α : Type u_1
β : Type u_2
γ : Type ?u.70879
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : DenselyOrdered α
inst✝¹ : ConditionallyCompleteLinearOrder β
inst✝ : OrderTopology β
f : α → β
a b c : α
h : ContinuousOn f [[a, b]]
⊢ sInf (f '' [[a, b]]) ≤ sSup (f '' [[a, b]])
α : Type u_1
β : Type u_2
γ : Type ?u.70879
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : DenselyOrdered α
inst✝¹ : ConditionallyCompleteLinearOrder β
inst✝ : OrderTopology β
f : α → β
a b c : α
h : ContinuousOn f [[a, b]]
⊢ f '' [[a, b]] = [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]]
refine' h.image_uIcc_eq_Icc.trans (uIcc_of_le _).symm
case refine'_1
α : Type u_1
β : Type u_2
γ : Type ?u.70879
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : DenselyOrdered α
inst✝¹ : ConditionallyCompleteLinearOrder β
inst✝ : OrderTopology β
f : α → β
a b c : α
h : ContinuousOn f [[a, b]]
⊢ BddBelow (Icc (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])))

case refine'_2
α : Type u_1
β : Type u_2
γ : Type ?u.70879
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : DenselyOrdered α
inst✝¹ : ConditionallyCompleteLinearOrder β
inst✝ : OrderTopology β
f : α → β
a b c : α
h : ContinuousOn f [[a, b]]
⊢ BddAbove (Icc (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])))
α : Type u_1
β : Type u_2
γ : Type ?u.70879
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : DenselyOrdered α
inst✝¹ : ConditionallyCompleteLinearOrder β
inst✝ : OrderTopology β
f : α → β
a b c : α
h : ContinuousOn f [[a, b]]
⊢ sInf (f '' [[a, b]]) ≤ sSup (f '' [[a, b]])
refine' csInf_le_csSup _ _ (nonempty_uIcc.image _) <;> rw [h.image_uIcc_eq_Icc]
no goals
case refine'_1
α : Type u_1
β : Type u_2
γ : Type ?u.70879
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : DenselyOrdered α
inst✝¹ : ConditionallyCompleteLinearOrder β
inst✝ : OrderTopology β
f : α → β
a b c : α
h : ContinuousOn f [[a, b]]
⊢ BddBelow (Icc (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])))

case refine'_2
α : Type u_1
β : Type u_2
γ : Type ?u.70879
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : DenselyOrdered α
inst✝¹ : ConditionallyCompleteLinearOrder β
inst✝ : OrderTopology β
f : α → β
a b c : α
h : ContinuousOn f [[a, b]]
⊢ BddAbove (Icc (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])))
exacts [bddBelow_Icc, bddAbove_Icc]