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leandojo-lean4-formal-informal-strings-split

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Path.trans_continuous_family X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b c : ι → X γ₁ : (t : ι) → Path (a t) (b t) h₁ : Continuous ↿γ₁ γ₂ : (t : ι) → Path (b t) (c t) h₂ : Continuous ↿γ₂ ⊢ Continuous ↿fun t => trans (γ₁ t) (γ₂ t) have h₁' := Path.continuous_uncurry_extend_of_continuous_family γ₁ h₁ X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b c : ι → X γ₁ : (t : ι) → Path (a t) (b t) h₁ : Continuous ↿γ₁ γ₂ : (t : ι) → Path (b t) (c t) h₂ : Continuous ↿γ₂ h₁' : Continuous ↿fun t => extend (γ₁ t) ⊢ Continuous ↿fun t => trans (γ₁ t) (γ₂ t) have h₂' := Path.continuous_uncurry_extend_of_continuous_family γ₂ h₂ ** X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b c : ι → X γ₁ : (t : ι) → Path (a t) (b t) h₁ : Continuous ↿γ₁ γ₂ : (t : ι) → Path (b t) (c t) h₂ : Continuous ↿γ₂ h₁' : Continuous ↿fun t => extend (γ₁ t) h₂' : Continuous ↿fun t => extend (γ₂ t) ⊢ Continuous fun p => ↑{ toContinuousMap := ContinuousMap.mk fun x => if ↑x ≤ 1 / 2 then extend (γ₁ p.1) (2 * ↑x) else extend (γ₂ p.1) (2 * ↑x - 1), source' := (_ : ContinuousMap.toFun (ContinuousMap.mk fun x => if ↑x ≤ 1 / 2 then extend (γ₁ p.1) (2 * ↑x) else extend (γ₂ p.1) (2 * ↑x - 1)) 0 = a p.1), target' := (_ : ContinuousMap.toFun (ContinuousMap.mk fun x => if ↑x ≤ 1 / 2 then extend (γ₁ p.1) (2 * ↑x) else extend (γ₂ p.1) (2 * ↑x - 1)) 1 = c p.1) } p.2 refine' Continuous.if_le _ _ (continuous_subtype_val.comp continuous_snd) continuous_const _ case refine'_1 X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b c : ι → X γ₁ : (t : ι) → Path (a t) (b t) h₁ : Continuous ↿γ₁ γ₂ : (t : ι) → Path (b t) (c t) h₂ : Continuous ↿γ₂ h₁' : Continuous ↿fun t => extend (γ₁ t) h₂' : Continuous ↿fun t => extend (γ₂ t) ⊢ Continuous fun p => extend (γ₁ p.1) (2 * ↑p.2) ** change Continuous ((fun p : ι × ℝ => (γ₁ p.1).extend p.2) ∘ Prod.map id (fun x => 2 * x : I → ℝ)) ** case refine'_1 X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b c : ι → X γ₁ : (t : ι) → Path (a t) (b t) h₁ : Continuous ↿γ₁ γ₂ : (t : ι) → Path (b t) (c t) h₂ : Continuous ↿γ₂ h₁' : Continuous ↿fun t => extend (γ₁ t) h₂' : Continuous ↿fun t => extend (γ₂ t) ⊢ Continuous ((fun p => extend (γ₁ p.1) p.2) ∘ Prod.map id fun x => 2 * ↑x) exact h₁'.comp (continuous_id.prod_map <\| continuous_const.mul continuous_subtype_val) case refine'_2 X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b c : ι → X γ₁ : (t : ι) → Path (a t) (b t) h₁ : Continuous ↿γ₁ γ₂ : (t : ι) → Path (b t) (c t) h₂ : Continuous ↿γ₂ h₁' : Continuous ↿fun t => extend (γ₁ t) h₂' : Continuous ↿fun t => extend (γ₂ t) ⊢ Continuous fun p => extend (γ₂ p.1) (2 * ↑p.2 - 1) ** change Continuous ((fun p : ι × ℝ => (γ₂ p.1).extend p.2) ∘ Prod.map id (fun x => 2 * x - 1 : I → ℝ)) ** case refine'_2 X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b c : ι → X γ₁ : (t : ι) → Path (a t) (b t) h₁ : Continuous ↿γ₁ γ₂ : (t : ι) → Path (b t) (c t) h₂ : Continuous ↿γ₂ h₁' : Continuous ↿fun t => extend (γ₁ t) h₂' : Continuous ↿fun t => extend (γ₂ t) ⊢ Continuous ((fun p => extend (γ₂ p.1) p.2) ∘ Prod.map id fun x => 2 * ↑x - 1) exact h₂'.comp (continuous_id.prod_map <\| (continuous_const.mul continuous_subtype_val).sub continuous_const) case refine'_3 X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b c : ι → X γ₁ : (t : ι) → Path (a t) (b t) h₁ : Continuous ↿γ₁ γ₂ : (t : ι) → Path (b t) (c t) h₂ : Continuous ↿γ₂ h₁' : Continuous ↿fun t => extend (γ₁ t) h₂' : Continuous ↿fun t => extend (γ₂ t) ⊢ ∀ (x : ι × ↑I), ↑x.2 = 1 / 2 → extend (γ₁ x.1) (2 * ↑x.2) = extend (γ₂ x.1) (2 * ↑x.2 - 1) rintro st hst case refine'_3 X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b c : ι → X γ₁ : (t : ι) → Path (a t) (b t) h₁ : Continuous ↿γ₁ γ₂ : (t : ι) → Path (b t) (c t) h₂ : Continuous ↿γ₂ h₁' : Continuous ↿fun t => extend (γ₁ t) h₂' : Continuous ↿fun t => extend (γ₂ t) st : ι × ↑I hst : ↑st.2 = 1 / 2 ⊢ extend (γ₁ st.1) (2 * ↑st.2) = extend (γ₂ st.1) (2 * ↑st.2 - 1) simp [hst, mul_inv_cancel (two_ne_zero' ℝ)] Qed
Continuous.path_trans X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y f : Y → Path x y g : Y → Path y z ⊢ Continuous f → Continuous g → Continuous fun t => trans (f t) (g t) intro hf hg X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y f : Y → Path x y g : Y → Path y z hf : Continuous f hg : Continuous g ⊢ Continuous fun t => trans (f t) (g t) apply continuous_uncurry_iff.mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y f : Y → Path x y g : Y → Path y z hf : Continuous f hg : Continuous g ⊢ Continuous ↿fun t => trans (f t) (g t) exact trans_continuous_family _ (continuous_uncurry_iff.mpr hf) _ (continuous_uncurry_iff.mpr hg) ** Qed
Path.trans_prod_eq_prod_trans X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y a₁ a₂ a₃ : X b₁ b₂ b₃ : Y γ₁ : Path a₁ a₂ δ₁ : Path a₂ a₃ γ₂ : Path b₁ b₂ δ₂ : Path b₂ b₃ ⊢ trans (Path.prod γ₁ γ₂) (Path.prod δ₁ δ₂) = Path.prod (trans γ₁ δ₁) (trans γ₂ δ₂) ext t <;> unfold Path.trans <;> simp only [Path.coe_mk_mk, Path.prod_coe, Function.comp_apply] <;> split_ifs <;> rfl ** Qed
Path.trans_pi_eq_pi_trans X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y χ : ι → Type u_4 inst✝ : (i : ι) → TopologicalSpace (χ i) as bs cs : (i : ι) → χ i γ₀ : (i : ι) → Path (as i) (bs i) γ₁ : (i : ι) → Path (bs i) (cs i) ⊢ trans (Path.pi γ₀) (Path.pi γ₁) = Path.pi fun i => trans (γ₀ i) (γ₁ i) ext t i case a.h.h X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y χ : ι → Type u_4 inst✝ : (i : ι) → TopologicalSpace (χ i) as bs cs : (i : ι) → χ i γ₀ : (i : ι) → Path (as i) (bs i) γ₁ : (i : ι) → Path (bs i) (cs i) t : ↑I i : ι ⊢ ↑(trans (Path.pi γ₀) (Path.pi γ₁)) t i = ↑(Path.pi fun i => trans (γ₀ i) (γ₁ i)) t i unfold Path.trans ** case a.h.h X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y χ : ι → Type u_4 inst✝ : (i : ι) → TopologicalSpace (χ i) as bs cs : (i : ι) → χ i γ₀ : (i : ι) → Path (as i) (bs i) γ₁ : (i : ι) → Path (bs i) (cs i) t : ↑I i : ι ⊢ ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend (Path.pi γ₀) (2 * t) else extend (Path.pi γ₁) (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend (Path.pi γ₀) (2 * x) else extend (Path.pi γ₁) (2 * x - 1)) ∘ Subtype.val) 0 = fun i => as i), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend (Path.pi γ₀) (2 * x) else extend (Path.pi γ₁) (2 * x - 1)) ∘ Subtype.val) 1 = fun i => cs i) } t i = ↑(Path.pi fun i => { toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend (γ₀ i) (2 * t) else extend (γ₁ i) (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend (γ₀ i) (2 * x) else extend (γ₁ i) (2 * x - 1)) ∘ Subtype.val) 0 = as i), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend (γ₀ i) (2 * x) else extend (γ₁ i) (2 * x - 1)) ∘ Subtype.val) 1 = cs i) }) t i simp only [Path.coe_mk_mk, Function.comp_apply, pi_coe] case a.h.h X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y χ : ι → Type u_4 inst✝ : (i : ι) → TopologicalSpace (χ i) as bs cs : (i : ι) → χ i γ₀ : (i : ι) → Path (as i) (bs i) γ₁ : (i : ι) → Path (bs i) (cs i) t : ↑I i : ι ⊢ ite (↑t ≤ 1 / 2) (extend (Path.pi γ₀) (2 * ↑t)) (extend (Path.pi γ₁) (2 * ↑t - 1)) i = if ↑t ≤ 1 / 2 then extend (γ₀ i) (2 * ↑t) else extend (γ₁ i) (2 * ↑t - 1) split_ifs <;> rfl Qed
Path.truncate_range X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t₀ t₁ : ℝ ⊢ range ↑(truncate γ t₀ t₁) ⊆ range ↑γ rw [← γ.extend_range] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t₀ t₁ : ℝ ⊢ range ↑(truncate γ t₀ t₁) ⊆ range (extend γ) simp only [range_subset_iff, SetCoe.exists, SetCoe.forall] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t₀ t₁ : ℝ ⊢ ∀ (x : ℝ) (h : x ∈ I), ↑(truncate γ t₀ t₁) { val := x, property := h } ∈ range (extend γ) intro x _hx X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ✝ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t₀ t₁ x : ℝ _hx : x ∈ I ⊢ ↑(truncate γ t₀ t₁) { val := x, property := _hx } ∈ range (extend γ) simp only [FunLike.coe, Path.truncate, mem_range_self] ** Qed
Path.truncate_const_continuous_family X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t : ℝ ⊢ Continuous ↿(truncate γ t) have key : Continuous (fun x => (t, x) : ℝ × I → ℝ × ℝ × I) := by continuity X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t : ℝ key : Continuous fun x => (t, x) ⊢ Continuous ↿(truncate γ t) exact γ.truncate_continuous_family.comp key X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t : ℝ ⊢ Continuous fun x => (t, x) continuity ** Qed
Path.truncate_self X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t : ℝ ⊢ extend γ (min t t) = extend γ t rw [min_self] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t : ℝ ⊢ truncate γ t t = cast (refl (extend γ t)) (_ : extend γ (min t t) = extend γ t) (_ : extend γ t = extend γ t) ext x case a.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ✝ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t : ℝ x : ↑I ⊢ ↑(truncate γ t t) x = ↑(cast (refl (extend γ t)) (_ : extend γ (min t t) = extend γ t) (_ : extend γ t = extend γ t)) x rw [cast_coe] case a.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ✝ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t : ℝ x : ↑I ⊢ ↑(truncate γ t t) x = ↑(refl (extend γ t)) x simp only [truncate, FunLike.coe, refl, min_def, max_def] case a.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ✝ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t : ℝ x : ↑I ⊢ extend γ (if (if ↑x ≤ t then t else ↑x) ≤ t then if ↑x ≤ t then t else ↑x else t) = extend γ t split_ifs with h₁ h₂ <;> congr ** Qed
Path.truncate_zero_zero X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b ⊢ extend γ (min 0 0) = a rw [min_self, γ.extend_zero] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b ⊢ truncate γ 0 0 = cast (refl a) (_ : extend γ (min 0 0) = a) (_ : extend γ 0 = a) convert γ.truncate_self 0 ** Qed
Path.truncate_one_one X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b ⊢ extend γ (min 1 1) = b rw [min_self, γ.extend_one] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b ⊢ truncate γ 1 1 = cast (refl b) (_ : extend γ (min 1 1) = b) (_ : extend γ 1 = b) convert γ.truncate_self 1 ** Qed
Path.truncate_zero_one X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b ⊢ extend γ (min 0 1) = a simp [zero_le_one, extend_zero] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b ⊢ extend γ 1 = b simp X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b ⊢ truncate γ 0 1 = cast γ (_ : extend γ (min 0 1) = a) (_ : extend γ 1 = b) ext x case a.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ✝ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b x : ↑I ⊢ ↑(truncate γ 0 1) x = ↑(cast γ (_ : extend γ (min 0 1) = a) (_ : extend γ 1 = b)) x rw [cast_coe] case a.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ✝ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b x : ↑I ⊢ ↑(truncate γ 0 1) x = ↑γ x have : ↑x ∈ (Icc 0 1 : Set ℝ) := x.2 case a.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ✝ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b x : ↑I this : ↑x ∈ Icc 0 1 ⊢ ↑(truncate γ 0 1) x = ↑γ x rw [truncate, coe_mk_mk, max_eq_left this.1, min_eq_left this.2, extend_extends'] ** Qed
Path.reparam_id X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y ⊢ reparam γ id (_ : Continuous id) (_ : id 0 = id 0) (_ : id 1 = id 1) = γ ext case a.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y x✝ : ↑I ⊢ ↑(reparam γ id (_ : Continuous id) (_ : id 0 = id 0) (_ : id 1 = id 1)) x✝ = ↑γ x✝ rfl ** Qed
Path.range_reparam X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 ⊢ range ↑(reparam γ f hfcont hf₀ hf₁) = range ↑γ change range (γ ∘ f) = range γ X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 this : range f = univ ⊢ range (↑γ ∘ f) = range ↑γ rw [range_comp, this, image_univ] X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 ⊢ range f = univ rw [range_iff_surjective] X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 ⊢ Surjective f intro t X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 t : ↑I ⊢ ∃ a, f a = t have h₁ : Continuous (Set.IccExtend (zero_le_one' ℝ) f) := by continuity X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 t : ↑I h₁ : Continuous (IccExtend (_ : 0 ≤ 1) f) ⊢ ∃ a, f a = t have := intermediate_value_Icc (zero_le_one' ℝ) h₁.continuousOn X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 t : ↑I ⊢ Continuous (IccExtend (_ : 0 ≤ 1) f) continuity X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 t : ↑I h₁ : Continuous (IccExtend (_ : 0 ≤ 1) f) this : Icc (IccExtend (_ : 0 ≤ 1) f 0) (IccExtend (_ : 0 ≤ 1) f 1) ⊆ IccExtend (_ : 0 ≤ 1) f '' Icc 0 1 ⊢ ∃ a, f a = t rw [IccExtend_left, IccExtend_right, Icc.mk_zero, Icc.mk_one, hf₀, hf₁] at this X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 t : ↑I h₁ : Continuous (IccExtend (_ : 0 ≤ 1) f) this : Icc 0 1 ⊆ IccExtend (_ : 0 ≤ 1) f '' Icc 0 1 ⊢ ∃ a, f a = t rcases this t.2 with ⟨w, hw₁, hw₂⟩ case intro.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 t : ↑I h₁ : Continuous (IccExtend (_ : 0 ≤ 1) f) this : Icc 0 1 ⊆ IccExtend (_ : 0 ≤ 1) f '' Icc 0 1 w : ℝ hw₁ : w ∈ Icc 0 1 hw₂ : IccExtend (_ : 0 ≤ 1) f w = t ⊢ ∃ a, f a = t rw [IccExtend_of_mem _ _ hw₁] at hw₂ case intro.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 t : ↑I h₁ : Continuous (IccExtend (_ : 0 ≤ 1) f) this : Icc 0 1 ⊆ IccExtend (_ : 0 ≤ 1) f '' Icc 0 1 w : ℝ hw₁ : w ∈ Icc 0 1 hw₂ : f { val := w, property := hw₁ } = t ⊢ ∃ a, f a = t exact ⟨_, hw₂⟩ ** Qed
Path.refl_reparam X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 ⊢ reparam (refl x) f hfcont hf₀ hf₁ = refl x ext case a.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 x✝ : ↑I ⊢ ↑(reparam (refl x) f hfcont hf₀ hf₁) x✝ = ↑(refl x) x✝ simp ** Qed
JoinedIn.mem X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : JoinedIn F x y ⊢ x ∈ F ∧ y ∈ F rcases h with ⟨γ, γ_in⟩ case intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X γ : Path x y γ_in : ∀ (t : ↑I), ↑γ t ∈ F ⊢ x ∈ F ∧ y ∈ F have : γ 0 ∈ F ∧ γ 1 ∈ F := by constructor <;> apply γ_in case intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X γ : Path x y γ_in : ∀ (t : ↑I), ↑γ t ∈ F this : ↑γ 0 ∈ F ∧ ↑γ 1 ∈ F ⊢ x ∈ F ∧ y ∈ F simpa using this X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X γ : Path x y γ_in : ∀ (t : ↑I), ↑γ t ∈ F ⊢ ↑γ 0 ∈ F ∧ ↑γ 1 ∈ F constructor <;> apply γ_in ** Qed
JoinedIn.joined_subtype X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : JoinedIn F x y ⊢ Continuous fun t => { val := ↑(somePath h) t, property := (_ : ↑(somePath h) t ∈ F) } continuity X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : JoinedIn F x y ⊢ ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(somePath h) t, property := (_ : ↑(somePath h) t ∈ F) }) 0 = { val := x, property := (_ : x ∈ F) } simp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : JoinedIn F x y ⊢ ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(somePath h) t, property := (_ : ↑(somePath h) t ∈ F) }) 1 = { val := y, property := (_ : y ∈ F) } simp ** Qed
joinedIn_iff_joined X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X x_in : x ∈ F y_in : y ∈ F h : Joined { val := x, property := x_in } { val := y, property := y_in } ⊢ ∀ (t : ↑I), ↑(Path.map (Joined.somePath h) (_ : Continuous Subtype.val)) t ∈ F simp ** Qed
joinedIn_univ X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ JoinedIn univ x y ↔ Joined x y simp [JoinedIn, Joined, exists_true_iff_nonempty] ** Qed
JoinedIn.symm X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : JoinedIn F x y ⊢ JoinedIn F y x cases' h.mem with hx hy case intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : JoinedIn F x y hx : x ∈ F hy : y ∈ F ⊢ JoinedIn F y x simp_all [joinedIn_iff_joined] case intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X hx✝ : x ∈ F hy✝ : y ∈ F h : Joined { val := x, property := (_ : x ∈ F) } { val := y, property := (_ : y ∈ F) } hx : x ∈ F hy : y ∈ F ⊢ Joined { val := y, property := (_ : y ∈ F) } { val := x, property := (_ : x ∈ F) } exact h.symm ** Qed
JoinedIn.trans X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X hxy : JoinedIn F x y hyz : JoinedIn F y z ⊢ JoinedIn F x z cases' hxy.mem with hx hy case intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X hxy : JoinedIn F x y hyz : JoinedIn F y z hx : x ∈ F hy : y ∈ F ⊢ JoinedIn F x z cases' hyz.mem with hx hy case intro.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X hxy : JoinedIn F x y hyz : JoinedIn F y z hx✝ : x ∈ F hy✝ hx : y ∈ F hy : z ∈ F ⊢ JoinedIn F x z simp_all [joinedIn_iff_joined] case intro.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X hx✝¹ : x ∈ F hy✝¹ : y ∈ F hy✝ : z ∈ F hxy : Joined { val := x, property := (_ : x ∈ F) } { val := y, property := (_ : y ∈ F) } hyz : Joined { val := y, property := (_ : y ∈ F) } { val := z, property := (_ : z ∈ F) } hx✝ : x ∈ F hx : y ∈ F hy : z ∈ F ⊢ Joined { val := x, property := (_ : x ∈ F) } { val := z, property := (_ : z ∈ F) } exact hxy.trans hyz ** Qed
pathComponent_congr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : x ∈ pathComponent y ⊢ pathComponent x = pathComponent y ext z case h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z✝ : X ι : Type u_3 F : Set X h : x ∈ pathComponent y z : X ⊢ z ∈ pathComponent x ↔ z ∈ pathComponent y constructor case h.mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z✝ : X ι : Type u_3 F : Set X h : x ∈ pathComponent y z : X ⊢ z ∈ pathComponent x → z ∈ pathComponent y intro h' case h.mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z✝ : X ι : Type u_3 F : Set X h : x ∈ pathComponent y z : X h' : z ∈ pathComponent x ⊢ z ∈ pathComponent y rw [pathComponent_symm] case h.mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z✝ : X ι : Type u_3 F : Set X h : x ∈ pathComponent y z : X h' : z ∈ pathComponent x ⊢ y ∈ pathComponent z exact (h.trans h').symm case h.mpr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z✝ : X ι : Type u_3 F : Set X h : x ∈ pathComponent y z : X ⊢ z ∈ pathComponent y → z ∈ pathComponent x intro h' case h.mpr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z✝ : X ι : Type u_3 F : Set X h : x ∈ pathComponent y z : X h' : z ∈ pathComponent y ⊢ z ∈ pathComponent x rw [pathComponent_symm] at h' ⊢ case h.mpr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z✝ : X ι : Type u_3 F : Set X h : x ∈ pathComponent y z : X h' : y ∈ pathComponent z ⊢ x ∈ pathComponent z exact h'.trans h ** Qed
pathComponent_subset_component X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z : X ι : Type u_3 F : Set X x y : X h : y ∈ pathComponent x ⊢ ↑(Joined.somePath h) 0 = x simp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z : X ι : Type u_3 F : Set X x y : X h : y ∈ pathComponent x ⊢ ↑(Joined.somePath h) 1 = y simp ** Qed
pathComponentIn_univ X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X x : X ⊢ pathComponentIn x univ = pathComponent x simp [pathComponentIn, pathComponent, JoinedIn, Joined, exists_true_iff_nonempty] ** Qed
