Theorem: Monotone.le_leftLim
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
⊢ f x ≤ leftLim 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
⊢ f x ≤ leftLim f y

tactic: rcases eq_or_ne (𝓝[<] y) ⊥ 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
⊢ f x ≤ leftLim 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 y] y ≠ ⊥
⊢ f x ≤ leftLim f y

tactic: refine' le_csSup ⟨f y, _⟩ (mem_image_of_mem _ 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 y] y ≠ ⊥
⊢ f x ≤ sSup (f '' Iio y)

tactic: simp only [upperBounds, mem_image, mem_Iio, forall_exists_index, and_imp,
  forall_apply_eq_imp_iff₂, mem_setOf_eq]
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 y] y ≠ ⊥
⊢ f y ∈ upperBounds (f '' Iio y)

tactic: intro z hz
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 y] y ≠ ⊥
⊢ ∀ (a : α), a < y → f a ≤ f y

tactic: exact hf hz.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 y] y ≠ ⊥
z : α
hz : z < y
⊢ f z ≤ f y

tactic: rw [leftLim_eq_of_eq_bot _ 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 y] y = ⊥
⊢ f x ≤ leftLim f y

tactic: exact hf h.le
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 y] y = ⊥
⊢ f x ≤ f y