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SetTheory.PGame.Domineering.card_of_mem_right ** b : Board m : ℤ × ℤ h : m ∈ right b ⊢ 2 ≤ Finset.card b ** have w₁ : m ∈ b := (Finset.mem_inter.1 h).1 ** b : Board m : ℤ × ℤ h : m ∈ right b w₁ : m ∈ b ⊢ 2 ≤ Finset.card b ** have w₂ := fst_pred_mem_erase_of_mem_right h ** b : Board m : ℤ × ℤ h : m ∈ right b w₁ : m ∈ b w₂ : (m.1 - 1, m.2) ∈ Finset.erase b m ⊢ 2 ≤ Finset.card b ** have i₁ := Finset.card_erase_lt_of_mem w₁ ** b : Board m : ℤ × ℤ h : m ∈ right b w₁ : m ∈ b w₂ : (m.1 - 1, m.2) ∈ Finset.erase b m i₁ : Finset.card (Finset.erase b m) < Finset.card b ⊢ 2 ≤ Finset.card b ** have i₂ := Nat.lt_of_le_of_lt (Nat.zero_le _) (Finset.card_erase_lt_of_mem w₂) ** b : Board m : ℤ × ℤ h : m ∈ right b w₁ : m ∈ b w₂ : (m.1 - 1, m.2) ∈ Finset.erase b m i₁ : Finset.card (Finset.erase b m) < Finset.card b i₂ : 0 < Finset.card (Finset.erase b m) ⊢ 2 ≤ Finset.card b ** exact Nat.lt_of_le_of_lt i₂ i₁ ** Qed
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SetTheory.PGame.Domineering.moveLeft_card ** b : Board m : ℤ × ℤ h : m ∈ left b ⊢ Finset.card (moveLeft b m) + 2 = Finset.card b ** dsimp [moveLeft] ** b : Board m : ℤ × ℤ h : m ∈ left b ⊢ Finset.card (Finset.erase (Finset.erase b m) (m.1, m.2 - 1)) + 2 = Finset.card b ** rw [Finset.card_erase_of_mem (snd_pred_mem_erase_of_mem_left h)] ** b : Board m : ℤ × ℤ h : m ∈ left b ⊢ Finset.card (Finset.erase b m) - 1 + 2 = Finset.card b ** rw [Finset.card_erase_of_mem (Finset.mem_of_mem_inter_left h)] ** b : Board m : ℤ × ℤ h : m ∈ left b ⊢ Finset.card b - 1 - 1 + 2 = Finset.card b ** exact tsub_add_cancel_of_le (card_of_mem_left h) ** Qed
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SetTheory.PGame.Domineering.moveRight_card ** b : Board m : ℤ × ℤ h : m ∈ right b ⊢ Finset.card (moveRight b m) + 2 = Finset.card b ** dsimp [moveRight] ** b : Board m : ℤ × ℤ h : m ∈ right b ⊢ Finset.card (Finset.erase (Finset.erase b m) (m.1 - 1, m.2)) + 2 = Finset.card b ** rw [Finset.card_erase_of_mem (fst_pred_mem_erase_of_mem_right h)] ** b : Board m : ℤ × ℤ h : m ∈ right b ⊢ Finset.card (Finset.erase b m) - 1 + 2 = Finset.card b ** rw [Finset.card_erase_of_mem (Finset.mem_of_mem_inter_left h)] ** b : Board m : ℤ × ℤ h : m ∈ right b ⊢ Finset.card b - 1 - 1 + 2 = Finset.card b ** exact tsub_add_cancel_of_le (card_of_mem_right h) ** Qed
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SetTheory.PGame.Domineering.moveLeft_smaller ** b : Board m : ℤ × ℤ h : m ∈ left b ⊢ Finset.card (moveLeft b m) / 2 < Finset.card b / 2 ** simp [← moveLeft_card h, lt_add_one] ** Qed
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SetTheory.PGame.Domineering.moveRight_smaller ** b : Board m : ℤ × ℤ h : m ∈ right b ⊢ Finset.card (moveRight b m) / 2 < Finset.card b / 2 ** simp [← moveRight_card h, lt_add_one] ** Qed
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Lists'.toList_cons ** α : Type u_1 a : Lists α l : Lists' α true ⊢ toList (cons a l) = a :: toList l ** simp ** Qed
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Lists'.to_ofList ** α : Type u_1 l : List (Lists α) ⊢ toList (ofList l) = l ** induction l <;> simp [*] ** Qed
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Lists'.of_toList ** α : Type u_1 b : Bool h : true = b l : Lists' α b ⊢ Lists' α true ** rw [h] ** α : Type u_1 b : Bool h : true = b l : Lists' α b ⊢ Lists' α b ** exact l ** α : Type u_1 b : Bool h : true = b l : Lists' α b ⊢ let l' := Eq.mpr (_ : Lists' α true = Lists' α b) l; ofList (toList l') = l' ** induction l with
| atom => cases h
| nil => simp
| cons' b a _ IH =>
intro l'
simpa [cons] using IH rfl ** case atom α : Type u_1 b : Bool a✝ : α h : true = false ⊢ let l' := Eq.mpr (_ : Lists' α true = Lists' α false) (atom a✝); ofList (toList l') = l' ** cases h ** case nil α : Type u_1 b : Bool h : true = true ⊢ let l' := Eq.mpr (_ : Lists' α true = Lists' α true) nil; ofList (toList l') = l' ** simp ** case cons' α : Type u_1 b✝¹ b✝ : Bool b : Lists' α b✝ a : Lists' α true a_ih✝ : ∀ (h : true = b✝), let l' := Eq.mpr (_ : Lists' α true = Lists' α b✝) b; ofList (toList l') = l' IH : ∀ (h : true = true), let l' := Eq.mpr (_ : Lists' α true = Lists' α true) a; ofList (toList l') = l' h : true = true ⊢ let l' := Eq.mpr (_ : Lists' α true = Lists' α true) (cons' b a); ofList (toList l') = l' ** intro l' ** case cons' α : Type u_1 b✝¹ b✝ : Bool b : Lists' α b✝ a : Lists' α true a_ih✝ : ∀ (h : true = b✝), let l' := Eq.mpr (_ : Lists' α true = Lists' α b✝) b; ofList (toList l') = l' IH : ∀ (h : true = true), let l' := Eq.mpr (_ : Lists' α true = Lists' α true) a; ofList (toList l') = l' h : true = true l' : Lists' α true := Eq.mpr (_ : Lists' α true = Lists' α true) (cons' b a) ⊢ ofList (toList l') = l' ** simpa [cons] using IH rfl ** Qed
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Lists'.mem_cons ** α : Type u_1 a y : Lists α l : Lists' α true ⊢ a ∈ cons y l ↔ a ~ y ∨ a ∈ l ** simp [mem_def, or_and_right, exists_or] ** Qed
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Lists'.cons_subset ** α : Type u_1 a : Lists α l₁ l₂ : Lists' α true ⊢ cons a l₁ ⊆ l₂ ↔ a ∈ l₂ ∧ l₁ ⊆ l₂ ** refine' ⟨fun h => _, fun ⟨⟨a', m, e⟩, s⟩ => Subset.cons e m s⟩ ** α : Type u_1 a : Lists α l₁ l₂ : Lists' α true h : cons a l₁ ⊆ l₂ ⊢ a ∈ l₂ ∧ l₁ ⊆ l₂ ** generalize h' : Lists'.cons a l₁ = l₁' at h ** α : Type u_1 a : Lists α l₁ l₂ l₁' : Lists' α true h' : cons a l₁ = l₁' h : l₁' ⊆ l₂ ⊢ a ∈ l₂ ∧ l₁ ⊆ l₂ ** cases' h with l a' a'' l l' e m s ** case cons α : Type u_1 a : Lists α l₁ l₂ : Lists' α true a' a'' : Lists α l : Lists' α true e : a' ~ a'' h' : cons a l₁ = cons a' l m : a'' ∈ toList l₂ s : Subset l l₂ ⊢ a ∈ l₂ ∧ l₁ ⊆ l₂ ** cases a ** case cons.mk α : Type u_1 l₁ l₂ : Lists' α true a' a'' : Lists α l : Lists' α true e : a' ~ a'' m : a'' ∈ toList l₂ s : Subset l l₂ fst✝ : Bool snd✝ : Lists' α fst✝ h' : cons { fst := fst✝, snd := snd✝ } l₁ = cons a' l ⊢ { fst := fst✝, snd := snd✝ } ∈ l₂ ∧ l₁ ⊆ l₂ ** cases a' ** case cons.mk.mk α : Type u_1 l₁ l₂ : Lists' α true a'' : Lists α l : Lists' α true m : a'' ∈ toList l₂ s : Subset l l₂ fst✝¹ : Bool snd✝¹ : Lists' α fst✝¹ fst✝ : Bool snd✝ : Lists' α fst✝ e : { fst := fst✝, snd := snd✝ } ~ a'' h' : cons { fst := fst✝¹, snd := snd✝¹ } l₁ = cons { fst := fst✝, snd := snd✝ } l ⊢ { fst := fst✝¹, snd := snd✝¹ } ∈ l₂ ∧ l₁ ⊆ l₂ ** cases h' ** case cons.mk.mk.refl α : Type u_1 l₁ l₂ : Lists' α true a'' : Lists α m : a'' ∈ toList l₂ fst✝ : Bool snd✝ : Lists' α fst✝ e : { fst := fst✝, snd := snd✝ } ~ a'' s : Subset l₁ l₂ ⊢ { fst := fst✝, snd := snd✝ } ∈ l₂ ∧ l₁ ⊆ l₂ ** exact ⟨⟨_, m, e⟩, s⟩ ** case nil α : Type u_1 a : Lists α l₁ l₂ : Lists' α true h' : cons a l₁ = nil ⊢ a ∈ l₂ ∧ l₁ ⊆ l₂ ** cases a ** case nil.mk α : Type u_1 l₁ l₂ : Lists' α true fst✝ : Bool snd✝ : Lists' α fst✝ h' : cons { fst := fst✝, snd := snd✝ } l₁ = nil ⊢ { fst := fst✝, snd := snd✝ } ∈ l₂ ∧ l₁ ⊆ l₂ ** cases h' ** Qed
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Lists'.ofList_subset ** α : Type u_1 l₁ l₂ : List (Lists α) h : l₁ ⊆ l₂ ⊢ ofList l₁ ⊆ ofList l₂ ** induction' l₁ with _ _ l₁_ih ** case cons α : Type u_1 l₁ l₂ : List (Lists α) h✝ : l₁ ⊆ l₂ head✝ : Lists α tail✝ : List (Lists α) l₁_ih : tail✝ ⊆ l₂ → ofList tail✝ ⊆ ofList l₂ h : head✝ :: tail✝ ⊆ l₂ ⊢ ofList (head✝ :: tail✝) ⊆ ofList l₂ ** refine' Subset.cons (Lists.Equiv.refl _) _ (l₁_ih (List.subset_of_cons_subset h)) ** case cons α : Type u_1 l₁ l₂ : List (Lists α) h✝ : l₁ ⊆ l₂ head✝ : Lists α tail✝ : List (Lists α) l₁_ih : tail✝ ⊆ l₂ → ofList tail✝ ⊆ ofList l₂ h : head✝ :: tail✝ ⊆ l₂ ⊢ head✝ ∈ toList (ofList l₂) ** simp at h ** case cons α : Type u_1 l₁ l₂ : List (Lists α) h✝ : l₁ ⊆ l₂ head✝ : Lists α tail✝ : List (Lists α) l₁_ih : tail✝ ⊆ l₂ → ofList tail✝ ⊆ ofList l₂ h : head✝ ∈ l₂ ∧ tail✝ ⊆ l₂ ⊢ head✝ ∈ toList (ofList l₂) ** simp [h] ** case nil α : Type u_1 l₁ l₂ : List (Lists α) h✝ : l₁ ⊆ l₂ h : [] ⊆ l₂ ⊢ ofList [] ⊆ ofList l₂ ** exact Subset.nil ** Qed
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Lists'.Subset.refl ** α : Type u_1 l : Lists' α true ⊢ l ⊆ l ** rw [← Lists'.of_toList l] ** α : Type u_1 l : Lists' α true ⊢ ofList (toList l) ⊆ ofList (toList l) ** exact ofList_subset (List.Subset.refl _) ** Qed
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Lists'.subset_nil ** α : Type u_1 l : Lists' α true ⊢ l ⊆ nil → l = nil ** rw [← of_toList l] ** α : Type u_1 l : Lists' α true ⊢ ofList (toList l) ⊆ nil → ofList (toList l) = nil ** induction toList l <;> intro h ** case nil α : Type u_1 l : Lists' α true h : ofList [] ⊆ nil ⊢ ofList [] = nil ** rfl ** case cons α : Type u_1 l : Lists' α true head✝ : Lists α tail✝ : List (Lists α) tail_ih✝ : ofList tail✝ ⊆ nil → ofList tail✝ = nil h : ofList (head✝ :: tail✝) ⊆ nil ⊢ ofList (head✝ :: tail✝) = nil ** rcases cons_subset.1 h with ⟨⟨_, ⟨⟩, _⟩, _⟩ ** Qed
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Lists'.mem_of_subset' ** α : Type u_1 a : Lists α x✝ : Lists' α true h : a ∈ toList nil ⊢ a ∈ x✝ ** cases h ** α : Type u_1 a : Lists α b✝ : Bool a0 : Lists' α b✝ l0 l₂ : Lists' α true s : cons' a0 l0 ⊆ l₂ h : a ∈ toList (cons' a0 l0) ⊢ a ∈ l₂ ** cases' s with _ _ _ _ _ e m s ** case cons α : Type u_1 a : Lists α l0 l₂ : Lists' α true a✝ a'✝ : Lists α e : a✝ ~ a'✝ h : a ∈ toList (cons' a✝.snd l0) m : a'✝ ∈ toList l₂ s : Subset l0 l₂ ⊢ a ∈ l₂ ** simp only [toList, Sigma.eta, List.find?, List.mem_cons] at h ** case cons α : Type u_1 a : Lists α l0 l₂ : Lists' α true a✝ a'✝ : Lists α e : a✝ ~ a'✝ m : a'✝ ∈ toList l₂ s : Subset l0 l₂ h : a = a✝ ∨ a ∈ toList l0 ⊢ a ∈ l₂ ** rcases h with (rfl | h) ** case cons.inl α : Type u_1 a : Lists α l0 l₂ : Lists' α true a'✝ : Lists α m : a'✝ ∈ toList l₂ s : Subset l0 l₂ e : a ~ a'✝ ⊢ a ∈ l₂ ** exact ⟨_, m, e⟩ ** case cons.inr α : Type u_1 a : Lists α l0 l₂ : Lists' α true a✝ a'✝ : Lists α e : a✝ ~ a'✝ m : a'✝ ∈ toList l₂ s : Subset l0 l₂ h : a ∈ toList l0 ⊢ a ∈ l₂ ** exact mem_of_subset' s h ** Qed
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Lists'.subset_def ** α : Type u_1 l₁ l₂ : Lists' α true H : ∀ (a : Lists α), a ∈ toList l₁ → a ∈ l₂ ⊢ l₁ ⊆ l₂ ** rw [← of_toList l₁] ** α : Type u_1 l₁ l₂ : Lists' α true H : ∀ (a : Lists α), a ∈ toList l₁ → a ∈ l₂ ⊢ ofList (toList l₁) ⊆ l₂ ** revert H ** α : Type u_1 l₁ l₂ : Lists' α true ⊢ (∀ (a : Lists α), a ∈ toList l₁ → a ∈ l₂) → ofList (toList l₁) ⊆ l₂ ** induction' toList l₁ with h t t_ih <;> intro H ** case nil α : Type u_1 l₁ l₂ : Lists' α true H : ∀ (a : Lists α), a ∈ [] → a ∈ l₂ ⊢ ofList [] ⊆ l₂ ** exact Subset.nil ** case cons α : Type u_1 l₁ l₂ : Lists' α true h : Lists α t : List (Lists α) t_ih : (∀ (a : Lists α), a ∈ t → a ∈ l₂) → ofList t ⊆ l₂ H : ∀ (a : Lists α), a ∈ h :: t → a ∈ l₂ ⊢ ofList (h :: t) ⊆ l₂ ** simp only [ofList, List.find?, List.mem_cons, forall_eq_or_imp] at * ** case cons α : Type u_1 l₁ l₂ : Lists' α true h : Lists α t : List (Lists α) t_ih : (∀ (a : Lists α), a ∈ t → a ∈ l₂) → ofList t ⊆ l₂ H : h ∈ l₂ ∧ ∀ (a : Lists α), a ∈ t → a ∈ l₂ ⊢ cons h (ofList t) ⊆ l₂ ** exact cons_subset.2 ⟨H.1, t_ih H.2⟩ ** Qed
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Lists.to_ofList ** α : Type u_1 l : List (Lists α) ⊢ toList (ofList l) = l ** simp [ofList, of'] ** Qed
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Lists.of_toList ** α : Type u_1 l : Lists' α true x✝ : IsList { fst := true, snd := l } ⊢ ofList (toList { fst := true, snd := l }) = { fst := true, snd := l } ** simp_all [ofList, of'] ** Qed
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Lists.Equiv.antisymm_iff ** α : Type u_1 l₁ l₂ : Lists' α true ⊢ of' l₁ ~ of' l₂ ↔ l₁ ⊆ l₂ ∧ l₂ ⊆ l₁ ** refine' ⟨fun h => _, fun ⟨h₁, h₂⟩ => Equiv.antisymm h₁ h₂⟩ ** α : Type u_1 l₁ l₂ : Lists' α true h : of' l₁ ~ of' l₂ ⊢ l₁ ⊆ l₂ ∧ l₂ ⊆ l₁ ** cases' h with _ _ _ h₁ h₂ ** case refl α : Type u_1 l₁ : Lists' α true ⊢ l₁ ⊆ l₁ ∧ l₁ ⊆ l₁ ** simp [Lists'.Subset.refl] ** case antisymm α : Type u_1 l₁ l₂ : Lists' α true h₁ : Lists'.Subset l₁ l₂ h₂ : Lists'.Subset l₂ l₁ ⊢ l₁ ⊆ l₂ ∧ l₂ ⊆ l₁ ** exact ⟨h₁, h₂⟩ ** Qed
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Lists.equiv_atom ** α : Type u_1 a : α l : Lists α h : atom a ~ l ⊢ atom a = l ** cases h ** case refl α : Type u_1 a : α ⊢ atom a = atom a ** rfl ** Qed
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Lists.Equiv.symm ** α : Type u_1 l₁ l₂ : Lists α h : l₁ ~ l₂ ⊢ l₂ ~ l₁ ** cases' h with _ _ _ h₁ h₂ <;> [rfl; exact Equiv.antisymm h₂ h₁] ** Qed
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Lists.Equiv.trans ** α : Type u_1 ⊢ ∀ {l₁ l₂ l₃ : Lists α}, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ ** let trans := fun l₁ : Lists α => ∀ ⦃l₂ l₃⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ ** α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ ⊢ ∀ {l₁ l₂ l₃ : Lists α}, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ ** suffices PProd (∀ l₁, trans l₁) (∀ (l : Lists' α true), ∀ l' ∈ l.toList, trans l') by exact this.1 ** α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ ⊢ PProd (∀ (l₁ : Lists α), trans l₁) (∀ (l : Lists' α true) (l' : Lists α), l' ∈ Lists'.toList l → trans l') ** apply inductionMut ** α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ this : PProd (∀ (l₁ : Lists α), trans l₁) (∀ (l : Lists' α true) (l' : Lists α), l' ∈ Lists'.toList l → trans l') ⊢ ∀ {l₁ l₂ l₃ : Lists α}, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ ** exact this.1 ** case C0 α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ ⊢ ∀ (a : α), trans (atom a) ** intro a l₂ l₃ h₁ h₂ ** case C0 α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ a : α l₂ l₃ : Lists α h₁ : atom a ~ l₂ h₂ : l₂ ~ l₃ ⊢ atom a ~ l₃ ** rwa [← equiv_atom.1 h₁] at h₂ ** case C1 α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ ⊢ ∀ (l : Lists' α true), (∀ (l' : Lists α), l' ∈ Lists'.toList l → trans l') → trans (of' l) ** intro l₁ IH l₂ l₃ h₁ h₂ ** case C1 α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ l₁ : Lists' α true IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l₁ → trans l' l₂ l₃ : Lists α h₁ : of' l₁ ~ l₂ h₂ : l₂ ~ l₃ ⊢ of' l₁ ~ l₃ ** have h₁' := h₁ ** case C1 α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ l₁ : Lists' α true IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l₁ → trans l' l₂ l₃ : Lists α h₁ : of' l₁ ~ l₂ h₂ : l₂ ~ l₃ h₁' : of' l₁ ~ l₂ ⊢ of' l₁ ~ l₃ ** have h₂' := h₂ ** case C1 α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ l₁ : Lists' α true IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l₁ → trans l' l₂ l₃ : Lists α h₁ : of' l₁ ~ l₂ h₂ : l₂ ~ l₃ h₁' : of' l₁ ~ l₂ h₂' : l₂ ~ l₃ ⊢ of' l₁ ~ l₃ ** cases' h₁ with _ _ l₂ ** case C1.antisymm α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ l₁ : Lists' α true IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l₁ → trans l' l₃ : Lists α l₂ : Lists' α true a✝¹ : Lists'.Subset l₁ l₂ a✝ : Lists'.Subset l₂ l₁ h₂ : { fst := true, snd := l₂ } ~ l₃ h₁' : of' l₁ ~ { fst := true, snd := l₂ } h₂' : { fst := true, snd := l₂ } ~ l₃ ⊢ of' l₁ ~ l₃ ** cases' h₂ with _ _ l₃ ** case C1.antisymm.antisymm α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ l₁ : Lists' α true IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l₁ → trans l' l₂ : Lists' α true a✝³ : Lists'.Subset l₁ l₂ a✝² : Lists'.Subset l₂ l₁ h₁' : of' l₁ ~ { fst := true, snd := l₂ } l₃ : Lists' α true a✝¹ : Lists'.Subset l₂ l₃ a✝ : Lists'.Subset l₃ l₂ h₂' : { fst := true, snd := l₂ } ~ { fst := true, snd := l₃ } ⊢ of' l₁ ~ { fst := true, snd := l₃ } ** cases' Equiv.antisymm_iff.1 h₁' with hl₁ hr₁ ** case C1.antisymm.antisymm.intro α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ l₁ : Lists' α true IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l₁ → trans l' l₂ : Lists' α true a✝³ : Lists'.Subset l₁ l₂ a✝² : Lists'.Subset l₂ l₁ h₁' : of' l₁ ~ { fst := true, snd := l₂ } l₃ : Lists' α true a✝¹ : Lists'.Subset l₂ l₃ a✝ : Lists'.Subset l₃ l₂ h₂' : { fst := true, snd := l₂ } ~ { fst := true, snd := l₃ } hl₁ : l₁ ⊆ l₂ hr₁ : l₂ ⊆ l₁ ⊢ of' l₁ ~ { fst := true, snd := l₃ } ** cases' Equiv.antisymm_iff.1 h₂' with hl₂ hr₂ ** case C1.antisymm.antisymm.intro.intro α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ l₁ : Lists' α true IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l₁ → trans l' l₂ : Lists' α true a✝³ : Lists'.Subset l₁ l₂ a✝² : Lists'.Subset l₂ l₁ h₁' : of' l₁ ~ { fst := true, snd := l₂ } l₃ : Lists' α true a✝¹ : Lists'.Subset l₂ l₃ a✝ : Lists'.Subset l₃ l₂ h₂' : { fst := true, snd := l₂ } ~ { fst := true, snd := l₃ } hl₁ : l₁ ⊆ l₂ hr₁ : l₂ ⊆ l₁ hl₂ : l₂ ⊆ l₃ hr₂ : l₃ ⊆ l₂ ⊢ of' l₁ ~ { fst := true, snd := l₃ } ** apply Equiv.antisymm_iff.2 ** case C1.antisymm.antisymm.intro.intro α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ l₁ : Lists' α true IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l₁ → trans l' l₂ : Lists' α true a✝³ : Lists'.Subset l₁ l₂ a✝² : Lists'.Subset l₂ l₁ h₁' : of' l₁ ~ { fst := true, snd := l₂ } l₃ : Lists' α true a✝¹ : Lists'.Subset l₂ l₃ a✝ : Lists'.Subset l₃ l₂ h₂' : { fst := true, snd := l₂ } ~ { fst := true, snd := l₃ } hl₁ : l₁ ⊆ l₂ hr₁ : l₂ ⊆ l₁ hl₂ : l₂ ⊆ l₃ hr₂ : l₃ ⊆ l₂ ⊢ l₁ ⊆ l₃ ∧ l₃ ⊆ l₁ ** constructor <;> apply Lists'.subset_def.2 ** case C1.refl α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ l₁ : Lists' α true IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l₁ → trans l' l₃ : Lists α h₂ : of' l₁ ~ l₃ h₁' : of' l₁ ~ of' l₁ h₂' : of' l₁ ~ l₃ ⊢ of' l₁ ~ l₃ ** exact h₂ ** case C1.antisymm.refl α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ l₁ : Lists' α true IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l₁ → trans l' l₂ : Lists' α true a✝¹ : Lists'.Subset l₁ l₂ a✝ : Lists'.Subset l₂ l₁ h₁' : of' l₁ ~ { fst := true, snd := l₂ } h₂' : { fst := true, snd := l₂ } ~ { fst := true, snd := l₂ } ⊢ of' l₁ ~ { fst := true, snd := l₂ } ** exact h₁' ** case C1.antisymm.antisymm.intro.intro.left α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ l₁ : Lists' α true IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l₁ → trans l' l₂ : Lists' α true a✝³ : Lists'.Subset l₁ l₂ a✝² : Lists'.Subset l₂ l₁ h₁' : of' l₁ ~ { fst := true, snd := l₂ } l₃ : Lists' α true a✝¹ : Lists'.Subset l₂ l₃ a✝ : Lists'.Subset l₃ l₂ h₂' : { fst := true, snd := l₂ } ~ { fst := true, snd := l₃ } hl₁ : l₁ ⊆ l₂ hr₁ : l₂ ⊆ l₁ hl₂ : l₂ ⊆ l₃ hr₂ : l₃ ⊆ l₂ ⊢ ∀ (a : Lists α), a ∈ Lists'.toList l₁ → a ∈ l₃ ** intro a₁ m₁ ** case C1.antisymm.antisymm.intro.intro.left α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ l₁ : Lists' α true IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l₁ → trans l' l₂ : Lists' α true a✝³ : Lists'.Subset l₁ l₂ a✝² : Lists'.Subset l₂ l₁ h₁' : of' l₁ ~ { fst := true, snd := l₂ } l₃ : Lists' α true a✝¹ : Lists'.Subset l₂ l₃ a✝ : Lists'.Subset l₃ l₂ h₂' : { fst := true, snd := l₂ } ~ { fst := true, snd := l₃ } hl₁ : l₁ ⊆ l₂ hr₁ : l₂ ⊆ l₁ hl₂ : l₂ ⊆ l₃ hr₂ : l₃ ⊆ l₂ a₁ : Lists