isPathConnected_iff_eq X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ IsPathConnected F ↔ ∃ x, x ∈ F ∧ pathComponentIn x F = F constructor <;> rintro ⟨x, x_in, h⟩ <;> use x, x_in case right X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X x : X x_in : x ∈ F h : ∀ {y : X}, y ∈ F → JoinedIn F x y ⊢ pathComponentIn x F = F ext y case right.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z : X ι : Type u_3 F : Set X x : X x_in : x ∈ F h : ∀ {y : X}, y ∈ F → JoinedIn F x y y : X ⊢ y ∈ pathComponentIn x F ↔ y ∈ F exact ⟨fun hy => hy.mem.2, h⟩ case right X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X x : X x_in : x ∈ F h : pathComponentIn x F = F ⊢ ∀ {y : X}, y ∈ F → JoinedIn F x y intro y y_in case right X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z : X ι : Type u_3 F : Set X x : X x_in : x ∈ F h : pathComponentIn x F = F y : X y_in : y ∈ F ⊢ JoinedIn F x y rwa [← h] at y_in ** Qed
IsPathConnected.image' X : Type u_1 Y✝ : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y✝ x y z : X ι : Type u_3 F : Set X Y : Type u_4 inst✝ : TopologicalSpace Y hF : IsPathConnected F f : X → Y hf : ContinuousOn f F ⊢ IsPathConnected (f '' F) rcases hF with ⟨x, x_in, hx⟩ case intro.intro X : Type u_1 Y✝ : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y✝ x✝ y z : X ι : Type u_3 F : Set X Y : Type u_4 inst✝ : TopologicalSpace Y f : X → Y hf : ContinuousOn f F x : X x_in : x ∈ F hx : ∀ {y : X}, y ∈ F → JoinedIn F x y ⊢ IsPathConnected (f '' F) use f x, mem_image_of_mem f x_in case right X : Type u_1 Y✝ : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y✝ x✝ y z : X ι : Type u_3 F : Set X Y : Type u_4 inst✝ : TopologicalSpace Y f : X → Y hf : ContinuousOn f F x : X x_in : x ∈ F hx : ∀ {y : X}, y ∈ F → JoinedIn F x y ⊢ ∀ {y : Y}, y ∈ f '' F → JoinedIn (f '' F) (f x) y rintro _ ⟨y, y_in, rfl⟩ case right.intro.intro X : Type u_1 Y✝ : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y✝ x✝ y✝ z : X ι : Type u_3 F : Set X Y : Type u_4 inst✝ : TopologicalSpace Y f : X → Y hf : ContinuousOn f F x : X x_in : x ∈ F hx : ∀ {y : X}, y ∈ F → JoinedIn F x y y : X y_in : y ∈ F ⊢ JoinedIn (f '' F) (f x) (f y) refine ⟨(hx y_in).somePath.map' ?_, fun t ↦ ⟨_, (hx y_in).somePath_mem t, rfl⟩⟩ case right.intro.intro X : Type u_1 Y✝ : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y✝ x✝ y✝ z : X ι : Type u_3 F : Set X Y : Type u_4 inst✝ : TopologicalSpace Y f : X → Y hf : ContinuousOn f F x : X x_in : x ∈ F hx : ∀ {y : X}, y ∈ F → JoinedIn F x y y : X y_in : y ∈ F ⊢ ContinuousOn f (range ↑(JoinedIn.somePath (_ : JoinedIn F x y))) exact hf.mono (range_subset_iff.2 (hx y_in).somePath_mem) ** Qed
isPathConnected_singleton X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X x : X ⊢ IsPathConnected {x} refine ⟨x, rfl, ?_⟩ X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X x : X ⊢ ∀ {y : X}, y ∈ {x} → JoinedIn {x} x y rintro y rfl X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y✝ z : X ι : Type u_3 F : Set X y : X ⊢ JoinedIn {y} y y exact JoinedIn.refl rfl ** Qed
IsPathConnected.union X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F U V : Set X hU : IsPathConnected U hV : IsPathConnected V hUV : Set.Nonempty (U ∩ V) ⊢ IsPathConnected (U ∪ V) rcases hUV with ⟨x, xU, xV⟩ case intro.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F U V : Set X hU : IsPathConnected U hV : IsPathConnected V x : X xU : x ∈ U xV : x ∈ V ⊢ IsPathConnected (U ∪ V) use x, Or.inl xU case right X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F U V : Set X hU : IsPathConnected U hV : IsPathConnected V x : X xU : x ∈ U xV : x ∈ V ⊢ ∀ {y : X}, y ∈ U ∪ V → JoinedIn (U ∪ V) x y rintro y (yU \| yV) case right.inl X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z : X ι : Type u_3 F U V : Set X hU : IsPathConnected U hV : IsPathConnected V x : X xU : x ∈ U xV : x ∈ V y : X yU : y ∈ U ⊢ JoinedIn (U ∪ V) x y exact (hU.joinedIn x xU y yU).mono (subset_union_left U V) case right.inr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z : X ι : Type u_3 F U V : Set X hU : IsPathConnected U hV : IsPathConnected V x : X xU : x ∈ U xV : x ∈ V y : X yV : y ∈ V ⊢ JoinedIn (U ∪ V) x y exact (hV.joinedIn x xV y yV).mono (subset_union_right U V) ** Qed
IsPathConnected.preimage_coe X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F U W : Set X hW : IsPathConnected W hWU : W ⊆ U ⊢ IsPathConnected (Subtype.val ⁻¹' W) rcases hW with ⟨x, x_in, hx⟩ case intro.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F U W : Set X hWU : W ⊆ U x : X x_in : x ∈ W hx : ∀ {y : X}, y ∈ W → JoinedIn W x y ⊢ IsPathConnected (Subtype.val ⁻¹' W) use ⟨x, hWU x_in⟩, by simp [x_in] case right X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F U W : Set X hWU : W ⊆ U x : X x_in : x ∈ W hx : ∀ {y : X}, y ∈ W → JoinedIn W x y ⊢ ∀ {y : { x // x ∈ U }}, y ∈ Subtype.val ⁻¹' W → JoinedIn (Subtype.val ⁻¹' W) { val := x, property := (_ : x ∈ U) } y rintro ⟨y, hyU⟩ hyW case right.mk X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z : X ι : Type u_3 F U W : Set X hWU : W ⊆ U x : X x_in : x ∈ W hx : ∀ {y : X}, y ∈ W → JoinedIn W x y y : X hyU : y ∈ U hyW : { val := y, property := hyU } ∈ Subtype.val ⁻¹' W ⊢ JoinedIn (Subtype.val ⁻¹' W) { val := x, property := (_ : x ∈ U) } { val := y, property := hyU } exact ⟨(hx hyW).joined_subtype.somePath.map (continuous_inclusion hWU), by simp⟩ X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F U W : Set X hWU : W ⊆ U x : X x_in : x ∈ W hx : ∀ {y : X}, y ∈ W → JoinedIn W x y ⊢ { val := x, property := (_ : x ∈ U) } ∈ Subtype.val ⁻¹' W simp [x_in] X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z : X ι : Type u_3 F U W : Set X hWU : W ⊆ U x : X x_in : x ∈ W hx : ∀ {y : X}, y ∈ W → JoinedIn W x y y : X hyU : y ∈ U hyW : { val := y, property := hyU } ∈ Subtype.val ⁻¹' W ⊢ ∀ (t : ↑I), ↑(Path.map (Joined.somePath (_ : Joined { val := x, property := (_ : x ∈ W) } { val := ↑{ val := y, property := hyU }, property := (_ : ↑{ val := y, property := hyU } ∈ W) })) (_ : Continuous (inclusion hWU))) t ∈ Subtype.val ⁻¹' W simp ** Qed
IsPathConnected.exists_path_through_family X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s ⊢ ∃ γ, range ↑γ ⊆ s ∧ ∀ (i : Fin (n + 1)), p i ∈ range ↑γ let p' : ℕ → X := fun k => if h : k < n + 1 then p ⟨k, h⟩ else p ⟨0, n.zero_lt_succ⟩ case intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s ⊢ ∃ γ, range ↑γ ⊆ s ∧ ∀ (i : Fin (n + 1)), p i ∈ range ↑γ have hpp' : ∀ k < n + 1, p k = p' k := by intro k hk simp only [hk, dif_pos] congr ext rw [Fin.val_cast_of_lt hk] case intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hpp' : ∀ (k : ℕ), k < n + 1 → p ↑k = p' k ⊢ ∃ γ, range ↑γ ⊆ s ∧ ∀ (i : Fin (n + 1)), p i ∈ range ↑γ use γ.cast (hpp' 0 n.zero_lt_succ) (hpp' n n.lt_succ_self) case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hpp' : ∀ (k : ℕ), k < n + 1 → p ↑k = p' k ⊢ range ↑(Path.cast γ (_ : p ↑0 = p' 0) (_ : p ↑n = p' n)) ⊆ s ∧ ∀ (i : Fin (n + 1)), p i ∈ range ↑(Path.cast γ (_ : p ↑0 = p' 0) (_ : p ↑n = p' n)) simp only [γ.cast_coe] case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hpp' : ∀ (k : ℕ), k < n + 1 → p ↑k = p' k ⊢ range ↑γ ⊆ s ∧ ∀ (i : Fin (n + 1)), p i ∈ range ↑γ refine' And.intro hγ.2 _ case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hpp' : ∀ (k : ℕ), k < n + 1 → p ↑k = p' k ⊢ ∀ (i : Fin (n + 1)), p i ∈ range ↑γ rintro ⟨i, hi⟩ case h.mk X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hpp' : ∀ (k : ℕ), k < n + 1 → p ↑k = p' k i : ℕ hi : i < n + 1 ⊢ p { val := i, isLt := hi } ∈ range ↑γ suffices p ⟨i, hi⟩ = p' i by convert hγ.1 i (Nat.le_of_lt_succ hi) case h.mk X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hpp' : ∀ (k : ℕ), k < n + 1 → p ↑k = p' k i : ℕ hi : i < n + 1 ⊢ p { val := i, isLt := hi } = p' i rw [← hpp' i hi] case h.mk X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hpp' : ∀ (k : ℕ), k < n + 1 → p ↑k = p' k i : ℕ hi : i < n + 1 ⊢ p { val := i, isLt := hi } = p ↑i suffices i = i % n.succ by congr case h.mk X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hpp' : ∀ (k : ℕ), k < n + 1 → p ↑k = p' k i : ℕ hi : i < n + 1 ⊢ i = i % Nat.succ n rw [Nat.mod_eq_of_lt hi] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } ⊢ ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s have hp' : ∀ i ≤ n, p' i ∈ s := by intro i hi simp [Nat.lt_succ_of_le hi, hp] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } hp' : ∀ (i : ℕ), i ≤ n → p' i ∈ s ⊢ ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s clear_value p' X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X hp' : ∀ (i : ℕ), i ≤ n → p' i ∈ s ⊢ ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s clear hp p X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp' : ∀ (i : ℕ), i ≤ n → p' i ∈ s ⊢ ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s induction' n with n hn X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } ⊢ ∀ (i : ℕ), i ≤ n → p' i ∈ s intro i hi X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } i : ℕ hi : i ≤ n ⊢ p' i ∈ s simp [Nat.lt_succ_of_le hi, hp] case zero X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n → p' i ∈ s hp' : ∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ s ⊢ ∃ γ, (∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s use Path.refl (p' 0) case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n → p' i ∈ s hp' : ∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ s ⊢ (∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ range ↑(Path.refl (p' 0))) ∧ range ↑(Path.refl (p' 0)) ⊆ s constructor case h.left X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n → p' i ∈ s hp' : ∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ s ⊢ ∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ range ↑(Path.refl (p' 0)) rintro i hi case h.left X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n → p' i ∈ s hp' : ∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ s i : ℕ hi : i ≤ Nat.zero ⊢ p' i ∈ range ↑(Path.refl (p' 0)) rw [le_zero_iff.mp hi] case h.left X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n → p' i ∈ s hp' : ∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ s i : ℕ hi : i ≤ Nat.zero ⊢ p' 0 ∈ range ↑(Path.refl (p' 0)) exact ⟨0, rfl⟩ case h.right X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n → p' i ∈ s hp' : ∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ s ⊢ range ↑(Path.refl (p' 0)) ⊆ s rw [range_subset_iff] case h.right X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n → p' i ∈ s hp' : ∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ s ⊢ ∀ (y : ↑I), ↑(Path.refl (p' 0)) y ∈ s rintro _x case h.right X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n → p' i ∈ s hp' : ∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ s _x : ↑I ⊢ ↑(Path.refl (p' 0)) _x ∈ s exact hp' 0 le_rfl case succ X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s ⊢ ∃ γ, (∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s rcases hn fun i hi => hp' i <\| Nat.le_succ_of_le hi with ⟨γ₀, hγ₀⟩ case succ.intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s ⊢ ∃ γ, (∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s rcases h.joinedIn (p' n) (hp' n n.le_succ) (p' <\| n + 1) (hp' (n + 1) <\| le_rfl) with ⟨γ₁, hγ₁⟩ case succ.intro.intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s ⊢ ∃ γ, (∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s let γ : Path (p' 0) (p' <\| n + 1) := γ₀.trans γ₁ case succ.intro.intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ ⊢ ∃ γ, (∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s use γ case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ ⊢ (∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s have range_eq : range γ = range γ₀ ∪ range γ₁ := γ₀.trans_range γ₁ case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ ⊢ (∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s constructor case h.left X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ ⊢ ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ range ↑γ rintro i hi case h.left X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ i : ℕ hi : i ≤ Nat.succ n ⊢ p' i ∈ range ↑γ by_cases hi' : i ≤ n case pos X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ i : ℕ hi : i ≤ Nat.succ n hi' : i ≤ n ⊢ p' i ∈ range ↑γ rw [range_eq] case pos X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ i : ℕ hi : i ≤ Nat.succ n hi' : i ≤ n ⊢ p' i ∈ range ↑γ₀ ∪ range ↑γ₁ left case pos.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ i : ℕ hi : i ≤ Nat.succ n hi' : i ≤ n ⊢ p' i ∈ range ↑γ₀ exact hγ₀.1 i hi' case neg X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ i : ℕ hi : i ≤ Nat.succ n hi' : ¬i ≤ n ⊢ p' i ∈ range ↑γ rw [not_le, ← Nat.succ_le_iff] at hi' case neg X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ i : ℕ hi : i ≤ Nat.succ n hi' : Nat.succ n ≤ i ⊢ p' i ∈ range ↑γ have : i = n.succ := le_antisymm hi hi' case neg X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ i : ℕ hi : i ≤ Nat.succ n hi' : Nat.succ n ≤ i this : i = Nat.succ n ⊢ p' i ∈ range ↑γ rw [this] case neg X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ i : ℕ hi : i ≤ Nat.succ n hi' : Nat.succ n ≤ i this : i = Nat.succ n ⊢ p' (Nat.succ n) ∈ range ↑γ use 1 case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ i : ℕ hi : i ≤ Nat.succ n hi' : Nat.succ n ≤ i this : i = Nat.succ n ⊢ ↑γ 1 = p' (Nat.succ n) exact γ.target case h.right X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ ⊢ range ↑γ ⊆ s rw [range_eq] case h.right X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ ⊢ range ↑γ₀ ∪ range ↑γ₁ ⊆ s apply union_subset hγ₀.2 case h.right X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ ⊢ range ↑γ₁ ⊆ s rw [range_subset_iff] case h.right X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ ⊢ ∀ (y : ↑I), ↑γ₁ y ∈ s exact hγ₁ X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s ⊢ ∀ (k : ℕ), k < n + 1 → p ↑k = p' k intro k hk X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s k : ℕ hk : k < n + 1 ⊢ p ↑k = p' k simp only [hk, dif_pos] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s k : ℕ hk : k < n + 1 ⊢ p ↑k = p { val := k, isLt := (_ : k < n + 1) } congr case e_a X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s k : ℕ hk : k < n + 1 ⊢ ↑k = { val := k, isLt := (_ : k < n + 1) } ext case e_a.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s k : ℕ hk : k < n + 1 ⊢ ↑↑k = ↑{ val := k, isLt := (_ : k < n + 1) } rw [Fin.val_cast_of_lt hk] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hpp' : ∀ (k : ℕ), k < n + 1 → p ↑k = p' k i : ℕ hi : i < n + 1 this : p { val := i, isLt := hi } = p' i ⊢ p { val := i, isLt := hi } ∈ range ↑γ convert hγ.1 i (Nat.le_of_lt_succ hi) X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hpp' : ∀ (k : ℕ), k < n + 1 → p ↑k = p' k i : ℕ hi : i < n + 1 this : i = i % Nat.succ n ⊢ p { val := i, isLt := hi } = p ↑i congr ** Qed
IsPathConnected.exists_path_through_family' X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s ⊢ ∃ γ t, (∀ (t : ↑I), ↑γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), ↑γ (t i) = p i rcases h.exists_path_through_family p hp with ⟨γ, hγ⟩ case intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s γ : Path (p 0) (p ↑n) hγ : range ↑γ ⊆ s ∧ ∀ (i : Fin (n + 1)), p i ∈ range ↑γ ⊢ ∃ γ t, (∀ (t : ↑I), ↑γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), ↑γ (t i) = p i rcases hγ with ⟨h₁, h₂⟩ case intro.intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s γ : Path (p 0) (p ↑n) h₁ : range ↑γ ⊆ s h₂ : ∀ (i : Fin (n + 1)), p i ∈ range ↑γ ⊢ ∃ γ t, (∀ (t : ↑I), ↑γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), ↑γ (t i) = p i simp only [range, mem_setOf_eq] at h₂ case intro.intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s γ : Path (p 0) (p ↑n) h₁ : range ↑γ ⊆ s h₂ : ∀ (i : Fin (n + 1)), ∃ y, ↑γ y = p i ⊢ ∃ γ t, (∀ (t : ↑I), ↑γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), ↑γ (t i) = p i rw [range_subset_iff] at h₁ case intro.intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s γ : Path (p 0) (p ↑n) h₁ : ∀ (y : ↑I), ↑γ y ∈ s h₂ : ∀ (i : Fin (n + 1)), ∃ y, ↑γ y = p i ⊢ ∃ γ t, (∀ (t : ↑I), ↑γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), ↑γ (t i) = p i choose! t ht using h₂ case intro.intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s γ : Path (p 0) (p ↑n) h₁ : ∀ (y : ↑I), ↑γ y ∈ s t : Fin (n + 1) → ↑I ht : ∀ (i : Fin (n + 1)), ↑γ (t i) = p i ⊢ ∃ γ t, (∀ (t : ↑I), ↑γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), ↑γ (t i) = p i exact ⟨γ, t, h₁, ht⟩ ** Qed
pathConnectedSpace_iff_zerothHomotopy X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ PathConnectedSpace X ↔ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) letI := pathSetoid X X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X this : Setoid X := pathSetoid X ⊢ PathConnectedSpace X ↔ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) constructor case mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X this : Setoid X := pathSetoid X ⊢ PathConnectedSpace X → Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) intro h case mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X this : Setoid X := pathSetoid X h : PathConnectedSpace X ⊢ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) refine' ⟨(nonempty_quotient_iff _).mpr h.1, ⟨_⟩⟩ case mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X this : Setoid X := pathSetoid X h : PathConnectedSpace X ⊢ ∀ (a b : ZerothHomotopy X), a = b rintro ⟨x⟩ ⟨y⟩ case mp.mk.mk X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z : X ι : Type u_3 F : Set X this : Setoid X := pathSetoid X h : PathConnectedSpace X a✝ : ZerothHomotopy X x : X b✝ : ZerothHomotopy X y : X ⊢ Quot.mk Setoid.r x = Quot.mk Setoid.r y exact Quotient.sound (PathConnectedSpace.Joined x y) case mpr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X this : Setoid X := pathSetoid X ⊢ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) → PathConnectedSpace X unfold ZerothHomotopy case mpr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X this : Setoid X := pathSetoid X ⊢ Nonempty (Quotient (pathSetoid X)) ∧ Subsingleton (Quotient (pathSetoid X)) → PathConnectedSpace X rintro ⟨h, h'⟩ case mpr.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X this : Setoid X := pathSetoid X h : Nonempty (Quotient (pathSetoid X)) h' : Subsingleton (Quotient (pathSetoid X)) ⊢ PathConnectedSpace X exact ⟨(nonempty_quotient_iff _).mp h, fun x y => Quotient.exact <\| Subsingleton.elim ⟦x⟧ ⟦y⟧⟩ ** Qed
