File: Mathlib/Topology/Order/Monotone.lean

full_name: Monotone.map_sSup_of_continuousAt'
code: /-- A monotone function continuous at the supremum of a nonempty set sends this supremum to
the supremum of the image of this set. -/
theorem Monotone.map_sSup_of_continuousAt' {f : α → β} {A : Set α} (Cf : ContinuousAt f (sSup A))
    (Mf : Monotone f) (A_nonemp : A.Nonempty) (A_bdd : BddAbove A := by bddDefault) :
    f (sSup A) = sSup (f '' A) 
goal: f (sSup A) = sSup (f '' A)

full_name: Monotone.map_iSup_of_continuousAt'
code: /-- A monotone function continuous at the indexed supremum over a nonempty `Sort` sends this indexed
supremum to the indexed supremum of the composition. -/
theorem Monotone.map_iSup_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
    (Cf : ContinuousAt f (iSup g)) (Mf : Monotone f)
    (bdd : BddAbove (range g) := by bddDefault) : f (⨆ i, g i) = ⨆ i, f (g i) 
goal: f (⨆ i, g i) = ⨆ i, f (g i)

full_name: Monotone.map_sInf_of_continuousAt'
code: /-- A monotone function continuous at the infimum of a nonempty set sends this infimum to
the infimum of the image of this set. -/
theorem Monotone.map_sInf_of_continuousAt' {f : α → β} {A : Set α} (Cf : ContinuousAt f (sInf A))
    (Mf : Monotone f) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault) :
    f (sInf A) = sInf (f '' A) 
goal: f (sInf A) = sInf (f '' A)

full_name: Monotone.map_iInf_of_continuousAt'
code: /-- A monotone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed
infimum to the indexed infimum of the composition. -/
theorem Monotone.map_iInf_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
    (Cf : ContinuousAt f (iInf g)) (Mf : Monotone f)
    (bdd : BddBelow (range g) := by bddDefault) : f (⨅ i, g i) = ⨅ i, f (g i) 
goal: f (⨅ i, g i) = ⨅ i, f (g i)

full_name: Antitone.map_sInf_of_continuousAt'
code: /-- An antitone function continuous at the infimum of a nonempty set sends this infimum to
the supremum of the image of this set. -/
theorem Antitone.map_sInf_of_continuousAt' {f : α → β} {A : Set α} (Cf : ContinuousAt f (sInf A))
    (Af : Antitone f) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault) :
    f (sInf A) = sSup (f '' A) 
goal: f (sInf A) = sSup (f '' A)

full_name: Antitone.map_iInf_of_continuousAt'
code: /-- An antitone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed
infimum to the indexed supremum of the composition. -/
theorem Antitone.map_iInf_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
    (Cf : ContinuousAt f (iInf g)) (Af : Antitone f)
    (bdd : BddBelow (range g) := by bddDefault) : f (⨅ i, g i) = ⨆ i, f (g i) 
goal: f (⨅ i, g i) = ⨆ i, f (g i)

full_name: Antitone.map_sSup_of_continuousAt'
code: /-- An antitone function continuous at the supremum of a nonempty set sends this supremum to
the infimum of the image of this set. -/
theorem Antitone.map_sSup_of_continuousAt' {f : α → β} {A : Set α} (Cf : ContinuousAt f (sSup A))
    (Af : Antitone f) (A_nonemp : A.Nonempty) (A_bdd : BddAbove A := by bddDefault) :
    f (sSup A) = sInf (f '' A) 
goal: f (sSup A) = sInf (f '' A)

full_name: Antitone.map_iSup_of_continuousAt'
code: /-- An antitone function continuous at the indexed supremum over a nonempty `Sort` sends this
indexed supremum to the indexed infimum of the composition. -/
theorem Antitone.map_iSup_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
    (Cf : ContinuousAt f (iSup g)) (Af : Antitone f)
    (bdd : BddAbove (range g) := by bddDefault) : f (⨆ i, g i) = ⨅ i, f (g i) 
goal: f (⨆ i, g i) = ⨅ i, f (g i)

full_name: sSup_mem_closure
code: theorem sSup_mem_closure {s : Set α} (hs : s.Nonempty) : sSup s ∈ closure s 
goal: sSup s ∈ closure s

full_name: sInf_mem_closure
code: theorem sInf_mem_closure {s : Set α} (hs : s.Nonempty) : sInf s ∈ closure s 
goal: sInf s ∈ closure s

full_name: IsClosed.sSup_mem
code: theorem IsClosed.sSup_mem {s : Set α} (hs : s.Nonempty) (hc : IsClosed s) : sSup s ∈ s 
goal: sSup s ∈ s

full_name: IsClosed.sInf_mem
code: theorem IsClosed.sInf_mem {s : Set α} (hs : s.Nonempty) (hc : IsClosed s) : sInf s ∈ s 
goal: sInf s ∈ s