α m₁ : a₁ ∈ Lists'.toList l₁ ⊢ a₁ ∈ l₃ ** rcases Lists'.mem_of_subset' hl₁ m₁ with ⟨a₂, m₂, e₁₂⟩ ** case C1.antisymm.antisymm.intro.intro.left.intro.intro α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ l₁ : Lists' α true IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l₁ → trans l' l₂ : Lists' α true a✝³ : Lists'.Subset l₁ l₂ a✝² : Lists'.Subset l₂ l₁ h₁' : of' l₁ ~ { fst := true, snd := l₂ } l₃ : Lists' α true a✝¹ : Lists'.Subset l₂ l₃ a✝ : Lists'.Subset l₃ l₂ h₂' : { fst := true, snd := l₂ } ~ { fst := true, snd := l₃ } hl₁ : l₁ ⊆ l₂ hr₁ : l₂ ⊆ l₁ hl₂ : l₂ ⊆ l₃ hr₂ : l₃ ⊆ l₂ a₁ : Lists α m₁ : a₁ ∈ Lists'.toList l₁ a₂ : Lists α m₂ : a₂ ∈ Lists'.toList l₂ e₁₂ : a₁ ~ a₂ ⊢ a₁ ∈ l₃ ** rcases Lists'.mem_of_subset' hl₂ m₂ with ⟨a₃, m₃, e₂₃⟩ ** case C1.antisymm.antisymm.intro.intro.left.intro.intro.intro.intro α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ l₁ : Lists' α true IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l₁ → trans l' l₂ : Lists' α true a✝³ : Lists'.Subset l₁ l₂ a✝² : Lists'.Subset l₂ l₁ h₁' : of' l₁ ~ { fst := true, snd := l₂ } l₃ : Lists' α true a✝¹ : Lists'.Subset l₂ l₃ a✝ : Lists'.Subset l₃ l₂ h₂' : { fst := true, snd := l₂ } ~ { fst := true, snd := l₃ } hl₁ : l₁ ⊆ l₂ hr₁ : l₂ ⊆ l₁ hl₂ : l₂ ⊆ l₃ hr₂ : l₃ ⊆ l₂ a₁ : Lists α m₁ : a₁ ∈ Lists'.toList l₁ a₂ : Lists α m₂ : a₂ ∈ Lists'.toList l₂ e₁₂ : a₁ ~ a₂ a₃ : Lists α m₃ : a₃ ∈ Lists'.toList l₃ e₂₃ : a₂ ~ a₃ ⊢ a₁ ∈ l₃ ** exact ⟨a₃, m₃, IH _ m₁ e₁₂ e₂₃⟩ ** case C1.antisymm.antisymm.intro.intro.right α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ l₁ : Lists' α true IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l₁ → trans l' l₂ : Lists' α true a✝³ : Lists'.Subset l₁ l₂ a✝² : Lists'.Subset l₂ l₁ h₁' : of' l₁ ~ { fst := true, snd := l₂ } l₃ : Lists' α true a✝¹ : Lists'.Subset l₂ l₃ a✝ : Lists'.Subset l₃ l₂ h₂' : { fst := true, snd := l₂ } ~ { fst := true, snd := l₃ } hl₁ : l₁ ⊆ l₂ hr₁ : l₂ ⊆ l₁ hl₂ : l₂ ⊆ l₃ hr₂ : l₃ ⊆ l₂ ⊢ ∀ (a : Lists α), a ∈ Lists'.toList l₃ → a ∈ l₁ ** intro a₃ m₃ ** case C1.antisymm.antisymm.intro.intro.right α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ l₁ : Lists' α true IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l₁ → trans l' l₂ : Lists' α true a✝³ : Lists'.Subset l₁ l₂ a✝² : Lists'.Subset l₂ l₁ h₁' : of' l₁ ~ { fst := true, snd := l₂ } l₃ : Lists' α true a✝¹ : Lists'.Subset l₂ l₃ a✝ : Lists'.Subset l₃ l₂ h₂' : { fst := true, snd := l₂ } ~ { fst := true, snd := l₃ } hl₁ : l₁ ⊆ l₂ hr₁ : l₂ ⊆ l₁ hl₂ : l₂ ⊆ l₃ hr₂ : l₃ ⊆ l₂ a₃ : Lists α m₃ : a₃ ∈ Lists'.toList l₃ ⊢ a₃ ∈ l₁ ** rcases Lists'.mem_of_subset' hr₂ m₃ with ⟨a₂, m₂, e₃₂⟩ ** case C1.antisymm.antisymm.intro.intro.right.intro.intro α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ l₁ : Lists' α true IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l₁ → trans l' l₂ : Lists' α true a✝³ : Lists'.Subset l₁ l₂ a✝² : Lists'.Subset l₂ l₁ h₁' : of' l₁ ~ { fst := true, snd := l₂ } l₃ : Lists' α true a✝¹ : Lists'.Subset l₂ l₃ a✝ : Lists'.Subset l₃ l₂ h₂' : { fst := true, snd := l₂ } ~ { fst := true, snd := l₃ } hl₁ : l₁ ⊆ l₂ hr₁ : l₂ ⊆ l₁ hl₂ : l₂ ⊆ l₃ hr₂ : l₃ ⊆ l₂ a₃ : Lists α m₃ : a₃ ∈ Lists'.toList l₃ a₂ : Lists α m₂ : a₂ ∈ Lists'.toList l₂ e₃₂ : a₃ ~ a₂ ⊢ a₃ ∈ l₁ ** rcases Lists'.mem_of_subset' hr₁ m₂ with ⟨a₁, m₁, e₂₁⟩ ** case C1.antisymm.antisymm.intro.intro.right.intro.intro.intro.intro α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ l₁ : Lists' α true IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l₁ → trans l' l₂ : Lists' α true a✝³ : Lists'.Subset l₁ l₂ a✝² : Lists'.Subset l₂ l₁ h₁' : of' l₁ ~ { fst := true, snd := l₂ } l₃ : Lists' α true a✝¹ : Lists'.Subset l₂ l₃ a✝ : Lists'.Subset l₃ l₂ h₂' : { fst := true, snd := l₂ } ~ { fst := true, snd := l₃ } hl₁ : l₁ ⊆ l₂ hr₁ : l₂ ⊆ l₁ hl₂ : l₂ ⊆ l₃ hr₂ : l₃ ⊆ l₂ a₃ : Lists α m₃ : a₃ ∈ Lists'.toList l₃ a₂ : Lists α m₂ : a₂ ∈ Lists'.toList l₂ e₃₂ : a₃ ~ a₂ a₁ : Lists α m₁ : a₁ ∈ Lists'.toList l₁ e₂₁ : a₂ ~ a₁ ⊢ a₃ ∈ l₁ ** exact ⟨a₁, m₁, (IH _ m₁ e₂₁.symm e₃₂.symm).symm⟩ ** case D0 α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ ⊢ ∀ (l' : Lists α), l' ∈ Lists'.toList Lists'.nil → trans l' ** rintro _ ⟨⟩ ** case D1 α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ ⊢ ∀ (a : Lists α) (l : Lists' α true), trans a → (∀ (l' : Lists α), l' ∈ Lists'.toList l → trans l') → ∀ (l' : Lists α), l' ∈ Lists'.toList (Lists'.cons a l) → trans l' ** intro a l IH₁ IH ** case D1 α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ a : Lists α l : Lists' α true IH₁ : trans a IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l → trans l' ⊢ ∀ (l' : Lists α), l' ∈ Lists'.toList (Lists'.cons a l) → trans l' ** simp only [Lists'.toList, Sigma.eta, List.find?, List.mem_cons, forall_eq_or_imp] ** case D1 α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ a : Lists α l : Lists' α true IH₁ : trans a IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l → trans l' ⊢ (∀ ⦃l₂ l₃ : Lists α⦄, a ~ l₂ → l₂ ~ l₃ → a ~ l₃) ∧ ∀ (a : Lists α), a ∈ Lists'.toList l → ∀ ⦃l₂ l₃ : Lists α⦄, a ~ l₂ → l₂ ~ l₃ → a ~ l₃ ** constructor ** case D1.left α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ a : Lists α l : Lists' α true IH₁ : trans a IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l → trans l' ⊢ ∀ ⦃l₂ l₃ : Lists α⦄, a ~ l₂ → l₂ ~ l₃ → a ~ l₃ ** intros l₂ l₃ h₁ h₂ ** case D1.left α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ a : Lists α l : Lists' α true IH₁ : trans a IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l → trans l' l₂ l₃ : Lists α h₁ : a ~ l₂ h₂ : l₂ ~ l₃ ⊢ a ~ l₃ ** exact IH₁ h₁ h₂ ** case D1.right α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ a : Lists α l : Lists' α true IH₁ : trans a IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l → trans l' ⊢ ∀ (a : Lists α), a ∈ Lists'.toList l → ∀ ⦃l₂ l₃ : Lists α⦄, a ~ l₂ → l₂ ~ l₃ → a ~ l₃ ** intros a h₁ l₂ l₃ h₂ h₃ ** case D1.right α : Type u_1 trans : Lists α → Prop := fun l₁ => ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ a✝ : Lists α l : Lists' α true IH₁ : trans a✝ IH : ∀ (l' : Lists α), l' ∈ Lists'.toList l → trans l' a : Lists α h₁ : a ∈ Lists'.toList l l₂ l₃ : Lists α h₂ : a ~ l₂ h₃ : l₂ ~ l₃ ⊢ a ~ l₃ ** exact IH _ h₁ h₂ h₃ ** Qed
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Lists.sizeof_pos ** α : Type u_1 b : Bool l : Lists' α b ⊢ 0 < sizeOf l ** cases l <;> simp only [Lists'.atom.sizeOf_spec, Lists'.nil.sizeOf_spec, Lists'.cons'.sizeOf_spec,
true_or, add_pos_iff] ** Qed
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Lists.lt_sizeof_cons' ** α : Type u_1 b : Bool a : Lists' α b l : Lists' α true ⊢ sizeOf { fst := b, snd := a } < sizeOf (Lists'.cons' a l) ** simp only [Sigma.mk.sizeOf_spec, Lists'.cons'.sizeOf_spec, lt_add_iff_pos_right] ** α : Type u_1 b : Bool a : Lists' α b l : Lists' α true ⊢ 0 < sizeOf l ** apply sizeof_pos ** Qed
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TopCat.Presheaf.coveringOfPresieve.iSup_eq_of_mem_grothendieck ** X : TopCat U : Opens ↑X R : Presieve U hR : Sieve.generate R ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U ⊢ iSup (coveringOfPresieve U R) = U ** apply le_antisymm ** case a X : TopCat U : Opens ↑X R : Presieve U hR : Sieve.generate R ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U ⊢ U ≤ iSup (coveringOfPresieve U R) ** intro x hxU ** case a X : TopCat U : Opens ↑X R : Presieve U hR : Sieve.generate R ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U x : ↑X hxU : x ∈ ↑U ⊢ x ∈ ↑(iSup (coveringOfPresieve U R)) ** rw [Opens.coe_iSup, Set.mem_iUnion] ** case a X : TopCat U : Opens ↑X R : Presieve U hR : Sieve.generate R ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U x : ↑X hxU : x ∈ ↑U ⊢ ∃ i, x ∈ ↑(coveringOfPresieve U R i) ** obtain ⟨V, iVU, ⟨W, iVW, iWU, hiWU, -⟩, hxV⟩ := hR x hxU ** case a.intro.intro.intro.intro.intro.intro.intro X : TopCat U : Opens ↑X R : Presieve U hR : Sieve.generate R ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U x : ↑X hxU : x ∈ ↑U V : Opens ↑X iVU : V ⟶ U hxV : x ∈ V W : Opens ↑X iVW : V ⟶ W iWU : W ⟶ U hiWU : R iWU ⊢ ∃ i, x ∈ ↑(coveringOfPresieve U R i) ** exact ⟨⟨W, ⟨iWU, hiWU⟩⟩, iVW.le hxV⟩ ** case a X : TopCat U : Opens ↑X R : Presieve U hR : Sieve.generate R ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U ⊢ iSup (coveringOfPresieve U R) ≤ U ** refine' iSup_le _ ** case a X : TopCat U : Opens ↑X R : Presieve U hR : Sieve.generate R ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U ⊢ ∀ (i : (V : Opens ↑X) × { f // R f }), coveringOfPresieve U R i ≤ U ** intro f ** case a X : TopCat U : Opens ↑X R : Presieve U hR : Sieve.generate R ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U f : (V : Opens ↑X) × { f // R f } ⊢ coveringOfPresieve U R f ≤ U ** exact f.2.1.le ** Qed
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TopCat.Presheaf.covering_presieve_eq_self ** X : TopCat Y : Opens ↑X R : Presieve Y ⊢ presieveOfCoveringAux (coveringOfPresieve Y R) Y = R ** funext Z ** case h X : TopCat Y : Opens ↑X R : Presieve Y Z : Opens ↑X ⊢ presieveOfCoveringAux (coveringOfPresieve Y R) Y = R ** ext f ** case h.h X : TopCat Y : Opens ↑X R : Presieve Y Z : Opens ↑X f : Z ⟶ Y ⊢ f ∈ presieveOfCoveringAux (coveringOfPresieve Y R) Y ↔ f ∈ R ** exact ⟨fun ⟨⟨_, f', h⟩, rfl⟩ => by rwa [Subsingleton.elim f f'], fun h => ⟨⟨Z, f, h⟩, rfl⟩⟩ ** X : TopCat Y : Opens ↑X R : Presieve Y Z : Opens ↑X f : Z ⟶ Y x✝ : f ∈ presieveOfCoveringAux (coveringOfPresieve Y R) Y f' : Z ⟶ Y h : R f' ⊢ f ∈ R ** rwa [Subsingleton.elim f f'] ** Qed
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TopCat.Presheaf.presieveOfCovering.mem_grothendieckTopology ** X : TopCat ι : Type v U : ι → Opens ↑X ⊢ Sieve.generate (presieveOfCovering U) ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) (iSup U) ** intro x hx ** X : TopCat ι : Type v U : ι → Opens ↑X x : ↑X hx : x ∈ iSup U ⊢ ∃ U_1 f, (Sieve.generate (presieveOfCovering U)).arrows f ∧ x ∈ U_1 ** obtain ⟨i, hxi⟩ := Opens.mem_iSup.mp hx ** case intro X : TopCat ι : Type v U : ι → Opens ↑X x : ↑X hx : x ∈ iSup U i : ι hxi : x ∈ U i ⊢ ∃ U_1 f, (Sieve.generate (presieveOfCovering U)).arrows f ∧ x ∈ U_1 ** exact ⟨U i, Opens.leSupr U i, ⟨U i, 𝟙 _, Opens.leSupr U i, ⟨i, rfl⟩, Category.id_comp _⟩, hxi⟩ ** Qed
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TopCat.Opens.coverDense_iff_isBasis ** X : TopCat ι : Type u_1 inst✝ : Category.{u_2, u_1} ι B : ι ⥤ Opens ↑X ⊢ CoverDense (Opens.grothendieckTopology ↑X) B ↔ Opens.IsBasis (Set.range B.obj) ** rw [Opens.isBasis_iff_nbhd] ** X : TopCat ι : Type u_1 inst✝ : Category.{u_2, u_1} ι B : ι ⥤ Opens ↑X ⊢ CoverDense (Opens.grothendieckTopology ↑X) B ↔ ∀ {U : Opens ↑X} {x : ↑X}, x ∈ U → ∃ U', U' ∈ Set.range B.obj ∧ x ∈ U' ∧ U' ≤ U ** constructor ** case mp X : TopCat ι : Type u_1 inst✝ : Category.{u_2, u_1} ι B : ι ⥤ Opens ↑X ⊢ CoverDense (Opens.grothendieckTopology ↑X) B → ∀ {U : Opens ↑X} {x : ↑X}, x ∈ U → ∃ U', U' ∈ Set.range B.obj ∧ x ∈ U' ∧ U' ≤ U case mpr X : TopCat ι : Type u_1 inst✝ : Category.{u_2, u_1} ι B : ι ⥤ Opens ↑X ⊢ (∀ {U : Opens ↑X} {x : ↑X}, x ∈ U → ∃ U', U' ∈ Set.range B.obj ∧ x ∈ U' ∧ U' ≤ U) → CoverDense (Opens.grothendieckTopology ↑X) B ** intro hd U x hx ** case mp X : TopCat ι : Type u_1 inst✝ : Category.{u_2, u_1} ι B : ι ⥤ Opens ↑X hd : CoverDense (Opens.grothendieckTopology ↑X) B U : Opens ↑X x : ↑X hx : x ∈ U ⊢ ∃ U', U' ∈ Set.range B.obj ∧ x ∈ U' ∧ U' ≤ U case mpr X : TopCat ι : Type u_1 inst✝ : Category.{u_2, u_1} ι B : ι ⥤ Opens ↑X ⊢ (∀ {U : Opens ↑X} {x : ↑X}, x ∈ U → ∃ U', U' ∈ Set.range B.obj ∧ x ∈ U' ∧ U' ≤ U) → CoverDense (Opens.grothendieckTopology ↑X) B ** rcases hd.1 U x hx with ⟨V, f, ⟨i, f₁, f₂, _⟩, hV⟩ ** case mp.intro.intro.intro.intro.mk X : TopCat ι : Type u_1 inst✝ : Category.{u_2, u_1} ι B : ι ⥤ Opens ↑X hd : CoverDense (Opens.grothendieckTopology ↑X) B U : Opens ↑X x : ↑X hx : x ∈ U V : Opens ↑X f : V ⟶ U hV : x ∈ V i : ι f₁ : V ⟶ B.obj i f₂ : B.obj i ⟶ U fac✝ : f₁ ≫ f₂ = f ⊢ ∃ U', U' ∈ Set.range B.obj ∧ x ∈ U' ∧ U' ≤ U case mpr X : TopCat ι : Type u_1 inst✝ : Category.{u_2, u_1} ι B : ι ⥤ Opens ↑X ⊢ (∀ {U : Opens ↑X} {x : ↑X}, x ∈ U → ∃ U', U' ∈ Set.range B.obj ∧ x ∈ U' ∧ U' ≤ U) → CoverDense (Opens.grothendieckTopology ↑X) B ** exact ⟨B.obj i, ⟨i, rfl⟩, f₁.le hV, f₂.le⟩ ** case mpr X : TopCat ι : Type u_1 inst✝ : Category.{u_2, u_1} ι B : ι ⥤ Opens ↑X ⊢ (∀ {U : Opens ↑X} {x : ↑X}, x ∈ U → ∃ U', U' ∈ Set.range B.obj ∧ x ∈ U' ∧ U' ≤ U) → CoverDense (Opens.grothendieckTopology ↑X) B ** intro hb ** case mpr X : TopCat ι : Type u_1 inst✝ : Category.{u_2, u_1} ι B : ι ⥤ Opens ↑X hb : ∀ {U : Opens ↑X} {x : ↑X}, x ∈ U → ∃ U', U' ∈ Set.range B.obj ∧ x ∈ U' ∧ U' ≤ U ⊢ CoverDense (Opens.grothendieckTopology ↑X) B ** constructor ** case mpr.is_cover X : TopCat ι : Type u_1 inst✝ : Category.{u_2, u_1} ι B : ι ⥤ Opens ↑X hb : ∀ {U : Opens ↑X} {x : ↑X}, x ∈ U → ∃ U', U' ∈ Set.range B.obj ∧ x ∈ U' ∧ U' ≤ U ⊢ ∀ (U : Opens ↑X), Sieve.coverByImage B U ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U ** intro U x hx ** case mpr.is_cover X : TopCat ι : Type u_1 inst✝ : Category.{u_2, u_1} ι B : ι ⥤ Opens ↑X hb : ∀ {U : Opens ↑X} {x : ↑X}, x ∈ U → ∃ U', U' ∈ Set.range B.obj ∧ x ∈ U' ∧ U' ≤ U U : Opens ↑X x : ↑X hx : x ∈ U ⊢ ∃ U_1 f, (Sieve.coverByImage B U).arrows f ∧ x ∈ U_1 ** rcases hb hx with ⟨_, ⟨i, rfl⟩, hx, hi⟩ ** case mpr.is_cover.intro.intro.intro.intro X : TopCat ι : Type u_1 inst✝ : Category.{u_2, u_1} ι B : ι ⥤ Opens ↑X hb : ∀ {U : Opens ↑X} {x : ↑X}, x ∈ U → ∃ U', U' ∈ Set.range B.obj ∧ x ∈ U' ∧ U' ≤ U U : Opens ↑X x : ↑X hx✝ : x ∈ U i : ι hx : x ∈ B.obj i hi : B.obj i ≤ U ⊢ ∃ U_1 f, (Sieve.coverByImage B U).arrows f ∧ x ∈ U_1 ** exact ⟨B.obj i, ⟨⟨hi⟩⟩, ⟨⟨i, 𝟙 _, ⟨⟨hi⟩⟩, rfl⟩⟩, hx⟩ ** Qed
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OpenEmbedding.compatiblePreserving ** C : Type u inst✝ : Category.{v, u} C X Y : TopCat f : X ⟶ Y F : TopCat.Presheaf C Y hf : OpenEmbedding ↑f ⊢ CompatiblePreserving (Opens.grothendieckTopology ↑Y) (IsOpenMap.functor (_ : IsOpenMap ↑f)) ** haveI : Mono f := (TopCat.mono_iff_injective f).mpr hf.inj ** C : Type u inst✝ : Category.{v, u} C X Y : TopCat f : X ⟶ Y F : TopCat.Presheaf C Y hf : OpenEmbedding ↑f this : Mono f ⊢ CompatiblePreserving (Opens.grothendieckTopology ↑Y) (IsOpenMap.functor (_ : IsOpenMap ↑f)) ** apply compatiblePreservingOfDownwardsClosed ** case hF C : Type u inst✝ : Category.{v, u} C X Y : TopCat f : X ⟶ Y F : TopCat.Presheaf C Y hf : OpenEmbedding ↑f this : Mono f ⊢ {c : Opens ↑X} → {d : Opens ↑Y} → (d ⟶ (IsOpenMap.functor (_ : IsOpenMap ↑f)).obj c) → (c' : Opens ↑X) × ((IsOpenMap.functor (_ : IsOpenMap ↑f)).obj c' ≅ d) ** intro U V i ** case hF C : Type u inst✝ : Category.{v, u} C X Y : TopCat f : X ⟶ Y F : TopCat.Presheaf C Y hf : OpenEmbedding ↑f this : Mono f U : Opens ↑X V : Opens ↑Y i : V ⟶ (IsOpenMap.functor (_ : IsOpenMap ↑f)).obj U ⊢ (c' : Opens ↑X) × ((IsOpenMap.functor (_ : IsOpenMap ↑f)).obj c' ≅ V) ** refine' ⟨(Opens.map f).obj V, eqToIso <| Opens.ext <| Set.image_preimage_eq_of_subset fun x h ↦ _⟩ ** case hF C : Type u inst✝ : Category.{v, u} C X Y : TopCat f : X ⟶ Y F : TopCat.Presheaf C Y hf : OpenEmbedding ↑f this : Mono f U : Opens ↑X V : Opens ↑Y i : V ⟶ (IsOpenMap.functor (_ : IsOpenMap ↑f)).obj U x : ↑Y h : x ∈ V.1 ⊢ x ∈ Set.range fun x => ↑f x ** obtain ⟨_, _, rfl⟩ := i.le h ** case hF.intro.intro C : Type u inst✝ : Category.{v, u} C X Y : TopCat f : X ⟶ Y F : TopCat.Presheaf C Y hf : OpenEmbedding ↑f this : Mono f U : Opens ↑X V : Opens ↑Y i : V ⟶ (IsOpenMap.functor (_ : IsOpenMap ↑f)).obj U w✝ : (forget TopCat).obj X left✝ : w✝ ∈ ↑U h : ↑f w✝ ∈ V.1 ⊢ ↑f w✝ ∈ Set.range fun x => ↑f x ** exact ⟨_, rfl⟩ ** Qed
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IsOpenMap.coverPreserving ** C : Type u inst✝ : Category.{v, u} C X Y : TopCat f : X ⟶ Y F : TopCat.Presheaf C Y hf : IsOpenMap ↑f ⊢ CoverPreserving (Opens.grothendieckTopology ↑X) (Opens.grothendieckTopology ↑Y) (functor hf) ** constructor ** case cover_preserve C : Type u inst✝ : Category.{v, u} C X Y : TopCat f : X ⟶ Y F : TopCat.Presheaf C Y hf : IsOpenMap ↑f ⊢ ∀ {U : Opens ↑X} {S : Sieve U}, S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U → Sieve.functorPushforward (functor hf) S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑Y) ((functor hf).obj U) ** rintro U S hU _ ⟨x, hx, rfl⟩ ** case cover_preserve.intro.intro C : Type u inst✝ : Category.{v, u} C X Y : TopCat f : X ⟶ Y F : TopCat.Presheaf C Y hf : IsOpenMap ↑f U : Opens ↑X S : Sieve U hU : S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U x : (forget TopCat).obj X hx : x ∈ ↑U ⊢ ∃ U_1 f_1, (Sieve.functorPushforward (functor hf) S).arrows f_1 ∧ ↑f x ∈ U_1 ** obtain ⟨V, i, hV, hxV⟩ := hU x hx ** case cover_preserve.intro.intro.intro.intro.intro C : Type u inst✝ : Category.{v, u} C X Y : TopCat f : X ⟶ Y F : TopCat.Presheaf C Y hf : IsOpenMap ↑f U : Opens ↑X S : Sieve U hU : S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U x : (forget TopCat).obj X hx : x ∈ ↑U V : Opens ↑X i : V ⟶ U hV : S.arrows i hxV : x ∈ V ⊢ ∃ U_1 f_1, (Sieve.functorPushforward (functor hf) S).arrows f_1 ∧ ↑f x ∈ U_1 ** exact ⟨_, hf.functor.map i, ⟨_, i, 𝟙 _, hV, rfl⟩, Set.mem_image_of_mem f hxV⟩ ** Qed
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coverPreserving_opens_map ** C : Type u inst✝ : Category.{v, u} C X Y : TopCat f : X ⟶ Y F : TopCat.Presheaf C Y ⊢ CoverPreserving (Opens.grothendieckTopology ↑Y) (Opens.grothendieckTopology ↑X) (Opens.map f) ** constructor ** case cover_preserve C : Type u inst✝ : Category.{v, u} C X Y : TopCat f : X ⟶ Y F : TopCat.Presheaf C Y ⊢ ∀ {U : Opens ↑Y} {S : Sieve U}, S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑Y) U → Sieve.functorPushforward (Opens.map f) S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ((Opens.map f).obj U) ** intro U S hS x hx ** case cover_preserve C : Type u inst✝ : Category.{v, u} C X Y : TopCat f : X ⟶ Y F : TopCat.Presheaf C Y U : Opens ↑Y S : Sieve U hS : S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑Y) U x : ↑X hx : x ∈ (Opens.map f).obj U ⊢ ∃ U_1 f_1, (Sieve.functorPushforward (Opens.map f) S).arrows f_1 ∧ x ∈ U_1 ** obtain ⟨V, i, hi, hxV⟩ := hS (f x) hx ** case cover_preserve.intro.intro.intro C : Type u inst✝ : Category.{v, u} C X Y : TopCat f : X ⟶ Y F : TopCat.Presheaf C Y U : Opens ↑Y S : Sieve U hS : S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑Y) U x : ↑X hx : x ∈ (Opens.map f).obj U V : Opens ↑Y i : V ⟶ U hi : S.arrows i hxV : ↑f x ∈ V ⊢ ∃ U_1 f_1, (Sieve.functorPushforward (Opens.map f) S).arrows f_1 ∧ x ∈ U_1 ** exact ⟨_, (Opens.map f).map i, ⟨_, _, 𝟙 _, hi, Subsingleton.elim _ _⟩, hxV⟩ ** Qed