isPathConnected_iff_pathConnectedSpace X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ IsPathConnected F ↔ PathConnectedSpace ↑F rw [isPathConnected_iff] X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ (Set.Nonempty F ∧ ∀ (x : X), x ∈ F → ∀ (y : X), y ∈ F → JoinedIn F x y) ↔ PathConnectedSpace ↑F constructor case mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ (Set.Nonempty F ∧ ∀ (x : X), x ∈ F → ∀ (y : X), y ∈ F → JoinedIn F x y) → PathConnectedSpace ↑F rintro ⟨⟨x, x_in⟩, h⟩ case mp.intro.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X h : ∀ (x : X), x ∈ F → ∀ (y : X), y ∈ F → JoinedIn F x y x : X x_in : x ∈ F ⊢ PathConnectedSpace ↑F refine' ⟨⟨⟨x, x_in⟩⟩, _⟩ case mp.intro.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X h : ∀ (x : X), x ∈ F → ∀ (y : X), y ∈ F → JoinedIn F x y x : X x_in : x ∈ F ⊢ ∀ (x y : ↑F), Joined x y rintro ⟨y, y_in⟩ ⟨z, z_in⟩ case mp.intro.intro.mk.mk X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z✝ : X ι : Type u_3 F : Set X h : ∀ (x : X), x ∈ F → ∀ (y : X), y ∈ F → JoinedIn F x y x : X x_in : x ∈ F y : X y_in : y ∈ F z : X z_in : z ∈ F ⊢ Joined { val := y, property := y_in } { val := z, property := z_in } have H := h y y_in z z_in case mp.intro.intro.mk.mk X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z✝ : X ι : Type u_3 F : Set X h : ∀ (x : X), x ∈ F → ∀ (y : X), y ∈ F → JoinedIn F x y x : X x_in : x ∈ F y : X y_in : y ∈ F z : X z_in : z ∈ F H : JoinedIn F y z ⊢ Joined { val := y, property := y_in } { val := z, property := z_in } rwa [joinedIn_iff_joined y_in z_in] at H case mpr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ PathConnectedSpace ↑F → Set.Nonempty F ∧ ∀ (x : X), x ∈ F → ∀ (y : X), y ∈ F → JoinedIn F x y rintro ⟨⟨x, x_in⟩, H⟩ case mpr.mk.intro.mk X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X H : ∀ (x y : ↑F), Joined x y x : X x_in : x ∈ F ⊢ Set.Nonempty F ∧ ∀ (x : X), x ∈ F → ∀ (y : X), y ∈ F → JoinedIn F x y refine' ⟨⟨x, x_in⟩, fun y y_in z z_in => _⟩ case mpr.mk.intro.mk X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z✝ : X ι : Type u_3 F : Set X H : ∀ (x y : ↑F), Joined x y x : X x_in : x ∈ F y : X y_in : y ∈ F z : X z_in : z ∈ F ⊢ JoinedIn F y z rw [joinedIn_iff_joined y_in z_in] case mpr.mk.intro.mk X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z✝ : X ι : Type u_3 F : Set X H : ∀ (x y : ↑F), Joined x y x : X x_in : x ∈ F y : X y_in : y ∈ F z : X z_in : z ∈ F ⊢ Joined { val := y, property := y_in } { val := z, property := z_in } apply H ** Qed
pathConnectedSpace_iff_univ X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ PathConnectedSpace X ↔ IsPathConnected univ constructor case mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ PathConnectedSpace X → IsPathConnected univ intro h case mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : PathConnectedSpace X ⊢ IsPathConnected univ haveI := @PathConnectedSpace.Nonempty X _ _ case mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : PathConnectedSpace X this : Nonempty X ⊢ IsPathConnected univ inhabit X case mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : PathConnectedSpace X this : Nonempty X inhabited_h : Inhabited X ⊢ IsPathConnected univ refine' ⟨default, mem_univ _, _⟩ case mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : PathConnectedSpace X this : Nonempty X inhabited_h : Inhabited X ⊢ ∀ {y : X}, y ∈ univ → JoinedIn univ default y intros y _hy case mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y✝ z : X ι : Type u_3 F : Set X h : PathConnectedSpace X this : Nonempty X inhabited_h : Inhabited X y : X _hy : y ∈ univ ⊢ JoinedIn univ default y simpa using PathConnectedSpace.Joined default y case mpr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ IsPathConnected univ → PathConnectedSpace X intro h case mpr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : IsPathConnected univ ⊢ PathConnectedSpace X have h' := h.joinedIn case mpr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : IsPathConnected univ h' : ∀ (x : X), x ∈ univ → ∀ (y : X), y ∈ univ → JoinedIn univ x y ⊢ PathConnectedSpace X cases' h with x h case mpr.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X h' : ∀ (x : X), x ∈ univ → ∀ (y : X), y ∈ univ → JoinedIn univ x y x : X h : x ∈ univ ∧ ∀ {y : X}, y ∈ univ → JoinedIn univ x y ⊢ PathConnectedSpace X exact ⟨⟨x⟩, by simpa using h'⟩ X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X h' : ∀ (x : X), x ∈ univ → ∀ (y : X), y ∈ univ → JoinedIn univ x y x : X h : x ∈ univ ∧ ∀ {y : X}, y ∈ univ → JoinedIn univ x y ⊢ ∀ (x y : X), Joined x y simpa using h' ** Qed
isPathConnected_range X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X f : X → Y hf : Continuous f ⊢ IsPathConnected (range f) rw [← image_univ] X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X f : X → Y hf : Continuous f ⊢ IsPathConnected (f '' univ) exact isPathConnected_univ.image hf ** Qed
Function.Surjective.pathConnectedSpace X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X f : X → Y hf : Surjective f hf' : Continuous f ⊢ PathConnectedSpace Y rw [pathConnectedSpace_iff_univ, ← hf.range_eq] X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X f : X → Y hf : Surjective f hf' : Continuous f ⊢ IsPathConnected (range f) exact isPathConnected_range hf' ** Qed
pathConnectedSpace_iff_eq X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ PathConnectedSpace X ↔ ∃ x, pathComponent x = univ simp [pathConnectedSpace_iff_univ, isPathConnected_iff_eq] ** Qed
IsPathConnected.isConnected X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X hF : IsPathConnected F ⊢ IsConnected F rw [isConnected_iff_connectedSpace] X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X hF : IsPathConnected F ⊢ ConnectedSpace ↑F rw [isPathConnected_iff_pathConnectedSpace] at hF X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X hF : PathConnectedSpace ↑F ⊢ ConnectedSpace ↑F exact @PathConnectedSpace.connectedSpace _ _ hF ** Qed
PathConnectedSpace.exists_path_through_family X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X n : ℕ p : Fin (n + 1) → X ⊢ ∃ γ, ∀ (i : Fin (n + 1)), p i ∈ range ↑γ have : IsPathConnected (univ : Set X) := pathConnectedSpace_iff_univ.mp (by infer_instance) X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X n : ℕ p : Fin (n + 1) → X this : IsPathConnected univ ⊢ ∃ γ, ∀ (i : Fin (n + 1)), p i ∈ range ↑γ rcases this.exists_path_through_family p fun _i => True.intro with ⟨γ, -, h⟩ case intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X n : ℕ p : Fin (n + 1) → X this : IsPathConnected univ γ : Path (p 0) (p ↑n) h : ∀ (i : Fin (n + 1)), p i ∈ range ↑γ ⊢ ∃ γ, ∀ (i : Fin (n + 1)), p i ∈ range ↑γ exact ⟨γ, h⟩ X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X n : ℕ p : Fin (n + 1) → X ⊢ PathConnectedSpace X infer_instance ** Qed
PathConnectedSpace.exists_path_through_family' X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X n : ℕ p : Fin (n + 1) → X ⊢ ∃ γ t, ∀ (i : Fin (n + 1)), ↑γ (t i) = p i have : IsPathConnected (univ : Set X) := pathConnectedSpace_iff_univ.mp (by infer_instance) X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X n : ℕ p : Fin (n + 1) → X this : IsPathConnected univ ⊢ ∃ γ t, ∀ (i : Fin (n + 1)), ↑γ (t i) = p i rcases this.exists_path_through_family' p fun _i => True.intro with ⟨γ, t, -, h⟩ case intro.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X n : ℕ p : Fin (n + 1) → X this : IsPathConnected univ γ : Path (p 0) (p ↑n) t : Fin (n + 1) → ↑I h : ∀ (i : Fin (n + 1)), ↑γ (t i) = p i ⊢ ∃ γ t, ∀ (i : Fin (n + 1)), ↑γ (t i) = p i exact ⟨γ, t, h⟩ X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X n : ℕ p : Fin (n + 1) → X ⊢ PathConnectedSpace X infer_instance ** Qed
locPathConnected_of_bases X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X p : ι → Prop s : X → ι → Set X h : ∀ (x : X), HasBasis (𝓝 x) p (s x) h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i) ⊢ LocPathConnectedSpace X constructor case path_connected_basis X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X p : ι → Prop s : X → ι → Set X h : ∀ (x : X), HasBasis (𝓝 x) p (s x) h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i) ⊢ ∀ (x : X), HasBasis (𝓝 x) (fun s => s ∈ 𝓝 x ∧ IsPathConnected s) id intro x case path_connected_basis X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X p : ι → Prop s : X → ι → Set X h : ∀ (x : X), HasBasis (𝓝 x) p (s x) h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i) x : X ⊢ HasBasis (𝓝 x) (fun s => s ∈ 𝓝 x ∧ IsPathConnected s) id apply (h x).to_hasBasis case path_connected_basis.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X p : ι → Prop s : X → ι → Set X h : ∀ (x : X), HasBasis (𝓝 x) p (s x) h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i) x : X ⊢ ∀ (i : ι), p i → ∃ i', (i' ∈ 𝓝 x ∧ IsPathConnected i') ∧ id i' ⊆ s x i intro i pi case path_connected_basis.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X p : ι → Prop s : X → ι → Set X h : ∀ (x : X), HasBasis (𝓝 x) p (s x) h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i) x : X i : ι pi : p i ⊢ ∃ i', (i' ∈ 𝓝 x ∧ IsPathConnected i') ∧ id i' ⊆ s x i exact ⟨s x i, ⟨(h x).mem_of_mem pi, h' x i pi⟩, by rfl⟩ X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X p : ι → Prop s : X → ι → Set X h : ∀ (x : X), HasBasis (𝓝 x) p (s x) h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i) x : X i : ι pi : p i ⊢ id (s x i) ⊆ s x i rfl case path_connected_basis.h' X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X p : ι → Prop s : X → ι → Set X h : ∀ (x : X), HasBasis (𝓝 x) p (s x) h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i) x : X ⊢ ∀ (i' : Set X), i' ∈ 𝓝 x ∧ IsPathConnected i' → ∃ i, p i ∧ s x i ⊆ id i' rintro U ⟨U_in, _hU⟩ case path_connected_basis.h'.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X p : ι → Prop s : X → ι → Set X h : ∀ (x : X), HasBasis (𝓝 x) p (s x) h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i) x : X U : Set X U_in : U ∈ 𝓝 x _hU : IsPathConnected U ⊢ ∃ i, p i ∧ s x i ⊆ id U rcases (h x).mem_iff.mp U_in with ⟨i, pi, hi⟩ case path_connected_basis.h'.intro.intro.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X p : ι → Prop s : X → ι → Set X h : ∀ (x : X), HasBasis (𝓝 x) p (s x) h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i) x : X U : Set X U_in : U ∈ 𝓝 x _hU : IsPathConnected U i : ι pi : p i hi : s x i ⊆ U ⊢ ∃ i, p i ∧ s x i ⊆ id U tauto ** Qed
pathConnectedSpace_iff_connectedSpace X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X ⊢ PathConnectedSpace X ↔ ConnectedSpace X constructor case mp X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X ⊢ PathConnectedSpace X → ConnectedSpace X intro h case mp X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X h : PathConnectedSpace X ⊢ ConnectedSpace X infer_instance case mpr X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X ⊢ ConnectedSpace X → PathConnectedSpace X intro hX case mpr X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X ⊢ PathConnectedSpace X rw [pathConnectedSpace_iff_eq] case mpr X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X ⊢ ∃ x, pathComponent x = univ use Classical.arbitrary X case h X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X ⊢ pathComponent (Classical.arbitrary X) = univ refine' IsClopen.eq_univ ⟨_, _⟩ (by simp) X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X ⊢ Set.Nonempty (pathComponent (Classical.arbitrary X)) simp case h.refine'_1 X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X ⊢ IsOpen (pathComponent (Classical.arbitrary X)) rw [isOpen_iff_mem_nhds] case h.refine'_1 X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X ⊢ ∀ (a : X), a ∈ pathComponent (Classical.arbitrary X) → pathComponent (Classical.arbitrary X) ∈ 𝓝 a intro y y_in case h.refine'_1 X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y✝ z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X y : X y_in : y ∈ pathComponent (Classical.arbitrary X) ⊢ pathComponent (Classical.arbitrary X) ∈ 𝓝 y rcases (path_connected_basis y).ex_mem with ⟨U, ⟨U_in, hU⟩⟩ case h.refine'_1.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y✝ z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X y : X y_in : y ∈ pathComponent (Classical.arbitrary X) U : Set X U_in : U ∈ 𝓝 y hU : IsPathConnected U ⊢ pathComponent (Classical.arbitrary X) ∈ 𝓝 y apply mem_of_superset U_in case h.refine'_1.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y✝ z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X y : X y_in : y ∈ pathComponent (Classical.arbitrary X) U : Set X U_in : U ∈ 𝓝 y hU : IsPathConnected U ⊢ U ⊆ pathComponent (Classical.arbitrary X) rw [← pathComponent_congr y_in] case h.refine'_1.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y✝ z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X y : X y_in : y ∈ pathComponent (Classical.arbitrary X) U : Set X U_in : U ∈ 𝓝 y hU : IsPathConnected U ⊢ U ⊆ pathComponent y exact hU.subset_pathComponent (mem_of_mem_nhds U_in) case h.refine'_2 X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X ⊢ IsClosed (pathComponent (Classical.arbitrary X)) rw [isClosed_iff_nhds] case h.refine'_2 X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X ⊢ ∀ (x : X), (∀ (U : Set X), U ∈ 𝓝 x → Set.Nonempty (U ∩ pathComponent (Classical.arbitrary X))) → x ∈ pathComponent (Classical.arbitrary X) intro y H case h.refine'_2 X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y✝ z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X y : X H : ∀ (U : Set X), U ∈ 𝓝 y → Set.Nonempty (U ∩ pathComponent (Classical.arbitrary X)) ⊢ y ∈ pathComponent (Classical.arbitrary X) rcases (path_connected_basis y).ex_mem with ⟨U, ⟨U_in, hU⟩⟩ case h.refine'_2.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y✝ z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X y : X H : ∀ (U : Set X), U ∈ 𝓝 y → Set.Nonempty (U ∩ pathComponent (Classical.arbitrary X)) U : Set X U_in : U ∈ 𝓝 y hU : IsPathConnected U ⊢ y ∈ pathComponent (Classical.arbitrary X) rcases H U U_in with ⟨z, hz, hz'⟩ case h.refine'_2.intro.intro.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y✝ z✝ : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X y : X H : ∀ (U : Set X), U ∈ 𝓝 y → Set.Nonempty (U ∩ pathComponent (Classical.arbitrary X)) U : Set X U_in : U ∈ 𝓝 y hU : IsPathConnected U z : X hz : z ∈ U hz' : z ∈ pathComponent (Classical.arbitrary X) ⊢ y ∈ pathComponent (Classical.arbitrary X) exact (hU.joinedIn z hz y <\| mem_of_mem_nhds U_in).joined.mem_pathComponent hz' ** Qed
locPathConnected_of_isOpen X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U ⊢ ∀ (x : ↑U), HasBasis (𝓝 x) (fun s => s ∈ 𝓝 x ∧ IsPathConnected s) id rintro ⟨x, x_in⟩ case mk X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x : X x_in : x ∈ U ⊢ HasBasis (𝓝 { val := x, property := x_in }) (fun s => s ∈ 𝓝 { val := x, property := x_in } ∧ IsPathConnected s) id rw [nhds_subtype_eq_comap] case mk X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x : X x_in : x ∈ U ⊢ HasBasis (comap Subtype.val (𝓝 x)) (fun s => s ∈ comap Subtype.val (𝓝 x) ∧ IsPathConnected s) id constructor case mk.mem_iff' X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x : X x_in : x ∈ U ⊢ ∀ (t : Set ↑U), t ∈ comap Subtype.val (𝓝 x) ↔ ∃ i, (i ∈ comap Subtype.val (𝓝 x) ∧ IsPathConnected i) ∧ id i ⊆ t intro V case mk.mem_iff' X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x : X x_in : x ∈ U V : Set ↑U ⊢ V ∈ comap Subtype.val (𝓝 x) ↔ ∃ i, (i ∈ comap Subtype.val (𝓝 x) ∧ IsPathConnected i) ∧ id i ⊆ V rw [(HasBasis.comap ((↑) : U → X) (pathConnected_subset_basis h x_in)).mem_iff] case mk.mem_iff' X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x : X x_in : x ∈ U V : Set ↑U ⊢ (∃ i, (i ∈ 𝓝 x ∧ IsPathConnected i ∧ i ⊆ U) ∧ Subtype.val ⁻¹' id i ⊆ V) ↔ ∃ i, (i ∈ comap Subtype.val (𝓝 x) ∧ IsPathConnected i) ∧ id i ⊆ V constructor case mk.mem_iff'.mp X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x : X x_in : x ∈ U V : Set ↑U ⊢ (∃ i, (i ∈ 𝓝 x ∧ IsPathConnected i ∧ i ⊆ U) ∧ Subtype.val ⁻¹' id i ⊆ V) → ∃ i, (i ∈ comap Subtype.val (𝓝 x) ∧ IsPathConnected i) ∧ id i ⊆ V rintro ⟨W, ⟨W_in, hW, hWU⟩, hWV⟩ case mk.mem_iff'.mp.intro.intro.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x : X x_in : x ∈ U V : Set ↑U W : Set X hWV : Subtype.val ⁻¹' id W ⊆ V W_in : W ∈ 𝓝 x hW : IsPathConnected W hWU : W ⊆ U ⊢ ∃ i, (i ∈ comap Subtype.val (𝓝 x) ∧ IsPathConnected i) ∧ id i ⊆ V exact ⟨Subtype.val ⁻¹' W, ⟨⟨preimage_mem_comap W_in, hW.preimage_coe hWU⟩, hWV⟩⟩ case mk.mem_iff'.mpr X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x : X x_in : x ∈ U V : Set ↑U ⊢ (∃ i, (i ∈ comap Subtype.val (𝓝 x) ∧ IsPathConnected i) ∧ id i ⊆ V) → ∃ i, (i ∈ 𝓝 x ∧ IsPathConnected i ∧ i ⊆ U) ∧ Subtype.val ⁻¹' id i ⊆ V rintro ⟨W, ⟨W_in, hW⟩, hWV⟩ case mk.mem_iff'.mpr.intro.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x : X x_in : x ∈ U V W : Set ↑U hWV : id W ⊆ V W_in : W ∈ comap Subtype.val (𝓝 x) hW : IsPathConnected W ⊢ ∃ i, (i ∈ 𝓝 x ∧ IsPathConnected i ∧ i ⊆ U) ∧ Subtype.val ⁻¹' id i ⊆ V refine' ⟨(↑) '' W, ⟨Filter.image_coe_mem_of_mem_comap (IsOpen.mem_nhds h x_in) W_in, hW.image continuous_subtype_val, Subtype.coe_image_subset U W⟩, _⟩ case mk.mem_iff'.mpr.intro.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x : X x_in : x ∈ U V W : Set ↑U hWV : id W ⊆ V W_in : W ∈ comap Subtype.val (𝓝 x) hW : IsPathConnected W ⊢ Subtype.val ⁻¹' id (Subtype.val '' W) ⊆ V rintro x ⟨y, ⟨y_in, hy⟩⟩ case mk.mem_iff'.mpr.intro.intro.intro.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝¹ y✝ z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x✝ : X x_in : x✝ ∈ U V W : Set ↑U hWV : id W ⊆ V W_in : W ∈ comap Subtype.val (𝓝 x✝) hW : IsPathConnected W x : { x // x ∈ U } y : ↑U y_in : y ∈ W hy : ↑y = ↑x ⊢ x ∈ V rw [← Subtype.coe_injective hy] case mk.mem_iff'.mpr.intro.intro.intro.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝¹ y✝ z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x✝ : X x_in : x✝ ∈ U V W : Set ↑U hWV : id W ⊆ V W_in : W ∈ comap Subtype.val (𝓝 x✝) hW : IsPathConnected W x : { x // x ∈ U } y : ↑U y_in : y ∈ W hy : ↑y = ↑x ⊢ y ∈ V tauto ** Qed
IsOpen.isConnected_iff_isPathConnected X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X U_op : IsOpen U ⊢ IsPathConnected U ↔ IsConnected U rw [isConnected_iff_connectedSpace, isPathConnected_iff_pathConnectedSpace] X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X U_op : IsOpen U ⊢ PathConnectedSpace ↑U ↔ ConnectedSpace ↑U haveI := locPathConnected_of_isOpen U_op X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X U_op : IsOpen U this : LocPathConnectedSpace ↑U ⊢ PathConnectedSpace ↑U ↔ ConnectedSpace ↑U exact pathConnectedSpace_iff_connectedSpace ** Qed
isPreconnected_of_forall α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t ⊢ IsPreconnected s rintro u v hu hv hs ⟨z, zs, zu⟩ ⟨y, ys, yv⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) have xs : x ∈ s := by rcases H y ys with ⟨t, ts, xt, -, -⟩ exact ts xt case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s ⊢ Set.Nonempty (s ∩ (u ∩ v)) cases hs xs case intro.intro.intro.intro.inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s h✝ : x ∈ u ⊢ Set.Nonempty (s ∩ (u ∩ v)) case intro.intro.intro.intro.inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s h✝ : x ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) case inl xu => rcases H y ys with ⟨t, ts, xt, yt, ht⟩ have := ht u v hu hv (ts.trans hs) ⟨x, xt, xu⟩ ⟨y, yt, yv⟩ exact this.imp fun z hz => ⟨ts hz.1, hz.2⟩ case intro.intro.intro.intro.inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s h✝ : x ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) case inr xv => rcases H z zs with ⟨t, ts, xt, zt, ht⟩ have := ht v u hv hu (ts.trans <\| by rwa [union_comm]) ⟨x, xt, xv⟩ ⟨z, zt, zu⟩ exact this.imp fun _ h => ⟨ts h.1, h.2.2, h.2.1⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v ⊢ x ∈ s rcases H y ys with ⟨t, ts, xt, -, -⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v t : Set α ts : t ⊆ s xt : x ∈ t ⊢ x ∈ s exact ts xt α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s xu : x ∈ u ⊢ Set.Nonempty (s ∩ (u ∩ v)) rcases H y ys with ⟨t, ts, xt, yt, ht⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s xu : x ∈ u t : Set α ts : t ⊆ s xt : x ∈ t yt : y ∈ t ht : IsPreconnected t ⊢ Set.Nonempty (s ∩ (u ∩ v)) have := ht u v hu hv (ts.trans hs) ⟨x, xt, xu⟩ ⟨y, yt, yv⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s xu : x ∈ u t : Set α ts : t ⊆ s xt : x ∈ t yt : y ∈ t ht : IsPreconnected t this : Set.Nonempty (t ∩ (u ∩ v)) ⊢ Set.Nonempty (s ∩ (u ∩ v)) exact this.imp fun z hz => ⟨ts hz.1, hz.2⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s xv : x ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) rcases H z zs with ⟨t, ts, xt, zt, ht⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s xv : x ∈ v t : Set α ts : t ⊆ s xt : x ∈ t zt : z ∈ t ht : IsPreconnected t ⊢ Set.Nonempty (s ∩ (u ∩ v)) have := ht v u hv hu (ts.trans <\| by rwa [union_comm]) ⟨x, xt, xv⟩ ⟨z, zt, zu⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s xv : x ∈ v t : Set α ts : t ⊆ s xt : x ∈ t zt : z ∈ t ht : IsPreconnected t this : Set.Nonempty (t ∩ (v ∩ u)) ⊢ Set.Nonempty (s ∩ (u ∩ v)) exact this.imp fun _ h => ⟨ts h.1, h.2.2, h.2.1⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s xv : x ∈ v t : Set α ts : t ⊆ s xt : x ∈ t zt : z ∈ t ht : IsPreconnected t ⊢ s ⊆ v ∪ u rwa [union_comm] ** Qed
