Theorem: CompactT2.Projective.extremallyDisconnected
File: Mathlib/Topology/ExtremallyDisconnected.lean

tactic: refine { open_closure := fun U hU => ?_ }
state_before: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
⊢ ExtremallyDisconnected X
state_after: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
⊢ IsOpen (closure U)

tactic: let Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
state_before: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
⊢ IsOpen (closure U)
state_after: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
⊢ IsOpen (closure U)

tactic: let Z₂ : Set (X × Bool) := closure U ×ˢ {false}
state_before: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
⊢ IsOpen (closure U)
state_after: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
⊢ IsOpen (closure U)

tactic: let Z : Set (X × Bool) := Z₁ ∪ Z₂
state_before: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
⊢ IsOpen (closure U)
state_after: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
⊢ IsOpen (closure U)

tactic: have hZ₁₂ : Disjoint Z₁ Z₂ := disjoint_left.2 fun x hx₁ hx₂ => by cases hx₁.2.symm.trans hx₂.2
state_before: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
⊢ IsOpen (closure U)
state_after: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
⊢ IsOpen (closure U)

tactic: have hZ₁ : IsClosed Z₁ := hU.isClosed_compl.prod (T1Space.t1 _)
state_before: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
⊢ IsOpen (closure U)
state_after: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
⊢ IsOpen (closure U)

tactic: have hZ₂ : IsClosed Z₂ := isClosed_closure.prod (T1Space.t1 false)
state_before: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
⊢ IsOpen (closure U)
state_after: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
⊢ IsOpen (closure U)

tactic: have hZ : IsClosed Z := hZ₁.union hZ₂
state_before: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
⊢ IsOpen (closure U)
state_after: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
⊢ IsOpen (closure U)

tactic: let f : Z → X := Prod.fst ∘ Subtype.val
state_before: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
⊢ IsOpen (closure U)
state_after: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
⊢ IsOpen (closure U)

tactic: have f_cont : Continuous f := continuous_fst.comp continuous_subtype_val
state_before: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
⊢ IsOpen (closure U)
state_after: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
⊢ IsOpen (closure U)

tactic: haveI : CompactSpace Z := isCompact_iff_compactSpace.mp hZ.isCompact
state_before: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
⊢ IsOpen (closure U)
state_after: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
⊢ IsOpen (closure U)

tactic: obtain ⟨g, hg, g_sec⟩ := h continuous_id f_cont f_sur
state_before: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
⊢ IsOpen (closure U)
state_after: case intro.intro
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
⊢ IsOpen (closure U)

tactic: let φ := Subtype.val ∘ g
state_before: case intro.intro
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
⊢ IsOpen (closure U)
state_after: case intro.intro
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
⊢ IsOpen (closure U)

tactic: have hφ : Continuous φ := continuous_subtype_val.comp hg
state_before: case intro.intro
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
⊢ IsOpen (closure U)
state_after: case intro.intro
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ : Continuous φ
⊢ IsOpen (closure U)

tactic: have hφ₁ : ∀ x, (φ x).1 = x := congr_fun g_sec
state_before: case intro.intro
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ : Continuous φ
⊢ IsOpen (closure U)
state_after: case intro.intro
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
⊢ IsOpen (closure U)

tactic: refine (closure_minimal ?_ <| hZ₂.preimage hφ).antisymm fun x hx => ?_
state_before: case intro.intro
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
⊢ closure U = φ ⁻¹' Z₂
state_after: case intro.intro.refine_1
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
⊢ U ⊆ φ ⁻¹' Z₂

case intro.intro.refine_2
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
x : X
hx : x ∈ φ ⁻¹' Z₂
⊢ x ∈ closure U

tactic: cases hx₁.2.symm.trans hx₂.2
state_before: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
x : X × Bool
hx₁ : x ∈ Z₁
hx₂ : x ∈ Z₂
⊢ False
state_after: no goals

tactic: intro x
state_before: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
⊢ Surjective f
state_after: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
x : X
⊢ ∃ a, f a = x

tactic: by_cases hx : x ∈ U
state_before: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
x : X
⊢ ∃ a, f a = x
state_after: case pos
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
x : X
hx : x ∈ U
⊢ ∃ a, f a = x

case neg
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
x : X
hx : x ∉ U
⊢ ∃ a, f a = x