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TopCat.Sheaf.extend_hom_app ** C : Type u inst✝ : Category.{v, u} C X : TopCat ι : Type u_1 B : ι → Opens ↑X F : Presheaf C X F' : Sheaf C X h : Opens.IsBasis (Set.range B) α : (inducedFunctor B).op ⋙ F ⟶ (inducedFunctor B).op ⋙ F'.val i : ι ⊢ (↑(restrictHomEquivHom F F' h) α).app (op (B i)) = α.app (op i) ** nth_rw 2 [← (restrictHomEquivHom F F' h).left_inv α] ** C : Type u inst✝ : Category.{v, u} C X : TopCat ι : Type u_1 B : ι → Opens ↑X F : Presheaf C X F' : Sheaf C X h : Opens.IsBasis (Set.range B) α : (inducedFunctor B).op ⋙ F ⟶ (inducedFunctor B).op ⋙ F'.val i : ι ⊢ (↑(restrictHomEquivHom F F' h) α).app (op (B i)) = (Equiv.invFun (restrictHomEquivHom F F' h) (Equiv.toFun (restrictHomEquivHom F F' h) α)).app (op i) ** rfl ** Qed
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TopCat.Sheaf.hom_ext ** C : Type u inst✝ : Category.{v, u} C X : TopCat ι : Type u_1 B : ι → Opens ↑X F : Presheaf C X F' : Sheaf C X h : Opens.IsBasis (Set.range B) α β : F ⟶ F'.val he : ∀ (i : ι), α.app (op (B i)) = β.app (op (B i)) ⊢ α = β ** apply (restrictHomEquivHom F F' h).symm.injective ** case a C : Type u inst✝ : Category.{v, u} C X : TopCat ι : Type u_1 B : ι → Opens ↑X F : Presheaf C X F' : Sheaf C X h : Opens.IsBasis (Set.range B) α β : F ⟶ F'.val he : ∀ (i : ι), α.app (op (B i)) = β.app (op (B i)) ⊢ ↑(restrictHomEquivHom F F' h).symm α = ↑(restrictHomEquivHom F F' h).symm β ** ext i ** case a.w.h C : Type u inst✝ : Category.{v, u} C X : TopCat ι : Type u_1 B : ι → Opens ↑X F : Presheaf C X F' : Sheaf C X h : Opens.IsBasis (Set.range B) α β : F ⟶ F'.val he : ∀ (i : ι), α.app (op (B i)) = β.app (op (B i)) i : (InducedCategory (Opens ↑X) B)ᵒᵖ ⊢ (↑(restrictHomEquivHom F F' h).symm α).app i = (↑(restrictHomEquivHom F F' h).symm β).app i ** exact he i.unop ** Qed
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ContinuousMap.embedding_sigmaMk_comp ** X : Type u_1 ι : Type u_2 Y : ι → Type u_3 inst✝² : TopologicalSpace X inst✝¹ : (i : ι) → TopologicalSpace (Y i) inst✝ : Nonempty X ⊢ Function.Injective fun g => comp (sigmaMk g.fst) g.snd ** rintro ⟨i, g⟩ ⟨i', g'⟩ h ** case mk.mk X : Type u_1 ι : Type u_2 Y : ι → Type u_3 inst✝² : TopologicalSpace X inst✝¹ : (i : ι) → TopologicalSpace (Y i) inst✝ : Nonempty X i : ι g : C(X, Y i) i' : ι g' : C(X, Y i') h : (fun g => comp (sigmaMk g.fst) g.snd) { fst := i, snd := g } = (fun g => comp (sigmaMk g.fst) g.snd) { fst := i', snd := g' } ⊢ { fst := i, snd := g } = { fst := i', snd := g' } ** obtain ⟨rfl, hg⟩ : i = i' ∧ HEq (⇑g) (⇑g') :=
Function.eq_of_sigmaMk_comp <| congr_arg FunLike.coe h ** case mk.mk.intro X : Type u_1 ι : Type u_2 Y : ι → Type u_3 inst✝² : TopologicalSpace X inst✝¹ : (i : ι) → TopologicalSpace (Y i) inst✝ : Nonempty X i : ι g g' : C(X, Y i) h : (fun g => comp (sigmaMk g.fst) g.snd) { fst := i, snd := g } = (fun g => comp (sigmaMk g.fst) g.snd) { fst := i, snd := g' } hg : HEq ↑g ↑g' ⊢ { fst := i, snd := g } = { fst := i, snd := g' } ** simpa using hg ** Qed
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Polynomial.aeval_continuousMap_apply ** R : Type u_1 α : Type u_2 inst✝³ : TopologicalSpace α inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R g : R[X] f : C(α, R) x : α ⊢ ↑(↑(aeval f) g) x = eval (↑f x) g ** refine' Polynomial.induction_on' g _ _ ** case refine'_1 R : Type u_1 α : Type u_2 inst✝³ : TopologicalSpace α inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R g : R[X] f : C(α, R) x : α ⊢ ∀ (p q : R[X]), ↑(↑(aeval f) p) x = eval (↑f x) p → ↑(↑(aeval f) q) x = eval (↑f x) q → ↑(↑(aeval f) (p + q)) x = eval (↑f x) (p + q) ** intro p q hp hq ** case refine'_1 R : Type u_1 α : Type u_2 inst✝³ : TopologicalSpace α inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R g : R[X] f : C(α, R) x : α p q : R[X] hp : ↑(↑(aeval f) p) x = eval (↑f x) p hq : ↑(↑(aeval f) q) x = eval (↑f x) q ⊢ ↑(↑(aeval f) (p + q)) x = eval (↑f x) (p + q) ** simp [hp, hq] ** case refine'_2 R : Type u_1 α : Type u_2 inst✝³ : TopologicalSpace α inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R g : R[X] f : C(α, R) x : α ⊢ ∀ (n : ℕ) (a : R), ↑(↑(aeval f) (↑(monomial n) a)) x = eval (↑f x) (↑(monomial n) a) ** intro n a ** case refine'_2 R : Type u_1 α : Type u_2 inst✝³ : TopologicalSpace α inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R g : R[X] f : C(α, R) x : α n : ℕ a : R ⊢ ↑(↑(aeval f) (↑(monomial n) a)) x = eval (↑f x) (↑(monomial n) a) ** simp [Pi.pow_apply] ** Qed
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polynomialFunctions_coe ** R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R X : Set R ⊢ ↑(polynomialFunctions X) = Set.range ↑(toContinuousMapOnAlgHom X) ** ext ** case h R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R X : Set R x✝ : C(↑X, R) ⊢ x✝ ∈ ↑(polynomialFunctions X) ↔ x✝ ∈ Set.range ↑(toContinuousMapOnAlgHom X) ** simp [polynomialFunctions] ** Qed
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polynomialFunctions_separatesPoints ** R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R X : Set R x y : ↑X h : x ≠ y ⊢ ∃ f, f ∈ (fun f => ↑f) '' ↑(polynomialFunctions X) ∧ f x ≠ f y ** refine' ⟨_, ⟨⟨_, ⟨⟨Polynomial.X, ⟨Algebra.mem_top, rfl⟩⟩, rfl⟩⟩, _⟩⟩ ** R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R X : Set R x y : ↑X h : x ≠ y ⊢ (fun f => ↑f) (↑↑(toContinuousMapOnAlgHom X) Polynomial.X) x ≠ (fun f => ↑f) (↑↑(toContinuousMapOnAlgHom X) Polynomial.X) y ** dsimp ** R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R X : Set R x y : ↑X h : x ≠ y ⊢ ¬eval (↑x) Polynomial.X = eval (↑y) Polynomial.X ** simp only [Polynomial.eval_X] ** R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R X : Set R x y : ↑X h : x ≠ y ⊢ ¬↑x = ↑y ** exact fun h' => h (Subtype.ext h') ** Qed
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polynomialFunctions.comap_compRightAlgHom_iccHomeoI ** R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b ⊢ Subalgebra.comap (compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) (polynomialFunctions I) = polynomialFunctions (Set.Icc a b) ** ext f ** case h R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) ⊢ f ∈ Subalgebra.comap (compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) (polynomialFunctions I) ↔ f ∈ polynomialFunctions (Set.Icc a b) ** fconstructor ** case h.mp R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) ⊢ f ∈ Subalgebra.comap (compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) (polynomialFunctions I) → f ∈ polynomialFunctions (Set.Icc a b) ** rintro ⟨p, ⟨-, w⟩⟩ ** case h.mp.intro.intro R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) f ⊢ f ∈ polynomialFunctions (Set.Icc a b) ** rw [FunLike.ext_iff] at w ** case h.mp.intro.intro R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) f w : ∀ (x : ↑I), ↑(↑↑(toContinuousMapOnAlgHom I) p) x = ↑(↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) f) x ⊢ f ∈ polynomialFunctions (Set.Icc a b) ** dsimp at w ** case h.mp.intro.intro R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) f w : ∀ (x : ↑I), Polynomial.eval (↑x) p = ↑f (↑(Homeomorph.symm (iccHomeoI a b h)) x) ⊢ f ∈ polynomialFunctions (Set.Icc a b) ** let q := p.comp ((b - a)⁻¹ • Polynomial.X + Polynomial.C (-a * (b - a)⁻¹)) ** case h.mp.intro.intro R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) f w : ∀ (x : ↑I), Polynomial.eval (↑x) p = ↑f (↑(Homeomorph.symm (iccHomeoI a b h)) x) q : ℝ[X] := Polynomial.comp p ((b - a)⁻¹ • X + ↑Polynomial.C (-a * (b - a)⁻¹)) ⊢ f ∈ polynomialFunctions (Set.Icc a b) ** refine' ⟨q, ⟨_, _⟩⟩ ** case h.mp.intro.intro.refine'_1 R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) f w : ∀ (x : ↑I), Polynomial.eval (↑x) p = ↑f (↑(Homeomorph.symm (iccHomeoI a b h)) x) q : ℝ[X] := Polynomial.comp p ((b - a)⁻¹ • X + ↑Polynomial.C (-a * (b - a)⁻¹)) ⊢ q ∈ ↑⊤.toSubsemiring ** simp ** case h.mp.intro.intro.refine'_2 R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) f w : ∀ (x : ↑I), Polynomial.eval (↑x) p = ↑f (↑(Homeomorph.symm (iccHomeoI a b h)) x) q : ℝ[X] := Polynomial.comp p ((b - a)⁻¹ • X + ↑Polynomial.C (-a * (b - a)⁻¹)) ⊢ ↑↑(toContinuousMapOnAlgHom (Set.Icc a b)) q = f ** ext x ** case h.mp.intro.intro.refine'_2.h R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) f w : ∀ (x : ↑I), Polynomial.eval (↑x) p = ↑f (↑(Homeomorph.symm (iccHomeoI a b h)) x) q : ℝ[X] := Polynomial.comp p ((b - a)⁻¹ • X + ↑Polynomial.C (-a * (b - a)⁻¹)) x : ↑(Set.Icc a b) ⊢ ↑(↑↑(toContinuousMapOnAlgHom (Set.Icc a b)) q) x = ↑f x ** simp only [neg_mul, RingHom.map_neg, RingHom.map_mul, AlgHom.coe_toRingHom, Polynomial.eval_X,
Polynomial.eval_neg, Polynomial.eval_C, Polynomial.eval_smul, smul_eq_mul,
Polynomial.eval_mul, Polynomial.eval_add, Polynomial.coe_aeval_eq_eval,
Polynomial.eval_comp, Polynomial.toContinuousMapOnAlgHom_apply,
Polynomial.toContinuousMapOn_apply, Polynomial.toContinuousMap_apply] ** case h.mp.intro.intro.refine'_2.h R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) f w : ∀ (x : ↑I), Polynomial.eval (↑x) p = ↑f (↑(Homeomorph.symm (iccHomeoI a b h)) x) q : ℝ[X] := Polynomial.comp p ((b - a)⁻¹ • X + ↑Polynomial.C (-a * (b - a)⁻¹)) x : ↑(Set.Icc a b) ⊢ Polynomial.eval ((b - a)⁻¹ * ↑x + -(a * (b - a)⁻¹)) p = ↑f x ** convert w ⟨_, _⟩ ** case h.e'_3.h.e'_6 R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) f w : ∀ (x : ↑I), Polynomial.eval (↑x) p = ↑f (↑(Homeomorph.symm (iccHomeoI a b h)) x) q : ℝ[X] := Polynomial.comp p ((b - a)⁻¹ • X + ↑Polynomial.C (-a * (b - a)⁻¹)) x : ↑(Set.Icc a b) ⊢ x = ↑(Homeomorph.symm (iccHomeoI a b h)) { val := (b - a)⁻¹ * ↑x + -(a * (b - a)⁻¹), property := ?h.mp.intro.intro.refine'_2.h.convert_2 } ** ext ** case h.e'_3.h.e'_6.a R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) f w : ∀ (x : ↑I), Polynomial.eval (↑x) p = ↑f (↑(Homeomorph.symm (iccHomeoI a b h)) x) q : ℝ[X] := Polynomial.comp p ((b - a)⁻¹ • X + ↑Polynomial.C (-a * (b - a)⁻¹)) x : ↑(Set.Icc a b) ⊢ ↑x = ↑(↑(Homeomorph.symm (iccHomeoI a b h)) { val := (b - a)⁻¹ * ↑x + -(a * (b - a)⁻¹), property := ?h.mp.intro.intro.refine'_2.h.convert_2 }) ** simp only [iccHomeoI_symm_apply_coe, Subtype.coe_mk] ** case h.e'_3.h.e'_6.a R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) f w : ∀ (x : ↑I), Polynomial.eval (↑x) p = ↑f (↑(Homeomorph.symm (iccHomeoI a b h)) x) q : ℝ[X] := Polynomial.comp p ((b - a)⁻¹ • X + ↑Polynomial.C (-a * (b - a)⁻¹)) x : ↑(Set.Icc a b) ⊢ ↑x = (b - a) * ((b - a)⁻¹ * ↑x + -(a * (b - a)⁻¹)) + a ** replace h : b - a ≠ 0 := sub_ne_zero_of_ne h.ne.symm ** case h.e'_3.h.e'_6.a R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h✝ : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h✝)))) f w : ∀ (x : ↑I), Polynomial.eval (↑x) p = ↑f (↑(Homeomorph.symm (iccHomeoI a b h✝)) x) q : ℝ[X] := Polynomial.comp p ((b - a)⁻¹ • X + ↑Polynomial.C (-a * (b - a)⁻¹)) x : ↑(Set.Icc a b) h : b - a ≠ 0 ⊢ ↑x = (b - a) * ((b - a)⁻¹ * ↑x + -(a * (b - a)⁻¹)) + a ** simp only [mul_add] ** case h.e'_3.h.e'_6.a R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h✝ : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h✝)))) f w : ∀ (x : ↑I), Polynomial.eval (↑x) p = ↑f (↑(Homeomorph.symm (iccHomeoI a b h✝)) x) q : ℝ[X] := Polynomial.comp p ((b - a)⁻¹ • X + ↑Polynomial.C (-a * (b - a)⁻¹)) x : ↑(Set.Icc a b) h : b - a ≠ 0 ⊢ ↑x = (b - a) * ((b - a)⁻¹ * ↑x) + (b - a) * -(a * (b - a)⁻¹) + a ** field_simp ** case h.e'_3.h.e'_6.a R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h✝ : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h✝)))) f w : ∀ (x : ↑I), Polynomial.eval (↑x) p = ↑f (↑(Homeomorph.symm (iccHomeoI a b h✝)) x) q : ℝ[X] := Polynomial.comp p ((b - a)⁻¹ • X + ↑Polynomial.C (-a * (b - a)⁻¹)) x : ↑(Set.Icc a b) h : b - a ≠ 0 ⊢ ↑x * (b - a) = (b - a) * ↑x + -((b - a) * a) + a * (b - a) ** ring ** case h.mp.intro.intro.refine'_2.h.convert_2 R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) f w : ∀ (x : ↑I), Polynomial.eval (↑x) p = ↑f (↑(Homeomorph.symm (iccHomeoI a b h)) x) q : ℝ[X] := Polynomial.comp p ((b - a)⁻¹ • X + ↑Polynomial.C (-a * (b - a)⁻¹)) x : ↑(Set.Icc a b) ⊢ (b - a)⁻¹ * ↑x + -(a * (b - a)⁻¹) ∈ I ** rw [mul_comm (b - a)⁻¹, ← neg_mul, ← add_mul, ← sub_eq_add_neg] ** case h.mp.intro.intro.refine'_2.h.convert_2 R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) f w : ∀ (x : ↑I), Polynomial.eval (↑x) p = ↑f (↑(Homeomorph.symm (iccHomeoI a b h)) x) q : ℝ[X] := Polynomial.comp p ((b - a)⁻¹ • X + ↑Polynomial.C (-a * (b - a)⁻¹)) x : ↑(Set.Icc a b) ⊢ (↑x - a) * (b - a)⁻¹ ∈ I ** have w₁ : 0 < (b - a)⁻¹ := inv_pos.mpr (sub_pos.mpr h) ** case h.mp.intro.intro.refine'_2.h.convert_2 R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) f w : ∀ (x : ↑I), Polynomial.eval (↑x) p = ↑f (↑(Homeomorph.symm (iccHomeoI a b h)) x) q : ℝ[X] := Polynomial.comp p ((b - a)⁻¹ • X + ↑Polynomial.C (-a * (b - a)⁻¹)) x : ↑(Set.Icc a b) w₁ : 0 < (b - a)⁻¹ ⊢ (↑x - a) * (b - a)⁻¹ ∈ I ** have w₂ : 0 ≤ (x : ℝ) - a := sub_nonneg.mpr x.2.1 ** case h.mp.intro.intro.refine'_2.h.convert_2 R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) f w : ∀ (x : ↑I), Polynomial.eval (↑x) p = ↑f (↑(Homeomorph.symm (iccHomeoI a b h)) x) q : ℝ[X] := Polynomial.comp p ((b - a)⁻¹ • X + ↑Polynomial.C (-a * (b - a)⁻¹)) x : ↑(Set.Icc a b) w₁ : 0 < (b - a)⁻¹ w₂ : 0 ≤ ↑x - a ⊢ (↑x - a) * (b - a)⁻¹ ∈ I ** have w₃ : (x : ℝ) - a ≤ b - a := sub_le_sub_right x.2.2 a ** case h.mp.intro.intro.refine'_2.h.convert_2 R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) f w : ∀ (x : ↑I), Polynomial.eval (↑x) p = ↑f (↑(Homeomorph.symm (iccHomeoI a b h)) x) q : ℝ[X] := Polynomial.comp p ((b - a)⁻¹ • X + ↑Polynomial.C (-a * (b - a)⁻¹)) x : ↑(Set.Icc a b) w₁ : 0 < (b - a)⁻¹ w₂ : 0 ≤ ↑x - a w₃ : ↑x - a ≤ b - a ⊢ (↑x - a) * (b - a)⁻¹ ∈ I ** fconstructor ** case h.mp.intro.intro.refine'_2.h.convert_2.left R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) f w : ∀ (x : ↑I), Polynomial.eval (↑x) p = ↑f (↑(Homeomorph.symm (iccHomeoI a b h)) x) q : ℝ[X] := Polynomial.comp p ((b - a)⁻¹ • X + ↑Polynomial.C (-a * (b - a)⁻¹)) x : ↑(Set.Icc a b) w₁ : 0 < (b - a)⁻¹ w₂ : 0 ≤ ↑x - a w₃ : ↑x - a ≤ b - a ⊢ 0 ≤ (↑x - a) * (b - a)⁻¹ ** exact mul_nonneg w₂ (le_of_lt w₁) ** case h.mp.intro.intro.refine'_2.h.convert_2.right R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) f w : ∀ (x : ↑I), Polynomial.eval (↑x) p = ↑f (↑(Homeomorph.symm (iccHomeoI a b h)) x) q : ℝ[X] := Polynomial.comp p ((b - a)⁻¹ • X + ↑Polynomial.C (-a * (b - a)⁻¹)) x : ↑(Set.Icc a b) w₁ : 0 < (b - a)⁻¹ w₂ : 0 ≤ ↑x - a w₃ : ↑x - a ≤ b - a ⊢ (↑x - a) * (b - a)⁻¹ ≤ 1 ** rw [← div_eq_mul_inv, div_le_one (sub_pos.mpr h)] ** case h.mp.intro.intro.refine'_2.h.convert_2.right R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) p : ℝ[X] w✝ : ↑↑(toContinuousMapOnAlgHom I) p = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) f w : ∀ (x : ↑I), Polynomial.eval (↑x) p = ↑f (↑(Homeomorph.symm (iccHomeoI a b h)) x) q : ℝ[X] := Polynomial.comp p ((b - a)⁻¹ • X + ↑Polynomial.C (-a * (b - a)⁻¹)) x : ↑(Set.Icc a b) w₁ : 0 < (b - a)⁻¹ w₂ : 0 ≤ ↑x - a w₃ : ↑x - a ≤ b - a ⊢ ↑x - a ≤ b - a ** exact w₃ ** case h.mpr R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b f : C(↑(Set.Icc a b), ℝ) ⊢ f ∈ polynomialFunctions (Set.Icc a b) → f ∈ Subalgebra.comap (compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) (polynomialFunctions I) ** rintro ⟨p, ⟨-, rfl⟩⟩ ** case h.mpr.intro.intro R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b p : ℝ[X] ⊢ ↑↑(toContinuousMapOnAlgHom (Set.Icc a b)) p ∈ Subalgebra.comap (compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) (polynomialFunctions I) ** let q := p.comp ((b - a) • Polynomial.X + Polynomial.C a) ** case h.mpr.intro.intro R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b p : ℝ[X] q : ℝ[X] := Polynomial.comp p ((b - a) • X + ↑Polynomial.C a) ⊢ ↑↑(toContinuousMapOnAlgHom (Set.Icc a b)) p ∈ Subalgebra.comap (compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) (polynomialFunctions I) ** refine' ⟨q, ⟨_, _⟩⟩ ** case h.mpr.intro.intro.refine'_1 R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b p : ℝ[X] q : ℝ[X] := Polynomial.comp p ((b - a) • X + ↑Polynomial.C a) ⊢ q ∈ ↑⊤.toSubsemiring ** simp ** case h.mpr.intro.intro.refine'_2 R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b p : ℝ[X] q : ℝ[X] := Polynomial.comp p ((b - a) • X + ↑Polynomial.C a) ⊢ ↑↑(toContinuousMapOnAlgHom I) q = ↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) (↑↑(toContinuousMapOnAlgHom (Set.Icc a b)) p) ** ext x ** case h.mpr.intro.intro.refine'_2.h R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R a b : ℝ h : a < b p : ℝ[X] q : ℝ[X] := Polynomial.comp p ((b - a) • X + ↑Polynomial.C a) x : ↑I ⊢ ↑(↑↑(toContinuousMapOnAlgHom I) q) x = ↑(↑↑(compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) (↑↑(toContinuousMapOnAlgHom (Set.Icc a b)) p)) x ** simp [mul_comm] ** Qed
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polynomialFunctions.eq_adjoin_X ** R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R s : Set R ⊢ polynomialFunctions s = Algebra.adjoin R {↑(toContinuousMapOnAlgHom s) X} ** refine le_antisymm ?_
(Algebra.adjoin_le fun _ h => ⟨X, trivial, (Set.mem_singleton_iff.1 h).symm⟩) ** R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R s : Set R ⊢ polynomialFunctions s ≤ Algebra.adjoin R {↑(toContinuousMapOnAlgHom s) X} ** rintro - ⟨p, -, rfl⟩ ** case intro.intro R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R s : Set R p : R[X] ⊢ ↑↑(toContinuousMapOnAlgHom s) p ∈ Algebra.adjoin R {↑(toContinuousMapOnAlgHom s) X} ** rw [AlgHom.coe_toRingHom] ** case intro.intro R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R s : Set R p : R[X] ⊢ ↑(toContinuousMapOnAlgHom s) p ∈ Algebra.adjoin R {↑(toContinuousMapOnAlgHom s) X} ** refine p.induction_on (fun r => ?_) (fun f g hf hg => ?_) fun n r hn => ?_ ** case intro.intro.refine_1 R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R s : Set R p : R[X] r : R ⊢ ↑(toContinuousMapOnAlgHom s) (↑Polynomial.C r) ∈ Algebra.adjoin R {↑(toContinuousMapOnAlgHom s) X} ** rw [Polynomial.C_eq_algebraMap, AlgHomClass.commutes] ** case intro.intro.refine_1 R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R s : Set R p : R[X] r : R ⊢ ↑(algebraMap R C(↑s, R)) r ∈ Algebra.adjoin R {↑(toContinuousMapOnAlgHom s) X} ** exact Subalgebra.algebraMap_mem _ r ** case intro.intro.refine_2 R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R s : Set R p f g : R[X] hf : ↑(toContinuousMapOnAlgHom s) f ∈ Algebra.adjoin R {↑(toContinuousMapOnAlgHom s) X} hg : ↑(toContinuousMapOnAlgHom s) g ∈ Algebra.adjoin R {↑(toContinuousMapOnAlgHom s) X} ⊢ ↑(toContinuousMapOnAlgHom s) (f + g) ∈ Algebra.adjoin R {↑(toContinuousMapOnAlgHom s) X} ** rw [map_add] ** case intro.intro.refine_2 R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R s : Set R p f g : R[X] hf : ↑(toContinuousMapOnAlgHom s) f ∈ Algebra.adjoin R {↑(toContinuousMapOnAlgHom s) X} hg : ↑(toContinuousMapOnAlgHom s) g ∈ Algebra.adjoin R {↑(toContinuousMapOnAlgHom s) X} ⊢ ↑(toContinuousMapOnAlgHom s) f + ↑(toContinuousMapOnAlgHom s) g ∈ Algebra.adjoin R {↑(toContinuousMapOnAlgHom s) X} ** exact add_mem hf hg ** case intro.intro.refine_3 R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R s : Set R p : R[X] n : ℕ r : R hn : ↑(toContinuousMapOnAlgHom s) (↑Polynomial.C r * X ^ n) ∈ Algebra.adjoin R {↑(toContinuousMapOnAlgHom s) X} ⊢ ↑(toContinuousMapOnAlgHom s) (↑Polynomial.C r * X ^ (n + 1)) ∈ Algebra.adjoin R {↑(toContinuousMapOnAlgHom s) X} ** rw [pow_succ', ← mul_assoc, map_mul] ** case intro.intro.refine_3 R : Type u_1 inst✝² : CommSemiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R s : Set R p : R[X] n : ℕ r : R hn : ↑(toContinuousMapOnAlgHom s) (↑Polynomial.C r * X ^ n) ∈ Algebra.adjoin R {↑(toContinuousMapOnAlgHom s) X} ⊢ ↑(toContinuousMapOnAlgHom s) (↑Polynomial.C r * X ^ n) * ↑(toContinuousMapOnAlgHom s) X ∈ Algebra.adjoin R {↑(toContinuousMapOnAlgHom s) X} ** exact mul_mem hn (Algebra.subset_adjoin <| Set.mem_singleton _) ** Qed
