Theorem: Monotone.leftLim_le
File: Mathlib/Topology/Algebra/Order/LeftRightLim.lean

tactic: letI : TopologicalSpace α := Preorder.topology α
state_before: α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
⊢ leftLim f x ≤ f y

tactic: haveI : OrderTopology α := ⟨rfl⟩
state_before: α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this : TopologicalSpace α := Preorder.topology α
⊢ leftLim f x ≤ f y

tactic: rcases eq_or_ne (𝓝[<] x) ⊥ with (h' | h')
state_before: α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
⊢ leftLim f x ≤ f y

tactic: haveI A : NeBot (𝓝[<] x) := neBot_iff.2 h'
state_before: case inr
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[Iio x] x ≠ ⊥
⊢ leftLim f x ≤ f y

tactic: rw [leftLim_eq_sSup hf h']
state_before: case inr
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[Iio x] x ≠ ⊥
A : NeBot (𝓝[Iio x] x)
⊢ leftLim f x ≤ f y

tactic: refine' csSup_le _ _
state_before: case inr
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[Iio x] x ≠ ⊥
A : NeBot (𝓝[Iio x] x)
⊢ sSup (f '' Iio x) ≤ f y

tactic: simpa [leftLim, h'] using hf h
state_before: case inl
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[Iio x] x = ⊥
⊢ leftLim f x ≤ f y

tactic: simp only [nonempty_image_iff]
state_before: case inr.refine'_1
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[Iio x] x ≠ ⊥
A : NeBot (𝓝[Iio x] x)
⊢ Set.Nonempty (f '' Iio x)

tactic: exact (forall_mem_nonempty_iff_neBot.2 A) _ self_mem_nhdsWithin
state_before: case inr.refine'_1
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[Iio x] x ≠ ⊥
A : NeBot (𝓝[Iio x] x)
⊢ Set.Nonempty (Iio x)

tactic: simp only [mem_image, mem_Iio, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
state_before: case inr.refine'_2
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[Iio x] x ≠ ⊥
A : NeBot (𝓝[Iio x] x)
⊢ ∀ (b : β), b ∈ f '' Iio x → b ≤ f y

tactic: intro z hz
state_before: case inr.refine'_2
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[Iio x] x ≠ ⊥
A : NeBot (𝓝[Iio x] x)
⊢ ∀ (a : α), a < x → f a ≤ f y

tactic: exact hf (hz.le.trans h)
state_before: case inr.refine'_2
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[Iio x] x ≠ ⊥
A : NeBot (𝓝[Iio x] x)
z : α
hz : z < x
⊢ f z ≤ f y