Theorem: ContinuousOn.strictMonoOn_of_injOn_Icc
File: Mathlib/Topology/Order/IntermediateValue.lean

tactic: letI := Icc.completeLinearOrder hab
state_before: α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hab : a ≤ b
hfab : f a ≤ f b
hf_c : ContinuousOn f (Icc a b)
hf_i : InjOn f (Icc a b)
⊢ StrictMonoOn f (Icc a b)
state_after: α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hab : a ≤ b
hfab : f a ≤ f b
hf_c : ContinuousOn f (Icc a b)
hf_i : InjOn f (Icc a b)
this : CompleteLinearOrder ↑(Icc a b) := Icc.completeLinearOrder hab
⊢ StrictMonoOn f (Icc a b)

tactic: refine StrictMono.of_restrict ?_
state_before: α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hab : a ≤ b
hfab : f a ≤ f b
hf_c : ContinuousOn f (Icc a b)
hf_i : InjOn f (Icc a b)
this : CompleteLinearOrder ↑(Icc a b) := Icc.completeLinearOrder hab
⊢ StrictMonoOn f (Icc a b)
state_after: α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hab : a ≤ b
hfab : f a ≤ f b
hf_c : ContinuousOn f (Icc a b)
hf_i : InjOn f (Icc a b)
this : CompleteLinearOrder ↑(Icc a b) := Icc.completeLinearOrder hab
⊢ StrictMono ((Icc a b).restrict f)

tactic: set g : Icc a b → δ := Set.restrict (Icc a b) f
state_before: α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hab : a ≤ b
hfab : f a ≤ f b
hf_c : ContinuousOn f (Icc a b)
hf_i : InjOn f (Icc a b)
this : CompleteLinearOrder ↑(Icc a b) := Icc.completeLinearOrder hab
⊢ StrictMono ((Icc a b).restrict f)
state_after: α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hab : a ≤ b
hfab : f a ≤ f b
hf_c : ContinuousOn f (Icc a b)
hf_i : InjOn f (Icc a b)
this : CompleteLinearOrder ↑(Icc a b) := Icc.completeLinearOrder hab
g : ↑(Icc a b) → δ := (Icc a b).restrict f
⊢ StrictMono g

tactic: have hgab : g ⊥ ≤ g ⊤ := by aesop
state_before: α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hab : a ≤ b
hfab : f a ≤ f b
hf_c : ContinuousOn f (Icc a b)
hf_i : InjOn f (Icc a b)
this : CompleteLinearOrder ↑(Icc a b) := Icc.completeLinearOrder hab
g : ↑(Icc a b) → δ := (Icc a b).restrict f
⊢ StrictMono g
state_after: α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hab : a ≤ b
hfab : f a ≤ f b
hf_c : ContinuousOn f (Icc a b)
hf_i : InjOn f (Icc a b)
this : CompleteLinearOrder ↑(Icc a b) := Icc.completeLinearOrder hab
g : ↑(Icc a b) → δ := (Icc a b).restrict f
hgab : g ⊥ ≤ g ⊤
⊢ StrictMono g

tactic: exact Continuous.strictMono_of_inj_boundedOrder (f := g) hf_c.restrict hgab hf_i.injective
state_before: α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hab : a ≤ b
hfab : f a ≤ f b
hf_c : ContinuousOn f (Icc a b)
hf_i : InjOn f (Icc a b)
this : CompleteLinearOrder ↑(Icc a b) := Icc.completeLinearOrder hab
g : ↑(Icc a b) → δ := (Icc a b).restrict f
hgab : g ⊥ ≤ g ⊤
⊢ StrictMono g
state_after: no goals

tactic: aesop
state_before: α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hab : a ≤ b
hfab : f a ≤ f b
hf_c : ContinuousOn f (Icc a b)
hf_i : InjOn f (Icc a b)
this : CompleteLinearOrder ↑(Icc a b) := Icc.completeLinearOrder hab
g : ↑(Icc a b) → δ := (Icc a b).restrict f
⊢ g ⊥ ≤ g ⊤
state_after: no goals