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polynomialFunctions.le_equalizer ** R : Type u_1 inst✝⁴ : CommSemiring R inst✝³ : TopologicalSpace R inst✝² : TopologicalSemiring R A : Type u_2 inst✝¹ : Semiring A inst✝ : Algebra R A s : Set R φ ψ : C(↑s, R) →ₐ[R] A h : ↑φ (↑(toContinuousMapOnAlgHom s) X) = ↑ψ (↑(toContinuousMapOnAlgHom s) X) ⊢ polynomialFunctions s ≤ AlgHom.equalizer φ ψ ** rw [polynomialFunctions.eq_adjoin_X s] ** R : Type u_1 inst✝⁴ : CommSemiring R inst✝³ : TopologicalSpace R inst✝² : TopologicalSemiring R A : Type u_2 inst✝¹ : Semiring A inst✝ : Algebra R A s : Set R φ ψ : C(↑s, R) →ₐ[R] A h : ↑φ (↑(toContinuousMapOnAlgHom s) X) = ↑ψ (↑(toContinuousMapOnAlgHom s) X) ⊢ Algebra.adjoin R {↑(toContinuousMapOnAlgHom s) X} ≤ AlgHom.equalizer φ ψ ** exact φ.adjoin_le_equalizer ψ fun x hx => (Set.mem_singleton_iff.1 hx).symm ▸ h ** Qed
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polynomialFunctions.starClosure_eq_adjoin_X ** R : Type u_1 inst✝⁴ : CommSemiring R inst✝³ : TopologicalSpace R inst✝² : TopologicalSemiring R inst✝¹ : StarRing R inst✝ : ContinuousStar R s : Set R ⊢ Subalgebra.starClosure (polynomialFunctions s) = adjoin R {↑(toContinuousMapOnAlgHom s) X} ** rw [polynomialFunctions.eq_adjoin_X s, adjoin_eq_starClosure_adjoin] ** Qed
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polynomialFunctions.starClosure_le_equalizer ** R : Type u_1 inst✝⁷ : CommSemiring R inst✝⁶ : TopologicalSpace R inst✝⁵ : TopologicalSemiring R A : Type u_2 inst✝⁴ : StarRing R inst✝³ : ContinuousStar R inst✝² : Semiring A inst✝¹ : StarRing A inst✝ : Algebra R A s : Set R φ ψ : C(↑s, R) →⋆ₐ[R] A h : ↑φ (↑(toContinuousMapOnAlgHom s) X) = ↑ψ (↑(toContinuousMapOnAlgHom s) X) ⊢ Subalgebra.starClosure (polynomialFunctions s) ≤ StarAlgHom.equalizer φ ψ ** rw [polynomialFunctions.starClosure_eq_adjoin_X s] ** R : Type u_1 inst✝⁷ : CommSemiring R inst✝⁶ : TopologicalSpace R inst✝⁵ : TopologicalSemiring R A : Type u_2 inst✝⁴ : StarRing R inst✝³ : ContinuousStar R inst✝² : Semiring A inst✝¹ : StarRing A inst✝ : Algebra R A s : Set R φ ψ : C(↑s, R) →⋆ₐ[R] A h : ↑φ (↑(toContinuousMapOnAlgHom s) X) = ↑ψ (↑(toContinuousMapOnAlgHom s) X) ⊢ adjoin R {↑(toContinuousMapOnAlgHom s) X} ≤ StarAlgHom.equalizer φ ψ ** exact StarAlgHom.adjoin_le_equalizer φ ψ fun x hx => (Set.mem_singleton_iff.1 hx).symm ▸ h ** Qed
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IsCompact.compl_mem_sets ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α hs : IsCompact s f : Filter α hf : ∀ (a : α), a ∈ s → sᶜ ∈ 𝓝 a ⊓ f ⊢ sᶜ ∈ f ** contrapose! hf ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α hs : IsCompact s f : Filter α hf : ¬sᶜ ∈ f ⊢ ∃ a, a ∈ s ∧ ¬sᶜ ∈ 𝓝 a ⊓ f ** simp only [not_mem_iff_inf_principal_compl, compl_compl, inf_assoc] at hf ⊢ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α hs : IsCompact s f : Filter α hf : NeBot (f ⊓ 𝓟 s) ⊢ ∃ a, a ∈ s ∧ NeBot (𝓝 a ⊓ (f ⊓ 𝓟 s)) ** exact @hs _ hf inf_le_right ** Qed
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IsCompact.compl_mem_sets_of_nhdsWithin ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α hs : IsCompact s f : Filter α hf : ∀ (a : α), a ∈ s → ∃ t, t ∈ 𝓝[s] a ∧ tᶜ ∈ f ⊢ sᶜ ∈ f ** refine' hs.compl_mem_sets fun a ha => _ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α hs : IsCompact s f : Filter α hf : ∀ (a : α), a ∈ s → ∃ t, t ∈ 𝓝[s] a ∧ tᶜ ∈ f a : α ha : a ∈ s ⊢ sᶜ ∈ 𝓝 a ⊓ f ** rcases hf a ha with ⟨t, ht, hst⟩ ** case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α hs : IsCompact s f : Filter α hf : ∀ (a : α), a ∈ s → ∃ t, t ∈ 𝓝[s] a ∧ tᶜ ∈ f a : α ha : a ∈ s t : Set α ht : t ∈ 𝓝[s] a hst : tᶜ ∈ f ⊢ sᶜ ∈ 𝓝 a ⊓ f ** replace ht := mem_inf_principal.1 ht ** case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α hs : IsCompact s f : Filter α hf : ∀ (a : α), a ∈ s → ∃ t, t ∈ 𝓝[s] a ∧ tᶜ ∈ f a : α ha : a ∈ s t : Set α hst : tᶜ ∈ f ht : {x | x ∈ s → x ∈ t} ∈ 𝓝 a ⊢ sᶜ ∈ 𝓝 a ⊓ f ** apply mem_inf_of_inter ht hst ** case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α hs : IsCompact s f : Filter α hf : ∀ (a : α), a ∈ s → ∃ t, t ∈ 𝓝[s] a ∧ tᶜ ∈ f a : α ha : a ∈ s t : Set α hst : tᶜ ∈ f ht : {x | x ∈ s → x ∈ t} ∈ 𝓝 a ⊢ {x | x ∈ s → x ∈ t} ∩ tᶜ ⊆ sᶜ ** rintro x ⟨h₁, h₂⟩ hs ** case intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α hs✝ : IsCompact s f : Filter α hf : ∀ (a : α), a ∈ s → ∃ t, t ∈ 𝓝[s] a ∧ tᶜ ∈ f a : α ha : a ∈ s t : Set α hst : tᶜ ∈ f ht : {x | x ∈ s → x ∈ t} ∈ 𝓝 a x : α h₁ : x ∈ {x | x ∈ s → x ∈ t} h₂ : x ∈ tᶜ hs : x ∈ s ⊢ False ** exact h₂ (h₁ hs) ** Qed
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IsCompact.induction_on ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t s : Set α hs : IsCompact s p : Set α → Prop he : p ∅ hmono : ∀ ⦃s t : Set α⦄, s ⊆ t → p t → p s hunion : ∀ ⦃s t : Set α⦄, p s → p t → p (s ∪ t) hnhds : ∀ (x : α), x ∈ s → ∃ t, t ∈ 𝓝[s] x ∧ p t ⊢ p s ** let f : Filter α :=
{ sets := { t | p tᶜ }
univ_sets := by simpa
sets_of_superset := fun ht₁ ht => hmono (compl_subset_compl.2 ht) ht₁
inter_sets := fun ht₁ ht₂ => by simp [compl_inter, hunion ht₁ ht₂] } ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t s : Set α hs : IsCompact s p : Set α → Prop he : p ∅ hmono : ∀ ⦃s t : Set α⦄, s ⊆ t → p t → p s hunion : ∀ ⦃s t : Set α⦄, p s → p t → p (s ∪ t) hnhds : ∀ (x : α), x ∈ s → ∃ t, t ∈ 𝓝[s] x ∧ p t f : Filter α := { sets := {t | p tᶜ}, univ_sets := (_ : univ ∈ {t | p tᶜ}), sets_of_superset := (_ : ∀ {x y : Set α}, x ∈ {t | p tᶜ} → x ⊆ y → p yᶜ), inter_sets := (_ : ∀ {x y : Set α}, x ∈ {t | p tᶜ} → y ∈ {t | p tᶜ} → p (x ∩ y)ᶜ) } ⊢ p s ** have : sᶜ ∈ f := hs.compl_mem_sets_of_nhdsWithin (by simpa using hnhds) ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t s : Set α hs : IsCompact s p : Set α → Prop he : p ∅ hmono : ∀ ⦃s t : Set α⦄, s ⊆ t → p t → p s hunion : ∀ ⦃s t : Set α⦄, p s → p t → p (s ∪ t) hnhds : ∀ (x : α), x ∈ s → ∃ t, t ∈ 𝓝[s] x ∧ p t f : Filter α := { sets := {t | p tᶜ}, univ_sets := (_ : univ ∈ {t | p tᶜ}), sets_of_superset := (_ : ∀ {x y : Set α}, x ∈ {t | p tᶜ} → x ⊆ y → p yᶜ), inter_sets := (_ : ∀ {x y : Set α}, x ∈ {t | p tᶜ} → y ∈ {t | p tᶜ} → p (x ∩ y)ᶜ) } this : sᶜ ∈ f ⊢ p s ** rwa [← compl_compl s] ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t s : Set α hs : IsCompact s p : Set α → Prop he : p ∅ hmono : ∀ ⦃s t : Set α⦄, s ⊆ t → p t → p s hunion : ∀ ⦃s t : Set α⦄, p s → p t → p (s ∪ t) hnhds : ∀ (x : α), x ∈ s → ∃ t, t ∈ 𝓝[s] x ∧ p t ⊢ univ ∈ {t | p tᶜ} ** simpa ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t s : Set α hs : IsCompact s p : Set α → Prop he : p ∅ hmono : ∀ ⦃s t : Set α⦄, s ⊆ t → p t → p s hunion : ∀ ⦃s t : Set α⦄, p s → p t → p (s ∪ t) hnhds : ∀ (x : α), x ∈ s → ∃ t, t ∈ 𝓝[s] x ∧ p t x✝ y✝ : Set α ht₁ : x✝ ∈ {t | p tᶜ} ht₂ : y✝ ∈ {t | p tᶜ} ⊢ x✝ ∩ y✝ ∈ {t | p tᶜ} ** simp [compl_inter, hunion ht₁ ht₂] ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t s : Set α hs : IsCompact s p : Set α → Prop he : p ∅ hmono : ∀ ⦃s t : Set α⦄, s ⊆ t → p t → p s hunion : ∀ ⦃s t : Set α⦄, p s → p t → p (s ∪ t) hnhds : ∀ (x : α), x ∈ s → ∃ t, t ∈ 𝓝[s] x ∧ p t f : Filter α := { sets := {t | p tᶜ}, univ_sets := (_ : univ ∈ {t | p tᶜ}), sets_of_superset := (_ : ∀ {x y : Set α}, x ∈ {t | p tᶜ} → x ⊆ y → p yᶜ), inter_sets := (_ : ∀ {x y : Set α}, x ∈ {t | p tᶜ} → y ∈ {t | p tᶜ} → p (x ∩ y)ᶜ) } ⊢ ∀ (a : α), a ∈ s → ∃ t, t ∈ 𝓝[s] a ∧ tᶜ ∈ f ** simpa using hnhds ** Qed
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IsCompact.inter_right ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α hs : IsCompact s ht : IsClosed t ⊢ IsCompact (s ∩ t) ** intro f hnf hstf ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α hs : IsCompact s ht : IsClosed t f : Filter α hnf : NeBot f hstf : f ≤ 𝓟 (s ∩ t) ⊢ ∃ a, a ∈ s ∩ t ∧ ClusterPt a f ** obtain ⟨a, hsa, ha⟩ : ∃ a ∈ s, ClusterPt a f :=
hs (le_trans hstf (le_principal_iff.2 (inter_subset_left _ _))) ** case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α hs : IsCompact s ht : IsClosed t f : Filter α hnf : NeBot f hstf : f ≤ 𝓟 (s ∩ t) a : α hsa : a ∈ s ha : ClusterPt a f ⊢ ∃ a, a ∈ s ∩ t ∧ ClusterPt a f ** have : a ∈ t := ht.mem_of_nhdsWithin_neBot <|
ha.mono <| le_trans hstf (le_principal_iff.2 (inter_subset_right _ _)) ** case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α hs : IsCompact s ht : IsClosed t f : Filter α hnf : NeBot f hstf : f ≤ 𝓟 (s ∩ t) a : α hsa : a ∈ s ha : ClusterPt a f this : a ∈ t ⊢ ∃ a, a ∈ s ∩ t ∧ ClusterPt a f ** exact ⟨a, ⟨hsa, this⟩, ha⟩ ** Qed
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IsCompact.image_of_continuousOn ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : α → β hs : IsCompact s hf : ContinuousOn f s ⊢ IsCompact (f '' s) ** intro l lne ls ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : α → β hs : IsCompact s hf : ContinuousOn f s l : Filter β lne : NeBot l ls : l ≤ 𝓟 (f '' s) ⊢ ∃ a, a ∈ f '' s ∧ ClusterPt a l ** have : NeBot (l.comap f ⊓ 𝓟 s) :=
comap_inf_principal_neBot_of_image_mem lne (le_principal_iff.1 ls) ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : α → β hs : IsCompact s hf : ContinuousOn f s l : Filter β lne : NeBot l ls : l ≤ 𝓟 (f '' s) this : NeBot (comap f l ⊓ 𝓟 s) ⊢ ∃ a, a ∈ f '' s ∧ ClusterPt a l ** obtain ⟨a, has, ha⟩ : ∃ a ∈ s, ClusterPt a (l.comap f ⊓ 𝓟 s) := @hs _ this inf_le_right ** case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : α → β hs : IsCompact s hf : ContinuousOn f s l : Filter β lne : NeBot l ls : l ≤ 𝓟 (f '' s) this : NeBot (comap f l ⊓ 𝓟 s) a : α has : a ∈ s ha : ClusterPt a (comap f l ⊓ 𝓟 s) ⊢ ∃ a, a ∈ f '' s ∧ ClusterPt a l ** haveI := ha.neBot ** case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : α → β hs : IsCompact s hf : ContinuousOn f s l : Filter β lne : NeBot l ls : l ≤ 𝓟 (f '' s) this✝ : NeBot (comap f l ⊓ 𝓟 s) a : α has : a ∈ s ha : ClusterPt a (comap f l ⊓ 𝓟 s) this : NeBot (𝓝 a ⊓ (comap f l ⊓ 𝓟 s)) ⊢ ∃ a, a ∈ f '' s ∧ ClusterPt a l ** use f a, mem_image_of_mem f has ** case right α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : α → β hs : IsCompact s hf : ContinuousOn f s l : Filter β lne : NeBot l ls : l ≤ 𝓟 (f '' s) this✝ : NeBot (comap f l ⊓ 𝓟 s) a : α has : a ∈ s ha : ClusterPt a (comap f l ⊓ 𝓟 s) this : NeBot (𝓝 a ⊓ (comap f l ⊓ 𝓟 s)) ⊢ ClusterPt (f a) l ** have : Tendsto f (𝓝 a ⊓ (comap f l ⊓ 𝓟 s)) (𝓝 (f a) ⊓ l) := by
convert (hf a has).inf (@tendsto_comap _ _ f l) using 1
rw [nhdsWithin]
ac_rfl ** case right α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : α → β hs : IsCompact s hf : ContinuousOn f s l : Filter β lne : NeBot l ls : l ≤ 𝓟 (f '' s) this✝¹ : NeBot (comap f l ⊓ 𝓟 s) a : α has : a ∈ s ha : ClusterPt a (comap f l ⊓ 𝓟 s) this✝ : NeBot (𝓝 a ⊓ (comap f l ⊓ 𝓟 s)) this : Tendsto f (𝓝 a ⊓ (comap f l ⊓ 𝓟 s)) (𝓝 (f a) ⊓ l) ⊢ ClusterPt (f a) l ** exact this.neBot ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : α → β hs : IsCompact s hf : ContinuousOn f s l : Filter β lne : NeBot l ls : l ≤ 𝓟 (f '' s) this✝ : NeBot (comap f l ⊓ 𝓟 s) a : α has : a ∈ s ha : ClusterPt a (comap f l ⊓ 𝓟 s) this : NeBot (𝓝 a ⊓ (comap f l ⊓ 𝓟 s)) ⊢ Tendsto f (𝓝 a ⊓ (comap f l ⊓ 𝓟 s)) (𝓝 (f a) ⊓ l) ** convert (hf a has).inf (@tendsto_comap _ _ f l) using 1 ** case h.e'_4 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : α → β hs : IsCompact s hf : ContinuousOn f s l : Filter β lne : NeBot l ls : l ≤ 𝓟 (f '' s) this✝ : NeBot (comap f l ⊓ 𝓟 s) a : α has : a ∈ s ha : ClusterPt a (comap f l ⊓ 𝓟 s) this : NeBot (𝓝 a ⊓ (comap f l ⊓ 𝓟 s)) ⊢ 𝓝 a ⊓ (comap f l ⊓ 𝓟 s) = 𝓝[s] a ⊓ comap f l ** rw [nhdsWithin] ** case h.e'_4 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : α → β hs : IsCompact s hf : ContinuousOn f s l : Filter β lne : NeBot l ls : l ≤ 𝓟 (f '' s) this✝ : NeBot (comap f l ⊓ 𝓟 s) a : α has : a ∈ s ha : ClusterPt a (comap f l ⊓ 𝓟 s) this : NeBot (𝓝 a ⊓ (comap f l ⊓ 𝓟 s)) ⊢ 𝓝 a ⊓ (comap f l ⊓ 𝓟 s) = 𝓝 a ⊓ 𝓟 s ⊓ comap f l ** ac_rfl ** Qed
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isCompact_iff_ultrafilter_le_nhds ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ⊢ IsCompact s ↔ ∀ (f : Ultrafilter α), ↑f ≤ 𝓟 s → ∃ a, a ∈ s ∧ ↑f ≤ 𝓝 a ** refine' (forall_neBot_le_iff _).trans _ ** case refine'_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ⊢ Monotone fun f => ∃ a, a ∈ s ∧ ClusterPt a f ** rintro f g hle ⟨a, has, haf⟩ ** case refine'_1.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f g : Filter α hle : f ≤ g a : α has : a ∈ s haf : ClusterPt a f ⊢ ∃ a, a ∈ s ∧ ClusterPt a g ** exact ⟨a, has, haf.mono hle⟩ ** case refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ⊢ (∀ (f : Ultrafilter α), ↑f ≤ 𝓟 s → ∃ a, a ∈ s ∧ ClusterPt a ↑f) ↔ ∀ (f : Ultrafilter α), ↑f ≤ 𝓟 s → ∃ a, a ∈ s ∧ ↑f ≤ 𝓝 a ** simp only [Ultrafilter.clusterPt_iff] ** Qed
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IsCompact.elim_nhds_subcover ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α hs : IsCompact s U : α → Set α hU : ∀ (x : α), x ∈ s → U x ∈ 𝓝 x t : Finset ↑s ht : s ⊆ ⋃ x ∈ t, U ↑x ⊢ s ⊆ ⋃ x ∈ Finset.image Subtype.val t, U x ** rwa [Finset.set_biUnion_finset_image] ** Qed
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IsCompact.disjoint_nhdsSet_left ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α l : Filter α hs : IsCompact s ⊢ Disjoint (𝓝ˢ s) l ↔ ∀ (x : α), x ∈ s → Disjoint (𝓝 x) l ** refine' ⟨fun h x hx => h.mono_left <| nhds_le_nhdsSet hx, fun H => _⟩ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α l : Filter α hs : IsCompact s H : ∀ (x : α), x ∈ s → Disjoint (𝓝 x) l ⊢ Disjoint (𝓝ˢ s) l ** choose! U hxU hUl using fun x hx => (nhds_basis_opens x).disjoint_iff_left.1 (H x hx) ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α l : Filter α hs : IsCompact s H : ∀ (x : α), x ∈ s → Disjoint (𝓝 x) l U : α → Set α hxU : ∀ (x : α), x ∈ s → x ∈ U x ∧ IsOpen (U x) hUl : ∀ (x : α), x ∈ s → (U x)ᶜ ∈ l ⊢ Disjoint (𝓝ˢ s) l ** choose hxU hUo using hxU ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α l : Filter α hs : IsCompact s H : ∀ (x : α), x ∈ s → Disjoint (𝓝 x) l U : α → Set α hUl : ∀ (x : α), x ∈ s → (U x)ᶜ ∈ l hxU : ∀ (x : α), x ∈ s → x ∈ U x hUo : ∀ (x : α), x ∈ s → IsOpen (U x) ⊢ Disjoint (𝓝ˢ s) l ** rcases hs.elim_nhds_subcover U fun x hx => (hUo x hx).mem_nhds (hxU x hx) with ⟨t, hts, hst⟩ ** case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α l : Filter α hs : IsCompact s H : ∀ (x : α), x ∈ s → Disjoint (𝓝 x) l U : α → Set α hUl : ∀ (x : α), x ∈ s → (U x)ᶜ ∈ l hxU : ∀ (x : α), x ∈ s → x ∈ U x hUo : ∀ (x : α), x ∈ s → IsOpen (U x) t : Finset α hts : ∀ (x : α), x ∈ t → x ∈ s hst : s ⊆ ⋃ x ∈ t, U x ⊢ Disjoint (𝓝ˢ s) l ** refine (hasBasis_nhdsSet _).disjoint_iff_left.2
⟨⋃ x ∈ t, U x, ⟨isOpen_biUnion fun x hx => hUo x (hts x hx), hst⟩, ?_⟩ ** case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α l : Filter α hs : IsCompact s H : ∀ (x : α), x ∈ s → Disjoint (𝓝 x) l U : α → Set α hUl : ∀ (x : α), x ∈ s → (U x)ᶜ ∈ l hxU : ∀ (x : α), x ∈ s → x ∈ U x hUo : ∀ (x : α), x ∈ s → IsOpen (U x) t : Finset α hts : ∀ (x : α), x ∈ t → x ∈ s hst : s ⊆ ⋃ x ∈ t, U x ⊢ (⋃ x ∈ t, U x)ᶜ ∈ l ** rw [compl_iUnion₂, biInter_finset_mem] ** case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α l : Filter α hs : IsCompact s H : ∀ (x : α), x ∈ s → Disjoint (𝓝 x) l U : α → Set α hUl : ∀ (x : α), x ∈ s → (U x)ᶜ ∈ l hxU : ∀ (x : α), x ∈ s → x ∈ U x hUo : ∀ (x : α), x ∈ s → IsOpen (U x) t : Finset α hts : ∀ (x : α), x ∈ t → x ∈ s hst : s ⊆ ⋃ x ∈ t, U x ⊢ ∀ (i : α), i ∈ t → (U i)ᶜ ∈ l ** exact fun x hx => hUl x (hts x hx) ** Qed
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IsCompact.disjoint_nhdsSet_right ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α l : Filter α hs : IsCompact s ⊢ Disjoint l (𝓝ˢ s) ↔ ∀ (x : α), x ∈ s → Disjoint l (𝓝 x) ** simpa only [disjoint_comm] using hs.disjoint_nhdsSet_left ** Qed
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IsCompact.elim_directed_family_closed ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ι : Type v hι : Nonempty ι hs : IsCompact s Z : ι → Set α hZc : ∀ (i : ι), IsClosed (Z i) hsZ : s ∩ ⋂ i, Z i = ∅ hdZ : Directed (fun x x_1 => x ⊇ x_1) Z ⊢ s ⊆ ⋃ i, (compl ∘ Z) i ** simpa only [subset_def, not_forall, eq_empty_iff_forall_not_mem, mem_iUnion, exists_prop,
mem_inter_iff, not_and, iff_self_iff, mem_iInter, mem_compl_iff] using hsZ ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α ι : Type v hι : Nonempty ι hs : IsCompact s Z : ι → Set α hZc : ∀ (i : ι), IsClosed (Z i) hsZ : s ∩ ⋂ i, Z i = ∅ hdZ : Directed (fun x x_1 => x ⊇ x_1) Z t : ι ht : s ⊆ (compl ∘ Z) t ⊢ s ∩ Z t = ∅ ** simpa only [subset_def, not_forall, eq_empty_iff_forall_not_mem, mem_iUnion, exists_prop,
mem_inter_iff, not_and, iff_self_iff, mem_iInter, mem_compl_iff] using ht ** Qed
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IsCompact.elim_finite_subfamily_closed ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t s : Set α ι : Type v hs : IsCompact s Z : ι → Set α hZc : ∀ (i : ι), IsClosed (Z i) hsZ : s ∩ ⋂ i, Z i = ∅ ⊢ s ∩ ⋂ i, ⋂ i_1 ∈ i, Z i_1 = ∅ ** rwa [← iInter_eq_iInter_finset] ** Qed
