α : Type u
β : Type v
γ : Type w
inst✝⁷ : CompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : CompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderClosedTopology β
inst✝¹ : Nonempty γ
ι : Sort u_1
inst✝ : Nonempty ι
f : α → β
g : ι → α
Cf : ContinuousAt f (iSup g)
Mf : Monotone f
⊢ sSup (range (f ∘ g)) = sSup (range fun i => f (g i))
α : Type u
β : Type v
γ : Type w
inst✝⁷ : CompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : CompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderClosedTopology β
inst✝¹ : Nonempty γ
ι : Sort u_1
inst✝ : Nonempty ι
f : α → β
g : ι → α
Cf : ContinuousAt f (iSup g)
Mf : Monotone f
⊢ f (⨆ (i : ι), g i) = ⨆ (i : ι), f (g i)
rw [iSup, Mf.map_sSup_of_continuousAt' Cf (range_nonempty g), ← range_comp, iSup]
no goals
α : Type u
β : Type v
γ : Type w
inst✝⁷ : CompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : CompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderClosedTopology β
inst✝¹ : Nonempty γ
ι : Sort u_1
inst✝ : Nonempty ι
f : α → β
g : ι → α
Cf : ContinuousAt f (iSup g)
Mf : Monotone f
⊢ sSup (range (f ∘ g)) = sSup (range fun i => f (g i))
rfl