Module: Mathlib.Topology.Algebra.Semigroup

Goal state before:
pp: M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ ⋂₀ c = ⋂ i, ↑i

pp: case refine_1.inr.convert_1
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ Directed (fun x1 x2 ↦ x1 ⊇ x2) Subtype.val

pp: case refine_1.inr.convert_2
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ ∀ (i : ↑c), (↑i).Nonempty

pp: case refine_1.inr.convert_3
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ ∀ (i : ↑c), IsCompact ↑i

pp: case refine_1.inr.convert_4
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ ∀ (i : ↑c), IsClosed ↑i

Tactic:
· exact Set.sInter_eq_iInter

Goal state after:
case refine_1.inr.convert_1
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ Directed (fun x1 x2 ↦ x1 ⊇ x2) Subtype.val

case refine_1.inr.convert_2
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ ∀ (i : ↑c), (↑i).Nonempty

case refine_1.inr.convert_3
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ ∀ (i : ↑c), IsCompact ↑i

case refine_1.inr.convert_4
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ ∀ (i : ↑c), IsClosed ↑i