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LocallyFinite.finite_nonempty_inter_compact ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t : Set α ι : Type u_3 f : ι → Set α hf : LocallyFinite f s : Set α hs : IsCompact s ⊢ Set.Finite {i | Set.Nonempty (f i ∩ s)} ** choose U hxU hUf using hf ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t : Set α ι : Type u_3 f : ι → Set α s : Set α hs : IsCompact s U : α → Set α hxU : ∀ (x : α), U x ∈ 𝓝 x hUf : ∀ (x : α), Set.Finite {i | Set.Nonempty (f i ∩ U x)} ⊢ Set.Finite {i | Set.Nonempty (f i ∩ s)} ** rcases hs.elim_nhds_subcover U fun x _ => hxU x with ⟨t, -, hsU⟩ ** case intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ : Set α ι : Type u_3 f : ι → Set α s : Set α hs : IsCompact s U : α → Set α hxU : ∀ (x : α), U x ∈ 𝓝 x hUf : ∀ (x : α), Set.Finite {i | Set.Nonempty (f i ∩ U x)} t : Finset α hsU : s ⊆ ⋃ x ∈ t, U x ⊢ Set.Finite {i | Set.Nonempty (f i ∩ s)} ** refine' (t.finite_toSet.biUnion fun x _ => hUf x).subset _ ** case intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ : Set α ι : Type u_3 f : ι → Set α s : Set α hs : IsCompact s U : α → Set α hxU : ∀ (x : α), U x ∈ 𝓝 x hUf : ∀ (x : α), Set.Finite {i | Set.Nonempty (f i ∩ U x)} t : Finset α hsU : s ⊆ ⋃ x ∈ t, U x ⊢ {i | Set.Nonempty (f i ∩ s)} ⊆ ⋃ i ∈ ↑t, {i_1 | Set.Nonempty (f i_1 ∩ U i)} ** rintro i ⟨x, hx⟩ ** case intro.intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ : Set α ι : Type u_3 f : ι → Set α s : Set α hs : IsCompact s U : α → Set α hxU : ∀ (x : α), U x ∈ 𝓝 x hUf : ∀ (x : α), Set.Finite {i | Set.Nonempty (f i ∩ U x)} t : Finset α hsU : s ⊆ ⋃ x ∈ t, U x i : ι x : α hx : x ∈ f i ∩ s ⊢ i ∈ ⋃ i ∈ ↑t, {i_1 | Set.Nonempty (f i_1 ∩ U i)} ** rcases mem_iUnion₂.1 (hsU hx.2) with ⟨c, hct, hcx⟩ ** case intro.intro.intro.intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ : Set α ι : Type u_3 f : ι → Set α s : Set α hs : IsCompact s U : α → Set α hxU : ∀ (x : α), U x ∈ 𝓝 x hUf : ∀ (x : α), Set.Finite {i | Set.Nonempty (f i ∩ U x)} t : Finset α hsU : s ⊆ ⋃ x ∈ t, U x i : ι x : α hx : x ∈ f i ∩ s c : α hct : c ∈ t hcx : x ∈ U c ⊢ i ∈ ⋃ i ∈ ↑t, {i_1 | Set.Nonempty (f i_1 ∩ U i)} ** exact mem_biUnion hct ⟨x, hx.1, hcx⟩ ** Qed
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IsCompact.inter_iInter_nonempty ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t s : Set α ι : Type v hs : IsCompact s Z : ι → Set α hZc : ∀ (i : ι), IsClosed (Z i) hsZ : ∀ (t : Finset ι), Set.Nonempty (s ∩ ⋂ i ∈ t, Z i) ⊢ Set.Nonempty (s ∩ ⋂ i, Z i) ** simp only [nonempty_iff_ne_empty] at hsZ ⊢ ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t s : Set α ι : Type v hs : IsCompact s Z : ι → Set α hZc : ∀ (i : ι), IsClosed (Z i) hsZ : ∀ (t : Finset ι), s ∩ ⋂ i ∈ t, Z i ≠ ∅ ⊢ s ∩ ⋂ i, Z i ≠ ∅ ** apply mt (hs.elim_finite_subfamily_closed Z hZc) ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t s : Set α ι : Type v hs : IsCompact s Z : ι → Set α hZc : ∀ (i : ι), IsClosed (Z i) hsZ : ∀ (t : Finset ι), s ∩ ⋂ i ∈ t, Z i ≠ ∅ ⊢ ¬∃ t, s ∩ ⋂ i ∈ t, Z i = ∅ ** push_neg ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t s : Set α ι : Type v hs : IsCompact s Z : ι → Set α hZc : ∀ (i : ι), IsClosed (Z i) hsZ : ∀ (t : Finset ι), s ∩ ⋂ i ∈ t, Z i ≠ ∅ ⊢ ∀ (t : Finset ι), s ∩ ⋂ i ∈ t, Z i ≠ ∅ ** exact hsZ ** Qed
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IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ι : Type v hι : Nonempty ι Z : ι → Set α hZd : Directed (fun x x_1 => x ⊇ x_1) Z hZn : ∀ (i : ι), Set.Nonempty (Z i) hZc : ∀ (i : ι), IsCompact (Z i) hZcl : ∀ (i : ι), IsClosed (Z i) ⊢ Set.Nonempty (⋂ i, Z i) ** let i₀ := hι.some ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ι : Type v hι : Nonempty ι Z : ι → Set α hZd : Directed (fun x x_1 => x ⊇ x_1) Z hZn : ∀ (i : ι), Set.Nonempty (Z i) hZc : ∀ (i : ι), IsCompact (Z i) hZcl : ∀ (i : ι), IsClosed (Z i) i₀ : ι := Nonempty.some hι ⊢ Set.Nonempty (⋂ i, Z i) ** suffices (Z i₀ ∩ ⋂ i, Z i).Nonempty by
rwa [inter_eq_right.mpr (iInter_subset _ i₀)] at this ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ι : Type v hι : Nonempty ι Z : ι → Set α hZd : Directed (fun x x_1 => x ⊇ x_1) Z hZn : ∀ (i : ι), Set.Nonempty (Z i) hZc : ∀ (i : ι), IsCompact (Z i) hZcl : ∀ (i : ι), IsClosed (Z i) i₀ : ι := Nonempty.some hι ⊢ Set.Nonempty (Z i₀ ∩ ⋂ i, Z i) ** simp only [nonempty_iff_ne_empty] at hZn ⊢ ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ι : Type v hι : Nonempty ι Z : ι → Set α hZd : Directed (fun x x_1 => x ⊇ x_1) Z hZc : ∀ (i : ι), IsCompact (Z i) hZcl : ∀ (i : ι), IsClosed (Z i) i₀ : ι := Nonempty.some hι hZn : ∀ (i : ι), Z i ≠ ∅ ⊢ Z (Nonempty.some hι) ∩ ⋂ i, Z i ≠ ∅ ** apply mt ((hZc i₀).elim_directed_family_closed Z hZcl) ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ι : Type v hι : Nonempty ι Z : ι → Set α hZd : Directed (fun x x_1 => x ⊇ x_1) Z hZc : ∀ (i : ι), IsCompact (Z i) hZcl : ∀ (i : ι), IsClosed (Z i) i₀ : ι := Nonempty.some hι hZn : ∀ (i : ι), Z i ≠ ∅ ⊢ ¬(Directed (fun x x_1 => x ⊇ x_1) Z → ∃ i, Z i₀ ∩ Z i = ∅) ** push_neg ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ι : Type v hι : Nonempty ι Z : ι → Set α hZd : Directed (fun x x_1 => x ⊇ x_1) Z hZc : ∀ (i : ι), IsCompact (Z i) hZcl : ∀ (i : ι), IsClosed (Z i) i₀ : ι := Nonempty.some hι hZn : ∀ (i : ι), Z i ≠ ∅ ⊢ Directed (fun x x_1 => x ⊇ x_1) Z ∧ ∀ (i : ι), Z i₀ ∩ Z i ≠ ∅ ** simp only [← nonempty_iff_ne_empty] at hZn ⊢ ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ι : Type v hι : Nonempty ι Z : ι → Set α hZd : Directed (fun x x_1 => x ⊇ x_1) Z hZc : ∀ (i : ι), IsCompact (Z i) hZcl : ∀ (i : ι), IsClosed (Z i) i₀ : ι := Nonempty.some hι hZn : ∀ (i : ι), Set.Nonempty (Z i) ⊢ Directed (fun x x_1 => x ⊇ x_1) Z ∧ ∀ (i : ι), Set.Nonempty (Z (Nonempty.some hι) ∩ Z i) ** refine' ⟨hZd, fun i => _⟩ ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ι : Type v hι : Nonempty ι Z : ι → Set α hZd : Directed (fun x x_1 => x ⊇ x_1) Z hZc : ∀ (i : ι), IsCompact (Z i) hZcl : ∀ (i : ι), IsClosed (Z i) i₀ : ι := Nonempty.some hι hZn : ∀ (i : ι), Set.Nonempty (Z i) i : ι ⊢ Set.Nonempty (Z (Nonempty.some hι) ∩ Z i) ** rcases hZd i₀ i with ⟨j, hji₀, hji⟩ ** case intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ι : Type v hι : Nonempty ι Z : ι → Set α hZd : Directed (fun x x_1 => x ⊇ x_1) Z hZc : ∀ (i : ι), IsCompact (Z i) hZcl : ∀ (i : ι), IsClosed (Z i) i₀ : ι := Nonempty.some hι hZn : ∀ (i : ι), Set.Nonempty (Z i) i j : ι hji₀ : Z i₀ ⊇ Z j hji : Z i ⊇ Z j ⊢ Set.Nonempty (Z (Nonempty.some hι) ∩ Z i) ** exact (hZn j).mono (subset_inter hji₀ hji) ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ι : Type v hι : Nonempty ι Z : ι → Set α hZd : Directed (fun x x_1 => x ⊇ x_1) Z hZn : ∀ (i : ι), Set.Nonempty (Z i) hZc : ∀ (i : ι), IsCompact (Z i) hZcl : ∀ (i : ι), IsClosed (Z i) i₀ : ι := Nonempty.some hι this : Set.Nonempty (Z i₀ ∩ ⋂ i, Z i) ⊢ Set.Nonempty (⋂ i, Z i) ** rwa [inter_eq_right.mpr (iInter_subset _ i₀)] at this ** Qed
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IsCompact.elim_finite_subcover_image ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α b : Set ι c : ι → Set α hs : IsCompact s hc₁ : ∀ (i : ι), i ∈ b → IsOpen (c i) hc₂ : s ⊆ ⋃ i ∈ b, c i ⊢ ∃ b', b' ⊆ b ∧ Set.Finite b' ∧ s ⊆ ⋃ i ∈ b', c i ** simp only [Subtype.forall', biUnion_eq_iUnion] at hc₁ hc₂ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α b : Set ι c : ι → Set α hs : IsCompact s hc₁ : ∀ (x : { a // a ∈ b }), IsOpen (c ↑x) hc₂ : s ⊆ ⋃ x, c ↑x ⊢ ∃ b', b' ⊆ b ∧ Set.Finite b' ∧ s ⊆ ⋃ i ∈ b', c i ** rcases hs.elim_finite_subcover (fun i => c i : b → Set α) hc₁ hc₂ with ⟨d, hd⟩ ** case intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α b : Set ι c : ι → Set α hs : IsCompact s hc₁ : ∀ (x : { a // a ∈ b }), IsOpen (c ↑x) hc₂ : s ⊆ ⋃ x, c ↑x d : Finset ↑b hd : s ⊆ ⋃ i ∈ d, c ↑i ⊢ ∃ b', b' ⊆ b ∧ Set.Finite b' ∧ s ⊆ ⋃ i ∈ b', c i ** refine' ⟨Subtype.val '' d.toSet, _, d.finite_toSet.image _, _⟩ ** case intro.refine'_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α b : Set ι c : ι → Set α hs : IsCompact s hc₁ : ∀ (x : { a // a ∈ b }), IsOpen (c ↑x) hc₂ : s ⊆ ⋃ x, c ↑x d : Finset ↑b hd : s ⊆ ⋃ i ∈ d, c ↑i ⊢ Subtype.val '' ↑d ⊆ b ** simp ** case intro.refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α b : Set ι c : ι → Set α hs : IsCompact s hc₁ : ∀ (x : { a // a ∈ b }), IsOpen (c ↑x) hc₂ : s ⊆ ⋃ x, c ↑x d : Finset ↑b hd : s ⊆ ⋃ i ∈ d, c ↑i ⊢ s ⊆ ⋃ i ∈ Subtype.val '' ↑d, c i ** rwa [biUnion_image] ** Qed
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isCompact_of_finite_subcover ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α h : ∀ {ι : Type u} (U : ι → Set α), (∀ (i : ι), IsOpen (U i)) → s ⊆ ⋃ i, U i → ∃ t, s ⊆ ⋃ i ∈ t, U i f : Filter α hf : NeBot f hfs : f ≤ 𝓟 s ⊢ ∃ a, a ∈ s ∧ ClusterPt a f ** contrapose! h ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : Filter α hf : NeBot f hfs : f ≤ 𝓟 s h : ∀ (a : α), a ∈ s → ¬ClusterPt a f ⊢ Exists fun {ι} => ∃ U, (∀ (i : ι), IsOpen (U i)) ∧ s ⊆ ⋃ i, U i ∧ ∀ (t : Finset ι), ¬s ⊆ ⋃ i ∈ t, U i ** simp only [ClusterPt, not_neBot, ← disjoint_iff, SetCoe.forall',
(nhds_basis_opens _).disjoint_iff_left] at h ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : Filter α hf : NeBot f hfs : f ≤ 𝓟 s h : ∀ (x : ↑s), ∃ i, (↑x ∈ i ∧ IsOpen i) ∧ iᶜ ∈ f ⊢ Exists fun {ι} => ∃ U, (∀ (i : ι), IsOpen (U i)) ∧ s ⊆ ⋃ i, U i ∧ ∀ (t : Finset ι), ¬s ⊆ ⋃ i ∈ t, U i ** choose U hU hUf using h ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : Filter α hf : NeBot f hfs : f ≤ 𝓟 s U : ↑s → Set α hU : ∀ (x : ↑s), ↑x ∈ U x ∧ IsOpen (U x) hUf : ∀ (x : ↑s), (U x)ᶜ ∈ f ⊢ Exists fun {ι} => ∃ U, (∀ (i : ι), IsOpen (U i)) ∧ s ⊆ ⋃ i, U i ∧ ∀ (t : Finset ι), ¬s ⊆ ⋃ i ∈ t, U i ** refine ⟨s, U, fun x => (hU x).2, fun x hx => mem_iUnion.2 ⟨⟨x, hx⟩, (hU _).1⟩, fun t ht => ?_⟩ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α f : Filter α hf : NeBot f hfs : f ≤ 𝓟 s U : ↑s → Set α hU : ∀ (x : ↑s), ↑x ∈ U x ∧ IsOpen (U x) hUf : ∀ (x : ↑s), (U x)ᶜ ∈ f t : Finset ↑s ht : s ⊆ ⋃ i ∈ t, U i ⊢ False ** refine compl_not_mem (le_principal_iff.1 hfs) ?_ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α f : Filter α hf : NeBot f hfs : f ≤ 𝓟 s U : ↑s → Set α hU : ∀ (x : ↑s), ↑x ∈ U x ∧ IsOpen (U x) hUf : ∀ (x : ↑s), (U x)ᶜ ∈ f t : Finset ↑s ht : s ⊆ ⋃ i ∈ t, U i ⊢ sᶜ ∈ f ** refine mem_of_superset ((biInter_finset_mem t).2 fun x _ => hUf x) ?_ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α f : Filter α hf : NeBot f hfs : f ≤ 𝓟 s U : ↑s → Set α hU : ∀ (x : ↑s), ↑x ∈ U x ∧ IsOpen (U x) hUf : ∀ (x : ↑s), (U x)ᶜ ∈ f t : Finset ↑s ht : s ⊆ ⋃ i ∈ t, U i ⊢ ⋂ i ∈ t, (U i)ᶜ ⊆ sᶜ ** rw [subset_compl_comm, compl_iInter₂] ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α f : Filter α hf : NeBot f hfs : f ≤ 𝓟 s U : ↑s → Set α hU : ∀ (x : ↑s), ↑x ∈ U x ∧ IsOpen (U x) hUf : ∀ (x : ↑s), (U x)ᶜ ∈ f t : Finset ↑s ht : s ⊆ ⋃ i ∈ t, U i ⊢ s ⊆ ⋃ i ∈ t, (U i)ᶜᶜ ** simpa only [compl_compl] ** Qed
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isCompact_of_finite_subfamily_closed ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α h : ∀ {ι : Type u} (Z : ι → Set α), (∀ (i : ι), IsClosed (Z i)) → s ∩ ⋂ i, Z i = ∅ → ∃ t, s ∩ ⋂ i ∈ t, Z i = ∅ ι✝ : Type u U : ι✝ → Set α hUo : ∀ (i : ι✝), IsOpen (U i) hsU : s ⊆ ⋃ i, U i ⊢ ∃ t, s ⊆ ⋃ i ∈ t, U i ** rw [← disjoint_compl_right_iff_subset, compl_iUnion, disjoint_iff] at hsU ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α h : ∀ {ι : Type u} (Z : ι → Set α), (∀ (i : ι), IsClosed (Z i)) → s ∩ ⋂ i, Z i = ∅ → ∃ t, s ∩ ⋂ i ∈ t, Z i = ∅ ι✝ : Type u U : ι✝ → Set α hUo : ∀ (i : ι✝), IsOpen (U i) hsU : s ⊓ ⋂ i, (U i)ᶜ = ⊥ ⊢ ∃ t, s ⊆ ⋃ i ∈ t, U i ** rcases h (fun i => (U i)ᶜ) (fun i => (hUo _).isClosed_compl) hsU with ⟨t, ht⟩ ** case intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α h : ∀ {ι : Type u} (Z : ι → Set α), (∀ (i : ι), IsClosed (Z i)) → s ∩ ⋂ i, Z i = ∅ → ∃ t, s ∩ ⋂ i ∈ t, Z i = ∅ ι✝ : Type u U : ι✝ → Set α hUo : ∀ (i : ι✝), IsOpen (U i) hsU : s ⊓ ⋂ i, (U i)ᶜ = ⊥ t : Finset ι✝ ht : s ∩ ⋂ i ∈ t, (U i)ᶜ = ∅ ⊢ ∃ t, s ⊆ ⋃ i ∈ t, U i ** refine ⟨t, ?_⟩ ** case intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α h : ∀ {ι : Type u} (Z : ι → Set α), (∀ (i : ι), IsClosed (Z i)) → s ∩ ⋂ i, Z i = ∅ → ∃ t, s ∩ ⋂ i ∈ t, Z i = ∅ ι✝ : Type u U : ι✝ → Set α hUo : ∀ (i : ι✝), IsOpen (U i) hsU : s ⊓ ⋂ i, (U i)ᶜ = ⊥ t : Finset ι✝ ht : s ∩ ⋂ i ∈ t, (U i)ᶜ = ∅ ⊢ s ⊆ ⋃ i ∈ t, U i ** rwa [← disjoint_compl_right_iff_subset, compl_iUnion₂, disjoint_iff] ** Qed
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IsCompact.mem_nhdsSet_prod_of_forall ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t K : Set α l : Filter β s : Set (α × β) hK : IsCompact K hs : ∀ (x : α), x ∈ K → s ∈ 𝓝 x ×ˢ l ⊢ s ∈ 𝓝ˢ K ×ˢ l ** refine hK.induction_on (by simp) (fun t t' ht hs ↦ ?_) (fun t t' ht ht' ↦ ?_) fun x hx ↦ ?_ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t K : Set α l : Filter β s : Set (α × β) hK : IsCompact K hs : ∀ (x : α), x ∈ K → s ∈ 𝓝 x ×ˢ l ⊢ s ∈ 𝓝ˢ ∅ ×ˢ l ** simp ** case refine_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ K : Set α l : Filter β s : Set (α × β) hK : IsCompact K hs✝ : ∀ (x : α), x ∈ K → s ∈ 𝓝 x ×ˢ l t t' : Set α ht : t ⊆ t' hs : s ∈ 𝓝ˢ t' ×ˢ l ⊢ s ∈ 𝓝ˢ t ×ˢ l ** exact prod_mono (nhdsSet_mono ht) le_rfl hs ** case refine_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ K : Set α l : Filter β s : Set (α × β) hK : IsCompact K hs : ∀ (x : α), x ∈ K → s ∈ 𝓝 x ×ˢ l t t' : Set α ht : s ∈ 𝓝ˢ t ×ˢ l ht' : s ∈ 𝓝ˢ t' ×ˢ l ⊢ s ∈ 𝓝ˢ (t ∪ t') ×ˢ l ** simp [sup_prod, *] ** case refine_3 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t K : Set α l : Filter β s : Set (α × β) hK : IsCompact K hs : ∀ (x : α), x ∈ K → s ∈ 𝓝 x ×ˢ l x : α hx : x ∈ K ⊢ ∃ t, t ∈ 𝓝[K] x ∧ s ∈ 𝓝ˢ t ×ˢ l ** rcases ((nhds_basis_opens _).prod l.basis_sets).mem_iff.1 (hs x hx)
with ⟨⟨u, v⟩, ⟨⟨hx, huo⟩, hv⟩, hs⟩ ** case refine_3.intro.mk.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t K : Set α l : Filter β s : Set (α × β) hK : IsCompact K hs✝ : ∀ (x : α), x ∈ K → s ∈ 𝓝 x ×ˢ l x : α hx✝ : x ∈ K u : Set α v : Set β hs : (u, v).1 ×ˢ id (u, v).2 ⊆ s hv : (u, v).2 ∈ l hx : x ∈ (u, v).1 huo : IsOpen (u, v).1 ⊢ ∃ t, t ∈ 𝓝[K] x ∧ s ∈ 𝓝ˢ t ×ˢ l ** refine ⟨u, nhdsWithin_le_nhds (huo.mem_nhds hx), mem_of_superset ?_ hs⟩ ** case refine_3.intro.mk.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t K : Set α l : Filter β s : Set (α × β) hK : IsCompact K hs✝ : ∀ (x : α), x ∈ K → s ∈ 𝓝 x ×ˢ l x : α hx✝ : x ∈ K u : Set α v : Set β hs : (u, v).1 ×ˢ id (u, v).2 ⊆ s hv : (u, v).2 ∈ l hx : x ∈ (u, v).1 huo : IsOpen (u, v).1 ⊢ (u, v).1 ×ˢ id (u, v).2 ∈ 𝓝ˢ u ×ˢ l ** exact prod_mem_prod (huo.mem_nhdsSet.2 Subset.rfl) hv ** Qed
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IsCompact.nhdsSet_prod_eq_biSup ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t K : Set α hK : IsCompact K l : Filter β s : Set (α × β) hs : s ∈ ⨆ x ∈ K, 𝓝 x ×ˢ l ⊢ ∀ (x : α), x ∈ K → s ∈ 𝓝 x ×ˢ l ** simpa using hs ** Qed
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IsCompact.prod_nhdsSet_eq_biSup ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α K : Set β hK : IsCompact K l : Filter α ⊢ l ×ˢ 𝓝ˢ K = ⨆ y ∈ K, l ×ˢ 𝓝 y ** simp only [prod_comm (f := l), hK.nhdsSet_prod_eq_biSup, map_iSup] ** Qed
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IsCompact.mem_prod_nhdsSet_of_forall ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t : Set α K : Set β l : Filter α s : Set (α × β) hK : IsCompact K hs : ∀ (y : β), y ∈ K → s ∈ l ×ˢ 𝓝 y ⊢ s ∈ ⨆ y ∈ K, l ×ˢ 𝓝 y ** simpa using hs ** Qed
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IsCompact.eventually_forall_of_forall_eventually ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α x₀ : α K : Set β hK : IsCompact K P : α → β → Prop hP : ∀ (y : β), y ∈ K → ∀ᶠ (z : α × β) in 𝓝 (x₀, y), P z.1 z.2 ⊢ ∀ᶠ (x : α) in 𝓝 x₀, ∀ (y : β), y ∈ K → P x y ** simp only [nhds_prod_eq, ← eventually_iSup, ← hK.prod_nhdsSet_eq_biSup] at hP ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α x₀ : α K : Set β hK : IsCompact K P : α → β → Prop hP : ∀ᶠ (z : α × β) in 𝓝 x₀ ×ˢ 𝓝ˢ K, P z.1 z.2 ⊢ ∀ᶠ (x : α) in 𝓝 x₀, ∀ (y : β), y ∈ K → P x y ** exact hP.curry.mono fun _ h ↦ h.self_of_nhdsSet ** Qed
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isCompact_singleton ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α a : α f : Filter α hf : NeBot f hfa : f ≤ 𝓟 {a} ⊢ 𝓟 {a} ≤ 𝓝 a ** simpa only [principal_singleton] using pure_le_nhds a ** Qed
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Set.Finite.isCompact_biUnion ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t : Set α s : Set ι f : ι → Set α hs : Set.Finite s hf : ∀ (i : ι), i ∈ s → IsCompact (f i) l : Ultrafilter α hl : ↑l ≤ 𝓟 (⋃ i ∈ s, f i) ⊢ ∃ a, a ∈ ⋃ i ∈ s, f i ∧ ↑l ≤ 𝓝 a ** rw [le_principal_iff, Ultrafilter.mem_coe, Ultrafilter.finite_biUnion_mem_iff hs] at hl ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t : Set α s : Set ι f : ι → Set α hs : Set.Finite s hf : ∀ (i : ι), i ∈ s → IsCompact (f i) l : Ultrafilter α hl : ∃ i, i ∈ s ∧ f i ∈ l ⊢ ∃ a, a ∈ ⋃ i ∈ s, f i ∧ ↑l ≤ 𝓝 a ** rcases hl with ⟨i, his, hi⟩ ** case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t : Set α s : Set ι f : ι → Set α hs : Set.Finite s hf : ∀ (i : ι), i ∈ s → IsCompact (f i) l : Ultrafilter α i : ι his : i ∈ s hi : f i ∈ l ⊢ ∃ a, a ∈ ⋃ i ∈ s, f i ∧ ↑l ≤ 𝓝 a ** rcases (hf i his).ultrafilter_le_nhds _ (le_principal_iff.2 hi) with ⟨x, hxi, hlx⟩ ** case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t : Set α s : Set ι f : ι → Set α hs : Set.Finite s hf : ∀ (i : ι), i ∈ s → IsCompact (f i) l : Ultrafilter α i : ι his : i ∈ s hi : f i ∈ l x : α hxi : x ∈ f i hlx : ↑l ≤ 𝓝 x ⊢ ∃ a, a ∈ ⋃ i ∈ s, f i ∧ ↑l ≤ 𝓝 a ** exact ⟨x, mem_iUnion₂.2 ⟨i, his, hxi⟩, hlx⟩ ** Qed
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Set.Finite.isCompact_sUnion ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α S : Set (Set α) hf : Set.Finite S hc : ∀ (s : Set α), s ∈ S → IsCompact s ⊢ IsCompact (⋃₀ S) ** rw [sUnion_eq_biUnion] ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α S : Set (Set α) hf : Set.Finite S hc : ∀ (s : Set α), s ∈ S → IsCompact s ⊢ IsCompact (⋃ i ∈ S, i) ** exact hf.isCompact_biUnion hc ** Qed
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IsCompact.finite_of_discrete ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t : Set α inst✝ : DiscreteTopology α s : Set α hs : IsCompact s ⊢ Set.Finite s ** have : ∀ x : α, ({x} : Set α) ∈ 𝓝 x := by simp [nhds_discrete] ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t : Set α inst✝ : DiscreteTopology α s : Set α hs : IsCompact s this : ∀ (x : α), {x} ∈ 𝓝 x ⊢ Set.Finite s ** rcases hs.elim_nhds_subcover (fun x => {x}) fun x _ => this x with ⟨t, _, hst⟩ ** case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t✝ : Set α inst✝ : DiscreteTopology α s : Set α hs : IsCompact s this : ∀ (x : α), {x} ∈ 𝓝 x t : Finset α left✝ : ∀ (x : α), x ∈ t → x ∈ s hst : s ⊆ ⋃ x ∈ t, {x} ⊢ Set.Finite s ** simp only [← t.set_biUnion_coe, biUnion_of_singleton] at hst ** case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t✝ : Set α inst✝ : DiscreteTopology α s : Set α hs : IsCompact s this : ∀ (x : α), {x} ∈ 𝓝 x t : Finset α left✝ : ∀ (x : α), x ∈ t → x ∈ s hst : s ⊆ ↑t ⊢ Set.Finite s ** exact t.finite_toSet.subset hst ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t : Set α inst✝ : DiscreteTopology α s : Set α hs : IsCompact s ⊢ ∀ (x : α), {x} ∈ 𝓝 x ** simp [nhds_discrete] ** Qed
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IsCompact.union ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α hs : IsCompact s ht : IsCompact t ⊢ IsCompact (s ∪ t) ** rw [union_eq_iUnion] ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α hs : IsCompact s ht : IsCompact t ⊢ IsCompact (⋃ b, bif b then s else t) ** exact isCompact_iUnion fun b => by cases b <;> assumption ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α hs : IsCompact s ht : IsCompact t b : Bool ⊢ IsCompact (bif b then s else t) ** cases b <;> assumption ** Qed