full_name: Monotone.map_sSup_of_continuousAt
code: /-- A monotone function `f` sending `bot` to `bot` and continuous at the supremum of a set sends
this supremum to the supremum of the image of this set. -/
theorem Monotone.map_sSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
    (Mf : Monotone f) (fbot : f ⊥ = ⊥) : f (sSup s) = sSup (f '' s) 
goal: f (sSup s) = sSup (f '' s)

full_name: Monotone.map_iSup_of_continuousAt
code: /-- If a monotone function sending `bot` to `bot` is continuous at the indexed supremum over
a `Sort`, then it sends this indexed supremum to the indexed supremum of the composition. -/
theorem Monotone.map_iSup_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
    (Cf : ContinuousAt f (iSup g)) (Mf : Monotone f) (fbot : f ⊥ = ⊥) :
    f (⨆ i, g i) = ⨆ i, f (g i) 
goal: f (⨆ i, g i) = ⨆ i, f (g i)

full_name: Monotone.map_sInf_of_continuousAt
code: /-- A monotone function `f` sending `top` to `top` and continuous at the infimum of a set sends
this infimum to the infimum of the image of this set. -/
theorem Monotone.map_sInf_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s))
    (Mf : Monotone f) (ftop : f ⊤ = ⊤) : f (sInf s) = sInf (f '' s) 
goal: f (sInf s) = sInf (f '' s)

full_name: Monotone.map_iInf_of_continuousAt
code: /-- If a monotone function sending `top` to `top` is continuous at the indexed infimum over
a `Sort`, then it sends this indexed infimum to the indexed infimum of the composition. -/
theorem Monotone.map_iInf_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
    (Cf : ContinuousAt f (iInf g)) (Mf : Monotone f) (ftop : f ⊤ = ⊤) : f (iInf g) = iInf (f ∘ g) 
goal: f (iInf g) = iInf (f ∘ g)

full_name: Antitone.map_sSup_of_continuousAt
code: /-- An antitone function `f` sending `bot` to `top` and continuous at the supremum of a set sends
this supremum to the infimum of the image of this set. -/
theorem Antitone.map_sSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
    (Af : Antitone f) (fbot : f ⊥ = ⊤) : f (sSup s) = sInf (f '' s) 
goal: f (sSup s) = sInf (f '' s)

full_name: Antitone.map_iSup_of_continuousAt
code: /-- An antitone function sending `bot` to `top` is continuous at the indexed supremum over
a `Sort`, then it sends this indexed supremum to the indexed supremum of the composition. -/
theorem Antitone.map_iSup_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
    (Cf : ContinuousAt f (iSup g)) (Af : Antitone f) (fbot : f ⊥ = ⊤) :
    f (⨆ i, g i) = ⨅ i, f (g i) 
goal: f (⨆ i, g i) = ⨅ i, f (g i)

full_name: Antitone.map_sInf_of_continuousAt
code: /-- An antitone function `f` sending `top` to `bot` and continuous at the infimum of a set sends
this infimum to the supremum of the image of this set. -/
theorem Antitone.map_sInf_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s))
    (Af : Antitone f) (ftop : f ⊤ = ⊥) : f (sInf s) = sSup (f '' s) 
goal: f (sInf s) = sSup (f '' s)

full_name: Antitone.map_iInf_of_continuousAt
code: /-- If an antitone function sending `top` to `bot` is continuous at the indexed infimum over
a `Sort`, then it sends this indexed infimum to the indexed supremum of the composition. -/
theorem Antitone.map_iInf_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
    (Cf : ContinuousAt f (iInf g)) (Af : Antitone f) (ftop : f ⊤ = ⊥) : f (iInf g) = iSup (f ∘ g) 
goal: f (iInf g) = iSup (f ∘ g)

full_name: csSup_mem_closure
code: theorem csSup_mem_closure {s : Set α} (hs : s.Nonempty) (B : BddAbove s) : sSup s ∈ closure s 
goal: sSup s ∈ closure s

full_name: csInf_mem_closure
code: theorem csInf_mem_closure {s : Set α} (hs : s.Nonempty) (B : BddBelow s) : sInf s ∈ closure s 
goal: sInf s ∈ closure s

full_name: IsClosed.csSup_mem
code: theorem IsClosed.csSup_mem {s : Set α} (hc : IsClosed s) (hs : s.Nonempty) (B : BddAbove s) :
    sSup s ∈ s 
goal: sSup s ∈ s

full_name: IsClosed.csInf_mem
code: theorem IsClosed.csInf_mem {s : Set α} (hc : IsClosed s) (hs : s.Nonempty) (B : BddBelow s) :
    sInf s ∈ s 
goal: sInf s ∈ s

full_name: IsClosed.isLeast_csInf
code: theorem IsClosed.isLeast_csInf {s : Set α} (hc : IsClosed s) (hs : s.Nonempty) (B : BddBelow s) :
    IsLeast s (sInf s) 
goal: IsLeast s (sInf s)

full_name: IsClosed.isGreatest_csSup
code: theorem IsClosed.isGreatest_csSup {s : Set α} (hc : IsClosed s) (hs : s.Nonempty) (B : BddAbove s) :
    IsGreatest s (sSup s) 
goal: IsGreatest s (sSup s)

