Theorem: lowerSemicontinuousWithinAt_ciSup
File: Mathlib/Topology/Semicontinuous.lean

tactic: cases isEmpty_or_nonempty ι
state_before: α : Type u_1
inst✝³ : TopologicalSpace α
β : Type u_2
inst✝² : Preorder β
f✝ g : α → β
x : α
s t : Set α
y z : β
ι : Sort u_3
δ : Type u_4
δ' : Type u_5
inst✝¹ : CompleteLinearOrder δ
inst✝ : ConditionallyCompleteLinearOrder δ'
f : ι → α → δ'
bdd : ∀ᶠ (y : α) in 𝓝[s] x, BddAbove (range fun i => f i y)
h : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x
⊢ LowerSemicontinuousWithinAt (fun x' => ⨆ i, f i x') s x
state_after: case inl
α : Type u_1
inst✝³ : TopologicalSpace α
β : Type u_2
inst✝² : Preorder β
f✝ g : α → β
x : α
s t : Set α
y z : β
ι : Sort u_3
δ : Type u_4
δ' : Type u_5
inst✝¹ : CompleteLinearOrder δ
inst✝ : ConditionallyCompleteLinearOrder δ'
f : ι → α → δ'
bdd : ∀ᶠ (y : α) in 𝓝[s] x, BddAbove (range fun i => f i y)
h : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x
h✝ : IsEmpty ι
⊢ LowerSemicontinuousWithinAt (fun x' => ⨆ i, f i x') s x

case inr
α : Type u_1
inst✝³ : TopologicalSpace α
β : Type u_2
inst✝² : Preorder β
f✝ g : α → β
x : α
s t : Set α
y z : β
ι : Sort u_3
δ : Type u_4
δ' : Type u_5
inst✝¹ : CompleteLinearOrder δ
inst✝ : ConditionallyCompleteLinearOrder δ'
f : ι → α → δ'
bdd : ∀ᶠ (y : α) in 𝓝[s] x, BddAbove (range fun i => f i y)
h : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x
h✝ : Nonempty ι
⊢ LowerSemicontinuousWithinAt (fun x' => ⨆ i, f i x') s x

tactic: simpa only [iSup_of_empty'] using lowerSemicontinuousWithinAt_const
state_before: case inl
α : Type u_1
inst✝³ : TopologicalSpace α
β : Type u_2
inst✝² : Preorder β
f✝ g : α → β
x : α
s t : Set α
y z : β
ι : Sort u_3
δ : Type u_4
δ' : Type u_5
inst✝¹ : CompleteLinearOrder δ
inst✝ : ConditionallyCompleteLinearOrder δ'
f : ι → α → δ'
bdd : ∀ᶠ (y : α) in 𝓝[s] x, BddAbove (range fun i => f i y)
h : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x
h✝ : IsEmpty ι
⊢ LowerSemicontinuousWithinAt (fun x' => ⨆ i, f i x') s x
state_after: no goals

tactic: intro y hy
state_before: case inr
α : Type u_1
inst✝³ : TopologicalSpace α
β : Type u_2
inst✝² : Preorder β
f✝ g : α → β
x : α
s t : Set α
y z : β
ι : Sort u_3
δ : Type u_4
δ' : Type u_5
inst✝¹ : CompleteLinearOrder δ
inst✝ : ConditionallyCompleteLinearOrder δ'
f : ι → α → δ'
bdd : ∀ᶠ (y : α) in 𝓝[s] x, BddAbove (range fun i => f i y)
h : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x
h✝ : Nonempty ι
⊢ LowerSemicontinuousWithinAt (fun x' => ⨆ i, f i x') s x
state_after: case inr
α : Type u_1
inst✝³ : TopologicalSpace α
β : Type u_2
inst✝² : Preorder β
f✝ g : α → β
x : α
s t : Set α
y✝ z : β
ι : Sort u_3
δ : Type u_4
δ' : Type u_5
inst✝¹ : CompleteLinearOrder δ
inst✝ : ConditionallyCompleteLinearOrder δ'
f : ι → α → δ'
bdd : ∀ᶠ (y : α) in 𝓝[s] x, BddAbove (range fun i => f i y)
h : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x
h✝ : Nonempty ι
y : δ'
hy : y < (fun x' => ⨆ i, f i x') x
⊢ ∀ᶠ (x' : α) in 𝓝[s] x, y < (fun x' => ⨆ i, f i x') x'

tactic: rcases exists_lt_of_lt_ciSup hy with ⟨i, hi⟩
state_before: case inr
α : Type u_1
inst✝³ : TopologicalSpace α
β : Type u_2
inst✝² : Preorder β
f✝ g : α → β
x : α
s t : Set α
y✝ z : β
ι : Sort u_3
δ : Type u_4
δ' : Type u_5
inst✝¹ : CompleteLinearOrder δ
inst✝ : ConditionallyCompleteLinearOrder δ'
f : ι → α → δ'
bdd : ∀ᶠ (y : α) in 𝓝[s] x, BddAbove (range fun i => f i y)
h : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x
h✝ : Nonempty ι
y : δ'
hy : y < (fun x' => ⨆ i, f i x') x
⊢ ∀ᶠ (x' : α) in 𝓝[s] x, y < (fun x' => ⨆ i, f i x') x'
state_after: case inr.intro
α : Type u_1
inst✝³ : TopologicalSpace α
β : Type u_2
inst✝² : Preorder β
f✝ g : α → β
x : α
s t : Set α
y✝ z : β
ι : Sort u_3
δ : Type u_4
δ' : Type u_5
inst✝¹ : CompleteLinearOrder δ
inst✝ : ConditionallyCompleteLinearOrder δ'
f : ι → α → δ'
bdd : ∀ᶠ (y : α) in 𝓝[s] x, BddAbove (range fun i => f i y)
h : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x
h✝ : Nonempty ι
y : δ'
hy : y < (fun x' => ⨆ i, f i x') x
i : ι
hi : y < f i x
⊢ ∀ᶠ (x' : α) in 𝓝[s] x, y < (fun x' => ⨆ i, f i x') x'

tactic: filter_upwards [h i y hi, bdd] with y hy hy' using hy.trans_le (le_ciSup hy' i)
state_before: case inr.intro
α : Type u_1
inst✝³ : TopologicalSpace α
β : Type u_2
inst✝² : Preorder β
f✝ g : α → β
x : α
s t : Set α
y✝ z : β
ι : Sort u_3
δ : Type u_4
δ' : Type u_5
inst✝¹ : CompleteLinearOrder δ
inst✝ : ConditionallyCompleteLinearOrder δ'
f : ι → α → δ'
bdd : ∀ᶠ (y : α) in 𝓝[s] x, BddAbove (range fun i => f i y)
h : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x
h✝ : Nonempty ι
y : δ'
hy : y < (fun x' => ⨆ i, f i x') x
i : ι
hi : y < f i x
⊢ ∀ᶠ (x' : α) in 𝓝[s] x, y < (fun x' => ⨆ i, f i x') x'
state_after: no goals