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exists_subset_nhds_of_isCompact' ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α ι : Type u_3 inst✝ : Nonempty ι V : ι → Set α hV : Directed (fun x x_1 => x ⊇ x_1) V hV_cpct : ∀ (i : ι), IsCompact (V i) hV_closed : ∀ (i : ι), IsClosed (V i) U : Set α hU : ∀ (x : α), x ∈ ⋂ i, V i → U ∈ 𝓝 x ⊢ ∃ i, V i ⊆ U ** obtain ⟨W, hsubW, W_op, hWU⟩ := exists_open_set_nhds hU ** case intro.intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α ι : Type u_3 inst✝ : Nonempty ι V : ι → Set α hV : Directed (fun x x_1 => x ⊇ x_1) V hV_cpct : ∀ (i : ι), IsCompact (V i) hV_closed : ∀ (i : ι), IsClosed (V i) U : Set α hU : ∀ (x : α), x ∈ ⋂ i, V i → U ∈ 𝓝 x W : Set α hsubW : ⋂ i, V i ⊆ W W_op : IsOpen W hWU : W ⊆ U ⊢ ∃ i, V i ⊆ U ** suffices : ∃ i, V i ⊆ W ** case this α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α ι : Type u_3 inst✝ : Nonempty ι V : ι → Set α hV : Directed (fun x x_1 => x ⊇ x_1) V hV_cpct : ∀ (i : ι), IsCompact (V i) hV_closed : ∀ (i : ι), IsClosed (V i) U : Set α hU : ∀ (x : α), x ∈ ⋂ i, V i → U ∈ 𝓝 x W : Set α hsubW : ⋂ i, V i ⊆ W W_op : IsOpen W hWU : W ⊆ U ⊢ ∃ i, V i ⊆ W ** by_contra' H ** case this α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α ι : Type u_3 inst✝ : Nonempty ι V : ι → Set α hV : Directed (fun x x_1 => x ⊇ x_1) V hV_cpct : ∀ (i : ι), IsCompact (V i) hV_closed : ∀ (i : ι), IsClosed (V i) U : Set α hU : ∀ (x : α), x ∈ ⋂ i, V i → U ∈ 𝓝 x W : Set α hsubW : ⋂ i, V i ⊆ W W_op : IsOpen W hWU : W ⊆ U H : ∀ (i : ι), ¬V i ⊆ W ⊢ False ** replace H : ∀ i, (V i ∩ Wᶜ).Nonempty := fun i => Set.inter_compl_nonempty_iff.mpr (H i) ** case this α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α ι : Type u_3 inst✝ : Nonempty ι V : ι → Set α hV : Directed (fun x x_1 => x ⊇ x_1) V hV_cpct : ∀ (i : ι), IsCompact (V i) hV_closed : ∀ (i : ι), IsClosed (V i) U : Set α hU : ∀ (x : α), x ∈ ⋂ i, V i → U ∈ 𝓝 x W : Set α hsubW : ⋂ i, V i ⊆ W W_op : IsOpen W hWU : W ⊆ U H : ∀ (i : ι), Set.Nonempty (V i ∩ Wᶜ) ⊢ False ** have : (⋂ i, V i ∩ Wᶜ).Nonempty := by
refine'
IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed _ (fun i j => _) H
(fun i => (hV_cpct i).inter_right W_op.isClosed_compl) fun i =>
(hV_closed i).inter W_op.isClosed_compl
rcases hV i j with ⟨k, hki, hkj⟩
refine' ⟨k, ⟨fun x => _, fun x => _⟩⟩ <;> simp only [and_imp, mem_inter_iff, mem_compl_iff] <;>
tauto ** case this α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α ι : Type u_3 inst✝ : Nonempty ι V : ι → Set α hV : Directed (fun x x_1 => x ⊇ x_1) V hV_cpct : ∀ (i : ι), IsCompact (V i) hV_closed : ∀ (i : ι), IsClosed (V i) U : Set α hU : ∀ (x : α), x ∈ ⋂ i, V i → U ∈ 𝓝 x W : Set α hsubW : ⋂ i, V i ⊆ W W_op : IsOpen W hWU : W ⊆ U H : ∀ (i : ι), Set.Nonempty (V i ∩ Wᶜ) this : Set.Nonempty (⋂ i, V i ∩ Wᶜ) ⊢ False ** have : ¬⋂ i : ι, V i ⊆ W := by simpa [← iInter_inter, inter_compl_nonempty_iff] ** case this α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α ι : Type u_3 inst✝ : Nonempty ι V : ι → Set α hV : Directed (fun x x_1 => x ⊇ x_1) V hV_cpct : ∀ (i : ι), IsCompact (V i) hV_closed : ∀ (i : ι), IsClosed (V i) U : Set α hU : ∀ (x : α), x ∈ ⋂ i, V i → U ∈ 𝓝 x W : Set α hsubW : ⋂ i, V i ⊆ W W_op : IsOpen W hWU : W ⊆ U H : ∀ (i : ι), Set.Nonempty (V i ∩ Wᶜ) this✝ : Set.Nonempty (⋂ i, V i ∩ Wᶜ) this : ¬⋂ i, V i ⊆ W ⊢ False ** contradiction ** case intro.intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α ι : Type u_3 inst✝ : Nonempty ι V : ι → Set α hV : Directed (fun x x_1 => x ⊇ x_1) V hV_cpct : ∀ (i : ι), IsCompact (V i) hV_closed : ∀ (i : ι), IsClosed (V i) U : Set α hU : ∀ (x : α), x ∈ ⋂ i, V i → U ∈ 𝓝 x W : Set α hsubW : ⋂ i, V i ⊆ W W_op : IsOpen W hWU : W ⊆ U this : ∃ i, V i ⊆ W ⊢ ∃ i, V i ⊆ U ** exact this.imp fun i hi => hi.trans hWU ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α ι : Type u_3 inst✝ : Nonempty ι V : ι → Set α hV : Directed (fun x x_1 => x ⊇ x_1) V hV_cpct : ∀ (i : ι), IsCompact (V i) hV_closed : ∀ (i : ι), IsClosed (V i) U : Set α hU : ∀ (x : α), x ∈ ⋂ i, V i → U ∈ 𝓝 x W : Set α hsubW : ⋂ i, V i ⊆ W W_op : IsOpen W hWU : W ⊆ U H : ∀ (i : ι), Set.Nonempty (V i ∩ Wᶜ) ⊢ Set.Nonempty (⋂ i, V i ∩ Wᶜ) ** refine'
IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed _ (fun i j => _) H
(fun i => (hV_cpct i).inter_right W_op.isClosed_compl) fun i =>
(hV_closed i).inter W_op.isClosed_compl ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α ι : Type u_3 inst✝ : Nonempty ι V : ι → Set α hV : Directed (fun x x_1 => x ⊇ x_1) V hV_cpct : ∀ (i : ι), IsCompact (V i) hV_closed : ∀ (i : ι), IsClosed (V i) U : Set α hU : ∀ (x : α), x ∈ ⋂ i, V i → U ∈ 𝓝 x W : Set α hsubW : ⋂ i, V i ⊆ W W_op : IsOpen W hWU : W ⊆ U H : ∀ (i : ι), Set.Nonempty (V i ∩ Wᶜ) i j : ι ⊢ ∃ z, (fun x x_1 => x ⊇ x_1) (V i ∩ Wᶜ) (V z ∩ Wᶜ) ∧ (fun x x_1 => x ⊇ x_1) (V j ∩ Wᶜ) (V z ∩ Wᶜ) ** rcases hV i j with ⟨k, hki, hkj⟩ ** case intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α ι : Type u_3 inst✝ : Nonempty ι V : ι → Set α hV : Directed (fun x x_1 => x ⊇ x_1) V hV_cpct : ∀ (i : ι), IsCompact (V i) hV_closed : ∀ (i : ι), IsClosed (V i) U : Set α hU : ∀ (x : α), x ∈ ⋂ i, V i → U ∈ 𝓝 x W : Set α hsubW : ⋂ i, V i ⊆ W W_op : IsOpen W hWU : W ⊆ U H : ∀ (i : ι), Set.Nonempty (V i ∩ Wᶜ) i j k : ι hki : V i ⊇ V k hkj : V j ⊇ V k ⊢ ∃ z, (fun x x_1 => x ⊇ x_1) (V i ∩ Wᶜ) (V z ∩ Wᶜ) ∧ (fun x x_1 => x ⊇ x_1) (V j ∩ Wᶜ) (V z ∩ Wᶜ) ** refine' ⟨k, ⟨fun x => _, fun x => _⟩⟩ <;> simp only [and_imp, mem_inter_iff, mem_compl_iff] <;>
tauto ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α ι : Type u_3 inst✝ : Nonempty ι V : ι → Set α hV : Directed (fun x x_1 => x ⊇ x_1) V hV_cpct : ∀ (i : ι), IsCompact (V i) hV_closed : ∀ (i : ι), IsClosed (V i) U : Set α hU : ∀ (x : α), x ∈ ⋂ i, V i → U ∈ 𝓝 x W : Set α hsubW : ⋂ i, V i ⊆ W W_op : IsOpen W hWU : W ⊆ U H : ∀ (i : ι), Set.Nonempty (V i ∩ Wᶜ) this : Set.Nonempty (⋂ i, V i ∩ Wᶜ) ⊢ ¬⋂ i, V i ⊆ W ** simpa [← iInter_inter, inter_compl_nonempty_iff] ** Qed
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isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) U : Set α ⊢ IsCompact U ∧ IsOpen U ↔ ∃ s, Set.Finite s ∧ U = ⋃ i ∈ s, b i ** constructor ** case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) U : Set α ⊢ IsCompact U ∧ IsOpen U → ∃ s, Set.Finite s ∧ U = ⋃ i ∈ s, b i ** rintro ⟨h₁, h₂⟩ ** case mp.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) U : Set α h₁ : IsCompact U h₂ : IsOpen U ⊢ ∃ s, Set.Finite s ∧ U = ⋃ i ∈ s, b i ** obtain ⟨β, f, e, hf⟩ := hb.open_eq_iUnion h₂ ** case mp.intro.intro.intro.intro α : Type u β✝ : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β✝ s t : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) U : Set α h₁ : IsCompact U h₂ : IsOpen U β : Type u f : β → Set α e : U = ⋃ i, f i hf : ∀ (i : β), f i ∈ range b ⊢ ∃ s, Set.Finite s ∧ U = ⋃ i ∈ s, b i ** choose f' hf' using hf ** case mp.intro.intro.intro.intro α : Type u β✝ : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β✝ s t : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) U : Set α h₁ : IsCompact U h₂ : IsOpen U β : Type u f : β → Set α e : U = ⋃ i, f i f' : β → ι hf' : ∀ (i : β), b (f' i) = f i ⊢ ∃ s, Set.Finite s ∧ U = ⋃ i ∈ s, b i ** have : b ∘ f' = f := funext hf' ** case mp.intro.intro.intro.intro α : Type u β✝ : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β✝ s t : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) U : Set α h₁ : IsCompact U h₂ : IsOpen U β : Type u f : β → Set α e : U = ⋃ i, f i f' : β → ι hf' : ∀ (i : β), b (f' i) = f i this : b ∘ f' = f ⊢ ∃ s, Set.Finite s ∧ U = ⋃ i ∈ s, b i ** subst this ** case mp.intro.intro.intro.intro α : Type u β✝ : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β✝ s t : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) U : Set α h₁ : IsCompact U h₂ : IsOpen U β : Type u f' : β → ι e : U = ⋃ i, (b ∘ f') i hf' : ∀ (i : β), b (f' i) = (b ∘ f') i ⊢ ∃ s, Set.Finite s ∧ U = ⋃ i ∈ s, b i ** obtain ⟨t, ht⟩ :=
h₁.elim_finite_subcover (b ∘ f') (fun i => hb.isOpen (Set.mem_range_self _)) (by rw [e]) ** case mp.intro.intro.intro.intro.intro α : Type u β✝ : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β✝ s t✝ : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) U : Set α h₁ : IsCompact U h₂ : IsOpen U β : Type u f' : β → ι e : U = ⋃ i, (b ∘ f') i hf' : ∀ (i : β), b (f' i) = (b ∘ f') i t : Finset β ht : U ⊆ ⋃ i ∈ t, (b ∘ f') i ⊢ ∃ s, Set.Finite s ∧ U = ⋃ i ∈ s, b i ** refine' ⟨t.image f', Set.Finite.intro inferInstance, le_antisymm _ _⟩ ** α : Type u β✝ : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β✝ s t : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) U : Set α h₁ : IsCompact U h₂ : IsOpen U β : Type u f' : β → ι e : U = ⋃ i, (b ∘ f') i hf' : ∀ (i : β), b (f' i) = (b ∘ f') i ⊢ U ⊆ ⋃ i, (b ∘ f') i ** rw [e] ** case mp.intro.intro.intro.intro.intro.refine'_1 α : Type u β✝ : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β✝ s t✝ : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) U : Set α h₁ : IsCompact U h₂ : IsOpen U β : Type u f' : β → ι e : U = ⋃ i, (b ∘ f') i hf' : ∀ (i : β), b (f' i) = (b ∘ f') i t : Finset β ht : U ⊆ ⋃ i ∈ t, (b ∘ f') i ⊢ U ≤ ⋃ i ∈ ↑(Finset.image f' t), b i ** refine' Set.Subset.trans ht _ ** case mp.intro.intro.intro.intro.intro.refine'_1 α : Type u β✝ : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β✝ s t✝ : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) U : Set α h₁ : IsCompact U h₂ : IsOpen U β : Type u f' : β → ι e : U = ⋃ i, (b ∘ f') i hf' : ∀ (i : β), b (f' i) = (b ∘ f') i t : Finset β ht : U ⊆ ⋃ i ∈ t, (b ∘ f') i ⊢ ⋃ i ∈ t, (b ∘ f') i ⊆ ⋃ i ∈ ↑(Finset.image f' t), b i ** simp only [Set.iUnion_subset_iff] ** case mp.intro.intro.intro.intro.intro.refine'_1 α : Type u β✝ : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β✝ s t✝ : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) U : Set α h₁ : IsCompact U h₂ : IsOpen U β : Type u f' : β → ι e : U = ⋃ i, (b ∘ f') i hf' : ∀ (i : β), b (f' i) = (b ∘ f') i t : Finset β ht : U ⊆ ⋃ i ∈ t, (b ∘ f') i ⊢ ∀ (i : β), i ∈ t → (b ∘ f') i ⊆ ⋃ i ∈ ↑(Finset.image f' t), b i ** intro i hi ** case mp.intro.intro.intro.intro.intro.refine'_1 α : Type u β✝ : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β✝ s t✝ : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) U : Set α h₁ : IsCompact U h₂ : IsOpen U β : Type u f' : β → ι e : U = ⋃ i, (b ∘ f') i hf' : ∀ (i : β), b (f' i) = (b ∘ f') i t : Finset β ht : U ⊆ ⋃ i ∈ t, (b ∘ f') i i : β hi : i ∈ t ⊢ (b ∘ f') i ⊆ ⋃ i ∈ ↑(Finset.image f' t), b i ** erw [← Set.iUnion_subtype (fun x : ι => x ∈ t.image f') fun i => b i.1] ** case mp.intro.intro.intro.intro.intro.refine'_1 α : Type u β✝ : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β✝ s t✝ : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) U : Set α h₁ : IsCompact U h₂ : IsOpen U β : Type u f' : β → ι e : U = ⋃ i, (b ∘ f') i hf' : ∀ (i : β), b (f' i) = (b ∘ f') i t : Finset β ht : U ⊆ ⋃ i ∈ t, (b ∘ f') i i : β hi : i ∈ t ⊢ (b ∘ f') i ⊆ ⋃ x, b ↑x ** exact Set.subset_iUnion (fun i : t.image f' => b i) ⟨_, Finset.mem_image_of_mem _ hi⟩ ** case mp.intro.intro.intro.intro.intro.refine'_2 α : Type u β✝ : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β✝ s t✝ : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) U : Set α h₁ : IsCompact U h₂ : IsOpen U β : Type u f' : β → ι e : U = ⋃ i, (b ∘ f') i hf' : ∀ (i : β), b (f' i) = (b ∘ f') i t : Finset β ht : U ⊆ ⋃ i ∈ t, (b ∘ f') i ⊢ ⋃ i ∈ ↑(Finset.image f' t), b i ≤ U ** apply Set.iUnion₂_subset ** case mp.intro.intro.intro.intro.intro.refine'_2.h α : Type u β✝ : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β✝ s t✝ : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) U : Set α h₁ : IsCompact U h₂ : IsOpen U β : Type u f' : β → ι e : U = ⋃ i, (b ∘ f') i hf' : ∀ (i : β), b (f' i) = (b ∘ f') i t : Finset β ht : U ⊆ ⋃ i ∈ t, (b ∘ f') i ⊢ ∀ (i : ι), i ∈ ↑(Finset.image f' t) → b i ⊆ U ** rintro i hi ** case mp.intro.intro.intro.intro.intro.refine'_2.h α : Type u β✝ : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β✝ s t✝ : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) U : Set α h₁ : IsCompact U h₂ : IsOpen U β : Type u f' : β → ι e : U = ⋃ i, (b ∘ f') i hf' : ∀ (i : β), b (f' i) = (b ∘ f') i t : Finset β ht : U ⊆ ⋃ i ∈ t, (b ∘ f') i i : ι hi : i ∈ ↑(Finset.image f' t) ⊢ b i ⊆ U ** obtain ⟨j, -, rfl⟩ := Finset.mem_image.mp hi ** case mp.intro.intro.intro.intro.intro.refine'_2.h.intro.intro α : Type u β✝ : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β✝ s t✝ : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) U : Set α h₁ : IsCompact U h₂ : IsOpen U β : Type u f' : β → ι e : U = ⋃ i, (b ∘ f') i hf' : ∀ (i : β), b (f' i) = (b ∘ f') i t : Finset β ht : U ⊆ ⋃ i ∈ t, (b ∘ f') i j : β hi : f' j ∈ ↑(Finset.image f' t) ⊢ b (f' j) ⊆ U ** rw [e] ** case mp.intro.intro.intro.intro.intro.refine'_2.h.intro.intro α : Type u β✝ : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β✝ s t✝ : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) U : Set α h₁ : IsCompact U h₂ : IsOpen U β : Type u f' : β → ι e : U = ⋃ i, (b ∘ f') i hf' : ∀ (i : β), b (f' i) = (b ∘ f') i t : Finset β ht : U ⊆ ⋃ i ∈ t, (b ∘ f') i j : β hi : f' j ∈ ↑(Finset.image f' t) ⊢ b (f' j) ⊆ ⋃ i, (b ∘ f') i ** exact Set.subset_iUnion (b ∘ f') j ** case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) U : Set α ⊢ (∃ s, Set.Finite s ∧ U = ⋃ i ∈ s, b i) → IsCompact U ∧ IsOpen U ** rintro ⟨s, hs, rfl⟩ ** case mpr.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) s : Set ι hs : Set.Finite s ⊢ IsCompact (⋃ i ∈ s, b i) ∧ IsOpen (⋃ i ∈ s, b i) ** constructor ** case mpr.intro.intro.left α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) s : Set ι hs : Set.Finite s ⊢ IsCompact (⋃ i ∈ s, b i) ** exact hs.isCompact_biUnion fun i _ => hb' i ** case mpr.intro.intro.right α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t : Set α b : ι → Set α hb : IsTopologicalBasis (range b) hb' : ∀ (i : ι), IsCompact (b i) s : Set ι hs : Set.Finite s ⊢ IsOpen (⋃ i ∈ s, b i) ** exact isOpen_biUnion fun i _ => hb.isOpen (Set.mem_range_self _) ** Qed
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Filter.cocompact_eq_cofinite ** α✝ : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α✝ inst✝² : TopologicalSpace β s t : Set α✝ α : Type u_3 inst✝¹ : TopologicalSpace α inst✝ : DiscreteTopology α ⊢ cocompact α = cofinite ** simp only [cocompact, hasBasis_cofinite.eq_biInf, isCompact_iff_finite] ** Qed
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Filter.Tendsto.isCompact_insert_range_of_cocompact ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : α → β b : β hf : Tendsto f (cocompact α) (𝓝 b) hfc : Continuous f ⊢ IsCompact (insert b (range f)) ** intro l hne hle ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : α → β b : β hf : Tendsto f (cocompact α) (𝓝 b) hfc : Continuous f l : Filter β hne : NeBot l hle : l ≤ 𝓟 (insert b (range f)) ⊢ ∃ a, a ∈ insert b (range f) ∧ ClusterPt a l ** by_cases hb : ClusterPt b l ** case neg α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : α → β b : β hf : Tendsto f (cocompact α) (𝓝 b) hfc : Continuous f l : Filter β hne : NeBot l hle : l ≤ 𝓟 (insert b (range f)) hb : ¬ClusterPt b l ⊢ ∃ a, a ∈ insert b (range f) ∧ ClusterPt a l ** simp only [clusterPt_iff, not_forall, ← not_disjoint_iff_nonempty_inter, not_not] at hb ** case neg α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : α → β b : β hf : Tendsto f (cocompact α) (𝓝 b) hfc : Continuous f l : Filter β hne : NeBot l hle : l ≤ 𝓟 (insert b (range f)) hb : ∃ x h x_1 h, Disjoint x x_1 ⊢ ∃ a, a ∈ insert b (range f) ∧ ClusterPt a l ** rcases hb with ⟨s, hsb, t, htl, hd⟩ ** case neg.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ : Set α f : α → β b : β hf : Tendsto f (cocompact α) (𝓝 b) hfc : Continuous f l : Filter β hne : NeBot l hle : l ≤ 𝓟 (insert b (range f)) s : Set β hsb : s ∈ 𝓝 b t : Set β htl : t ∈ l hd : Disjoint s t ⊢ ∃ a, a ∈ insert b (range f) ∧ ClusterPt a l ** rcases mem_cocompact.1 (hf hsb) with ⟨K, hKc, hKs⟩ ** case neg.intro.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ : Set α f : α → β b : β hf : Tendsto f (cocompact α) (𝓝 b) hfc : Continuous f l : Filter β hne : NeBot l hle : l ≤ 𝓟 (insert b (range f)) s : Set β hsb : s ∈ 𝓝 b t : Set β htl : t ∈ l hd : Disjoint s t K : Set α hKc : IsCompact K hKs : Kᶜ ⊆ f ⁻¹' s ⊢ ∃ a, a ∈ insert b (range f) ∧ ClusterPt a l ** have : f '' K ∈ l := by
filter_upwards [htl, le_principal_iff.1 hle] with y hyt hyf
rcases hyf with (rfl | ⟨x, rfl⟩)
exacts [(hd.le_bot ⟨mem_of_mem_nhds hsb, hyt⟩).elim,
mem_image_of_mem _ (not_not.1 fun hxK => hd.le_bot ⟨hKs hxK, hyt⟩)] ** case neg.intro.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ : Set α f : α → β b : β hf : Tendsto f (cocompact α) (𝓝 b) hfc : Continuous f l : Filter β hne : NeBot l hle : l ≤ 𝓟 (insert b (range f)) s : Set β hsb : s ∈ 𝓝 b t : Set β htl : t ∈ l hd : Disjoint s t K : Set α hKc : IsCompact K hKs : Kᶜ ⊆ f ⁻¹' s this : f '' K ∈ l ⊢ ∃ a, a ∈ insert b (range f) ∧ ClusterPt a l ** rcases hKc.image hfc (le_principal_iff.2 this) with ⟨y, hy, hyl⟩ ** case neg.intro.intro.intro.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ : Set α f : α → β b : β hf : Tendsto f (cocompact α) (𝓝 b) hfc : Continuous f l : Filter β hne : NeBot l hle : l ≤ 𝓟 (insert b (range f)) s : Set β hsb : s ∈ 𝓝 b t : Set β htl : t ∈ l hd : Disjoint s t K : Set α hKc : IsCompact K hKs : Kᶜ ⊆ f ⁻¹' s this : f '' K ∈ l y : β hy : y ∈ f '' K hyl : ClusterPt y l ⊢ ∃ a, a ∈ insert b (range f) ∧ ClusterPt a l ** exact ⟨y, Or.inr <| image_subset_range _ _ hy, hyl⟩ ** case pos α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : α → β b : β hf : Tendsto f (cocompact α) (𝓝 b) hfc : Continuous f l : Filter β hne : NeBot l hle : l ≤ 𝓟 (insert b (range f)) hb : ClusterPt b l ⊢ ∃ a, a ∈ insert b (range f) ∧ ClusterPt a l ** exact ⟨b, Or.inl rfl, hb⟩ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ : Set α f : α → β b : β hf : Tendsto f (cocompact α) (𝓝 b) hfc : Continuous f l : Filter β hne : NeBot l hle : l ≤ 𝓟 (insert b (range f)) s : Set β hsb : s ∈ 𝓝 b t : Set β htl : t ∈ l hd : Disjoint s t K : Set α hKc : IsCompact K hKs : Kᶜ ⊆ f ⁻¹' s ⊢ f '' K ∈ l ** filter_upwards [htl, le_principal_iff.1 hle] with y hyt hyf ** case h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ : Set α f : α → β b : β hf : Tendsto f (cocompact α) (𝓝 b) hfc : Continuous f l : Filter β hne : NeBot l hle : l ≤ 𝓟 (insert b (range f)) s : Set β hsb : s ∈ 𝓝 b t : Set β htl : t ∈ l hd : Disjoint s t K : Set α hKc : IsCompact K hKs : Kᶜ ⊆ f ⁻¹' s y : β hyt : y ∈ t hyf : y ∈ insert b (range f) ⊢ y ∈ f '' K ** rcases hyf with (rfl | ⟨x, rfl⟩) ** case h.inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ : Set α f : α → β hfc : Continuous f l : Filter β hne : NeBot l s t : Set β htl : t ∈ l hd : Disjoint s t K : Set α hKc : IsCompact K hKs : Kᶜ ⊆ f ⁻¹' s y : β hyt : y ∈ t hf : Tendsto f (cocompact α) (𝓝 y) hle : l ≤ 𝓟 (insert y (range f)) hsb : s ∈ 𝓝 y ⊢ y ∈ f '' K case h.inr.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ : Set α f : α → β b : β hf : Tendsto f (cocompact α) (𝓝 b) hfc : Continuous f l : Filter β hne : NeBot l hle : l ≤ 𝓟 (insert b (range f)) s : Set β hsb : s ∈ 𝓝 b t : Set β htl : t ∈ l hd : Disjoint s t K : Set α hKc : IsCompact K hKs : Kᶜ ⊆ f ⁻¹' s x : α hyt : f x ∈ t ⊢ f x ∈ f '' K ** exacts [(hd.le_bot ⟨mem_of_mem_nhds hsb, hyt⟩).elim,