full_name: Monotone.map_csSup_of_continuousAt
code: /-- If a monotone function is continuous at the supremum of a nonempty bounded above set `s`,
then it sends this supremum to the supremum of the image of `s`. -/
theorem Monotone.map_csSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
    (Mf : Monotone f) (ne : s.Nonempty) (H : BddAbove s) : f (sSup s) = sSup (f '' s) 
goal: f (sSup s) = sSup (f '' s)

full_name: Monotone.map_ciSup_of_continuousAt
code: /-- If a monotone function is continuous at the indexed supremum of a bounded function on
a nonempty `Sort`, then it sends this supremum to the supremum of the composition. -/
theorem Monotone.map_ciSup_of_continuousAt {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨆ i, g i))
    (Mf : Monotone f) (H : BddAbove (range g)) : f (⨆ i, g i) = ⨆ i, f (g i) 
goal: f (⨆ i, g i) = ⨆ i, f (g i)

full_name: Monotone.map_csInf_of_continuousAt
code: /-- If a monotone function is continuous at the infimum of a nonempty bounded below set `s`,
then it sends this infimum to the infimum of the image of `s`. -/
theorem Monotone.map_csInf_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s))
    (Mf : Monotone f) (ne : s.Nonempty) (H : BddBelow s) : f (sInf s) = sInf (f '' s) 
goal: f (sInf s) = sInf (f '' s)

full_name: Monotone.map_ciInf_of_continuousAt
code: /-- A continuous monotone function sends indexed infimum to indexed infimum in conditionally
complete linear order, under a boundedness assumption. -/
theorem Monotone.map_ciInf_of_continuousAt {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨅ i, g i))
    (Mf : Monotone f) (H : BddBelow (range g)) : f (⨅ i, g i) = ⨅ i, f (g i) 
goal: f (⨅ i, g i) = ⨅ i, f (g i)

full_name: Antitone.map_csSup_of_continuousAt
code: /-- If an antitone function is continuous at the supremum of a nonempty bounded above set `s`,
then it sends this supremum to the infimum of the image of `s`. -/
theorem Antitone.map_csSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
    (Af : Antitone f) (ne : s.Nonempty) (H : BddAbove s) : f (sSup s) = sInf (f '' s) 
goal: f (sSup s) = sInf (f '' s)

full_name: Antitone.map_ciSup_of_continuousAt
code: /-- If an antitone function is continuous at the indexed supremum of a bounded function on
a nonempty `Sort`, then it sends this supremum to the infimum of the composition. -/
theorem Antitone.map_ciSup_of_continuousAt {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨆ i, g i))
    (Af : Antitone f) (H : BddAbove (range g)) : f (⨆ i, g i) = ⨅ i, f (g i) 
goal: f (⨆ i, g i) = ⨅ i, f (g i)

full_name: Antitone.map_csInf_of_continuousAt
code: /-- If an antitone function is continuous at the infimum of a nonempty bounded below set `s`,
then it sends this infimum to the supremum of the image of `s`. -/
theorem Antitone.map_csInf_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s))
    (Af : Antitone f) (ne : s.Nonempty) (H : BddBelow s) : f (sInf s) = sSup (f '' s) 
goal: f (sInf s) = sSup (f '' s)

full_name: Antitone.map_ciInf_of_continuousAt
code: /-- A continuous antitone function sends indexed infimum to indexed supremum in conditionally
complete linear order, under a boundedness assumption. -/
theorem Antitone.map_ciInf_of_continuousAt {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨅ i, g i))
    (Af : Antitone f) (H : BddBelow (range g)) : f (⨅ i, g i) = ⨆ i, f (g i) 
goal: f (⨅ i, g i) = ⨆ i, f (g i)

full_name: Monotone.tendsto_nhdsWithin_Iio
code: /-- A monotone map has a limit to the left of any point `x`, equal to `sSup (f '' (Iio x))`. -/
theorem Monotone.tendsto_nhdsWithin_Iio {α β : Type*} [LinearOrder α] [TopologicalSpace α]
    [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
    {f : α → β} (Mf : Monotone f) (x : α) : Tendsto f (𝓝[<] x) (𝓝 (sSup (f '' Iio x))) 
goal: Filter.Tendsto f (𝓝[<] x) (𝓝 (sSup (f '' Set.Iio x)))

full_name: Monotone.tendsto_nhdsWithin_Ioi
code: /-- A monotone map has a limit to the right of any point `x`, equal to `sInf (f '' (Ioi x))`. -/
theorem Monotone.tendsto_nhdsWithin_Ioi {α β : Type*} [LinearOrder α] [TopologicalSpace α]
    [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
    {f : α → β} (Mf : Monotone f) (x : α) : Tendsto f (𝓝[>] x) (𝓝 (sInf (f '' Ioi x))) 
goal: Filter.Tendsto f (𝓝[>] x) (𝓝 (sInf (f '' Set.Ioi x)))