isPreconnected_of_forall_pair α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α H : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t ⊢ IsPreconnected s rcases eq_empty_or_nonempty s with (rfl \| ⟨x, hx⟩) case inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α H : ∀ (x : α), x ∈ ∅ → ∀ (y : α), y ∈ ∅ → ∃ t, t ⊆ ∅ ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t ⊢ IsPreconnected ∅ case inr.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α H : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t x : α hx : x ∈ s ⊢ IsPreconnected s exacts [isPreconnected_empty, isPreconnected_of_forall x fun y => H x hx y] ** Qed
isPreconnected_sUnion α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α c : Set (Set α) H1 : ∀ (s : Set α), s ∈ c → x ∈ s H2 : ∀ (s : Set α), s ∈ c → IsPreconnected s ⊢ IsPreconnected (⋃₀ c) apply isPreconnected_of_forall x α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α c : Set (Set α) H1 : ∀ (s : Set α), s ∈ c → x ∈ s H2 : ∀ (s : Set α), s ∈ c → IsPreconnected s ⊢ ∀ (y : α), y ∈ ⋃₀ c → ∃ t, t ⊆ ⋃₀ c ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t rintro y ⟨s, sc, ys⟩ case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v : Set α x : α c : Set (Set α) H1 : ∀ (s : Set α), s ∈ c → x ∈ s H2 : ∀ (s : Set α), s ∈ c → IsPreconnected s y : α s : Set α sc : s ∈ c ys : y ∈ s ⊢ ∃ t, t ⊆ ⋃₀ c ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t exact ⟨s, subset_sUnion_of_mem sc, H1 s sc, ys, H2 s sc⟩ ** Qed
IsPreconnected.union α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α x : α s t : Set α H1 : x ∈ s H2 : x ∈ t H3 : IsPreconnected s H4 : IsPreconnected t ⊢ ∀ (s_1 : Set α), s_1 ∈ {s, t} → x ∈ s_1 rintro r (rfl \| rfl \| h) <;> assumption α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α x : α s t : Set α H1 : x ∈ s H2 : x ∈ t H3 : IsPreconnected s H4 : IsPreconnected t ⊢ ∀ (s_1 : Set α), s_1 ∈ {s, t} → IsPreconnected s_1 rintro r (rfl \| rfl \| h) <;> assumption ** Qed
IsPreconnected.union' α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s t : Set α H : Set.Nonempty (s ∩ t) hs : IsPreconnected s ht : IsPreconnected t ⊢ IsPreconnected (s ∪ t) rcases H with ⟨x, hxs, hxt⟩ case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s t : Set α hs : IsPreconnected s ht : IsPreconnected t x : α hxs : x ∈ s hxt : x ∈ t ⊢ IsPreconnected (s ∪ t) exact hs.union x hxs hxt ht ** Qed
IsConnected.union α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s t : Set α H : Set.Nonempty (s ∩ t) Hs : IsConnected s Ht : IsConnected t ⊢ IsConnected (s ∪ t) rcases H with ⟨x, hx⟩ case intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s t : Set α Hs : IsConnected s Ht : IsConnected t x : α hx : x ∈ s ∩ t ⊢ IsConnected (s ∪ t) refine' ⟨⟨x, mem_union_left t (mem_of_mem_inter_left hx)⟩, _⟩ case intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s t : Set α Hs : IsConnected s Ht : IsConnected t x : α hx : x ∈ s ∩ t ⊢ IsPreconnected (s ∪ t) exact Hs.isPreconnected.union x (mem_of_mem_inter_left hx) (mem_of_mem_inter_right hx) Ht.isPreconnected ** Qed
IsPreconnected.sUnion_directed α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α S : Set (Set α) K : DirectedOn (fun x x_1 => x ⊆ x_1) S H : ∀ (s : Set α), s ∈ S → IsPreconnected s ⊢ IsPreconnected (⋃₀ S) rintro u v hu hv Huv ⟨a, ⟨s, hsS, has⟩, hau⟩ ⟨b, ⟨t, htS, hbt⟩, hbv⟩ 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✝ : TopologicalSpace α s✝ t✝ u✝ v✝ : Set α S : Set (Set α) K : DirectedOn (fun x x_1 => x ⊆ x_1) S H : ∀ (s : Set α), s ∈ S → IsPreconnected s u v : Set α hu : IsOpen u hv : IsOpen v Huv : ⋃₀ S ⊆ u ∪ v a : α hau : a ∈ u s : Set α hsS : s ∈ S has : a ∈ s b : α hbv : b ∈ v t : Set α htS : t ∈ S hbt : b ∈ t ⊢ Set.Nonempty (⋃₀ S ∩ (u ∩ v)) obtain ⟨r, hrS, hsr, htr⟩ : ∃ r ∈ S, s ⊆ r ∧ t ⊆ r := K s hsS t htS case intro.intro.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✝ : TopologicalSpace α s✝ t✝ u✝ v✝ : Set α S : Set (Set α) K : DirectedOn (fun x x_1 => x ⊆ x_1) S H : ∀ (s : Set α), s ∈ S → IsPreconnected s u v : Set α hu : IsOpen u hv : IsOpen v Huv : ⋃₀ S ⊆ u ∪ v a : α hau : a ∈ u s : Set α hsS : s ∈ S has : a ∈ s b : α hbv : b ∈ v t : Set α htS : t ∈ S hbt : b ∈ t r : Set α hrS : r ∈ S hsr : s ⊆ r htr : t ⊆ r ⊢ Set.Nonempty (⋃₀ S ∩ (u ∩ v)) have Hnuv : (r ∩ (u ∩ v)).Nonempty := H _ hrS u v hu hv ((subset_sUnion_of_mem hrS).trans Huv) ⟨a, hsr has, hau⟩ ⟨b, htr hbt, hbv⟩ case intro.intro.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✝ : TopologicalSpace α s✝ t✝ u✝ v✝ : Set α S : Set (Set α) K : DirectedOn (fun x x_1 => x ⊆ x_1) S H : ∀ (s : Set α), s ∈ S → IsPreconnected s u v : Set α hu : IsOpen u hv : IsOpen v Huv : ⋃₀ S ⊆ u ∪ v a : α hau : a ∈ u s : Set α hsS : s ∈ S has : a ∈ s b : α hbv : b ∈ v t : Set α htS : t ∈ S hbt : b ∈ t r : Set α hrS : r ∈ S hsr : s ⊆ r htr : t ⊆ r Hnuv : Set.Nonempty (r ∩ (u ∩ v)) ⊢ Set.Nonempty (⋃₀ S ∩ (u ∩ v)) have Kruv : r ∩ (u ∩ v) ⊆ ⋃₀ S ∩ (u ∩ v) := inter_subset_inter_left _ (subset_sUnion_of_mem hrS) case intro.intro.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✝ : TopologicalSpace α s✝ t✝ u✝ v✝ : Set α S : Set (Set α) K : DirectedOn (fun x x_1 => x ⊆ x_1) S H : ∀ (s : Set α), s ∈ S → IsPreconnected s u v : Set α hu : IsOpen u hv : IsOpen v Huv : ⋃₀ S ⊆ u ∪ v a : α hau : a ∈ u s : Set α hsS : s ∈ S has : a ∈ s b : α hbv : b ∈ v t : Set α htS : t ∈ S hbt : b ∈ t r : Set α hrS : r ∈ S hsr : s ⊆ r htr : t ⊆ r Hnuv : Set.Nonempty (r ∩ (u ∩ v)) Kruv : r ∩ (u ∩ v) ⊆ ⋃₀ S ∩ (u ∩ v) ⊢ Set.Nonempty (⋃₀ S ∩ (u ∩ v)) exact Hnuv.mono Kruv ** Qed
IsPreconnected.biUnion_of_reflTransGen α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j ⊢ IsPreconnected (⋃ n ∈ t, s n) let R := fun i j : ι => (s i ∩ s j).Nonempty ∧ i ∈ t α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t ⊢ IsPreconnected (⋃ n ∈ t, s n) have P : ∀ i, i ∈ t → ∀ j, j ∈ t → ReflTransGen R i j → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) := fun i hi j hj h => by induction h case refl => refine ⟨{i}, singleton_subset_iff.mpr hi, mem_singleton i, mem_singleton i, ?_⟩ rw [biUnion_singleton] exact H i hi case tail j k _ hjk ih => obtain ⟨p, hpt, hip, hjp, hp⟩ := ih hjk.2 refine ⟨insert k p, insert_subset_iff.mpr ⟨hj, hpt⟩, mem_insert_of_mem k hip, mem_insert k p, ?_⟩ rw [biUnion_insert] refine (H k hj).union' (hjk.1.mono ?_) hp rw [inter_comm] exact inter_subset_inter_right _ (subset_biUnion_of_mem hjp) α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t P : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen R i j → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) ⊢ IsPreconnected (⋃ n ∈ t, s n) refine' isPreconnected_of_forall_pair _ α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t P : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen R i j → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) ⊢ ∀ (x : α), x ∈ ⋃ n ∈ t, s n → ∀ (y : α), y ∈ ⋃ n ∈ t, s n → ∃ t_1, t_1 ⊆ ⋃ n ∈ t, s n ∧ x ∈ t_1 ∧ y ∈ t_1 ∧ IsPreconnected t_1 intro x hx y hy α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t P : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen R i j → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) x : α hx : x ∈ ⋃ n ∈ t, s n y : α hy : y ∈ ⋃ n ∈ t, s n ⊢ ∃ t_1, t_1 ⊆ ⋃ n ∈ t, s n ∧ x ∈ t_1 ∧ y ∈ t_1 ∧ IsPreconnected t_1 obtain ⟨i : ι, hi : i ∈ t, hxi : x ∈ s i⟩ := mem_iUnion₂.1 hx case intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t P : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen R i j → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) x : α hx : x ∈ ⋃ n ∈ t, s n y : α hy : y ∈ ⋃ n ∈ t, s n i : ι hi : i ∈ t hxi : x ∈ s i ⊢ ∃ t_1, t_1 ⊆ ⋃ n ∈ t, s n ∧ x ∈ t_1 ∧ y ∈ t_1 ∧ IsPreconnected t_1 obtain ⟨j : ι, hj : j ∈ t, hyj : y ∈ s j⟩ := mem_iUnion₂.1 hy case intro.intro.intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t P : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen R i j → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) x : α hx : x ∈ ⋃ n ∈ t, s n y : α hy : y ∈ ⋃ n ∈ t, s n i : ι hi : i ∈ t hxi : x ∈ s i j : ι hj : j ∈ t hyj : y ∈ s j ⊢ ∃ t_1, t_1 ⊆ ⋃ n ∈ t, s n ∧ x ∈ t_1 ∧ y ∈ t_1 ∧ IsPreconnected t_1 obtain ⟨p, hpt, hip, hjp, hp⟩ := P i hi j hj (K i hi j hj) 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✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t P : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen R i j → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) x : α hx : x ∈ ⋃ n ∈ t, s n y : α hy : y ∈ ⋃ n ∈ t, s n i : ι hi : i ∈ t hxi : x ∈ s i j : ι hj : j ∈ t hyj : y ∈ s j p : Set ι hpt : p ⊆ t hip : i ∈ p hjp : j ∈ p hp : IsPreconnected (⋃ j ∈ p, s j) ⊢ ∃ t_1, t_1 ⊆ ⋃ n ∈ t, s n ∧ x ∈ t_1 ∧ y ∈ t_1 ∧ IsPreconnected t_1 exact ⟨⋃ j ∈ p, s j, biUnion_subset_biUnion_left hpt, mem_biUnion hip hxi, mem_biUnion hjp hyj, hp⟩ α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j : ι hj : j ∈ t h : ReflTransGen R i j ⊢ ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) induction h case refl α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j : ι hj : i ∈ t ⊢ ∃ p, p ⊆ t ∧ i ∈ p ∧ i ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) case tail α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j b✝ c✝ : ι a✝¹ : ReflTransGen R i b✝ a✝ : R b✝ c✝ a_ih✝ : b✝ ∈ t → ∃ p, p ⊆ t ∧ i ∈ p ∧ b✝ ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) hj : c✝ ∈ t ⊢ ∃ p, p ⊆ t ∧ i ∈ p ∧ c✝ ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) case refl => refine ⟨{i}, singleton_subset_iff.mpr hi, mem_singleton i, mem_singleton i, ?_⟩ rw [biUnion_singleton] exact H i hi case tail α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j b✝ c✝ : ι a✝¹ : ReflTransGen R i b✝ a✝ : R b✝ c✝ a_ih✝ : b✝ ∈ t → ∃ p, p ⊆ t ∧ i ∈ p ∧ b✝ ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) hj : c✝ ∈ t ⊢ ∃ p, p ⊆ t ∧ i ∈ p ∧ c✝ ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) case tail j k _ hjk ih => obtain ⟨p, hpt, hip, hjp, hp⟩ := ih hjk.2 refine ⟨insert k p, insert_subset_iff.mpr ⟨hj, hpt⟩, mem_insert_of_mem k hip, mem_insert k p, ?_⟩ rw [biUnion_insert] refine (H k hj).union' (hjk.1.mono ?_) hp rw [inter_comm] exact inter_subset_inter_right _ (subset_biUnion_of_mem hjp) α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j : ι hj : i ∈ t ⊢ ∃ p, p ⊆ t ∧ i ∈ p ∧ i ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) refine ⟨{i}, singleton_subset_iff.mpr hi, mem_singleton i, mem_singleton i, ?_⟩ α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j : ι hj : i ∈ t ⊢ IsPreconnected (⋃ j ∈ {i}, s j) rw [biUnion_singleton] α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j : ι hj : i ∈ t ⊢ IsPreconnected (s i) exact H i hi α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j✝ j k : ι a✝ : ReflTransGen R i j hjk : R j k ih : j ∈ t → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) hj : k ∈ t ⊢ ∃ p, p ⊆ t ∧ i ∈ p ∧ k ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) obtain ⟨p, hpt, hip, hjp, hp⟩ := ih hjk.2 case intro.intro.intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j✝ j k : ι a✝ : ReflTransGen R i j hjk : R j k ih : j ∈ t → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) hj : k ∈ t p : Set ι hpt : p ⊆ t hip : i ∈ p hjp : j ∈ p hp : IsPreconnected (⋃ j ∈ p, s j) ⊢ ∃ p, p ⊆ t ∧ i ∈ p ∧ k ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) refine ⟨insert k p, insert_subset_iff.mpr ⟨hj, hpt⟩, mem_insert_of_mem k hip, mem_insert k p, ?_⟩ case intro.intro.intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j✝ j k : ι a✝ : ReflTransGen R i j hjk : R j k ih : j ∈ t → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) hj : k ∈ t p : Set ι hpt : p ⊆ t hip : i ∈ p hjp : j ∈ p hp : IsPreconnected (⋃ j ∈ p, s j) ⊢ IsPreconnected (⋃ j ∈ insert k p, s j) rw [biUnion_insert] case intro.intro.intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j✝ j k : ι a✝ : ReflTransGen R i j hjk : R j k ih : j ∈ t → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) hj : k ∈ t p : Set ι hpt : p ⊆ t hip : i ∈ p hjp : j ∈ p hp : IsPreconnected (⋃ j ∈ p, s j) ⊢ IsPreconnected (s k ∪ ⋃ x ∈ p, s x) refine (H k hj).union' (hjk.1.mono ?_) hp case intro.intro.intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j✝ j k : ι a✝ : ReflTransGen R i j hjk : R j k ih : j ∈ t → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) hj : k ∈ t p : Set ι hpt : p ⊆ t hip : i ∈ p hjp : j ∈ p hp : IsPreconnected (⋃ j ∈ p, s j) ⊢ s j ∩ s k ⊆ s k ∩ ⋃ x ∈ p, s x rw [inter_comm] case intro.intro.intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j✝ j k : ι a✝ : ReflTransGen R i j hjk : R j k ih : j ∈ t → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) hj : k ∈ t p : Set ι hpt : p ⊆ t hip : i ∈ p hjp : j ∈ p hp : IsPreconnected (⋃ j ∈ p, s j) ⊢ s k ∩ s j ⊆ s k ∩ ⋃ x ∈ p, s x exact inter_subset_inter_right _ (subset_biUnion_of_mem hjp) ** Qed
IsPreconnected.iUnion_of_reflTransGen α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t u v : Set α ι : Type u_3 s : ι → Set α H : ∀ (i : ι), IsPreconnected (s i) K : ∀ (i j : ι), ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j)) i j ⊢ IsPreconnected (⋃ n, s n) rw [← biUnion_univ] α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t u v : Set α ι : Type u_3 s : ι → Set α H : ∀ (i : ι), IsPreconnected (s i) K : ∀ (i j : ι), ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j)) i j ⊢ IsPreconnected (⋃ x ∈ univ, s x) exact IsPreconnected.biUnion_of_reflTransGen (fun i _ => H i) fun i _ j _ => by simpa [mem_univ] using K i j α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t u v : Set α ι : Type u_3 s : ι → Set α H : ∀ (i : ι), IsPreconnected (s i) K : ∀ (i j : ι), ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j)) i j i : ι x✝¹ : i ∈ univ j : ι x✝ : j ∈ univ ⊢ ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ univ) i j simpa [mem_univ] using K i j ** Qed
IsPreconnected.iUnion_of_chain α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α s✝ t u v : Set α inst✝² : LinearOrder β inst✝¹ : SuccOrder β inst✝ : IsSuccArchimedean β s : β → Set α H : ∀ (n : β), IsPreconnected (s n) K : ∀ (n : β), Set.Nonempty (s n ∩ s (succ n)) i✝ j i : β x✝ : i ∈ Ico j i✝ ⊢ Set.Nonempty (s (succ i) ∩ s i) rw [inter_comm] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α s✝ t u v : Set α inst✝² : LinearOrder β inst✝¹ : SuccOrder β inst✝ : IsSuccArchimedean β s : β → Set α H : ∀ (n : β), IsPreconnected (s n) K : ∀ (n : β), Set.Nonempty (s n ∩ s (succ n)) i✝ j i : β x✝ : i ∈ Ico j i✝ ⊢ Set.Nonempty (s i ∩ s (succ i)) exact K i ** Qed
IsConnected.iUnion_of_chain α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝⁴ : TopologicalSpace α s✝ t u v : Set α inst✝³ : LinearOrder β inst✝² : SuccOrder β inst✝¹ : IsSuccArchimedean β inst✝ : Nonempty β s : β → Set α H : ∀ (n : β), IsConnected (s n) K : ∀ (n : β), Set.Nonempty (s n ∩ s (succ n)) i✝ j i : β x✝ : i ∈ Ico j i✝ ⊢ Set.Nonempty (s (succ i) ∩ s i) rw [inter_comm] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝⁴ : TopologicalSpace α s✝ t u v : Set α inst✝³ : LinearOrder β inst✝² : SuccOrder β inst✝¹ : IsSuccArchimedean β inst✝ : Nonempty β s : β → Set α H : ∀ (n : β), IsConnected (s n) K : ∀ (n : β), Set.Nonempty (s n ∩ s (succ n)) i✝ j i : β x✝ : i ∈ Ico j i✝ ⊢ Set.Nonempty (s i ∩ s (succ i)) exact K i ** Qed
IsPreconnected.biUnion_of_chain α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α s✝ t✝ u v : Set α inst✝² : LinearOrder β inst✝¹ : SuccOrder β inst✝ : IsSuccArchimedean β s : β → Set α t : Set β ht : OrdConnected t H : ∀ (n : β), n ∈ t → IsPreconnected (s n) K : ∀ (n : β), n ∈ t → succ n ∈ t → Set.Nonempty (s n ∩ s (succ n)) ⊢ IsPreconnected (⋃ n ∈ t, s n) have h1 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → k ∈ t := fun hi hj hk => ht.out hi hj (Ico_subset_Icc_self hk) α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α s✝ t✝ u v : Set α inst✝² : LinearOrder β inst✝¹ : SuccOrder β inst✝ : IsSuccArchimedean β s : β → Set α t : Set β ht : OrdConnected t H : ∀ (n : β), n ∈ t → IsPreconnected (s n) K : ∀ (n : β), n ∈ t → succ n ∈ t → Set.Nonempty (s n ∩ s (succ n)) h1 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → k ∈ t ⊢ IsPreconnected (⋃ n ∈ t, s n) have h2 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → succ k ∈ t := fun hi hj hk => ht.out hi hj ⟨hk.1.trans <\| le_succ _, succ_le_of_lt hk.2⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α s✝ t✝ u v : Set α inst✝² : LinearOrder β inst✝¹ : SuccOrder β inst✝ : IsSuccArchimedean β s : β → Set α t : Set β ht : OrdConnected t H : ∀ (n : β), n ∈ t → IsPreconnected (s n) K : ∀ (n : β), n ∈ t → succ n ∈ t → Set.Nonempty (s n ∩ s (succ n)) h1 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → k ∈ t h2 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → succ k ∈ t ⊢ IsPreconnected (⋃ n ∈ t, s n) have h3 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → (s k ∩ s (succ k)).Nonempty := fun hi hj hk => K _ (h1 hi hj hk) (h2 hi hj hk) α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α s✝ t✝ u v : Set α inst✝² : LinearOrder β inst✝¹ : SuccOrder β inst✝ : IsSuccArchimedean β s : β → Set α t : Set β ht : OrdConnected t H : ∀ (n : β), n ∈ t → IsPreconnected (s n) K : ∀ (n : β), n ∈ t → succ n ∈ t → Set.Nonempty (s n ∩ s (succ n)) h1 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → k ∈ t h2 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → succ k ∈ t h3 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → Set.Nonempty (s k ∩ s (succ k)) ⊢ IsPreconnected (⋃ n ∈ t, s n) refine' IsPreconnected.biUnion_of_reflTransGen H fun i hi j hj => _ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α s✝ t✝ u v : Set α inst✝² : LinearOrder β inst✝¹ : SuccOrder β inst✝ : IsSuccArchimedean β s : β → Set α t : Set β ht : OrdConnected t H : ∀ (n : β), n ∈ t → IsPreconnected (s n) K : ∀ (n : β), n ∈ t → succ n ∈ t → Set.Nonempty (s n ∩ s (succ n)) h1 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → k ∈ t h2 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → succ k ∈ t h3 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → Set.Nonempty (s k ∩ s (succ k)) i : β hi : i ∈ t j : β hj : j ∈ t ⊢ ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j exact reflTransGen_of_succ _ (fun k hk => ⟨h3 hi hj hk, h1 hi hj hk⟩) fun k hk => ⟨by rw [inter_comm]; exact h3 hj hi hk, h2 hj hi hk⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α s✝ t✝ u v : Set α inst✝² : LinearOrder β inst✝¹ : SuccOrder β inst✝ : IsSuccArchimedean β s : β → Set α t : Set β ht : OrdConnected t H : ∀ (n : β), n ∈ t → IsPreconnected (s n) K : ∀ (n : β), n ∈ t → succ n ∈ t → Set.Nonempty (s n ∩ s (succ n)) h1 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → k ∈ t h2 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → succ k ∈ t h3 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → Set.Nonempty (s k ∩ s (succ k)) i : β hi : i ∈ t j : β hj : j ∈ t k : β hk : k ∈ Ico j i ⊢ Set.Nonempty (s (succ k) ∩ s k) rw [inter_comm] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α s✝ t✝ u v : Set α inst✝² : LinearOrder β inst✝¹ : SuccOrder β inst✝ : IsSuccArchimedean β s : β → Set α t : Set β ht : OrdConnected t H : ∀ (n : β), n ∈ t → IsPreconnected (s n) K : ∀ (n : β), n ∈ t → succ n ∈ t → Set.Nonempty (s n ∩ s (succ n)) h1 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → k ∈ t h2 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → succ k ∈ t h3 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → Set.Nonempty (s k ∩ s (succ k)) i : β hi : i ∈ t j : β hj : j ∈ t k : β hk : k ∈ Ico j i ⊢ Set.Nonempty (s k ∩ s (succ k)) exact h3 hj hi hk ** Qed
