Theorem: exists_idempotent_of_compact_t2_of_continuous_mul_left
File: Mathlib/Topology/Algebra/Semigroup.lean

Proof steps:
state_before: M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_mul_left : ∀ (r : M), Continuous fun x => x * r
⊢ ∃ m, m * m = m
state_after: M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_mul_left : ∀ (r : M), Continuous fun x => x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
⊢ ∃ m, m * m = m
tactic: let S : Set (Set M) :=
    { N | IsClosed N ∧ N.Nonempty ∧ ∀ (m) (_ : m ∈ N) (m') (_ : m' ∈ N), m * m' ∈ N }
state_before: M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_mul_left : ∀ (r : M), Continuous fun x => x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
⊢ ∃ m, m * m = m
state_after: case intro.intro.intro.intro.intro
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_mul_left : ∀ (r : M), Continuous fun x => x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
N_minimal : ∀ N' ∈ S, N' ⊆ N → N' = N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
⊢ ∃ m, m * m = m

M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_mul_left : ∀ (r : M), Continuous fun x => x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
⊢ ∃ N ∈ S, ∀ N' ∈ S, N' ⊆ N → N' = N
tactic: rsuffices ⟨N, ⟨N_closed, ⟨m, hm⟩, N_mul⟩, N_minimal⟩ : ∃ N ∈ S, ∀ N' ∈ S, N' ⊆ N → N' = N
state_before: M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_mul_left : ∀ (r : M), Continuous fun x => x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
⊢ ∃ N ∈ S, ∀ N' ∈ S, N' ⊆ N → N' = N
state_after: M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_mul_left : ∀ (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 x x_1 => x ⊆ x_1) c
⊢ ∃ lb ∈ S, ∀ s ∈ c, lb ⊆ s
tactic: refine zorn_superset _ fun c hcs hc => ?_
state_before: M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_mul_left : ∀ (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 x x_1 => x ⊆ x_1) c
⊢ ∃ lb ∈ S, ∀ s ∈ c, lb ⊆ s
state_after: case refine_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_mul_left : ∀ (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 x x_1 => x ⊆ x_1) c
⊢ (⋂₀ c).Nonempty

case refine_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_mul_left : ∀ (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 x x_1 => x ⊆ x_1) c
m : M
hm : m ∈ ⋂₀ c
m' : M
hm' : m' ∈ ⋂₀ c
⊢ m * m' ∈ ⋂₀ c
tactic: refine
    ⟨⋂₀ c, ⟨isClosed_sInter fun t ht => (hcs ht).1, ?_, fun m hm m' hm' => ?_⟩, fun s hs =>
      Set.sInter_subset_of_mem hs⟩