib/Topology/Algebra/Order/MonotoneContinuity.lean

tactic: have ha : a ∈ Ici a := left_mem_Ici
state_before: α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : LinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
s : Set α
a : α
h_mono : MonotoneOn f s
hs : s ∈ 𝓝[Ici a] a
hfs : ∀ (b : β), b > f a → ∃ c, c ∈ s ∧ f c ∈ Ioo (f a) b
⊢ ContinuousWithinAt f (Ici a) a

tactic: have has : a ∈ s := mem_of_mem_nhdsWithin ha hs
state_before: α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : LinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
s : Set α
a : α
h_mono : MonotoneOn f s
hs : s ∈ 𝓝[Ici a] a
hfs : ∀ (b : β), b > f a → ∃ c, c ∈ s ∧ f c ∈ Ioo (f a) b
ha : a ∈ Ici a
⊢ ContinuousWithinAt f (Ici a) a

tactic: refine' tendsto_order.2 ⟨fun b hb => _, fun b hb => _⟩
state_before: α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : LinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
s : Set α
a : α
h_mono : MonotoneOn f s
hs : s ∈ 𝓝[Ici a] a
hfs : ∀ (b : β), b > f a → ∃ c, c ∈ s ∧ f c ∈ Ioo (f a) b
ha : a ∈ Ici a
has : a ∈ s
⊢ ContinuousWithinAt f (Ici a) a

tactic: filter_upwards [hs, @self_mem_nhdsWithin _ _ a (Ici a)] with _ hxs hxa using hb.trans_le
  (h_mono has hxs hxa)
state_before: case refine'_1
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : LinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
s : Set α
a : α
h_mono : MonotoneOn f s
hs : s ∈ 𝓝[Ici a] a
hfs : ∀ (b : β), b > f a → ∃ c, c ∈ s ∧ f c ∈ Ioo (f a) b
ha : a ∈ Ici a
has : a ∈ s
b : β
hb : b < f a
⊢ ∀ᶠ (b_1 : α) in 𝓝[Ici a] a, b < f b_1

tactic: rcases hfs b hb with ⟨c, hcs, hac, hcb⟩
state_before: case refine'_2
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : LinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
s : Set α
a : α
h_mono : MonotoneOn f s
hs : s ∈ 𝓝[Ici a] a
hfs : ∀ (b : β), b > f a → ∃ c, c ∈ s ∧ f c ∈ Ioo (f a) b
ha : a ∈ Ici a
has : a ∈ s
b : β
hb : b > f a
⊢ ∀ᶠ (b_1 : α) in 𝓝[Ici a] a, f b_1 < b

tactic: have : a < c := not_le.1 fun h => hac.not_le <| h_mono hcs has h
state_before: case refine'_2.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : LinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
s : Set α
a : α
h_mono : MonotoneOn f s
hs : s ∈ 𝓝[Ici a] a
hfs : ∀ (b : β), b > f a → ∃ c, c ∈ s ∧ f c ∈ Ioo (f a) b
ha : a ∈ Ici a
has : a ∈ s
b : β
hb : b > f a
c : α
hcs : c ∈ s
hac : f a < f c
hcb : f c < b
⊢ ∀ᶠ (b_1 : α) in 𝓝[Ici a] a, f b_1 < b

tactic: filter_upwards [hs, Ico_mem_nhdsWithin_Ici (left_mem_Ico.2 this)]
state_before: case refine'_2.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : LinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
s : Set α
a : α
h_mono : MonotoneOn f s
hs : s ∈ 𝓝[Ici a] a
hfs : ∀ (b : β), b > f a → ∃ c, c ∈ s ∧ f c ∈ Ioo (f a) b
ha : a ∈ Ici a
has : a ∈ s
b : β
hb : b > f a
c : α
hcs : c ∈ s
hac : f a < f c
hcb : f c < b
this : a < c
⊢ ∀ᶠ (b_1 : α) in 𝓝[Ici a] a, f b_1 < b

tactic: rintro x hx ⟨_, hxc⟩
state_before: case h
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : LinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
s : Set α
a : α
h_mono : MonotoneOn f s
hs : s ∈ 𝓝[Ici a] a
hfs : ∀ (b : β), b > f a → ∃ c, c ∈ s ∧ f c ∈ Ioo (f a) b
ha : a ∈ Ici a
has : a ∈ s
b : β
hb : b > f a
c : α
hcs : c ∈ s
hac : f a < f c
hcb : f c < b
this : a < c
⊢ ∀ (a_1 : α), a_1 ∈ s → a_1 ∈ Ico a c → f a_1 < b

tactic: exact (h_mono hx hcs hxc.le).trans_lt hcb
state_before: case h.intro
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : LinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
s : Set α
a : α
h_mono : MonotoneOn f s
hs : s ∈ 𝓝[Ici a] a
hfs : ∀ (b : β), b > f a → ∃ c, c ∈ s ∧ f c ∈ Ioo (f a) b
ha : a ∈ Ici a
has : a ∈ s
b : β
hb : b > f a
c : α
hcs : c ∈ s
hac : f a < f c
hcb : f c < b
this : a < c
x : α
hx : x ∈ s
left✝ : a ≤ x
hxc : x < c
⊢ f x < b