IsPreconnected.image α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s ⊢ IsPreconnected (f '' s) rintro u v hu hv huv ⟨_, ⟨x, xs, rfl⟩, xu⟩ ⟨_, ⟨y, ys, rfl⟩, yv⟩ 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✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v huv : f '' s ⊆ u ∪ v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v ⊢ Set.Nonempty (f '' s ∩ (u ∩ v)) rcases continuousOn_iff'.1 hf u hu with ⟨u', hu', u'_eq⟩ case intro.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 s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v huv : f '' s ⊆ u ∪ v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s ⊢ Set.Nonempty (f '' s ∩ (u ∩ v)) rcases continuousOn_iff'.1 hf v hv with ⟨v', hv', v'_eq⟩ case intro.intro.intro.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 s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v huv : f '' s ⊆ u ∪ v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s ⊢ Set.Nonempty (f '' s ∩ (u ∩ v)) replace huv : s ⊆ u' ∪ v' case intro.intro.intro.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 s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s huv : s ⊆ u' ∪ v' ⊢ Set.Nonempty (f '' s ∩ (u ∩ v)) obtain ⟨z, hz⟩ : (s ∩ (u' ∩ v')).Nonempty := by refine H u' v' hu' hv' huv ⟨x, ?_⟩ ⟨y, ?_⟩ <;> rw [inter_comm] exacts [u'_eq ▸ ⟨xu, xs⟩, v'_eq ▸ ⟨yv, ys⟩] case intro.intro.intro.intro.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 s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s huv : s ⊆ u' ∪ v' z : α hz : z ∈ s ∩ (u' ∩ v') ⊢ Set.Nonempty (f '' s ∩ (u ∩ v)) rw [← inter_self s, inter_assoc, inter_left_comm s u', ← inter_assoc, inter_comm s, inter_comm s, ← u'_eq, ← v'_eq] at hz case intro.intro.intro.intro.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 s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s huv : s ⊆ u' ∪ v' z : α hz : z ∈ f ⁻¹' u ∩ s ∩ (f ⁻¹' v ∩ s) ⊢ Set.Nonempty (f '' s ∩ (u ∩ v)) exact ⟨f z, ⟨z, hz.1.2, rfl⟩, hz.1.1, hz.2.1⟩ case huv α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v huv : f '' s ⊆ u ∪ v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s ⊢ s ⊆ u' ∪ v' rw [image_subset_iff, preimage_union] at huv case huv α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v huv : s ⊆ f ⁻¹' u ∪ f ⁻¹' v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s ⊢ s ⊆ u' ∪ v' replace huv := subset_inter huv Subset.rfl case huv α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s huv : s ⊆ (f ⁻¹' u ∪ f ⁻¹' v) ∩ s ⊢ s ⊆ u' ∪ v' rw [inter_distrib_right, u'_eq, v'_eq, ← inter_distrib_right] at huv case huv α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s huv : s ⊆ (u' ∪ v') ∩ s ⊢ s ⊆ u' ∪ v' exact (subset_inter_iff.1 huv).1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s huv : s ⊆ u' ∪ v' ⊢ Set.Nonempty (s ∩ (u' ∩ v')) refine H u' v' hu' hv' huv ⟨x, ?_⟩ ⟨y, ?_⟩ <;> rw [inter_comm] case refine_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s huv : s ⊆ u' ∪ v' ⊢ x ∈ u' ∩ s case refine_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s huv : s ⊆ u' ∪ v' ⊢ y ∈ v' ∩ s exacts [u'_eq ▸ ⟨xu, xs⟩, v'_eq ▸ ⟨yv, ys⟩] ** Qed
isPreconnected_closed_iff α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α ⊢ IsPreconnected s → ∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t')) rintro h t t' ht ht' htt' ⟨x, xs, xt⟩ ⟨y, ys, yt'⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s : Set α h : IsPreconnected s t t' : Set α ht : IsClosed t ht' : IsClosed t' htt' : s ⊆ t ∪ t' x : α xs : x ∈ s xt : x ∈ t y : α ys : y ∈ s yt' : y ∈ t' ⊢ Set.Nonempty (s ∩ (t ∩ t')) rw [← not_disjoint_iff_nonempty_inter, ← subset_compl_iff_disjoint_right, compl_inter] case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s : Set α h : IsPreconnected s t t' : Set α ht : IsClosed t ht' : IsClosed t' htt' : s ⊆ t ∪ t' x : α xs : x ∈ s xt : x ∈ t y : α ys : y ∈ s yt' : y ∈ t' ⊢ ¬s ⊆ tᶜ ∪ t'ᶜ intro h' case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s : Set α h : IsPreconnected s t t' : Set α ht : IsClosed t ht' : IsClosed t' htt' : s ⊆ t ∪ t' x : α xs : x ∈ s xt : x ∈ t y : α ys : y ∈ s yt' : y ∈ t' h' : s ⊆ tᶜ ∪ t'ᶜ ⊢ False have xt' : x ∉ t' := (h' xs).resolve_left (absurd xt) case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s : Set α h : IsPreconnected s t t' : Set α ht : IsClosed t ht' : IsClosed t' htt' : s ⊆ t ∪ t' x : α xs : x ∈ s xt : x ∈ t y : α ys : y ∈ s yt' : y ∈ t' h' : s ⊆ tᶜ ∪ t'ᶜ xt' : ¬x ∈ t' ⊢ False have yt : y ∉ t := (h' ys).resolve_right (absurd yt') case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s : Set α h : IsPreconnected s t t' : Set α ht : IsClosed t ht' : IsClosed t' htt' : s ⊆ t ∪ t' x : α xs : x ∈ s xt : x ∈ t y : α ys : y ∈ s yt' : y ∈ t' h' : s ⊆ tᶜ ∪ t'ᶜ xt' : ¬x ∈ t' yt : ¬y ∈ t ⊢ False have := h _ _ ht.isOpen_compl ht'.isOpen_compl h' ⟨y, ys, yt⟩ ⟨x, xs, xt'⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s : Set α h : IsPreconnected s t t' : Set α ht : IsClosed t ht' : IsClosed t' htt' : s ⊆ t ∪ t' x : α xs : x ∈ s xt : x ∈ t y : α ys : y ∈ s yt' : y ∈ t' h' : s ⊆ tᶜ ∪ t'ᶜ xt' : ¬x ∈ t' yt : ¬y ∈ t this : Set.Nonempty (s ∩ (tᶜ ∩ t'ᶜ)) ⊢ False rw [← compl_union] at this case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s : Set α h : IsPreconnected s t t' : Set α ht : IsClosed t ht' : IsClosed t' htt' : s ⊆ t ∪ t' x : α xs : x ∈ s xt : x ∈ t y : α ys : y ∈ s yt' : y ∈ t' h' : s ⊆ tᶜ ∪ t'ᶜ xt' : ¬x ∈ t' yt : ¬y ∈ t this : Set.Nonempty (s ∩ (t ∪ t')ᶜ) ⊢ False exact this.ne_empty htt'.disjoint_compl_right.inter_eq α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α ⊢ (∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t'))) → IsPreconnected s rintro h u v hu hv huv ⟨x, xs, xu⟩ ⟨y, ys, yv⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t')) u v : Set α hu : IsOpen u hv : IsOpen v huv : s ⊆ u ∪ v x : α xs : x ∈ s xu : x ∈ u y : α ys : y ∈ s yv : y ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) rw [← not_disjoint_iff_nonempty_inter, ← subset_compl_iff_disjoint_right, compl_inter] case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t')) u v : Set α hu : IsOpen u hv : IsOpen v huv : s ⊆ u ∪ v x : α xs : x ∈ s xu : x ∈ u y : α ys : y ∈ s yv : y ∈ v ⊢ ¬s ⊆ uᶜ ∪ vᶜ intro h' case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t')) u v : Set α hu : IsOpen u hv : IsOpen v huv : s ⊆ u ∪ v x : α xs : x ∈ s xu : x ∈ u y : α ys : y ∈ s yv : y ∈ v h' : s ⊆ uᶜ ∪ vᶜ ⊢ False have xv : x ∉ v := (h' xs).elim (absurd xu) id case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t')) u v : Set α hu : IsOpen u hv : IsOpen v huv : s ⊆ u ∪ v x : α xs : x ∈ s xu : x ∈ u y : α ys : y ∈ s yv : y ∈ v h' : s ⊆ uᶜ ∪ vᶜ xv : ¬x ∈ v ⊢ False have yu : y ∉ u := (h' ys).elim id (absurd yv) case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t')) u v : Set α hu : IsOpen u hv : IsOpen v huv : s ⊆ u ∪ v x : α xs : x ∈ s xu : x ∈ u y : α ys : y ∈ s yv : y ∈ v h' : s ⊆ uᶜ ∪ vᶜ xv : ¬x ∈ v yu : ¬y ∈ u ⊢ False have := h _ _ hu.isClosed_compl hv.isClosed_compl h' ⟨y, ys, yu⟩ ⟨x, xs, xv⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t')) u v : Set α hu : IsOpen u hv : IsOpen v huv : s ⊆ u ∪ v x : α xs : x ∈ s xu : x ∈ u y : α ys : y ∈ s yv : y ∈ v h' : s ⊆ uᶜ ∪ vᶜ xv : ¬x ∈ v yu : ¬y ∈ u this : Set.Nonempty (s ∩ (uᶜ ∩ vᶜ)) ⊢ False rw [← compl_union] at this case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t')) u v : Set α hu : IsOpen u hv : IsOpen v huv : s ⊆ u ∪ v x : α xs : x ∈ s xu : x ∈ u y : α ys : y ∈ s yv : y ∈ v h' : s ⊆ uᶜ ∪ vᶜ xv : ¬x ∈ v yu : ¬y ∈ u this : Set.Nonempty (s ∩ (u ∪ v)ᶜ) ⊢ False exact this.ne_empty huv.disjoint_compl_right.inter_eq ** Qed
Inducing.isPreconnected_image α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set α f : α → β hf : Inducing f ⊢ IsPreconnected (f '' s) ↔ IsPreconnected s refine' ⟨fun h => _, fun h => h.image _ hf.continuous.continuousOn⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set α f : α → β hf : Inducing f h : IsPreconnected (f '' s) ⊢ IsPreconnected s rintro u v hu' hv' huv ⟨x, hxs, hxu⟩ ⟨y, hys, hyv⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α f : α → β hf : Inducing f h : IsPreconnected (f '' s) u v : Set α hu' : IsOpen u hv' : IsOpen v huv : s ⊆ u ∪ v x : α hxs : x ∈ s hxu : x ∈ u y : α hys : y ∈ s hyv : y ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) rcases hf.isOpen_iff.1 hu' with ⟨u, hu, rfl⟩ case 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 s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α f : α → β hf : Inducing f h : IsPreconnected (f '' s) v : Set α hv' : IsOpen v x : α hxs : x ∈ s y : α hys : y ∈ s hyv : y ∈ v u : Set β hu : IsOpen u hu' : IsOpen (f ⁻¹' u) huv : s ⊆ f ⁻¹' u ∪ v hxu : x ∈ f ⁻¹' u ⊢ Set.Nonempty (s ∩ (f ⁻¹' u ∩ v)) rcases hf.isOpen_iff.1 hv' with ⟨v, hv, 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✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α f : α → β hf : Inducing f h : IsPreconnected (f '' s) x : α hxs : x ∈ s y : α hys : y ∈ s u : Set β hu : IsOpen u hu' : IsOpen (f ⁻¹' u) hxu : x ∈ f ⁻¹' u v : Set β hv : IsOpen v hv' : IsOpen (f ⁻¹' v) hyv : y ∈ f ⁻¹' v huv : s ⊆ f ⁻¹' u ∪ f ⁻¹' v ⊢ Set.Nonempty (s ∩ (f ⁻¹' u ∩ f ⁻¹' v)) replace huv : f '' s ⊆ u ∪ v 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✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α f : α → β hf : Inducing f h : IsPreconnected (f '' s) x : α hxs : x ∈ s y : α hys : y ∈ s u : Set β hu : IsOpen u hu' : IsOpen (f ⁻¹' u) hxu : x ∈ f ⁻¹' u v : Set β hv : IsOpen v hv' : IsOpen (f ⁻¹' v) hyv : y ∈ f ⁻¹' v huv : f '' s ⊆ u ∪ v ⊢ Set.Nonempty (s ∩ (f ⁻¹' u ∩ f ⁻¹' v)) rcases h u v hu hv huv ⟨f x, mem_image_of_mem _ hxs, hxu⟩ ⟨f y, mem_image_of_mem _ hys, hyv⟩ with ⟨_, ⟨z, hzs, rfl⟩, hzuv⟩ case intro.intro.intro.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 s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α f : α → β hf : Inducing f h : IsPreconnected (f '' s) x : α hxs : x ∈ s y : α hys : y ∈ s u : Set β hu : IsOpen u hu' : IsOpen (f ⁻¹' u) hxu : x ∈ f ⁻¹' u v : Set β hv : IsOpen v hv' : IsOpen (f ⁻¹' v) hyv : y ∈ f ⁻¹' v huv : f '' s ⊆ u ∪ v z : α hzs : z ∈ s hzuv : f z ∈ u ∩ v ⊢ Set.Nonempty (s ∩ (f ⁻¹' u ∩ f ⁻¹' v)) exact ⟨z, hzs, hzuv⟩ case huv α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α f : α → β hf : Inducing f h : IsPreconnected (f '' s) x : α hxs : x ∈ s y : α hys : y ∈ s u : Set β hu : IsOpen u hu' : IsOpen (f ⁻¹' u) hxu : x ∈ f ⁻¹' u v : Set β hv : IsOpen v hv' : IsOpen (f ⁻¹' v) hyv : y ∈ f ⁻¹' v huv : s ⊆ f ⁻¹' u ∪ f ⁻¹' v ⊢ f '' s ⊆ u ∪ v rwa [image_subset_iff] ** Qed
IsPreconnected.preimage_of_open_map α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsOpenMap f hsf : s ⊆ range f u v : Set α hu : IsOpen u hv : IsOpen v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) ⊢ Set.Nonempty (f ⁻¹' s ∩ (u ∩ v)) replace hsf : f '' (f ⁻¹' s) = s := image_preimage_eq_of_subset hsf α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsOpenMap f u v : Set α hu : IsOpen u hv : IsOpen v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s ⊢ Set.Nonempty (f ⁻¹' s ∩ (u ∩ v)) obtain ⟨_, has, ⟨a, hau, rfl⟩, hav⟩ : (s ∩ (f '' u ∩ f '' v)).Nonempty α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsOpenMap f u v : Set α hu : IsOpen u hv : IsOpen v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s ⊢ Set.Nonempty (s ∩ (f '' u ∩ f '' v)) refine hs (f '' u) (f '' v) (hf u hu) (hf v hv) ?_ ?_ ?_ case refine_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsOpenMap f u v : Set α hu : IsOpen u hv : IsOpen v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s ⊢ s ⊆ f '' u ∪ f '' v simpa only [hsf, image_union] using image_subset f hsuv case refine_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsOpenMap f u v : Set α hu : IsOpen u hv : IsOpen v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s ⊢ Set.Nonempty (s ∩ f '' u) simpa only [image_preimage_inter] using hsu.image f case refine_3 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsOpenMap f u v : Set α hu : IsOpen u hv : IsOpen v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s ⊢ Set.Nonempty (s ∩ f '' v) simpa only [image_preimage_inter] using hsv.image f 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 s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsOpenMap f u v : Set α hu : IsOpen u hv : IsOpen v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s a : α hau : a ∈ u has : f a ∈ s hav : f a ∈ f '' v ⊢ Set.Nonempty (f ⁻¹' s ∩ (u ∩ v)) exact ⟨a, has, hau, hinj.mem_set_image.1 hav⟩ ** Qed
IsPreconnected.preimage_of_closed_map α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsClosedMap f hsf : s ⊆ range f u v : Set α hu : IsClosed u hv : IsClosed v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) ⊢ Set.Nonempty (f ⁻¹' s ∩ (u ∩ v)) replace hsf : f '' (f ⁻¹' s) = s := image_preimage_eq_of_subset hsf α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsClosedMap f u v : Set α hu : IsClosed u hv : IsClosed v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s ⊢ Set.Nonempty (f ⁻¹' s ∩ (u ∩ v)) obtain ⟨_, has, ⟨a, hau, rfl⟩, hav⟩ : (s ∩ (f '' u ∩ f '' v)).Nonempty α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsClosedMap f u v : Set α hu : IsClosed u hv : IsClosed v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s ⊢ Set.Nonempty (s ∩ (f '' u ∩ f '' v)) refine isPreconnected_closed_iff.1 hs (f '' u) (f '' v) (hf u hu) (hf v hv) ?_ ?_ ?_ case refine_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsClosedMap f u v : Set α hu : IsClosed u hv : IsClosed v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s ⊢ s ⊆ f '' u ∪ f '' v simpa only [hsf, image_union] using image_subset f hsuv case refine_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsClosedMap f u v : Set α hu : IsClosed u hv : IsClosed v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s ⊢ Set.Nonempty (s ∩ f '' u) simpa only [image_preimage_inter] using hsu.image f case refine_3 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsClosedMap f u v : Set α hu : IsClosed u hv : IsClosed v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s ⊢ Set.Nonempty (s ∩ f '' v) simpa only [image_preimage_inter] using hsv.image f 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 s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsClosedMap f u v : Set α hu : IsClosed u hv : IsClosed v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s a : α hau : a ∈ u has : f a ∈ s hav : f a ∈ f '' v ⊢ Set.Nonempty (f ⁻¹' s ∩ (u ∩ v)) exact ⟨a, has, hau, hinj.mem_set_image.1 hav⟩ ** Qed
IsPreconnected.subset_or_subset α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hu : IsOpen u hv : IsOpen v huv : Disjoint u v hsuv : s ⊆ u ∪ v hs : IsPreconnected s ⊢ s ⊆ u ∨ s ⊆ v specialize hs u v hu hv hsuv α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hu : IsOpen u hv : IsOpen v huv : Disjoint u v hsuv : s ⊆ u ∪ v hs : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) ⊢ s ⊆ u ∨ s ⊆ v obtain hsu \| hsu := (s ∩ u).eq_empty_or_nonempty case inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hu : IsOpen u hv : IsOpen v huv : Disjoint u v hsuv : s ⊆ u ∪ v hs : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) hsu : s ∩ u = ∅ ⊢ s ⊆ u ∨ s ⊆ v exact Or.inr ((Set.disjoint_iff_inter_eq_empty.2 hsu).subset_right_of_subset_union hsuv) case inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hu : IsOpen u hv : IsOpen v huv : Disjoint u v hsuv : s ⊆ u ∪ v hs : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) hsu : Set.Nonempty (s ∩ u) ⊢ s ⊆ u ∨ s ⊆ v replace hs := mt (hs hsu) case inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hu : IsOpen u hv : IsOpen v huv : Disjoint u v hsuv : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hs : ¬Set.Nonempty (s ∩ (u ∩ v)) → ¬Set.Nonempty (s ∩ v) ⊢ s ⊆ u ∨ s ⊆ v simp_rw [Set.not_nonempty_iff_eq_empty, ← Set.disjoint_iff_inter_eq_empty, disjoint_iff_inter_eq_empty.1 huv] at hs case inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hu : IsOpen u hv : IsOpen v huv : Disjoint u v hsuv : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hs : Disjoint s ∅ → Disjoint s v ⊢ s ⊆ u ∨ s ⊆ v exact Or.inl ((hs s.disjoint_empty).subset_left_of_subset_union hsuv) ** Qed
IsPreconnected.subset_left_of_subset_union α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hu : IsOpen u hv : IsOpen v huv : Disjoint u v hsuv : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hs : IsPreconnected s ⊢ Disjoint s v by_contra hsv α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hu : IsOpen u hv : IsOpen v huv : Disjoint u v hsuv : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hs : IsPreconnected s hsv : ¬Disjoint s v ⊢ False rw [not_disjoint_iff_nonempty_inter] at hsv α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hu : IsOpen u hv : IsOpen v huv : Disjoint u v hsuv : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hs : IsPreconnected s hsv : Set.Nonempty (s ∩ v) ⊢ False obtain ⟨x, _, hx⟩ := hs u v hu hv hsuv hsu hsv case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hu : IsOpen u hv : IsOpen v huv : Disjoint u v hsuv : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hs : IsPreconnected s hsv : Set.Nonempty (s ∩ v) x : α left✝ : x ∈ s hx : x ∈ u ∩ v ⊢ False exact Set.disjoint_iff.1 huv hx ** Qed
IsPreconnected.subset_clopen α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s t : Set α hs : IsPreconnected s ht : IsClopen t hne : Set.Nonempty (s ∩ t) ⊢ s ⊆ t ∪ tᶜ simp ** Qed
