case mp
M : Type u_1
G : Type ?u.17773
α : Type u_2
inst✝⁵ : Monoid M
inst✝⁴ : Group G
inst✝³ : TopologicalSpace α
inst✝² : MulAction M α
inst✝¹ : MulAction G α
inst✝ : ContinuousConstSMul M α
⊢ IsMinimal M α → ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ

case mpr
M : Type u_1
G : Type ?u.17773
α : Type u_2
inst✝⁵ : Monoid M
inst✝⁴ : Group G
inst✝³ : TopologicalSpace α
inst✝² : MulAction M α
inst✝¹ : MulAction G α
inst✝ : ContinuousConstSMul M α
⊢ (∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ) → IsMinimal M α
M : Type u_1
G : Type ?u.17773
α : Type u_2
inst✝⁵ : Monoid M
inst✝⁴ : Group G
inst✝³ : TopologicalSpace α
inst✝² : MulAction M α
inst✝¹ : MulAction G α
inst✝ : ContinuousConstSMul M α
⊢ IsMinimal M α ↔ ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ
constructor
case mpr.refine'_1
M : Type u_1
G : Type ?u.17773
α : Type u_2
inst✝⁵ : Monoid M
inst✝⁴ : Group G
inst✝³ : TopologicalSpace α
inst✝² : MulAction M α
inst✝¹ : MulAction G α
inst✝ : ContinuousConstSMul M α
H : ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ
x✝ : α
⊢ IsClosed (closure (orbit M x✝))

case mpr.refine'_2
M : Type u_1
G : Type ?u.17773
α : Type u_2
inst✝⁵ : Monoid M
inst✝⁴ : Group G
inst✝³ : TopologicalSpace α
inst✝² : MulAction M α
inst✝¹ : MulAction G α
inst✝ : ContinuousConstSMul M α
H : ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ
x✝ : α
⊢ ∀ (c : M), c • closure (orbit M x✝) ⊆ closure (orbit M x✝)

case mpr.refine'_3
M : Type u_1
G : Type ?u.17773
α : Type u_2
inst✝⁵ : Monoid M
inst✝⁴ : Group G
inst✝³ : TopologicalSpace α
inst✝² : MulAction M α
inst✝¹ : MulAction G α
inst✝ : ContinuousConstSMul M α
H : ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ
x✝ : α
⊢ ¬closure (orbit M x✝) = ∅
case mpr
M : Type u_1
G : Type ?u.17773
α : Type u_2
inst✝⁵ : Monoid M
inst✝⁴ : Group G
inst✝³ : TopologicalSpace α
inst✝² : MulAction M α
inst✝¹ : MulAction G α
inst✝ : ContinuousConstSMul M α
⊢ (∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ) → IsMinimal M α
refine' fun H ↦ ⟨fun _ ↦ dense_iff_closure_eq.2 <| (H _ _ _).resolve_left _⟩
no goals
case mpr.refine'_1
M : Type u_1
G : Type ?u.17773
α : Type u_2
inst✝⁵ : Monoid M
inst✝⁴ : Group G
inst✝³ : TopologicalSpace α
inst✝² : MulAction M α
inst✝¹ : MulAction G α
inst✝ : ContinuousConstSMul M α
H : ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ
x✝ : α
⊢ IsClosed (closure (orbit M x✝))

case mpr.refine'_2
M : Type u_1
G : Type ?u.17773
α : Type u_2
inst✝⁵ : Monoid M
inst✝⁴ : Group G
inst✝³ : TopologicalSpace α
inst✝² : MulAction M α
inst✝¹ : MulAction G α
inst✝ : ContinuousConstSMul M α
H : ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ
x✝ : α
⊢ ∀ (c : M), c • closure (orbit M x✝) ⊆ closure (orbit M x✝)

case mpr.refine'_3
M : Type u_1
G : Type ?u.17773
α : Type u_2
inst✝⁵ : Monoid M
inst✝⁴ : Group G
inst✝³ : TopologicalSpace α
inst✝² : MulAction M α
inst✝¹ : MulAction G α
inst✝ : ContinuousConstSMul M α
H : ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ
x✝ : α
⊢ ¬closure (orbit M x✝) = ∅
exacts [isClosed_closure, fun _ ↦ smul_closure_orbit_subset _ _,
  (orbit_nonempty _).closure.ne_empty]
case mp
M : Type u_1
G : Type ?u.17773
α : Type u_2
inst✝⁵ : Monoid M
inst✝⁴ : Group G
inst✝³ : TopologicalSpace α
inst✝² : MulAction M α
inst✝¹ : MulAction G α
inst✝ : ContinuousConstSMul M α
a✝ : IsMinimal M α
s✝ : Set α
⊢ IsClosed s✝ → (∀ (c : M), c • s✝ ⊆ s✝) → s✝ = ∅ ∨ s✝ = univ
case mp
M : Type u_1
G : Type ?u.17773
α : Type u_2
inst✝⁵ : Monoid M
inst✝⁴ : Group G
inst✝³ : TopologicalSpace α
inst✝² : MulAction M α
inst✝¹ : MulAction G α
inst✝ : ContinuousConstSMul M α
⊢ IsMinimal M α → ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ
intro _ _
no goals
case mp
M : Type u_1
G : Type ?u.17773
α : Type u_2
inst✝⁵ : Monoid M
inst✝⁴ : Group G
inst✝³ : TopologicalSpace α
inst✝² : MulAction M α
inst✝¹ : MulAction G α
inst✝ : ContinuousConstSMul M α
a✝ : IsMinimal M α
s✝ : Set α
⊢ IsClosed s✝ → (∀ (c : M), c • s✝ ⊆ s✝) → s✝ = ∅ ∨ s✝ = univ
exact eq_empty_or_univ_of_smul_invariant_closed M