tactic: exact ⟨⟨(x, false), Or.inr ⟨subset_closure hx, mem_singleton _⟩⟩, rfl⟩
state_before: case pos
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
x : X
hx : x ∈ U
⊢ ∃ a, f a = x
state_after: no goals

tactic: exact ⟨⟨(x, true), Or.inl ⟨hx, mem_singleton _⟩⟩, rfl⟩
state_before: case neg
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
x : X
hx : x ∉ U
⊢ ∃ a, f a = x
state_after: no goals

tactic: rw [this, preimage_comp, ← isClosed_compl_iff, ← preimage_compl,
  ← preimage_subtype_coe_eq_compl Subset.rfl]
state_before: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this✝ : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
this : closure U = φ ⁻¹' Z₂
⊢ IsOpen (closure U)
state_after: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this✝ : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
this : closure U = φ ⁻¹' Z₂
⊢ IsClosed (g ⁻¹' (Subtype.val ⁻¹' Z₁))

X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this✝ : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
this : closure U = φ ⁻¹' Z₂
⊢ (Z₁ ∪ Z₂) ∩ (Z₁ ∩ Z₂) = ∅

tactic: exact hZ₁.preimage hφ
state_before: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this✝ : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
this : closure U = φ ⁻¹' Z₂
⊢ IsClosed (g ⁻¹' (Subtype.val ⁻¹' Z₁))
state_after: no goals

tactic: rw [hZ₁₂.inter_eq, inter_empty]
state_before: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this✝ : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
this : closure U = φ ⁻¹' Z₂
⊢ (Z₁ ∪ Z₂) ∩ (Z₁ ∩ Z₂) = ∅
state_after: no goals

tactic: intro x hx
state_before: case intro.intro.refine_1
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
⊢ U ⊆ φ ⁻¹' Z₂
state_after: case intro.intro.refine_1
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
x : X
hx : x ∈ U
⊢ x ∈ φ ⁻¹' Z₂

tactic: have : φ x ∈ Z₁ ∪ Z₂ := (g x).2
state_before: case intro.intro.refine_1
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
x : X
hx : x ∈ U
⊢ x ∈ φ ⁻¹' Z₂
state_after: case intro.intro.refine_1
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this✝ : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
x : X
hx : x ∈ U
this : φ x ∈ Z₁ ∪ Z₂
⊢ x ∈ φ ⁻¹' Z₂

tactic: cases' this with hφ hφ
state_before: case intro.intro.refine_1
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this✝ : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
x : X
hx : x ∈ U
this : φ x ∈ Z₁ ∪ Z₂
⊢ x ∈ φ ⁻¹' Z₂
state_after: case intro.intro.refine_1.inl
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ✝ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
x : X
hx : x ∈ U
hφ : φ x ∈ Z₁
⊢ x ∈ φ ⁻¹' Z₂

case intro.intro.refine_1.inr
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ✝ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
x : X
hx : x ∈ U
hφ : φ x ∈ Z₂
⊢ x ∈ φ ⁻¹' Z₂

tactic: exact ((hφ₁ x ▸ hφ.1) hx).elim
state_before: case intro.intro.refine_1.inl
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ✝ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
x : X
hx : x ∈ U
hφ : φ x ∈ Z₁
⊢ x ∈ φ ⁻¹' Z₂
state_after: no goals

tactic: exact hφ
state_before: case intro.intro.refine_1.inr
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ✝ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
x : X
hx : x ∈ U
hφ : φ x ∈ Z₂
⊢ x ∈ φ ⁻¹' Z₂
state_after: no goals

tactic: rw [← hφ₁ x]
state_before: case intro.intro.refine_2
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
x : X
hx : x ∈ φ ⁻¹' Z₂
⊢ x ∈ closure U
state_after: case intro.intro.refine_2
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
x : X
hx : x ∈ φ ⁻¹' Z₂
⊢ (φ x).1 ∈ closure U

tactic: exact hx.1
state_before: case intro.intro.refine_2
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed Z
f : ↑Z → X := Prod.fst ∘ Subtype.val
f_cont : Continuous f
f_sur : Surjective f
this : CompactSpace ↑Z
g : X → ↑Z
hg : Continuous g
g_sec : f ∘ g = id
φ : X → X × Bool := Subtype.val ∘ g
hφ : Continuous φ
hφ₁ : ∀ (x : X), (φ x).1 = x
x : X
hx : x ∈ φ ⁻¹' Z₂
⊢ (φ x).1 ∈ closure U
state_after: no goals