IsPreconnected.subset_of_closure_inter_subset α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hs : IsPreconnected s hu : IsOpen u h'u : Set.Nonempty (s ∩ u) h : closure u ∩ s ⊆ u A : s ⊆ u ∪ (closure u)ᶜ ⊢ s ⊆ u apply hs.subset_left_of_subset_union hu isClosed_closure.isOpen_compl _ A h'u α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hs : IsPreconnected s hu : IsOpen u h'u : Set.Nonempty (s ∩ u) h : closure u ∩ s ⊆ u A : s ⊆ u ∪ (closure u)ᶜ ⊢ Disjoint u (closure u)ᶜ exact disjoint_compl_right.mono_right (compl_subset_compl.2 subset_closure) α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hs : IsPreconnected s hu : IsOpen u h'u : Set.Nonempty (s ∩ u) h : closure u ∩ s ⊆ u ⊢ s ⊆ u ∪ (closure u)ᶜ intro x hx α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hs : IsPreconnected s hu : IsOpen u h'u : Set.Nonempty (s ∩ u) h : closure u ∩ s ⊆ u x : α hx : x ∈ s ⊢ x ∈ u ∪ (closure u)ᶜ by_cases xu : x ∈ u case pos α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hs : IsPreconnected s hu : IsOpen u h'u : Set.Nonempty (s ∩ u) h : closure u ∩ s ⊆ u x : α hx : x ∈ s xu : x ∈ u ⊢ x ∈ u ∪ (closure u)ᶜ exact Or.inl xu case neg α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hs : IsPreconnected s hu : IsOpen u h'u : Set.Nonempty (s ∩ u) h : closure u ∩ s ⊆ u x : α hx : x ∈ s xu : ¬x ∈ u ⊢ x ∈ u ∪ (closure u)ᶜ right case neg.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hs : IsPreconnected s hu : IsOpen u h'u : Set.Nonempty (s ∩ u) h : closure u ∩ s ⊆ u x : α hx : x ∈ s xu : ¬x ∈ u ⊢ x ∈ (closure u)ᶜ intro h'x case neg.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hs : IsPreconnected s hu : IsOpen u h'u : Set.Nonempty (s ∩ u) h : closure u ∩ s ⊆ u x : α hx : x ∈ s xu : ¬x ∈ u h'x : x ∈ closure u ⊢ False exact xu (h (mem_inter h'x hx)) ** Qed
IsPreconnected.prod α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t✝ u v : Set α inst✝ : TopologicalSpace β s : Set α t : Set β hs : IsPreconnected s ht : IsPreconnected t ⊢ IsPreconnected (s ×ˢ t) apply isPreconnected_of_forall_pair case H α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t✝ u v : Set α inst✝ : TopologicalSpace β s : Set α t : Set β hs : IsPreconnected s ht : IsPreconnected t ⊢ ∀ (x : α × β), x ∈ s ×ˢ t → ∀ (y : α × β), y ∈ s ×ˢ t → ∃ t_1, t_1 ⊆ s ×ˢ t ∧ x ∈ t_1 ∧ y ∈ t_1 ∧ IsPreconnected t_1 rintro ⟨a₁, b₁⟩ ⟨ha₁, hb₁⟩ ⟨a₂, b₂⟩ ⟨ha₂, hb₂⟩ case H.mk.intro.mk.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t✝ u v : Set α inst✝ : TopologicalSpace β s : Set α t : Set β hs : IsPreconnected s ht : IsPreconnected t a₁ : α b₁ : β ha₁ : (a₁, b₁).1 ∈ s hb₁ : (a₁, b₁).2 ∈ t a₂ : α b₂ : β ha₂ : (a₂, b₂).1 ∈ s hb₂ : (a₂, b₂).2 ∈ t ⊢ ∃ t_1, t_1 ⊆ s ×ˢ t ∧ (a₁, b₁) ∈ t_1 ∧ (a₂, b₂) ∈ t_1 ∧ IsPreconnected t_1 refine' ⟨Prod.mk a₁ '' t ∪ flip Prod.mk b₂ '' s, _, .inl ⟨b₁, hb₁, rfl⟩, .inr ⟨a₂, ha₂, rfl⟩, _⟩ case H.mk.intro.mk.intro.refine'_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t✝ u v : Set α inst✝ : TopologicalSpace β s : Set α t : Set β hs : IsPreconnected s ht : IsPreconnected t a₁ : α b₁ : β ha₁ : (a₁, b₁).1 ∈ s hb₁ : (a₁, b₁).2 ∈ t a₂ : α b₂ : β ha₂ : (a₂, b₂).1 ∈ s hb₂ : (a₂, b₂).2 ∈ t ⊢ Prod.mk a₁ '' t ∪ flip Prod.mk b₂ '' s ⊆ s ×ˢ t rintro _ (⟨y, hy, rfl⟩ \| ⟨x, hx, rfl⟩) case H.mk.intro.mk.intro.refine'_1.inl.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t✝ u v : Set α inst✝ : TopologicalSpace β s : Set α t : Set β hs : IsPreconnected s ht : IsPreconnected t a₁ : α b₁ : β ha₁ : (a₁, b₁).1 ∈ s hb₁ : (a₁, b₁).2 ∈ t a₂ : α b₂ : β ha₂ : (a₂, b₂).1 ∈ s hb₂ : (a₂, b₂).2 ∈ t y : β hy : y ∈ t ⊢ (a₁, y) ∈ s ×ˢ t case H.mk.intro.mk.intro.refine'_1.inr.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t✝ u v : Set α inst✝ : TopologicalSpace β s : Set α t : Set β hs : IsPreconnected s ht : IsPreconnected t a₁ : α b₁ : β ha₁ : (a₁, b₁).1 ∈ s hb₁ : (a₁, b₁).2 ∈ t a₂ : α b₂ : β ha₂ : (a₂, b₂).1 ∈ s hb₂ : (a₂, b₂).2 ∈ t x : α hx : x ∈ s ⊢ flip Prod.mk b₂ x ∈ s ×ˢ t exacts [⟨ha₁, hy⟩, ⟨hx, hb₂⟩] case H.mk.intro.mk.intro.refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t✝ u v : Set α inst✝ : TopologicalSpace β s : Set α t : Set β hs : IsPreconnected s ht : IsPreconnected t a₁ : α b₁ : β ha₁ : (a₁, b₁).1 ∈ s hb₁ : (a₁, b₁).2 ∈ t a₂ : α b₂ : β ha₂ : (a₂, b₂).1 ∈ s hb₂ : (a₂, b₂).2 ∈ t ⊢ IsPreconnected (Prod.mk a₁ '' t ∪ flip Prod.mk b₂ '' s) exact (ht.image _ (Continuous.Prod.mk _).continuousOn).union (a₁, b₂) ⟨b₂, hb₂, rfl⟩ ⟨a₁, ha₁, rfl⟩ (hs.image _ (continuous_id.prod_mk continuous_const).continuousOn) ** Qed
isPreconnected_univ_pi α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) ⊢ IsPreconnected (pi univ s) rintro u v uo vo hsuv ⟨f, hfs, hfu⟩ ⟨g, hgs, hgv⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) rcases exists_finset_piecewise_mem_of_mem_nhds (uo.mem_nhds hfu) g with ⟨I, hI⟩ 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 s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I : Finset ι hI : Finset.piecewise I f g ∈ u ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) induction' I using Finset.induction_on with i I _ ihI case intro.intro.intro.intro.intro.empty α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I : Finset ι hI✝ : Finset.piecewise I f g ∈ u hI : Finset.piecewise ∅ f g ∈ u ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) refine' ⟨g, hgs, ⟨_, hgv⟩⟩ case intro.intro.intro.intro.intro.empty α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I : Finset ι hI✝ : Finset.piecewise I f g ∈ u hI : Finset.piecewise ∅ f g ∈ u ⊢ g ∈ u simpa using hI case intro.intro.intro.intro.intro.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : Finset.piecewise (insert i I) f g ∈ u ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) rw [Finset.piecewise_insert] at hI case intro.intro.intro.intro.intro.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) have := I.piecewise_mem_set_pi hfs hgs case intro.intro.intro.intro.intro.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) refine' (hsuv this).elim ihI fun h => _ case intro.intro.intro.intro.intro.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s h : Finset.piecewise I f g ∈ v ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) set S := update (I.piecewise f g) i '' s i case intro.intro.intro.intro.intro.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s h : Finset.piecewise I f g ∈ v S : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) have hsub : S ⊆ pi univ s := by refine' image_subset_iff.2 fun z hz => _ rwa [update_preimage_univ_pi] exact fun j _ => this j trivial case intro.intro.intro.intro.intro.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s h : Finset.piecewise I f g ∈ v S : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i hsub : S ⊆ pi univ s ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) have hconn : IsPreconnected S := (hs i).image _ (continuous_const.update i continuous_id).continuousOn case intro.intro.intro.intro.intro.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s h : Finset.piecewise I f g ∈ v S : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i hsub : S ⊆ pi univ s hconn : IsPreconnected S ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) have hSu : (S ∩ u).Nonempty := ⟨_, mem_image_of_mem _ (hfs _ trivial), hI⟩ case intro.intro.intro.intro.intro.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s h : Finset.piecewise I f g ∈ v S : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i hsub : S ⊆ pi univ s hconn : IsPreconnected S hSu : Set.Nonempty (S ∩ u) ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) have hSv : (S ∩ v).Nonempty := ⟨_, ⟨_, this _ trivial, update_eq_self _ _⟩, h⟩ case intro.intro.intro.intro.intro.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s h : Finset.piecewise I f g ∈ v S : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i hsub : S ⊆ pi univ s hconn : IsPreconnected S hSu : Set.Nonempty (S ∩ u) hSv : Set.Nonempty (S ∩ v) ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) refine' (hconn u v uo vo (hsub.trans hsuv) hSu hSv).mono _ case intro.intro.intro.intro.intro.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s h : Finset.piecewise I f g ∈ v S : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i hsub : S ⊆ pi univ s hconn : IsPreconnected S hSu : Set.Nonempty (S ∩ u) hSv : Set.Nonempty (S ∩ v) ⊢ S ∩ (u ∩ v) ⊆ pi univ s ∩ (u ∩ v) exact inter_subset_inter_left _ hsub α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s h : Finset.piecewise I f g ∈ v S : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i ⊢ S ⊆ pi univ s refine' image_subset_iff.2 fun z hz => _ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s h : Finset.piecewise I f g ∈ v S : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i z : π i hz : z ∈ s i ⊢ z ∈ update (Finset.piecewise I f g) i ⁻¹' pi univ s rwa [update_preimage_univ_pi] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s h : Finset.piecewise I f g ∈ v S : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i z : π i hz : z ∈ s i ⊢ ∀ (j : ι), j ≠ i → Finset.piecewise I f g j ∈ s j exact fun j _ => this j trivial ** Qed
isConnected_univ_pi α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) ⊢ IsConnected (pi univ s) ↔ ∀ (i : ι), IsConnected (s i) simp only [IsConnected, ← univ_pi_nonempty_iff, forall_and, and_congr_right_iff] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) ⊢ Set.Nonempty (pi univ s) → (IsPreconnected (pi univ s) ↔ ∀ (x : ι), IsPreconnected (s x)) refine' fun hne => ⟨fun hc i => _, isPreconnected_univ_pi⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hne : Set.Nonempty (pi univ s) hc : IsPreconnected (pi univ s) i : ι ⊢ IsPreconnected (s i) rw [← eval_image_univ_pi hne] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hne : Set.Nonempty (pi univ s) hc : IsPreconnected (pi univ s) i : ι ⊢ IsPreconnected ((fun f => f i) '' pi univ s) exact hc.image _ (continuous_apply _).continuousOn ** Qed
Sigma.isConnected_iff α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : Set ((i : ι) × π i) ⊢ IsConnected s ↔ ∃ i t, IsConnected t ∧ s = mk i '' t refine' ⟨fun hs => _, _⟩ case refine'_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : Set ((i : ι) × π i) hs : IsConnected s ⊢ ∃ i t, IsConnected t ∧ s = mk i '' t obtain ⟨⟨i, x⟩, hx⟩ := hs.nonempty case refine'_1.intro.mk α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : Set ((i : ι) × π i) hs : IsConnected s i : ι x : π i hx : { fst := i, snd := x } ∈ s ⊢ ∃ i t, IsConnected t ∧ s = mk i '' t have : s ⊆ range (Sigma.mk i) := hs.isPreconnected.subset_clopen isClopen_range_sigmaMk ⟨⟨i, x⟩, hx, x, rfl⟩ case refine'_1.intro.mk α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : Set ((i : ι) × π i) hs : IsConnected s i : ι x : π i hx : { fst := i, snd := x } ∈ s this : s ⊆ range (mk i) ⊢ ∃ i t, IsConnected t ∧ s = mk i '' t exact ⟨i, Sigma.mk i ⁻¹' s, hs.preimage_of_openMap sigma_mk_injective isOpenMap_sigmaMk this, (Set.image_preimage_eq_of_subset this).symm⟩ case refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : Set ((i : ι) × π i) ⊢ (∃ i t, IsConnected t ∧ s = mk i '' t) → IsConnected s rintro ⟨i, t, ht, rfl⟩ case refine'_2.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) i : ι t : Set (π i) ht : IsConnected t ⊢ IsConnected (mk i '' t) exact ht.image _ continuous_sigmaMk.continuousOn ** Qed
Sigma.isPreconnected_iff α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α hι : Nonempty ι inst✝ : (i : ι) → TopologicalSpace (π i) s : Set ((i : ι) × π i) ⊢ IsPreconnected s ↔ ∃ i t, IsPreconnected t ∧ s = mk i '' t refine' ⟨fun hs => _, _⟩ case refine'_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α hι : Nonempty ι inst✝ : (i : ι) → TopologicalSpace (π i) s : Set ((i : ι) × π i) hs : IsPreconnected s ⊢ ∃ i t, IsPreconnected t ∧ s = mk i '' t obtain rfl \| h := s.eq_empty_or_nonempty case refine'_1.inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t u v : Set α hι : Nonempty ι inst✝ : (i : ι) → TopologicalSpace (π i) hs : IsPreconnected ∅ ⊢ ∃ i t, IsPreconnected t ∧ ∅ = mk i '' t exact ⟨Classical.choice hι, ∅, isPreconnected_empty, (Set.image_empty _).symm⟩ case refine'_1.inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α hι : Nonempty ι inst✝ : (i : ι) → TopologicalSpace (π i) s : Set ((i : ι) × π i) hs : IsPreconnected s h : Set.Nonempty s ⊢ ∃ i t, IsPreconnected t ∧ s = mk i '' t obtain ⟨a, t, ht, rfl⟩ := Sigma.isConnected_iff.1 ⟨h, hs⟩ case refine'_1.inr.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α hι : Nonempty ι inst✝ : (i : ι) → TopologicalSpace (π i) a : ι t : Set (π a) ht : IsConnected t hs : IsPreconnected (mk a '' t) h : Set.Nonempty (mk a '' t) ⊢ ∃ i t_1, IsPreconnected t_1 ∧ mk a '' t = mk i '' t_1 refine' ⟨a, t, ht.isPreconnected, rfl⟩ case refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α hι : Nonempty ι inst✝ : (i : ι) → TopologicalSpace (π i) s : Set ((i : ι) × π i) ⊢ (∃ i t, IsPreconnected t ∧ s = mk i '' t) → IsPreconnected s rintro ⟨a, t, ht, rfl⟩ case refine'_2.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α hι : Nonempty ι inst✝ : (i : ι) → TopologicalSpace (π i) a : ι t : Set (π a) ht : IsPreconnected t ⊢ IsPreconnected (mk a '' t) exact ht.image _ continuous_sigmaMk.continuousOn ** Qed
Sum.isConnected_iff α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) ⊢ IsConnected s ↔ (∃ t, IsConnected t ∧ s = inl '' t) ∨ ∃ t, IsConnected t ∧ s = inr '' t refine' ⟨fun hs => _, _⟩ case refine'_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsConnected s ⊢ (∃ t, IsConnected t ∧ s = inl '' t) ∨ ∃ t, IsConnected t ∧ s = inr '' t obtain ⟨x \| x, hx⟩ := hs.nonempty case refine'_1.intro.inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsConnected s x : α hx : inl x ∈ s ⊢ (∃ t, IsConnected t ∧ s = inl '' t) ∨ ∃ t, IsConnected t ∧ s = inr '' t have h : s ⊆ range Sum.inl := hs.isPreconnected.subset_clopen isClopen_range_inl ⟨.inl x, hx, x, rfl⟩ case refine'_1.intro.inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsConnected s x : α hx : inl x ∈ s h : s ⊆ range inl ⊢ (∃ t, IsConnected t ∧ s = inl '' t) ∨ ∃ t, IsConnected t ∧ s = inr '' t refine' Or.inl ⟨Sum.inl ⁻¹' s, _, _⟩ case refine'_1.intro.inl.refine'_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsConnected s x : α hx : inl x ∈ s h : s ⊆ range inl ⊢ IsConnected (inl ⁻¹' s) exact hs.preimage_of_openMap Sum.inl_injective isOpenMap_inl h case refine'_1.intro.inl.refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsConnected s x : α hx : inl x ∈ s h : s ⊆ range inl ⊢ s = inl '' (inl ⁻¹' s) exact (image_preimage_eq_of_subset h).symm case refine'_1.intro.inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsConnected s x : β hx : inr x ∈ s ⊢ (∃ t, IsConnected t ∧ s = inl '' t) ∨ ∃ t, IsConnected t ∧ s = inr '' t have h : s ⊆ range Sum.inr := hs.isPreconnected.subset_clopen isClopen_range_inr ⟨.inr x, hx, x, rfl⟩ case refine'_1.intro.inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsConnected s x : β hx : inr x ∈ s h : s ⊆ range inr ⊢ (∃ t, IsConnected t ∧ s = inl '' t) ∨ ∃ t, IsConnected t ∧ s = inr '' t refine' Or.inr ⟨Sum.inr ⁻¹' s, _, _⟩ case refine'_1.intro.inr.refine'_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsConnected s x : β hx : inr x ∈ s h : s ⊆ range inr ⊢ IsConnected (inr ⁻¹' s) exact hs.preimage_of_openMap Sum.inr_injective isOpenMap_inr h case refine'_1.intro.inr.refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsConnected s x : β hx : inr x ∈ s h : s ⊆ range inr ⊢ s = inr '' (inr ⁻¹' s) exact (image_preimage_eq_of_subset h).symm case refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) ⊢ ((∃ t, IsConnected t ∧ s = inl '' t) ∨ ∃ t, IsConnected t ∧ s = inr '' t) → IsConnected s rintro (⟨t, ht, rfl⟩ \| ⟨t, ht, rfl⟩) case refine'_2.inl.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β t : Set α ht : IsConnected t ⊢ IsConnected (inl '' t) exact ht.image _ continuous_inl.continuousOn case refine'_2.inr.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β t : Set β ht : IsConnected t ⊢ IsConnected (inr '' t) exact ht.image _ continuous_inr.continuousOn ** Qed
Sum.isPreconnected_iff α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) ⊢ IsPreconnected s ↔ (∃ t, IsPreconnected t ∧ s = inl '' t) ∨ ∃ t, IsPreconnected t ∧ s = inr '' t refine' ⟨fun hs => _, _⟩ case refine'_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsPreconnected s ⊢ (∃ t, IsPreconnected t ∧ s = inl '' t) ∨ ∃ t, IsPreconnected t ∧ s = inr '' t obtain rfl \| h := s.eq_empty_or_nonempty case refine'_1.inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsPreconnected s h : Set.Nonempty s ⊢ (∃ t, IsPreconnected t ∧ s = inl '' t) ∨ ∃ t, IsPreconnected t ∧ s = inr '' t obtain ⟨t, ht, rfl⟩ \| ⟨t, ht, rfl⟩ := Sum.isConnected_iff.1 ⟨h, hs⟩ case refine'_1.inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t u v : Set α inst✝ : TopologicalSpace β hs : IsPreconnected ∅ ⊢ (∃ t, IsPreconnected t ∧ ∅ = inl '' t) ∨ ∃ t, IsPreconnected t ∧ ∅ = inr '' t exact Or.inl ⟨∅, isPreconnected_empty, (Set.image_empty _).symm⟩ case refine'_1.inr.inl.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β t : Set α ht : IsConnected t hs : IsPreconnected (inl '' t) h : Set.Nonempty (inl '' t) ⊢ (∃ t_1, IsPreconnected t_1 ∧ inl '' t = inl '' t_1) ∨ ∃ t_1, IsPreconnected t_1 ∧ inl '' t = inr '' t_1 exact Or.inl ⟨t, ht.isPreconnected, rfl⟩ case refine'_1.inr.inr.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β t : Set β ht : IsConnected t hs : IsPreconnected (inr '' t) h : Set.Nonempty (inr '' t) ⊢ (∃ t_1, IsPreconnected t_1 ∧ inr '' t = inl '' t_1) ∨ ∃ t_1, IsPreconnected t_1 ∧ inr '' t = inr '' t_1 exact Or.inr ⟨t, ht.isPreconnected, rfl⟩ case refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) ⊢ ((∃ t, IsPreconnected t ∧ s = inl '' t) ∨ ∃ t, IsPreconnected t ∧ s = inr '' t) → IsPreconnected s rintro (⟨t, ht, rfl⟩ \| ⟨t, ht, rfl⟩) case refine'_2.inl.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β t : Set α ht : IsPreconnected t ⊢ IsPreconnected (inl '' t) exact ht.image _ continuous_inl.continuousOn case refine'_2.inr.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β t : Set β ht : IsPreconnected t ⊢ IsPreconnected (inr '' t) exact ht.image _ continuous_inr.continuousOn ** Qed
mem_connectedComponentIn α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hx : x ∈ F ⊢ x ∈ connectedComponentIn F x simp [connectedComponentIn_eq_image hx, mem_connectedComponent, hx] ** Qed
connectedComponentIn_nonempty_iff α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α ⊢ Set.Nonempty (connectedComponentIn F x) ↔ x ∈ F rw [connectedComponentIn] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α ⊢ Set.Nonempty (if h : x ∈ F then Subtype.val '' connectedComponent { val := x, property := h } else ∅) ↔ x ∈ F split_ifs <;> simp [connectedComponent_nonempty, ] * Qed
connectedComponentIn_subset α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v F : Set α x : α ⊢ connectedComponentIn F x ⊆ F rw [connectedComponentIn] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v F : Set α x : α ⊢ (if h : x ∈ F then Subtype.val '' connectedComponent { val := x, property := h } else ∅) ⊆ F split_ifs <;> simp ** Qed
isPreconnected_connectedComponentIn α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α ⊢ IsPreconnected (connectedComponentIn F x) rw [connectedComponentIn] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α ⊢ IsPreconnected (if h : x ∈ F then Subtype.val '' connectedComponent { val := x, property := h } else ∅) split_ifs case pos α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α h✝ : x ∈ F ⊢ IsPreconnected (Subtype.val '' connectedComponent { val := x, property := h✝ }) exact inducing_subtype_val.isPreconnected_image.mpr isPreconnected_connectedComponent case neg α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α h✝ : ¬x ∈ F ⊢ IsPreconnected ∅ exact isPreconnected_empty ** Qed