mem_image_of_mem _ (not_not.1 fun hxK => hd.le_bot ⟨hKs hxK, hyt⟩)] ** Qed
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Filter.Tendsto.isCompact_insert_range_of_cofinite ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : ι → α a : α hf : Tendsto f cofinite (𝓝 a) ⊢ IsCompact (insert a (range f)) ** letI : TopologicalSpace ι := ⊥ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : ι → α a : α hf : Tendsto f cofinite (𝓝 a) this : TopologicalSpace ι := ⊥ ⊢ IsCompact (insert a (range f)) ** haveI h : DiscreteTopology ι := ⟨rfl⟩ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : ι → α a : α hf : Tendsto f cofinite (𝓝 a) this : TopologicalSpace ι := ⊥ h : DiscreteTopology ι ⊢ IsCompact (insert a (range f)) ** rw [← cocompact_eq_cofinite ι] at hf ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : ι → α a : α this : TopologicalSpace ι := ⊥ hf : Tendsto f (cocompact ι) (𝓝 a) h : DiscreteTopology ι ⊢ IsCompact (insert a (range f)) ** exact hf.isCompact_insert_range_of_cocompact continuous_of_discreteTopology ** Qed
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Filter.hasBasis_coclosedCompact ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ⊢ HasBasis (coclosedCompact α) (fun s => IsClosed s ∧ IsCompact s) compl ** simp only [Filter.coclosedCompact, iInf_and'] ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ⊢ HasBasis (⨅ s, ⨅ (_ : IsClosed s ∧ IsCompact s), 𝓟 sᶜ) (fun s => IsClosed s ∧ IsCompact s) compl ** refine' hasBasis_biInf_principal' _ ⟨∅, isClosed_empty, isCompact_empty⟩ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ⊢ ∀ (i : Set α), IsClosed i ∧ IsCompact i → ∀ (j : Set α), IsClosed j ∧ IsCompact j → ∃ k, (IsClosed k ∧ IsCompact k) ∧ kᶜ ⊆ iᶜ ∧ kᶜ ⊆ jᶜ ** rintro s ⟨hs₁, hs₂⟩ t ⟨ht₁, ht₂⟩ ** case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ s : Set α hs₁ : IsClosed s hs₂ : IsCompact s t : Set α ht₁ : IsClosed t ht₂ : IsCompact t ⊢ ∃ k, (IsClosed k ∧ IsCompact k) ∧ kᶜ ⊆ sᶜ ∧ kᶜ ⊆ tᶜ ** exact ⟨s ∪ t, ⟨⟨hs₁.union ht₁, hs₂.union ht₂⟩, compl_subset_compl.2 (subset_union_left _ _),
compl_subset_compl.2 (subset_union_right _ _)⟩⟩ ** Qed
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Filter.mem_coclosedCompact ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ⊢ s ∈ coclosedCompact α ↔ ∃ t, IsClosed t ∧ IsCompact t ∧ tᶜ ⊆ s ** simp only [hasBasis_coclosedCompact.mem_iff, and_assoc] ** Qed
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Filter.mem_coclosed_compact' ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ⊢ s ∈ coclosedCompact α ↔ ∃ t, IsClosed t ∧ IsCompact t ∧ sᶜ ⊆ t ** simp only [mem_coclosedCompact, compl_subset_comm] ** Qed
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Bornology.inCompact.isBounded_iff ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ⊢ IsBounded s ↔ ∃ t, IsCompact t ∧ s ⊆ t ** change sᶜ ∈ Filter.cocompact α ↔ _ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ⊢ sᶜ ∈ cocompact α ↔ ∃ t, IsCompact t ∧ s ⊆ t ** rw [Filter.mem_cocompact] ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ⊢ (∃ t, IsCompact t ∧ tᶜ ⊆ sᶜ) ↔ ∃ t, IsCompact t ∧ s ⊆ t ** simp ** Qed
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IsCompact.nhdsSet_prod_eq ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ s : Set α t : Set β hs : IsCompact s ht : IsCompact t ⊢ 𝓝ˢ (s ×ˢ t) = 𝓝ˢ s ×ˢ 𝓝ˢ t ** simp_rw [hs.nhdsSet_prod_eq_biSup, ht.prod_nhdsSet_eq_biSup, nhdsSet, sSup_image, biSup_prod,
nhds_prod_eq] ** Qed
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generalized_tube_lemma ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ s : Set α hs : IsCompact s t : Set β ht : IsCompact t n : Set (α × β) hn : IsOpen n hp : s ×ˢ t ⊆ n ⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ u ×ˢ v ⊆ n ** rw [← hn.mem_nhdsSet, hs.nhdsSet_prod_eq ht,
((hasBasis_nhdsSet _).prod (hasBasis_nhdsSet _)).mem_iff] at hp ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ s : Set α hs : IsCompact s t : Set β ht : IsCompact t n : Set (α × β) hn : IsOpen n hp : ∃ i, ((IsOpen i.1 ∧ s ⊆ i.1) ∧ IsOpen i.2 ∧ t ⊆ i.2) ∧ i.1 ×ˢ i.2 ⊆ n ⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ u ×ˢ v ⊆ n ** rcases hp with ⟨⟨u, v⟩, ⟨⟨huo, hsu⟩, hvo, htv⟩, hn⟩ ** case intro.mk.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ s : Set α hs : IsCompact s t : Set β ht : IsCompact t n : Set (α × β) hn✝ : IsOpen n u : Set α v : Set β hn : (u, v).1 ×ˢ (u, v).2 ⊆ n huo : IsOpen (u, v).1 hsu : s ⊆ (u, v).1 hvo : IsOpen (u, v).2 htv : t ⊆ (u, v).2 ⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ u ×ˢ v ⊆ n ** exact ⟨u, v, huo, hvo, hsu, htv, hn⟩ ** Qed
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cluster_point_of_compact ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β s t : Set α inst✝¹ : CompactSpace α f : Filter α inst✝ : NeBot f ⊢ ∃ x, ClusterPt x f ** simpa using isCompact_univ (show f ≤ 𝓟 univ by simp) ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β s t : Set α inst✝¹ : CompactSpace α f : Filter α inst✝ : NeBot f ⊢ f ≤ 𝓟 univ ** simp ** Qed
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CompactSpace.elim_nhds_subcover ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α U : α → Set α hU : ∀ (x : α), U x ∈ 𝓝 x ⊢ ∃ t, ⋃ x ∈ t, U x = ⊤ ** obtain ⟨t, -, s⟩ := IsCompact.elim_nhds_subcover isCompact_univ U fun x _ => hU x ** case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t✝ : Set α inst✝ : CompactSpace α U : α → Set α hU : ∀ (x : α), U x ∈ 𝓝 x t : Finset α s : univ ⊆ ⋃ x ∈ t, U x ⊢ ∃ t, ⋃ x ∈ t, U x = ⊤ ** exact ⟨t, top_unique s⟩ ** Qed
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compactSpace_of_finite_subfamily_closed ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α h : ∀ {ι : Type u} (Z : ι → Set α), (∀ (i : ι), IsClosed (Z i)) → ⋂ i, Z i = ∅ → ∃ t, ⋂ i ∈ t, Z i = ∅ ι✝ : Type u Z : ι✝ → Set α ⊢ (∀ (i : ι✝), IsClosed (Z i)) → univ ∩ ⋂ i, Z i = ∅ → ∃ t, univ ∩ ⋂ i ∈ t, Z i = ∅ ** simpa using h Z ** Qed
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exists_nhds_ne_neBot ** α✝ : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝⁴ : TopologicalSpace α✝ inst✝³ : TopologicalSpace β s t : Set α✝ α : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : Infinite α ⊢ ∃ z, NeBot (𝓝[{z}ᶜ] z) ** by_contra' H ** α✝ : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝⁴ : TopologicalSpace α✝ inst✝³ : TopologicalSpace β s t : Set α✝ α : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : Infinite α H : ∀ (z : α), ¬NeBot (𝓝[{z}ᶜ] z) ⊢ False ** simp_rw [not_neBot] at H ** α✝ : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝⁴ : TopologicalSpace α✝ inst✝³ : TopologicalSpace β s t : Set α✝ α : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : Infinite α H : ∀ (z : α), 𝓝[{z}ᶜ] z = ⊥ ⊢ False ** haveI := discreteTopology_iff_nhds_ne.2 H ** α✝ : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝⁴ : TopologicalSpace α✝ inst✝³ : TopologicalSpace β s t : Set α✝ α : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : Infinite α H : ∀ (z : α), 𝓝[{z}ᶜ] z = ⊥ this : DiscreteTopology α ⊢ False ** exact Infinite.not_finite (finite_of_compact_of_discrete : Finite α) ** Qed
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LocallyFinite.finite_nonempty_of_compact ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α ι : Type u_3 inst✝ : CompactSpace α f : ι → Set α hf : LocallyFinite f ⊢ Set.Finite {i | Set.Nonempty (f i)} ** simpa only [inter_univ] using hf.finite_nonempty_inter_compact isCompact_univ ** Qed
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LocallyFinite.finite_of_compact ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α ι : Type u_3 inst✝ : CompactSpace α f : ι → Set α hf : LocallyFinite f hne : ∀ (i : ι), Set.Nonempty (f i) ⊢ Set.Finite univ ** simpa only [hne] using hf.finite_nonempty_of_compact ** Qed
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Filter.comap_cocompact_le ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : α → β hf : Continuous f ⊢ comap f (cocompact β) ≤ cocompact α ** rw [(Filter.hasBasis_cocompact.comap f).le_basis_iff Filter.hasBasis_cocompact] ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : α → β hf : Continuous f ⊢ ∀ (i' : Set α), IsCompact i' → ∃ i, IsCompact i ∧ f ⁻¹' iᶜ ⊆ i'ᶜ ** intro t ht ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α f : α → β hf : Continuous f t : Set α ht : IsCompact t ⊢ ∃ i, IsCompact i ∧ f ⁻¹' iᶜ ⊆ tᶜ ** refine' ⟨f '' t, ht.image hf, _⟩ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α f : α → β hf : Continuous f t : Set α ht : IsCompact t ⊢ f ⁻¹' (f '' t)ᶜ ⊆ tᶜ ** simpa using t.subset_preimage_image f ** Qed
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isCompact_range ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α f : α → β hf : Continuous f ⊢ IsCompact (range f) ** rw [← image_univ] ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α f : α → β hf : Continuous f ⊢ IsCompact (f '' univ) ** exact isCompact_univ.image hf ** Qed
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isClosedMap_snd_of_compactSpace ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝⁴ : TopologicalSpace α inst✝³ : TopologicalSpace β s✝ t : Set α X : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X Y : Type u_4 inst✝ : TopologicalSpace Y s : Set (X × Y) hs : IsClosed s ⊢ IsClosed (Prod.snd '' s) ** rw [← isOpen_compl_iff, isOpen_iff_mem_nhds] ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝⁴ : TopologicalSpace α inst✝³ : TopologicalSpace β s✝ t : Set α X : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X Y : Type u_4 inst✝ : TopologicalSpace Y s : Set (X × Y) hs : IsClosed s ⊢ ∀ (a : Y), a ∈ (Prod.snd '' s)ᶜ → (Prod.snd '' s)ᶜ ∈ 𝓝 a ** intro y hy ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝⁴ : TopologicalSpace α inst✝³ : TopologicalSpace β s✝ t : Set α X : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X Y : Type u_4 inst✝ : TopologicalSpace Y s : Set (X × Y) hs : IsClosed s y : Y hy : y ∈ (Prod.snd '' s)ᶜ ⊢ (Prod.snd '' s)ᶜ ∈ 𝓝 y ** have : univ ×ˢ {y} ⊆ sᶜ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝⁴ : TopologicalSpace α inst✝³ : TopologicalSpace β s✝ t : Set α X : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X Y : Type u_4 inst✝ : TopologicalSpace Y s : Set (X × Y) hs : IsClosed s y : Y hy : y ∈ (Prod.snd '' s)ᶜ this : univ ×ˢ {y} ⊆ sᶜ ⊢ (Prod.snd '' s)ᶜ ∈ 𝓝 y ** rcases generalized_tube_lemma isCompact_univ isCompact_singleton hs.isOpen_compl this
with ⟨U, V, -, hVo, hU, hV, hs⟩ ** case intro.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝⁴ : TopologicalSpace α inst✝³ : TopologicalSpace β s✝ t : Set α X : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X Y : Type u_4 inst✝ : TopologicalSpace Y s : Set (X × Y) hs✝ : IsClosed s y : Y hy : y ∈ (Prod.snd '' s)ᶜ this : univ ×ˢ {y} ⊆ sᶜ U : Set X V : Set Y hVo : IsOpen V hU : univ ⊆ U hV : {y} ⊆ V hs : U ×ˢ V ⊆ sᶜ ⊢ (Prod.snd '' s)ᶜ ∈ 𝓝 y ** refine mem_nhds_iff.2 ⟨V, ?_, hVo, hV rfl⟩ ** case intro.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝⁴ : TopologicalSpace α inst✝³ : TopologicalSpace β s✝ t : Set α X : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X Y : Type u_4 inst✝ : TopologicalSpace Y s : Set (X × Y) hs✝ : IsClosed s y : Y hy : y ∈ (Prod.snd '' s)ᶜ this : univ ×ˢ {y} ⊆ sᶜ U : Set X V : Set Y hVo : IsOpen V hU : univ ⊆ U hV : {y} ⊆ V hs : U ×ˢ V ⊆ sᶜ ⊢ V ⊆ (Prod.snd '' s)ᶜ ** rintro _ hzV ⟨z, hzs, rfl⟩ ** case intro.intro.intro.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝⁴ : TopologicalSpace α inst✝³ : TopologicalSpace β s✝ t : Set α X : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X Y : Type u_4 inst✝ : TopologicalSpace Y s : Set (X × Y) hs✝ : IsClosed s y : Y hy : y ∈ (Prod.snd '' s)ᶜ this : univ ×ˢ {y} ⊆ sᶜ U : Set X V : Set Y hVo : IsOpen V hU : univ ⊆ U hV : {y} ⊆ V hs : U ×ˢ V ⊆ sᶜ z : X × Y hzs : z ∈ s hzV : z.2 ∈ V ⊢ False ** exact hs ⟨hU trivial, hzV⟩ hzs ** case this α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝⁴ : TopologicalSpace α inst✝³ : TopologicalSpace β s✝ t : Set α X : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X Y : Type u_4 inst✝ : TopologicalSpace Y s : Set (X × Y) hs : IsClosed s y : Y hy : y ∈ (Prod.snd '' s)ᶜ ⊢ univ ×ˢ {y} ⊆ sᶜ ** exact fun (x, y') ⟨_, rfl⟩ hs => hy ⟨(x, y'), hs, rfl⟩ ** Qed
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Inducing.isCompact_iff ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t : Set α f : α → β hf : Inducing f s : Set α ⊢ IsCompact s ↔ IsCompact (f '' s) ** refine ⟨fun hs => hs.image hf.continuous, fun hs F F_ne_bot F_le => ?_⟩ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t : Set α f : α → β hf : Inducing f s : Set α hs : IsCompact (f '' s) F : Filter α F_ne_bot : NeBot F F_le : F ≤ 𝓟 s ⊢ ∃ a, a ∈ s ∧ ClusterPt a F ** obtain ⟨_, ⟨x, x_in : x ∈ s, rfl⟩, hx : ClusterPt (f x) (map f F)⟩ :=
hs ((map_mono F_le).trans_eq map_principal) ** case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t : Set α f : α → β hf : Inducing f s : Set α hs : IsCompact (f '' s) F : Filter α F_ne_bot : NeBot F F_le : F ≤ 𝓟 s x : α x_in : x ∈ s hx : ClusterPt (f x) (map f F) ⊢ ∃ a, a ∈ s ∧ ClusterPt a F ** exact ⟨x, x_in, hf.mapClusterPt_iff.1 hx⟩ ** Qed
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Inducing.isCompact_preimage ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : α → β hf : Inducing f hf' : IsClosed (range f) K : Set β hK : IsCompact K ⊢ IsCompact (f ⁻¹' K) ** replace hK := hK.inter_right hf' ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α f : α → β hf : Inducing f hf' : IsClosed (range f) K : Set β hK : IsCompact (K ∩ range f) ⊢ IsCompact (f ⁻¹' K) ** rwa [hf.isCompact_iff, image_preimage_eq_inter_range] ** Qed
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isCompact_iff_isCompact_univ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t s : Set α ⊢ IsCompact s ↔ IsCompact univ ** rw [Subtype.isCompact_iff, image_univ, Subtype.range_coe] ** Qed
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exists_nhds_ne_inf_principal_neBot ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t s : Set α hs : IsCompact s hs' : Set.Infinite s ⊢ ∃ z, z ∈ s ∧ NeBot (𝓝[{z}ᶜ] z ⊓ 𝓟 s) ** by_contra' H ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t s : Set α hs : IsCompact s hs' : Set.Infinite s H : ∀ (z : α), z ∈ s → ¬NeBot (𝓝[{z}ᶜ] z ⊓ 𝓟 s) ⊢ False ** simp_rw [not_neBot] at H ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t s : Set α hs : IsCompact s hs' : Set.Infinite s H : ∀ (z : α), z ∈ s → 𝓝[{z}ᶜ] z ⊓ 𝓟 s = ⊥ ⊢ False ** exact hs' (hs.finite <| discreteTopology_subtype_iff.mpr H) ** Qed
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ClosedEmbedding.compactSpace ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α h : CompactSpace β f : α → β hf : ClosedEmbedding f ⊢ IsCompact univ ** rw [hf.toInducing.isCompact_iff, image_univ] ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α h : CompactSpace β f : α → β hf : ClosedEmbedding f ⊢ IsCompact (range f) ** exact hf.closed_range.isCompact ** Qed
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IsCompact.prod ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ s : Set α t : Set β hs : IsCompact s ht : IsCompact t ⊢ IsCompact (s ×ˢ t) ** rw [isCompact_iff_ultrafilter_le_nhds] at hs ht ⊢ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ s : Set α t : Set β hs : ∀ (f : Ultrafilter α), ↑f ≤ 𝓟 s → ∃ a, a ∈ s ∧ ↑f ≤ 𝓝 a ht : ∀ (f : Ultrafilter β), ↑f ≤ 𝓟 t → ∃ a, a ∈ t ∧ ↑f ≤ 𝓝 a ⊢ ∀ (f : Ultrafilter (α × β)), ↑f ≤ 𝓟 (s ×ˢ t) → ∃ a, a ∈ s ×ˢ t ∧ ↑f ≤ 𝓝 a ** intro f hfs ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ s : Set α t : Set β hs : ∀ (f : Ultrafilter α), ↑f ≤ 𝓟 s → ∃ a, a ∈ s ∧ ↑f ≤ 𝓝 a ht : ∀ (f : Ultrafilter β), ↑f ≤ 𝓟 t → ∃ a, a ∈ t ∧ ↑f ≤ 𝓝 a f : Ultrafilter (α × β) hfs : ↑f ≤ 𝓟 (s ×ˢ t) ⊢ ∃ a, a ∈ s ×ˢ t ∧ ↑f ≤ 𝓝 a ** rw [le_principal_iff] at hfs ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ s : Set α t : Set β hs : ∀ (f : Ultrafilter α), ↑f ≤ 𝓟 s → ∃ a, a ∈ s ∧ ↑f ≤ 𝓝 a ht : ∀ (f : Ultrafilter β), ↑f ≤ 𝓟 t → ∃ a, a ∈ t ∧ ↑f ≤ 𝓝 a f : Ultrafilter (α × β) hfs : s ×ˢ t ∈ ↑f ⊢ ∃ a, a ∈ s ×ˢ t ∧ ↑f ≤ 𝓝 a ** obtain ⟨a : α, sa : a ∈ s, ha : map Prod.fst f.1 ≤ 𝓝 a⟩ :=
hs (f.map Prod.fst) (le_principal_iff.2 <| mem_map.2 <| mem_of_superset hfs fun x => And.left) ** case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ s : Set α t : Set β hs : ∀ (f : Ultrafilter α), ↑f ≤ 𝓟 s → ∃ a, a ∈ s ∧ ↑f ≤ 𝓝 a ht : ∀ (f : Ultrafilter β), ↑f ≤ 𝓟 t → ∃ a, a ∈ t ∧ ↑f ≤ 𝓝 a f : Ultrafilter (α × β) hfs : s ×ˢ t ∈ ↑f a : α sa : a ∈ s ha : map Prod.fst ↑f ≤ 𝓝 a ⊢ ∃ a, a ∈ s ×ˢ t ∧ ↑f ≤ 𝓝 a ** obtain ⟨b : β, tb : b ∈ t, hb : map Prod.snd f.1 ≤ 𝓝 b⟩ :=
ht (f.map Prod.snd) (le_principal_iff.2 <| mem_map.2 <| mem_of_superset hfs fun x => And.right) ** case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ s : Set α t : Set β hs : ∀ (f : Ultrafilter α), ↑f ≤ 𝓟 s → ∃ a, a ∈ s ∧ ↑f ≤ 𝓝 a ht : ∀ (f : Ultrafilter β), ↑f ≤ 𝓟 t → ∃ a, a ∈ t ∧ ↑f ≤ 𝓝 a f : Ultrafilter (α × β) hfs : s ×ˢ t ∈ ↑f a : α sa : a ∈ s ha : map Prod.fst ↑f ≤ 𝓝 a b : β tb : b ∈ t hb : map Prod.snd ↑f ≤ 𝓝 b ⊢ ∃ a, a ∈ s ×ˢ t ∧ ↑f ≤ 𝓝 a ** rw [map_le_iff_le_comap] at ha hb ** case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ s : Set α t : Set β hs : ∀ (f : Ultrafilter α), ↑f ≤ 𝓟 s → ∃ a, a ∈ s ∧ ↑f ≤ 𝓝 a ht : ∀ (f : Ultrafilter β), ↑f ≤ 𝓟 t → ∃ a, a ∈ t ∧ ↑f ≤ 𝓝 a f : Ultrafilter (α × β) hfs : s ×ˢ t ∈ ↑f a : α sa : a ∈ s ha : ↑f ≤ comap Prod.fst (𝓝 a) b : β tb : b ∈ t hb : ↑f ≤ comap Prod.snd (𝓝 b) ⊢ ∃ a, a ∈ s ×ˢ t ∧ ↑f ≤ 𝓝 a ** refine' ⟨⟨a, b⟩, ⟨sa, tb⟩, _⟩ ** case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ s : Set α t : Set β hs : ∀ (f : Ultrafilter α), ↑f ≤ 𝓟 s → ∃ a, a ∈ s ∧ ↑f ≤ 𝓝 a ht : ∀ (f : Ultrafilter β), ↑f ≤ 𝓟 t → ∃ a, a ∈ t ∧ ↑f ≤ 𝓝 a f : Ultrafilter (α × β) hfs : s ×ˢ t ∈ ↑f a : α sa : a ∈ s ha : ↑f ≤ comap Prod.fst (𝓝 a) b : β tb : b ∈ t hb : ↑f ≤ comap Prod.snd (𝓝 b) ⊢ ↑f ≤ 𝓝 (a, b) ** rw [nhds_prod_eq] ** case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s✝ t✝ s : Set α t : Set β hs : ∀ (f : Ultrafilter α), ↑f ≤ 𝓟 s → ∃ a, a ∈ s ∧ ↑f ≤ 𝓝 a ht : ∀ (f : Ultrafilter β), ↑f ≤ 𝓟 t → ∃ a, a ∈ t ∧ ↑f ≤ 𝓝 a f : Ultrafilter (α × β) hfs : s ×ˢ t ∈ ↑f a : α sa : a ∈ s ha : ↑f ≤ comap Prod.fst (𝓝 a) b : β tb : b ∈ t hb : ↑f ≤ comap Prod.snd (𝓝 b) ⊢ ↑f ≤ 𝓝 a ×ˢ 𝓝 b ** exact le_inf ha hb ** Qed