isConnected_connectedComponentIn_iff α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α ⊢ IsConnected (connectedComponentIn F x) ↔ x ∈ F simp_rw [← connectedComponentIn_nonempty_iff, IsConnected, isPreconnected_connectedComponentIn, and_true_iff] ** Qed
IsPreconnected.subset_connectedComponentIn α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hs : IsPreconnected s hxs : x ∈ s hsF : s ⊆ F ⊢ s ⊆ connectedComponentIn F x have : IsPreconnected (((↑) : F → α) ⁻¹' s) := by refine' inducing_subtype_val.isPreconnected_image.mp _ rwa [Subtype.image_preimage_coe, inter_eq_left.mpr hsF] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hs : IsPreconnected s hxs : x ∈ s hsF : s ⊆ F this : IsPreconnected (Subtype.val ⁻¹' s) ⊢ s ⊆ connectedComponentIn F x have h2xs : (⟨x, hsF hxs⟩ : F) ∈ (↑) ⁻¹' s := by rw [mem_preimage] exact hxs α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hs : IsPreconnected s hxs : x ∈ s hsF : s ⊆ F this : IsPreconnected (Subtype.val ⁻¹' s) h2xs : { val := x, property := (_ : x ∈ F) } ∈ Subtype.val ⁻¹' s ⊢ s ⊆ connectedComponentIn F x have := this.subset_connectedComponent h2xs α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hs : IsPreconnected s hxs : x ∈ s hsF : s ⊆ F this✝ : IsPreconnected (Subtype.val ⁻¹' s) h2xs : { val := x, property := (_ : x ∈ F) } ∈ Subtype.val ⁻¹' s this : Subtype.val ⁻¹' s ⊆ connectedComponent { val := x, property := (_ : x ∈ F) } ⊢ s ⊆ connectedComponentIn F x rw [connectedComponentIn_eq_image (hsF hxs)] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hs : IsPreconnected s hxs : x ∈ s hsF : s ⊆ F this✝ : IsPreconnected (Subtype.val ⁻¹' s) h2xs : { val := x, property := (_ : x ∈ F) } ∈ Subtype.val ⁻¹' s this : Subtype.val ⁻¹' s ⊆ connectedComponent { val := x, property := (_ : x ∈ F) } ⊢ s ⊆ Subtype.val '' connectedComponent { val := x, property := (_ : x ∈ F) } refine' Subset.trans _ (image_subset _ this) α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hs : IsPreconnected s hxs : x ∈ s hsF : s ⊆ F this✝ : IsPreconnected (Subtype.val ⁻¹' s) h2xs : { val := x, property := (_ : x ∈ F) } ∈ Subtype.val ⁻¹' s this : Subtype.val ⁻¹' s ⊆ connectedComponent { val := x, property := (_ : x ∈ F) } ⊢ s ⊆ Subtype.val '' (Subtype.val ⁻¹' s) rw [Subtype.image_preimage_coe, inter_eq_left.mpr hsF] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hs : IsPreconnected s hxs : x ∈ s hsF : s ⊆ F ⊢ IsPreconnected (Subtype.val ⁻¹' s) refine' inducing_subtype_val.isPreconnected_image.mp _ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hs : IsPreconnected s hxs : x ∈ s hsF : s ⊆ F ⊢ IsPreconnected (Subtype.val '' (Subtype.val ⁻¹' s)) rwa [Subtype.image_preimage_coe, inter_eq_left.mpr hsF] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hs : IsPreconnected s hxs : x ∈ s hsF : s ⊆ F this : IsPreconnected (Subtype.val ⁻¹' s) ⊢ { val := x, property := (_ : x ∈ F) } ∈ Subtype.val ⁻¹' s rw [mem_preimage] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hs : IsPreconnected s hxs : x ∈ s hsF : s ⊆ F this : IsPreconnected (Subtype.val ⁻¹' s) ⊢ ↑{ val := x, property := (_ : x ∈ F) } ∈ s exact hxs ** Qed
connectedComponentIn_eq α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x y : α F : Set α h : y ∈ connectedComponentIn F x ⊢ connectedComponentIn F x = connectedComponentIn F y have hx : x ∈ F := connectedComponentIn_nonempty_iff.mp ⟨y, h⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x y : α F : Set α h : y ∈ connectedComponentIn F x hx : x ∈ F ⊢ connectedComponentIn F x = connectedComponentIn F y simp_rw [connectedComponentIn_eq_image hx] at h ⊢ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x y : α F : Set α hx : x ∈ F h : y ∈ Subtype.val '' connectedComponent { val := x, property := hx } ⊢ Subtype.val '' connectedComponent { val := x, property := hx } = connectedComponentIn F y obtain ⟨⟨y, hy⟩, h2y, rfl⟩ := h case intro.mk.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hx : x ∈ F y : α hy : y ∈ F h2y : { val := y, property := hy } ∈ connectedComponent { val := x, property := hx } ⊢ Subtype.val '' connectedComponent { val := x, property := hx } = connectedComponentIn F ↑{ val := y, property := hy } simp_rw [connectedComponentIn_eq_image hy, connectedComponent_eq h2y] ** Qed
connectedComponentIn_mono α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F G : Set α h : F ⊆ G ⊢ connectedComponentIn F x ⊆ connectedComponentIn G x by_cases hx : x ∈ F case pos α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F G : Set α h : F ⊆ G hx : x ∈ F ⊢ connectedComponentIn F x ⊆ connectedComponentIn G x rw [connectedComponentIn_eq_image hx, connectedComponentIn_eq_image (h hx), ← show ((↑) : G → α) ∘ inclusion h = (↑) from rfl, image_comp] case pos α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F G : Set α h : F ⊆ G hx : x ∈ F ⊢ Subtype.val '' (inclusion h '' connectedComponent { val := x, property := hx }) ⊆ Subtype.val '' connectedComponent { val := x, property := (_ : x ∈ G) } exact image_subset _ ((continuous_inclusion h).image_connectedComponent_subset ⟨x, hx⟩) case neg α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F G : Set α h : F ⊆ G hx : ¬x ∈ F ⊢ connectedComponentIn F x ⊆ connectedComponentIn G x rw [connectedComponentIn_eq_empty hx] case neg α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F G : Set α h : F ⊆ G hx : ¬x ∈ F ⊢ ∅ ⊆ connectedComponentIn G x exact Set.empty_subset _ ** Qed
Function.Surjective.connectedSpace α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α s t u v : Set α inst✝¹ : ConnectedSpace α inst✝ : TopologicalSpace β f : α → β hf : Surjective f hf' : Continuous f ⊢ ConnectedSpace β rw [connectedSpace_iff_univ, ← hf.range_eq] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α s t u v : Set α inst✝¹ : ConnectedSpace α inst✝ : TopologicalSpace β f : α → β hf : Surjective f hf' : Continuous f ⊢ IsConnected (range f) exact isConnected_range hf' ** Qed
connectedSpace_iff_connectedComponent α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α ⊢ ConnectedSpace α ↔ ∃ x, connectedComponent x = univ constructor case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α ⊢ ConnectedSpace α → ∃ x, connectedComponent x = univ rintro ⟨⟨x⟩⟩ case mp.mk.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α toPreconnectedSpace✝ : PreconnectedSpace α x : α ⊢ ∃ x, connectedComponent x = univ exact ⟨x, eq_univ_of_univ_subset <\| isPreconnected_univ.subset_connectedComponent (mem_univ x)⟩ case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α ⊢ (∃ x, connectedComponent x = univ) → ConnectedSpace α rintro ⟨x, h⟩ case mpr.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α h : connectedComponent x = univ ⊢ ConnectedSpace α haveI : PreconnectedSpace α := ⟨by rw [← h]; exact isPreconnected_connectedComponent⟩ case mpr.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α h : connectedComponent x = univ this : PreconnectedSpace α ⊢ ConnectedSpace α exact ⟨⟨x⟩⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α h : connectedComponent x = univ ⊢ IsPreconnected univ rw [← h] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α h : connectedComponent x = univ ⊢ IsPreconnected (connectedComponent x) exact isPreconnected_connectedComponent ** Qed
preconnectedSpace_iff_connectedComponent α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α ⊢ PreconnectedSpace α ↔ ∀ (x : α), connectedComponent x = univ constructor case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α ⊢ PreconnectedSpace α → ∀ (x : α), connectedComponent x = univ intro h x case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : PreconnectedSpace α x : α ⊢ connectedComponent x = univ exact eq_univ_of_univ_subset <\| isPreconnected_univ.subset_connectedComponent (mem_univ x) case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α ⊢ (∀ (x : α), connectedComponent x = univ) → PreconnectedSpace α intro h case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : ∀ (x : α), connectedComponent x = univ ⊢ PreconnectedSpace α cases' isEmpty_or_nonempty α with hα hα case mpr.inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : ∀ (x : α), connectedComponent x = univ hα : IsEmpty α ⊢ PreconnectedSpace α exact ⟨by rw [univ_eq_empty_iff.mpr hα]; exact isPreconnected_empty⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : ∀ (x : α), connectedComponent x = univ hα : IsEmpty α ⊢ IsPreconnected univ rw [univ_eq_empty_iff.mpr hα] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : ∀ (x : α), connectedComponent x = univ hα : IsEmpty α ⊢ IsPreconnected ∅ exact isPreconnected_empty case mpr.inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : ∀ (x : α), connectedComponent x = univ hα : Nonempty α ⊢ PreconnectedSpace α exact ⟨by rw [← h (Classical.choice hα)]; exact isPreconnected_connectedComponent⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : ∀ (x : α), connectedComponent x = univ hα : Nonempty α ⊢ IsPreconnected univ rw [← h (Classical.choice hα)] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : ∀ (x : α), connectedComponent x = univ hα : Nonempty α ⊢ IsPreconnected (connectedComponent (Classical.choice hα)) exact isPreconnected_connectedComponent ** Qed
Continuous.exists_lift_sigma α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α s t u v : Set α inst✝¹ : ConnectedSpace α inst✝ : (i : ι) → TopologicalSpace (π i) f : α → (i : ι) × π i hf : Continuous f ⊢ ∃ i g, Continuous g ∧ f = Sigma.mk i ∘ g obtain ⟨i, hi⟩ : ∃ i, range f ⊆ range (.mk i) case intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α s t u v : Set α inst✝¹ : ConnectedSpace α inst✝ : (i : ι) → TopologicalSpace (π i) f : α → (i : ι) × π i hf : Continuous f i : ι hi : range f ⊆ range (Sigma.mk i) ⊢ ∃ i g, Continuous g ∧ f = Sigma.mk i ∘ g rcases range_subset_range_iff_exists_comp.1 hi with ⟨g, rfl⟩ case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α s t u v : Set α inst✝¹ : ConnectedSpace α inst✝ : (i : ι) → TopologicalSpace (π i) i : ι g : α → π i hf : Continuous (Sigma.mk i ∘ g) hi : range (Sigma.mk i ∘ g) ⊆ range (Sigma.mk i) ⊢ ∃ i_1 g_1, Continuous g_1 ∧ Sigma.mk i ∘ g = Sigma.mk i_1 ∘ g_1 refine ⟨i, g, ?_, rfl⟩ case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α s t u v : Set α inst✝¹ : ConnectedSpace α inst✝ : (i : ι) → TopologicalSpace (π i) i : ι g : α → π i hf : Continuous (Sigma.mk i ∘ g) hi : range (Sigma.mk i ∘ g) ⊆ range (Sigma.mk i) ⊢ Continuous g rwa [← embedding_sigmaMk.continuous_iff] at hf α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α s t u v : Set α inst✝¹ : ConnectedSpace α inst✝ : (i : ι) → TopologicalSpace (π i) f : α → (i : ι) × π i hf : Continuous f ⊢ ∃ i, range f ⊆ range (Sigma.mk i) rcases Sigma.isConnected_iff.1 (isConnected_range hf) with ⟨i, s, -, hs⟩ case intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α s✝ t u v : Set α inst✝¹ : ConnectedSpace α inst✝ : (i : ι) → TopologicalSpace (π i) f : α → (i : ι) × π i hf : Continuous f i : ι s : Set (π i) hs : range f = Sigma.mk i '' s ⊢ ∃ i, range f ⊆ range (Sigma.mk i) exact ⟨i, hs.trans_subset (image_subset_range _ _)⟩ ** Qed
nonempty_inter α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t✝ u v : Set α inst✝ : PreconnectedSpace α s t : Set α ⊢ IsOpen s → IsOpen t → s ∪ t = univ → Set.Nonempty s → Set.Nonempty t → Set.Nonempty (s ∩ t) simpa only [univ_inter, univ_subset_iff] using @PreconnectedSpace.isPreconnected_univ α _ _ s t ** Qed
isClopen_iff α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : PreconnectedSpace α s : Set α hs : IsClopen s h : ¬(s = ∅ ∨ s = univ) h2 : sᶜ = ∅ ⊢ s = univ rw [← compl_compl s, h2, compl_empty] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : PreconnectedSpace α s : Set α ⊢ s = ∅ ∨ s = univ → IsClopen s rintro (rfl \| rfl) <;> [exact isClopen_empty; exact isClopen_univ] ** Qed
nonempty_frontier_iff α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : PreconnectedSpace α s : Set α ⊢ Set.Nonempty (frontier s) ↔ Set.Nonempty s ∧ s ≠ univ simp only [nonempty_iff_ne_empty, Ne.def, frontier_eq_empty_iff, not_or] ** Qed
Subtype.preconnectedSpace α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α h : IsPreconnected s ⊢ IsPreconnected univ rwa [← inducing_subtype_val.isPreconnected_image, image_univ, Subtype.range_val] ** Qed
isPreconnected_iff_preconnectedSpace α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α h : PreconnectedSpace ↑s ⊢ IsPreconnected s simpa using isPreconnected_univ.image ((↑) : s → α) continuous_subtype_val.continuousOn ** Qed
isPreconnected_iff_subset_of_disjoint α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α ⊢ IsPreconnected s ↔ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v constructor <;> intro h case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α h : IsPreconnected s ⊢ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v intro u v hu hv hs huv case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : IsPreconnected s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ ⊢ s ⊆ u ∨ s ⊆ v specialize h u v hu hv hs case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) ⊢ s ⊆ u ∨ s ⊆ v contrapose! huv case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) huv : ¬s ⊆ u ∧ ¬s ⊆ v ⊢ s ∩ (u ∩ v) ≠ ∅ rw [← nonempty_iff_ne_empty] case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) huv : ¬s ⊆ u ∧ ¬s ⊆ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) simp [not_subset] at huv case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) huv : (∃ a, a ∈ s ∧ ¬a ∈ u) ∧ ∃ a, a ∈ s ∧ ¬a ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) rcases huv with ⟨⟨x, hxs, hxu⟩, ⟨y, hys, hyv⟩⟩ case mp.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) x : α hxs : x ∈ s hxu : ¬x ∈ u y : α hys : y ∈ s hyv : ¬y ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) have hxv : x ∈ v := or_iff_not_imp_left.mp (hs hxs) hxu case mp.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) x : α hxs : x ∈ s hxu : ¬x ∈ u y : α hys : y ∈ s hyv : ¬y ∈ v hxv : x ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) have hyu : y ∈ u := or_iff_not_imp_right.mp (hs hys) hyv case mp.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) x : α hxs : x ∈ s hxu : ¬x ∈ u y : α hys : y ∈ s hyv : ¬y ∈ v hxv : x ∈ v hyu : y ∈ u ⊢ Set.Nonempty (s ∩ (u ∩ v)) exact h ⟨y, hys, hyu⟩ ⟨x, hxs, hxv⟩ case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v ⊢ IsPreconnected s intro u v hu hv hs hsu hsv case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) ⊢ Set.Nonempty (s ∩ (u ∩ v)) rw [nonempty_iff_ne_empty] case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) ⊢ s ∩ (u ∩ v) ≠ ∅ intro H case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) H : s ∩ (u ∩ v) = ∅ ⊢ False specialize h u v hu hv hs H case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) H : s ∩ (u ∩ v) = ∅ h : s ⊆ u ∨ s ⊆ v ⊢ False contrapose H case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) h : s ⊆ u ∨ s ⊆ v H : ¬False ⊢ ¬s ∩ (u ∩ v) = ∅ apply Nonempty.ne_empty case mpr.a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) h : s ⊆ u ∨ s ⊆ v H : ¬False ⊢ Set.Nonempty (s ∩ (u ∩ v)) cases' h with h h case mpr.a.inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) H : ¬False h : s ⊆ u ⊢ Set.Nonempty (s ∩ (u ∩ v)) rcases hsv with ⟨x, hxs, hxv⟩ case mpr.a.inl.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) H : ¬False h : s ⊆ u x : α hxs : x ∈ s hxv : x ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) exact ⟨x, hxs, ⟨h hxs, hxv⟩⟩ case mpr.a.inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) H : ¬False h : s ⊆ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) rcases hsu with ⟨x, hxs, hxu⟩ case mpr.a.inr.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsv : Set.Nonempty (s ∩ v) H : ¬False h : s ⊆ v x : α hxs : x ∈ s hxu : x ∈ u ⊢ Set.Nonempty (s ∩ (u ∩ v)) exact ⟨x, hxs, ⟨hxu, h hxs⟩⟩ ** Qed
isConnected_iff_sUnion_disjoint_open α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α ⊢ IsConnected s ↔ ∀ (U : Finset (Set α)), (∀ (u v : Set α), u ∈ U → v ∈ U → Set.Nonempty (s ∩ (u ∩ v)) → u = v) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u rw [IsConnected, isPreconnected_iff_subset_of_disjoint] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α ⊢ (Set.Nonempty s ∧ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v) ↔ ∀ (U : Finset (Set α)), (∀ (u v : Set α), u ∈ U → v ∈ U → Set.Nonempty (s ∩ (u ∩ v)) → u = v) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u refine ⟨fun ⟨hne, h⟩ U hU hUo hsU => ?_, fun h => ⟨?_, fun u v hu hv hs hsuv => ?_⟩⟩ case refine_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α x✝ : Set.Nonempty s ∧ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v U : Finset (Set α) hU : ∀ (u v : Set α), u ∈ U → v ∈ U → Set.Nonempty (s ∩ (u ∩ v)) → u = v hUo : ∀ (u : Set α), u ∈ U → IsOpen u hsU : s ⊆ ⋃₀ ↑U hne : Set.Nonempty s h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v ⊢ ∃ u, u ∈ U ∧ s ⊆ u induction U using Finset.induction_on case refine_1.empty α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α x✝ : Set.Nonempty s ∧ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v hne : Set.Nonempty s h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v hU : ∀ (u v : Set α), u ∈ ∅ → v ∈ ∅ → Set.Nonempty (s ∩ (u ∩ v)) → u = v hUo : ∀ (u : Set α), u ∈ ∅ → IsOpen u hsU : s ⊆ ⋃₀ ↑∅ ⊢ ∃ u, u ∈ ∅ ∧ s ⊆ u case refine_1.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝¹ t u v s : Set α x✝ : Set.Nonempty s ∧ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v hne : Set.Nonempty s h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v a✝² : Set α s✝ : Finset (Set α) a✝¹ : ¬a✝² ∈ s✝ a✝ : (∀ (u v : Set α), u ∈ s✝ → v ∈ s✝ → Set.Nonempty (s ∩ (u ∩ v)) → u = v) → (∀ (u : Set α), u ∈ s✝ → IsOpen u) → s ⊆ ⋃₀ ↑s✝ → ∃ u, u ∈ s✝ ∧ s ⊆ u hU : ∀ (u v : Set α), u ∈ insert a✝² s✝ → v ∈ insert a✝² s✝ → Set.Nonempty (s ∩ (u ∩ v)) → u = v hUo : ∀ (u : Set α), u ∈ insert a✝² s✝ → IsOpen u hsU : s ⊆ ⋃₀ ↑(insert a✝² s✝) ⊢ ∃ u, u ∈ insert a✝² s✝ ∧ s ⊆ u case empty => exact absurd (by simpa using hsU) hne.not_subset_empty α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α x✝ : Set.Nonempty s ∧ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v hne : Set.Nonempty s h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v hU : ∀ (u v : Set α), u ∈ ∅ → v ∈ ∅ → Set.Nonempty (s ∩ (u ∩ v)) → u = v hUo : ∀ (u : Set α), u ∈ ∅ → IsOpen u hsU : s ⊆ ⋃₀ ↑∅ ⊢ ∃ u, u ∈ ∅ ∧ s ⊆ u exact absurd (by simpa using hsU) hne.not_subset_empty α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α x✝ : Set.Nonempty s ∧ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v hne : Set.Nonempty s h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v hU : ∀ (u v : Set α), u ∈ ∅ → v ∈ ∅ → Set.Nonempty (s ∩ (u ∩ v)) → u = v hUo : ∀ (u : Set α), u ∈ ∅ → IsOpen u hsU : s ⊆ ⋃₀ ↑∅ ⊢ s ⊆ ∅ simpa using hsU α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v s : Set α x✝ : Set.Nonempty s ∧ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v hne : Set.Nonempty s h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v u : Set α U : Finset (Set α) uU : ¬u ∈ U IH : (∀ (u v : Set α), u ∈ U → v ∈ U → Set.Nonempty (s ∩ (u ∩ v)) → u = v) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u hU : ∀ (u_1 v : Set α), u_1 ∈ insert u U → v ∈ insert u U → Set.Nonempty (s ∩ (u_1 ∩ v)) → u_1 = v hUo : ∀ (u_1 : Set α), u_1 ∈ insert u U → IsOpen u_1 hsU : s ⊆ ⋃₀ ↑(insert u U) ⊢ ∃ u_1, u_1 ∈ insert u U ∧ s ⊆ u_1 simp only [← ball_cond_comm, Finset.forall_mem_insert, Finset.exists_mem_insert, Finset.coe_insert, sUnion_insert, implies_true, true_and] at * α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v s : Set α hne : Set.Nonempty s u : Set α U : Finset (Set α) uU : ¬u ∈ U x✝ : Set.Nonempty s ∧ ∀ (a : Set α), IsOpen a → ∀ (b : Set α), IsOpen b → s ⊆ a ∪ b → s ∩ (a ∩ b) = ∅ → s ⊆ a ∨ s ⊆ b h : ∀ (a : Set α), IsOpen a → ∀ (b : Set α), IsOpen b → s ⊆ a ∪ b → s ∩ (a ∩ b) = ∅ → s ⊆ a ∨ s ⊆ b IH : (∀ (a : Set α), a ∈ U → ∀ (b : Set α), b ∈ U → Set.Nonempty (s ∩ (a ∩ b)) → a = b) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u hU : (∀ (x : Set α), x ∈ U → Set.Nonempty (s ∩ (u ∩ x)) → u = x) ∧ ∀ (x : Set α), x ∈ U → (Set.Nonempty (s ∩ (x ∩ u)) → x = u) ∧ ∀ (x_1 : Set α), x_1 ∈ U → Set.Nonempty (s ∩ (x ∩ x_1)) → x = x_1 hUo : IsOpen u ∧ ∀ (x : Set α), x ∈ U → IsOpen x hsU : s ⊆ u ∪ ⋃₀ ↑U ⊢ s ⊆ u ∨ ∃ x, x ∈ U ∧ s ⊆ x refine (h _ hUo.1 (⋃₀ ↑U) (isOpen_sUnion hUo.2) hsU ?