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Filter.coprod_cocompact ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ⊢ Filter.coprod (cocompact α) (cocompact β) = cocompact (α × β) ** ext S ** case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α S : Set (α × β) ⊢ S ∈ Filter.coprod (cocompact α) (cocompact β) ↔ S ∈ cocompact (α × β) ** simp only [mem_coprod_iff, exists_prop, mem_comap, Filter.mem_cocompact] ** case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α S : Set (α × β) ⊢ ((∃ t₁, (∃ t, IsCompact t ∧ tᶜ ⊆ t₁) ∧ Prod.fst ⁻¹' t₁ ⊆ S) ∧ ∃ t₂, (∃ t, IsCompact t ∧ tᶜ ⊆ t₂) ∧ Prod.snd ⁻¹' t₂ ⊆ S) ↔ ∃ t, IsCompact t ∧ tᶜ ⊆ S ** constructor ** case a.mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α S : Set (α × β) ⊢ ((∃ t₁, (∃ t, IsCompact t ∧ tᶜ ⊆ t₁) ∧ Prod.fst ⁻¹' t₁ ⊆ S) ∧ ∃ t₂, (∃ t, IsCompact t ∧ tᶜ ⊆ t₂) ∧ Prod.snd ⁻¹' t₂ ⊆ S) → ∃ t, IsCompact t ∧ tᶜ ⊆ S ** rintro ⟨⟨A, ⟨t, ht, hAt⟩, hAS⟩, B, ⟨t', ht', hBt'⟩, hBS⟩ ** case a.mp.intro.intro.intro.intro.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α S : Set (α × β) A : Set α hAS : Prod.fst ⁻¹' A ⊆ S t : Set α ht : IsCompact t hAt : tᶜ ⊆ A B : Set β hBS : Prod.snd ⁻¹' B ⊆ S t' : Set β ht' : IsCompact t' hBt' : t'ᶜ ⊆ B ⊢ ∃ t, IsCompact t ∧ tᶜ ⊆ S ** refine' ⟨t ×ˢ t', ht.prod ht', _⟩ ** case a.mp.intro.intro.intro.intro.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α S : Set (α × β) A : Set α hAS : Prod.fst ⁻¹' A ⊆ S t : Set α ht : IsCompact t hAt : tᶜ ⊆ A B : Set β hBS : Prod.snd ⁻¹' B ⊆ S t' : Set β ht' : IsCompact t' hBt' : t'ᶜ ⊆ B ⊢ (t ×ˢ t')ᶜ ⊆ S ** refine' Subset.trans _ (union_subset hAS hBS) ** case a.mp.intro.intro.intro.intro.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α S : Set (α × β) A : Set α hAS : Prod.fst ⁻¹' A ⊆ S t : Set α ht : IsCompact t hAt : tᶜ ⊆ A B : Set β hBS : Prod.snd ⁻¹' B ⊆ S t' : Set β ht' : IsCompact t' hBt' : t'ᶜ ⊆ B ⊢ (t ×ˢ t')ᶜ ⊆ Prod.fst ⁻¹' A ∪ Prod.snd ⁻¹' B ** rw [compl_subset_comm] at hAt hBt' ⊢ ** case a.mp.intro.intro.intro.intro.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α S : Set (α × β) A : Set α hAS : Prod.fst ⁻¹' A ⊆ S t : Set α ht : IsCompact t hAt : Aᶜ ⊆ t B : Set β hBS : Prod.snd ⁻¹' B ⊆ S t' : Set β ht' : IsCompact t' hBt' : Bᶜ ⊆ t' ⊢ (Prod.fst ⁻¹' A ∪ Prod.snd ⁻¹' B)ᶜ ⊆ t ×ˢ t' ** refine' Subset.trans (fun x => _) (Set.prod_mono hAt hBt') ** case a.mp.intro.intro.intro.intro.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α S : Set (α × β) A : Set α hAS : Prod.fst ⁻¹' A ⊆ S t : Set α ht : IsCompact t hAt : Aᶜ ⊆ t B : Set β hBS : Prod.snd ⁻¹' B ⊆ S t' : Set β ht' : IsCompact t' hBt' : Bᶜ ⊆ t' x : α × β ⊢ x ∈ (Prod.fst ⁻¹' A ∪ Prod.snd ⁻¹' B)ᶜ → x ∈ Aᶜ ×ˢ Bᶜ ** simp only [compl_union, mem_inter_iff, mem_prod, mem_preimage, mem_compl_iff] ** case a.mp.intro.intro.intro.intro.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α S : Set (α × β) A : Set α hAS : Prod.fst ⁻¹' A ⊆ S t : Set α ht : IsCompact t hAt : Aᶜ ⊆ t B : Set β hBS : Prod.snd ⁻¹' B ⊆ S t' : Set β ht' : IsCompact t' hBt' : Bᶜ ⊆ t' x : α × β ⊢ ¬x.1 ∈ A ∧ ¬x.2 ∈ B → ¬x.1 ∈ A ∧ ¬x.2 ∈ B ** tauto ** case a.mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α S : Set (α × β) ⊢ (∃ t, IsCompact t ∧ tᶜ ⊆ S) → (∃ t₁, (∃ t, IsCompact t ∧ tᶜ ⊆ t₁) ∧ Prod.fst ⁻¹' t₁ ⊆ S) ∧ ∃ t₂, (∃ t, IsCompact t ∧ tᶜ ⊆ t₂) ∧ Prod.snd ⁻¹' t₂ ⊆ S ** rintro ⟨t, ht, htS⟩ ** case a.mpr.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α S t : Set (α × β) ht : IsCompact t htS : tᶜ ⊆ S ⊢ (∃ t₁, (∃ t, IsCompact t ∧ tᶜ ⊆ t₁) ∧ Prod.fst ⁻¹' t₁ ⊆ S) ∧ ∃ t₂, (∃ t, IsCompact t ∧ tᶜ ⊆ t₂) ∧ Prod.snd ⁻¹' t₂ ⊆ S ** refine' ⟨⟨(Prod.fst '' t)ᶜ, _, _⟩, ⟨(Prod.snd '' t)ᶜ, _, _⟩⟩ ** case a.mpr.intro.intro.refine'_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α S t : Set (α × β) ht : IsCompact t htS : tᶜ ⊆ S ⊢ ∃ t_1, IsCompact t_1 ∧ t_1ᶜ ⊆ (Prod.fst '' t)ᶜ ** exact ⟨Prod.fst '' t, ht.image continuous_fst, Subset.rfl⟩ ** case a.mpr.intro.intro.refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α S t : Set (α × β) ht : IsCompact t htS : tᶜ ⊆ S ⊢ Prod.fst ⁻¹' (Prod.fst '' t)ᶜ ⊆ S ** rw [preimage_compl] ** case a.mpr.intro.intro.refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α S t : Set (α × β) ht : IsCompact t htS : tᶜ ⊆ S ⊢ (Prod.fst ⁻¹' (Prod.fst '' t))ᶜ ⊆ S ** rw [compl_subset_comm] at htS ⊢ ** case a.mpr.intro.intro.refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α S t : Set (α × β) ht : IsCompact t htS : Sᶜ ⊆ t ⊢ Sᶜ ⊆ Prod.fst ⁻¹' (Prod.fst '' t) ** exact htS.trans (subset_preimage_image Prod.fst _) ** case a.mpr.intro.intro.refine'_3 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α S t : Set (α × β) ht : IsCompact t htS : tᶜ ⊆ S ⊢ ∃ t_1, IsCompact t_1 ∧ t_1ᶜ ⊆ (Prod.snd '' t)ᶜ ** exact ⟨Prod.snd '' t, ht.image continuous_snd, Subset.rfl⟩ ** case a.mpr.intro.intro.refine'_4 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α S t : Set (α × β) ht : IsCompact t htS : tᶜ ⊆ S ⊢ Prod.snd ⁻¹' (Prod.snd '' t)ᶜ ⊆ S ** rw [preimage_compl] ** case a.mpr.intro.intro.refine'_4 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α S t : Set (α × β) ht : IsCompact t htS : tᶜ ⊆ S ⊢ (Prod.snd ⁻¹' (Prod.snd '' t))ᶜ ⊆ S ** rw [compl_subset_comm] at htS ⊢ ** case a.mpr.intro.intro.refine'_4 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t✝ : Set α S t : Set (α × β) ht : IsCompact t htS : Sᶜ ⊆ t ⊢ Sᶜ ⊆ Prod.snd ⁻¹' (Prod.snd '' t) ** exact htS.trans (subset_preimage_image Prod.snd _) ** Qed
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Prod.noncompactSpace_iff ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β s t : Set α ⊢ NoncompactSpace (α × β) ↔ NoncompactSpace α ∧ Nonempty β ∨ Nonempty α ∧ NoncompactSpace β ** simp [← Filter.cocompact_neBot_iff, ← Filter.coprod_cocompact, Filter.coprod_neBot_iff] ** Qed
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isCompact_pi_infinite ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) ⊢ (∀ (i : ι), IsCompact (s i)) → IsCompact {x | ∀ (i : ι), x i ∈ s i} ** simp only [isCompact_iff_ultrafilter_le_nhds, nhds_pi, Filter.pi, exists_prop, mem_setOf_eq,
le_iInf_iff, le_principal_iff] ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) ⊢ (∀ (i : ι) (f : Ultrafilter (π i)), s i ∈ ↑f → ∃ a, a ∈ s i ∧ ↑f ≤ 𝓝 a) → ∀ (f : Ultrafilter ((i : ι) → π i)), {x | ∀ (i : ι), x i ∈ s i} ∈ ↑f → ∃ a, (∀ (i : ι), a i ∈ s i) ∧ ∀ (i : ι), ↑f ≤ comap (Function.eval i) (𝓝 (a i)) ** intro h f hfs ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) h : ∀ (i : ι) (f : Ultrafilter (π i)), s i ∈ ↑f → ∃ a, a ∈ s i ∧ ↑f ≤ 𝓝 a f : Ultrafilter ((i : ι) → π i) hfs : {x | ∀ (i : ι), x i ∈ s i} ∈ ↑f ⊢ ∃ a, (∀ (i : ι), a i ∈ s i) ∧ ∀ (i : ι), ↑f ≤ comap (Function.eval i) (𝓝 (a i)) ** have : ∀ i : ι, ∃ a, a ∈ s i ∧ Tendsto (Function.eval i) f (𝓝 a) := by
refine fun i => h i (f.map _) (mem_map.2 ?_)
exact mem_of_superset hfs fun x hx => hx i ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) h : ∀ (i : ι) (f : Ultrafilter (π i)), s i ∈ ↑f → ∃ a, a ∈ s i ∧ ↑f ≤ 𝓝 a f : Ultrafilter ((i : ι) → π i) hfs : {x | ∀ (i : ι), x i ∈ s i} ∈ ↑f this : ∀ (i : ι), ∃ a, a ∈ s i ∧ Tendsto (Function.eval i) (↑f) (𝓝 a) ⊢ ∃ a, (∀ (i : ι), a i ∈ s i) ∧ ∀ (i : ι), ↑f ≤ comap (Function.eval i) (𝓝 (a i)) ** choose a ha using this ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) h : ∀ (i : ι) (f : Ultrafilter (π i)), s i ∈ ↑f → ∃ a, a ∈ s i ∧ ↑f ≤ 𝓝 a f : Ultrafilter ((i : ι) → π i) hfs : {x | ∀ (i : ι), x i ∈ s i} ∈ ↑f a : (i : ι) → π i ha : ∀ (i : ι), a i ∈ s i ∧ Tendsto (Function.eval i) (↑f) (𝓝 (a i)) ⊢ ∃ a, (∀ (i : ι), a i ∈ s i) ∧ ∀ (i : ι), ↑f ≤ comap (Function.eval i) (𝓝 (a i)) ** exact ⟨a, fun i => (ha i).left, fun i => (ha i).right.le_comap⟩ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) h : ∀ (i : ι) (f : Ultrafilter (π i)), s i ∈ ↑f → ∃ a, a ∈ s i ∧ ↑f ≤ 𝓝 a f : Ultrafilter ((i : ι) → π i) hfs : {x | ∀ (i : ι), x i ∈ s i} ∈ ↑f ⊢ ∀ (i : ι), ∃ a, a ∈ s i ∧ Tendsto (Function.eval i) (↑f) (𝓝 a) ** refine fun i => h i (f.map _) (mem_map.2 ?_) ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) h : ∀ (i : ι) (f : Ultrafilter (π i)), s i ∈ ↑f → ∃ a, a ∈ s i ∧ ↑f ≤ 𝓝 a f : Ultrafilter ((i : ι) → π i) hfs : {x | ∀ (i : ι), x i ∈ s i} ∈ ↑f i : ι ⊢ Function.eval i ⁻¹' s i ∈ ↑f ** exact mem_of_superset hfs fun x hx => hx i ** Qed
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isCompact_univ_pi ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) h : ∀ (i : ι), IsCompact (s i) ⊢ IsCompact (Set.pi univ s) ** convert isCompact_pi_infinite h ** case h.e'_3 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) h : ∀ (i : ι), IsCompact (s i) ⊢ Set.pi univ s = {x | ∀ (i : ι), x i ∈ s i} ** simp only [← mem_univ_pi, setOf_mem_eq] ** Qed
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Filter.coprodᵢ_cocompact ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β s t : Set α inst✝¹ : (i : ι) → TopologicalSpace (π i) δ : Type u_3 κ : δ → Type u_4 inst✝ : (d : δ) → TopologicalSpace (κ d) ⊢ (Filter.coprodᵢ fun d => cocompact (κ d)) = cocompact ((d : δ) → κ d) ** refine' le_antisymm (iSup_le fun i => Filter.comap_cocompact_le (continuous_apply i)) _ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β s t : Set α inst✝¹ : (i : ι) → TopologicalSpace (π i) δ : Type u_3 κ : δ → Type u_4 inst✝ : (d : δ) → TopologicalSpace (κ d) ⊢ cocompact ((d : δ) → κ d) ≤ Filter.coprodᵢ fun d => cocompact (κ d) ** refine' compl_surjective.forall.2 fun s H => _ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β s✝ t : Set α inst✝¹ : (i : ι) → TopologicalSpace (π i) δ : Type u_3 κ : δ → Type u_4 inst✝ : (d : δ) → TopologicalSpace (κ d) s : Set ((i : δ) → κ i) H : sᶜ ∈ Filter.coprodᵢ fun d => cocompact (κ d) ⊢ sᶜ ∈ cocompact ((d : δ) → κ d) ** simp only [compl_mem_coprodᵢ, Filter.mem_cocompact, compl_subset_compl, image_subset_iff] at H ⊢ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β s✝ t : Set α inst✝¹ : (i : ι) → TopologicalSpace (π i) δ : Type u_3 κ : δ → Type u_4 inst✝ : (d : δ) → TopologicalSpace (κ d) s : Set ((i : δ) → κ i) H : ∀ (i : δ), ∃ t, IsCompact t ∧ s ⊆ Function.eval i ⁻¹' t ⊢ ∃ t, IsCompact t ∧ s ⊆ t ** choose K hKc htK using H ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β s✝ t : Set α inst✝¹ : (i : ι) → TopologicalSpace (π i) δ : Type u_3 κ : δ → Type u_4 inst✝ : (d : δ) → TopologicalSpace (κ d) s : Set ((i : δ) → κ i) K : (i : δ) → Set (κ i) hKc : ∀ (i : δ), IsCompact (K i) htK : ∀ (i : δ), s ⊆ Function.eval i ⁻¹' K i ⊢ ∃ t, IsCompact t ∧ s ⊆ t ** exact ⟨Set.pi univ K, isCompact_univ_pi hKc, fun f hf i _ => htK i hf⟩ ** Qed
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IsClosed.exists_minimal_nonempty_closed_subset ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S ⊢ ∃ V, V ⊆ S ∧ Set.Nonempty V ∧ IsClosed V ∧ ∀ (V' : Set α), V' ⊆ V → Set.Nonempty V' → IsClosed V' → V' = V ** let opens := { U : Set α | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty } ** case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} U : Set α h : ∀ (a : Set α), a ∈ opens → U ⊆ a → a = U Uc : Sᶜ ⊆ U Uo : IsOpen U Ucne : Set.Nonempty Uᶜ ⊢ ∃ V, V ⊆ S ∧ Set.Nonempty V ∧ IsClosed V ∧ ∀ (V' : Set α), V' ⊆ V → Set.Nonempty V' → IsClosed V' → V' = V ** refine' ⟨Uᶜ, Set.compl_subset_comm.mp Uc, Ucne, Uo.isClosed_compl, _⟩ ** case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} U : Set α h : ∀ (a : Set α), a ∈ opens → U ⊆ a → a = U Uc : Sᶜ ⊆ U Uo : IsOpen U Ucne : Set.Nonempty Uᶜ ⊢ ∀ (V' : Set α), V' ⊆ Uᶜ → Set.Nonempty V' → IsClosed V' → V' = Uᶜ ** intro V' V'sub V'ne V'cls ** case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} U : Set α h : ∀ (a : Set α), a ∈ opens → U ⊆ a → a = U Uc : Sᶜ ⊆ U Uo : IsOpen U Ucne : Set.Nonempty Uᶜ V' : Set α V'sub : V' ⊆ Uᶜ V'ne : Set.Nonempty V' V'cls : IsClosed V' ⊢ V' = Uᶜ ** have : V'ᶜ = U := by
refine' h V'ᶜ ⟨_, isOpen_compl_iff.mpr V'cls, _⟩ (Set.subset_compl_comm.mp V'sub)
exact Set.Subset.trans Uc (Set.subset_compl_comm.mp V'sub)
simp only [compl_compl, V'ne] ** case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} U : Set α h : ∀ (a : Set α), a ∈ opens → U ⊆ a → a = U Uc : Sᶜ ⊆ U Uo : IsOpen U Ucne : Set.Nonempty Uᶜ V' : Set α V'sub : V' ⊆ Uᶜ V'ne : Set.Nonempty V' V'cls : IsClosed V' this : V'ᶜ = U ⊢ V' = Uᶜ ** rw [← this, compl_compl] ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c ⊢ ∃ ub, ub ∈ opens ∧ ∀ (s : Set α), s ∈ c → s ⊆ ub ** by_cases hcne : c.Nonempty ** case pos α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c hcne : Set.Nonempty c ⊢ ∃ ub, ub ∈ opens ∧ ∀ (s : Set α), s ∈ c → s ⊆ ub ** obtain ⟨U₀, hU₀⟩ := hcne ** case pos.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c U₀ : Set α hU₀ : U₀ ∈ c ⊢ ∃ ub, ub ∈ opens ∧ ∀ (s : Set α), s ∈ c → s ⊆ ub ** haveI : Nonempty { U // U ∈ c } := ⟨⟨U₀, hU₀⟩⟩ ** case pos.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c U₀ : Set α hU₀ : U₀ ∈ c this : Nonempty { U // U ∈ c } ⊢ ∃ ub, ub ∈ opens ∧ ∀ (s : Set α), s ∈ c → s ⊆ ub ** obtain ⟨U₀compl, -, -⟩ := hc hU₀ ** case pos.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c U₀ : Set α hU₀ : U₀ ∈ c this : Nonempty { U // U ∈ c } U₀compl : Sᶜ ⊆ U₀ ⊢ ∃ ub, ub ∈ opens ∧ ∀ (s : Set α), s ∈ c → s ⊆ ub ** use ⋃₀ c ** case h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c U₀ : Set α hU₀ : U₀ ∈ c this : Nonempty { U // U ∈ c } U₀compl : Sᶜ ⊆ U₀ ⊢ ⋃₀ c ∈ opens ∧ ∀ (s : Set α), s ∈ c → s ⊆ ⋃₀ c ** refine' ⟨⟨_, _, _⟩, fun U hU a ha => ⟨U, hU, ha⟩⟩ ** case h.refine'_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c U₀ : Set α hU₀ : U₀ ∈ c this : Nonempty { U // U ∈ c } U₀compl : Sᶜ ⊆ U₀ ⊢ Sᶜ ⊆ ⋃₀ c ** exact fun a ha => ⟨U₀, hU₀, U₀compl ha⟩ ** case h.refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c U₀ : Set α hU₀ : U₀ ∈ c this : Nonempty { U // U ∈ c } U₀compl : Sᶜ ⊆ U₀ ⊢ IsOpen (⋃₀ c) ** exact isOpen_sUnion fun _ h => (hc h).2.1 ** case h.refine'_3 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c U₀ : Set α hU₀ : U₀ ∈ c this : Nonempty { U // U ∈ c } U₀compl : Sᶜ ⊆ U₀ ⊢ Set.Nonempty (⋃₀ c)ᶜ ** convert_to (⋂ U : { U // U ∈ c }, U.1ᶜ).Nonempty ** case h.refine'_3 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c U₀ : Set α hU₀ : U₀ ∈ c this : Nonempty { U // U ∈ c } U₀compl : Sᶜ ⊆ U₀ ⊢ Set.Nonempty (⋂ U, (↑U)ᶜ) ** apply IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed ** case h.e'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c U₀ : Set α hU₀ : U₀ ∈ c this : Nonempty { U // U ∈ c } U₀compl : Sᶜ ⊆ U₀ ⊢ (⋃₀ c)ᶜ = ⋂ U, (↑U)ᶜ ** ext ** case h.e'_2.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c U₀ : Set α hU₀ : U₀ ∈ c this : Nonempty { U // U ∈ c } U₀compl : Sᶜ ⊆ U₀ x✝ : α ⊢ x✝ ∈ (⋃₀ c)ᶜ ↔ x✝ ∈ ⋂ U, (↑U)ᶜ ** simp only [not_exists, exists_prop, not_and, Set.mem_iInter, Subtype.forall,
mem_setOf_eq, mem_compl_iff, mem_sUnion] ** case h.refine'_3.hZd α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c U₀ : Set α hU₀ : U₀ ∈ c this : Nonempty { U // U ∈ c } U₀compl : Sᶜ ⊆ U₀ ⊢ Directed (fun x x_1 => x ⊇ x_1) fun i => (↑i)ᶜ ** rintro ⟨U, hU⟩ ⟨U', hU'⟩ ** case h.refine'_3.hZd.mk.mk α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c U₀ : Set α hU₀ : U₀ ∈ c this : Nonempty { U // U ∈ c } U₀compl : Sᶜ ⊆ U₀ U : Set α hU : U ∈ c U' : Set α hU' : U' ∈ c ⊢ ∃ z, (fun x x_1 => x ⊇ x_1) ((fun i => (↑i)ᶜ) { val := U, property := hU }) ((fun i => (↑i)ᶜ) z) ∧ (fun x x_1 => x ⊇ x_1) ((fun i => (↑i)ᶜ) { val := U', property := hU' }) ((fun i => (↑i)ᶜ) z) ** obtain ⟨V, hVc, hVU, hVU'⟩ := hz.directedOn U hU U' hU' ** case h.refine'_3.hZd.mk.mk.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c U₀ : Set α hU₀ : U₀ ∈ c this : Nonempty { U // U ∈ c } U₀compl : Sᶜ ⊆ U₀ U : Set α hU : U ∈ c U' : Set α hU' : U' ∈ c V : Set α hVc : V ∈ c hVU : U ⊆ V hVU' : U' ⊆ V ⊢ ∃ z, (fun x x_1 => x ⊇ x_1) ((fun i => (↑i)ᶜ) { val := U, property := hU }) ((fun i => (↑i)ᶜ) z) ∧ (fun x x_1 => x ⊇ x_1) ((fun i => (↑i)ᶜ) { val := U', property := hU' }) ((fun i => (↑i)ᶜ) z) ** exact ⟨⟨V, hVc⟩, Set.compl_subset_compl.mpr hVU, Set.compl_subset_compl.mpr hVU'⟩ ** case h.refine'_3.hZn α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c U₀ : Set α hU₀ : U₀ ∈ c this : Nonempty { U // U ∈ c } U₀compl : Sᶜ ⊆ U₀ ⊢ ∀ (i : { U // U ∈ c }), Set.Nonempty (↑i)ᶜ ** exact fun U => (hc U.2).2.2 ** case h.refine'_3.hZc α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c U₀ : Set α hU₀ : U₀ ∈ c this : Nonempty { U // U ∈ c } U₀compl : Sᶜ ⊆ U₀ ⊢ ∀ (i : { U // U ∈ c }), IsCompact (↑i)ᶜ ** exact fun U => (hc U.2).2.1.isClosed_compl.isCompact ** case h.refine'_3.hZcl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c U₀ : Set α hU₀ : U₀ ∈ c this : Nonempty { U // U ∈ c } U₀compl : Sᶜ ⊆ U₀ ⊢ ∀ (i : { U // U ∈ c }), IsClosed (↑i)ᶜ ** exact fun U => (hc U.2).2.1.isClosed_compl ** case neg α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c hcne : ¬Set.Nonempty c ⊢ ∃ ub, ub ∈ opens ∧ ∀ (s : Set α), s ∈ c → s ⊆ ub ** use Sᶜ ** case h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c hcne : ¬Set.Nonempty c ⊢ Sᶜ ∈ opens ∧ ∀ (s : Set α), s ∈ c → s ⊆ Sᶜ ** refine' ⟨⟨Set.Subset.refl _, isOpen_compl_iff.mpr hS, _⟩, fun U Uc => (hcne ⟨U, Uc⟩).elim⟩ ** case h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c hcne : ¬Set.Nonempty c ⊢ Set.Nonempty Sᶜᶜ ** rw [compl_compl] ** case h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} c : Set (Set α) hc : c ⊆ opens hz : IsChain (fun x x_1 => x ⊆ x_1) c hcne : ¬Set.Nonempty c ⊢ Set.Nonempty S ** exact hne ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} U : Set α h : ∀ (a : Set α), a ∈ opens → U ⊆ a → a = U Uc : Sᶜ ⊆ U Uo : IsOpen U Ucne : Set.Nonempty Uᶜ V' : Set α V'sub : V' ⊆ Uᶜ V'ne : Set.Nonempty V' V'cls : IsClosed V' ⊢ V'ᶜ = U ** refine' h V'ᶜ ⟨_, isOpen_compl_iff.mpr V'cls, _⟩ (Set.subset_compl_comm.mp V'sub) ** case refine'_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} U : Set α h : ∀ (a : Set α), a ∈ opens → U ⊆ a → a = U Uc : Sᶜ ⊆ U Uo : IsOpen U Ucne : Set.Nonempty Uᶜ V' : Set α V'sub : V' ⊆ Uᶜ V'ne : Set.Nonempty V' V'cls : IsClosed V' ⊢ Sᶜ ⊆ V'ᶜ case refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} U : Set α h : ∀ (a : Set α), a ∈ opens → U ⊆ a → a = U Uc : Sᶜ ⊆ U Uo : IsOpen U Ucne : Set.Nonempty Uᶜ V' : Set α V'sub : V' ⊆ Uᶜ V'ne : Set.Nonempty V' V'cls : IsClosed V' ⊢ Set.Nonempty V'ᶜᶜ ** exact Set.Subset.trans Uc (Set.subset_compl_comm.mp V'sub) ** case refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : CompactSpace α S : Set α hS : IsClosed S hne : Set.Nonempty S opens : Set (Set α) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ} U : Set α h : ∀ (a : Set α), a ∈ opens → U ⊆ a → a = U Uc : Sᶜ ⊆ U Uo : IsOpen U Ucne : Set.Nonempty Uᶜ V' : Set α V'sub : V' ⊆ Uᶜ V'ne : Set.Nonempty V' V'cls : IsClosed V' ⊢ Set.Nonempty V'ᶜᶜ ** simp only [compl_compl, V'ne] ** Qed
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