_).imp_right ?_ case refine_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v s : Set α hne : Set.Nonempty s u : Set α U : Finset (Set α) uU : ¬u ∈ U x✝ : Set.Nonempty s ∧ ∀ (a : Set α), IsOpen a → ∀ (b : Set α), IsOpen b → s ⊆ a ∪ b → s ∩ (a ∩ b) = ∅ → s ⊆ a ∨ s ⊆ b h : ∀ (a : Set α), IsOpen a → ∀ (b : Set α), IsOpen b → s ⊆ a ∪ b → s ∩ (a ∩ b) = ∅ → s ⊆ a ∨ s ⊆ b IH : (∀ (a : Set α), a ∈ U → ∀ (b : Set α), b ∈ U → Set.Nonempty (s ∩ (a ∩ b)) → a = b) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u hU : (∀ (x : Set α), x ∈ U → Set.Nonempty (s ∩ (u ∩ x)) → u = x) ∧ ∀ (x : Set α), x ∈ U → (Set.Nonempty (s ∩ (x ∩ u)) → x = u) ∧ ∀ (x_1 : Set α), x_1 ∈ U → Set.Nonempty (s ∩ (x ∩ x_1)) → x = x_1 hUo : IsOpen u ∧ ∀ (x : Set α), x ∈ U → IsOpen x hsU : s ⊆ u ∪ ⋃₀ ↑U ⊢ s ∩ (u ∩ ⋃₀ ↑U) = ∅ refine subset_empty_iff.1 fun x ⟨hxs, hxu, v, hvU, hxv⟩ => ?_ case refine_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hne : Set.Nonempty s u : Set α U : Finset (Set α) uU : ¬u ∈ U x✝¹ : Set.Nonempty s ∧ ∀ (a : Set α), IsOpen a → ∀ (b : Set α), IsOpen b → s ⊆ a ∪ b → s ∩ (a ∩ b) = ∅ → s ⊆ a ∨ s ⊆ b h : ∀ (a : Set α), IsOpen a → ∀ (b : Set α), IsOpen b → s ⊆ a ∪ b → s ∩ (a ∩ b) = ∅ → s ⊆ a ∨ s ⊆ b IH : (∀ (a : Set α), a ∈ U → ∀ (b : Set α), b ∈ U → Set.Nonempty (s ∩ (a ∩ b)) → a = b) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u hU : (∀ (x : Set α), x ∈ U → Set.Nonempty (s ∩ (u ∩ x)) → u = x) ∧ ∀ (x : Set α), x ∈ U → (Set.Nonempty (s ∩ (x ∩ u)) → x = u) ∧ ∀ (x_1 : Set α), x_1 ∈ U → Set.Nonempty (s ∩ (x ∩ x_1)) → x = x_1 hUo : IsOpen u ∧ ∀ (x : Set α), x ∈ U → IsOpen x hsU : s ⊆ u ∪ ⋃₀ ↑U x : α x✝ : x ∈ s ∩ (u ∩ ⋃₀ ↑U) hxs : x ∈ s hxu : x ∈ u v : Set α hvU : v ∈ ↑U hxv : x ∈ v ⊢ x ∈ ∅ exact ne_of_mem_of_not_mem hvU uU (hU.1 v hvU ⟨x, hxs, hxu, hxv⟩).symm case refine_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v s : Set α hne : Set.Nonempty s u : Set α U : Finset (Set α) uU : ¬u ∈ U x✝ : Set.Nonempty s ∧ ∀ (a : Set α), IsOpen a → ∀ (b : Set α), IsOpen b → s ⊆ a ∪ b → s ∩ (a ∩ b) = ∅ → s ⊆ a ∨ s ⊆ b h : ∀ (a : Set α), IsOpen a → ∀ (b : Set α), IsOpen b → s ⊆ a ∪ b → s ∩ (a ∩ b) = ∅ → s ⊆ a ∨ s ⊆ b IH : (∀ (a : Set α), a ∈ U → ∀ (b : Set α), b ∈ U → Set.Nonempty (s ∩ (a ∩ b)) → a = b) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u hU : (∀ (x : Set α), x ∈ U → Set.Nonempty (s ∩ (u ∩ x)) → u = x) ∧ ∀ (x : Set α), x ∈ U → (Set.Nonempty (s ∩ (x ∩ u)) → x = u) ∧ ∀ (x_1 : Set α), x_1 ∈ U → Set.Nonempty (s ∩ (x ∩ x_1)) → x = x_1 hUo : IsOpen u ∧ ∀ (x : Set α), x ∈ U → IsOpen x hsU : s ⊆ u ∪ ⋃₀ ↑U ⊢ s ⊆ ⋃₀ ↑U → ∃ x, x ∈ U ∧ s ⊆ x exact IH (fun u hu => (hU.2 u hu).2) hUo.2 case refine_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α h : ∀ (U : Finset (Set α)), (∀ (u v : Set α), u ∈ U → v ∈ U → Set.Nonempty (s ∩ (u ∩ v)) → u = v) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u ⊢ Set.Nonempty s simpa [subset_empty_iff, nonempty_iff_ne_empty] using h ∅ case refine_3 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (U : Finset (Set α)), (∀ (u v : Set α), u ∈ U → v ∈ U → Set.Nonempty (s ∩ (u ∩ v)) → u = v) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsuv : s ∩ (u ∩ v) = ∅ ⊢ s ⊆ u ∨ s ⊆ v rw [← not_nonempty_iff_eq_empty] at hsuv case refine_3 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (U : Finset (Set α)), (∀ (u v : Set α), u ∈ U → v ∈ U → Set.Nonempty (s ∩ (u ∩ v)) → u = v) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsuv : ¬Set.Nonempty (s ∩ (u ∩ v)) ⊢ s ⊆ u ∨ s ⊆ v have := hsuv case refine_3 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (U : Finset (Set α)), (∀ (u v : Set α), u ∈ U → v ∈ U → Set.Nonempty (s ∩ (u ∩ v)) → u = v) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsuv this : ¬Set.Nonempty (s ∩ (u ∩ v)) ⊢ s ⊆ u ∨ s ⊆ v rw [inter_comm u] at this case refine_3 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (U : Finset (Set α)), (∀ (u v : Set α), u ∈ U → v ∈ U → Set.Nonempty (s ∩ (u ∩ v)) → u = v) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsuv : ¬Set.Nonempty (s ∩ (u ∩ v)) this : ¬Set.Nonempty (s ∩ (v ∩ u)) ⊢ s ⊆ u ∨ s ⊆ v simpa [, or_imp, forall_and] using h {u, v} * Qed
isPreconnected_iff_subset_of_disjoint_closed α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α ⊢ IsPreconnected s ↔ ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v constructor <;> intro h case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : IsPreconnected s ⊢ ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v intro u v hu hv hs huv case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ : Set α h : IsPreconnected s u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ ⊢ s ⊆ u ∨ s ⊆ v rw [isPreconnected_closed_iff] at h case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ : Set α h : ∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t')) u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ ⊢ s ⊆ u ∨ s ⊆ v specialize h u v hu hv hs case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) ⊢ s ⊆ u ∨ s ⊆ v contrapose! huv case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) huv : ¬s ⊆ u ∧ ¬s ⊆ v ⊢ s ∩ (u ∩ v) ≠ ∅ rw [← nonempty_iff_ne_empty] case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) huv : ¬s ⊆ u ∧ ¬s ⊆ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) simp [not_subset] at huv case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) huv : (∃ a, a ∈ s ∧ ¬a ∈ u) ∧ ∃ a, a ∈ s ∧ ¬a ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) rcases huv with ⟨⟨x, hxs, hxu⟩, ⟨y, hys, hyv⟩⟩ case mp.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) x : α hxs : x ∈ s hxu : ¬x ∈ u y : α hys : y ∈ s hyv : ¬y ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) have hxv : x ∈ v := or_iff_not_imp_left.mp (hs hxs) hxu case mp.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) x : α hxs : x ∈ s hxu : ¬x ∈ u y : α hys : y ∈ s hyv : ¬y ∈ v hxv : x ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) have hyu : y ∈ u := or_iff_not_imp_right.mp (hs hys) hyv case mp.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) x : α hxs : x ∈ s hxu : ¬x ∈ u y : α hys : y ∈ s hyv : ¬y ∈ v hxv : x ∈ v hyu : y ∈ u ⊢ Set.Nonempty (s ∩ (u ∩ v)) exact h ⟨y, hys, hyu⟩ ⟨x, hxs, hxv⟩ case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v ⊢ IsPreconnected s rw [isPreconnected_closed_iff] case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v ⊢ ∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t')) intro u v hu hv hs hsu hsv case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ : Set α h : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) ⊢ Set.Nonempty (s ∩ (u ∩ v)) rw [nonempty_iff_ne_empty] case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ : Set α h : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) ⊢ s ∩ (u ∩ v) ≠ ∅ intro H case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ : Set α h : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) H : s ∩ (u ∩ v) = ∅ ⊢ False specialize h u v hu hv hs H case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) H : s ∩ (u ∩ v) = ∅ h : s ⊆ u ∨ s ⊆ v ⊢ False contrapose H case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) h : s ⊆ u ∨ s ⊆ v H : ¬False ⊢ ¬s ∩ (u ∩ v) = ∅ apply Nonempty.ne_empty case mpr.a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) h : s ⊆ u ∨ s ⊆ v H : ¬False ⊢ Set.Nonempty (s ∩ (u ∩ v)) cases' h with h h case mpr.a.inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) H : ¬False h : s ⊆ u ⊢ Set.Nonempty (s ∩ (u ∩ v)) rcases hsv with ⟨x, hxs, hxv⟩ case mpr.a.inl.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) H : ¬False h : s ⊆ u x : α hxs : x ∈ s hxv : x ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) exact ⟨x, hxs, ⟨h hxs, hxv⟩⟩ case mpr.a.inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) H : ¬False h : s ⊆ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) rcases hsu with ⟨x, hxs, hxu⟩ case mpr.a.inr.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v hsv : Set.Nonempty (s ∩ v) H : ¬False h : s ⊆ v x : α hxs : x ∈ s hxu : x ∈ u ⊢ Set.Nonempty (s ∩ (u ∩ v)) exact ⟨x, hxs, ⟨hxu, h hxs⟩⟩ ** Qed
isPreconnected_iff_subset_of_fully_disjoint_closed α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α hs : IsClosed s ⊢ IsPreconnected s ↔ ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v refine isPreconnected_iff_subset_of_disjoint_closed.trans ⟨?_, ?_⟩ <;> intro H u v hu hv hss huv case refine_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : IsClosed s H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hss : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ ⊢ s ⊆ u ∨ s ⊆ v have H1 := H (u ∩ s) (v ∩ s) case refine_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : IsClosed s H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hss : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ H1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∩ s ∨ s ⊆ v ∩ s ⊢ s ⊆ u ∨ s ⊆ v rw [subset_inter_iff, subset_inter_iff] at H1 case refine_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : IsClosed s H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hss : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ H1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∧ s ⊆ s ∨ s ⊆ v ∧ s ⊆ s ⊢ s ⊆ u ∨ s ⊆ v simp only [Subset.refl, and_true] at H1 case refine_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : IsClosed s H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hss : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ H1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∨ s ⊆ v ⊢ s ⊆ u ∨ s ⊆ v apply H1 (hu.inter hs) (hv.inter hs) case refine_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : IsClosed s H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hss : s ⊆ u ∪ v huv : Disjoint u v ⊢ s ⊆ u ∨ s ⊆ v apply H u v hu hv hss case refine_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : IsClosed s H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hss : s ⊆ u ∪ v huv : Disjoint u v ⊢ s ∩ (u ∩ v) = ∅ rw [huv.inter_eq, inter_empty] case refine_2.a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : IsClosed s H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hss : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ H1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∨ s ⊆ v ⊢ s ⊆ u ∩ s ∪ v ∩ s rw [← inter_distrib_right] case refine_2.a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : IsClosed s H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hss : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ H1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∨ s ⊆ v ⊢ s ⊆ (u ∪ v) ∩ s exact subset_inter hss Subset.rfl case refine_2.a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : IsClosed s H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hss : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ H1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∨ s ⊆ v ⊢ Disjoint (u ∩ s) (v ∩ s) rwa [disjoint_iff_inter_eq_empty, ← inter_inter_distrib_right, inter_comm] ** Qed
preimage_connectedComponent_connected α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β ⊢ IsConnected (f ⁻¹' connectedComponent t) have hf : Surjective f := Surjective.of_comp fun t : β => (connected_fibers t).1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f ⊢ IsConnected (f ⁻¹' connectedComponent t) refine ⟨Nonempty.preimage connectedComponent_nonempty hf, ?_⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f ⊢ IsPreconnected (f ⁻¹' connectedComponent t) have hT : IsClosed (f ⁻¹' connectedComponent t) := (hcl (connectedComponent t)).1 isClosed_connectedComponent α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) ⊢ IsPreconnected (f ⁻¹' connectedComponent t) rw [isPreconnected_iff_subset_of_fully_disjoint_closed hT] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) ⊢ ∀ (u v : Set α), IsClosed u → IsClosed v → f ⁻¹' connectedComponent t ⊆ u ∪ v → Disjoint u v → f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v intro u v hu hv huv uv_disj α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v ⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v let T₁ := { t' ∈ connectedComponent t \| f ⁻¹' {t'} ⊆ u } α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} ⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v let T₂ := { t' ∈ connectedComponent t \| f ⁻¹' {t'} ⊆ v } α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v ⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v have hT₁ : IsClosed T₁ := (hcl T₁).2 (T₁_u.symm ▸ IsClosed.inter hT hu) α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ ⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v have hT₂ : IsClosed T₂ := (hcl T₂).2 (T₂_v.symm ▸ IsClosed.inter hT hv) α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ ⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v have T_disjoint : Disjoint T₁ T₂ := by refine' Disjoint.of_preimage hf _ rw [T₁_u, T₂_v, disjoint_iff_inter_eq_empty, ← inter_inter_distrib_left, uv_disj.inter_eq, inter_empty] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ T_disjoint : Disjoint T₁ T₂ ⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v cases' (isPreconnected_iff_subset_of_fully_disjoint_closed isClosed_connectedComponent).1 isPreconnected_connectedComponent T₁ T₂ hT₁ hT₂ T_decomp T_disjoint with h h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} ⊢ ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v intro t' ht' α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} t' : β ht' : t' ∈ connectedComponent t ⊢ f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v apply isPreconnected_iff_subset_of_disjoint_closed.1 (connected_fibers t').2 u v hu hv case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} t' : β ht' : t' ∈ connectedComponent t ⊢ f ⁻¹' {t'} ∩ (u ∩ v) = ∅ rw [uv_disj.inter_eq, inter_empty] case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} t' : β ht' : t' ∈ connectedComponent t ⊢ f ⁻¹' {t'} ⊆ u ∪ v exact Subset.trans (preimage_mono (singleton_subset_iff.2 ht')) huv α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v ⊢ f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u apply eq_of_subset_of_subset case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v ⊢ f ⁻¹' connectedComponent t ∩ u ⊆ f ⁻¹' T₁ rintro a ⟨hat, hau⟩ case a.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v a : α hat : a ∈ f ⁻¹' connectedComponent t hau : a ∈ u ⊢ a ∈ f ⁻¹' T₁ constructor case a.intro.right α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v a : α hat : a ∈ f ⁻¹' connectedComponent t hau : a ∈ u ⊢ f ⁻¹' {f a} ⊆ u refine (fiber_decomp (f a) (mem_preimage.1 hat)).resolve_right fun h => ?_ case a.intro.right α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v a : α hat : a ∈ f ⁻¹' connectedComponent t hau : a ∈ u h : f ⁻¹' {f a} ⊆ v ⊢ False exact uv_disj.subset_compl_right hau (h rfl) case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v ⊢ f ⁻¹' T₁ ⊆ f ⁻¹' connectedComponent t ∩ u rw [← biUnion_preimage_singleton] case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v ⊢ ⋃ y ∈ T₁, f ⁻¹' {y} ⊆ f ⁻¹' connectedComponent t ∩ u refine' iUnion₂_subset fun t' ht' => subset_inter _ ht'.2 case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v t' : β ht' : t' ∈ T₁ ⊢ f ⁻¹' {t'} ⊆ f ⁻¹' connectedComponent t rw [hf.preimage_subset_preimage_iff, singleton_subset_iff] case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v t' : β ht' : t' ∈ T₁ ⊢ t' ∈ connectedComponent t exact ht'.1 case a.intro.left α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v a : α hat : a ∈ f ⁻¹' connectedComponent t hau : a ∈ u ⊢ f a ∈ connectedComponent t exact mem_preimage.1 hat α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u ⊢ f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v apply eq_of_subset_of_subset case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u ⊢ f ⁻¹' connectedComponent t ∩ v ⊆ f ⁻¹' T₂ rintro a ⟨hat, hav⟩ case a.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u a : α hat : a ∈ f ⁻¹' connectedComponent t hav : a ∈ v ⊢ a ∈ f ⁻¹' T₂ constructor case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u ⊢ f ⁻¹' T₂ ⊆ f ⁻¹' connectedComponent t ∩ v rw [← biUnion_preimage_singleton] case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u ⊢ ⋃ y ∈ T₂, f ⁻¹' {y} ⊆ f ⁻¹' connectedComponent t ∩ v refine' iUnion₂_subset fun t' ht' => subset_inter _ ht'.2 case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u t' : β ht' : t' ∈ T₂ ⊢ f ⁻¹' {t'} ⊆ f ⁻¹' connectedComponent t rw [hf.preimage_subset_preimage_iff, singleton_subset_iff] case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u t' : β ht' : t' ∈ T₂ ⊢ t' ∈ connectedComponent t exact ht'.1 case a.intro.left α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u a : α hat : a ∈ f ⁻¹' connectedComponent t hav : a ∈ v ⊢ f a ∈ connectedComponent t exact mem_preimage.1 hat case a.intro.right α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u a : α hat : a ∈ f ⁻¹' connectedComponent t hav : a ∈ v ⊢ f ⁻¹' {f a} ⊆ v refine (fiber_decomp (f a) (mem_preimage.1 hat)).resolve_left fun h => ?_ case a.intro.right α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u a : α hat : a ∈ f ⁻¹' connectedComponent t hav : a ∈ v h : f ⁻¹' {f a} ⊆ u ⊢ False exact uv_disj.subset_compl_left hav (h rfl) α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ t' : β ht' : t' ∈ connectedComponent t ⊢ t' ∈ T₁ ∪ T₂ rw [mem_union t' T₁ T₂] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ t' : β ht' : t' ∈ connectedComponent t ⊢ t' ∈ T₁ ∨ t' ∈ T₂ cases' fiber_decomp t' ht' with htu htv case inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ t' : β ht' : t' ∈ connectedComponent t htv : f ⁻¹' {t'} ⊆ v ⊢ t' ∈ T₁ ∨ t' ∈ T₂ right case inr.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ t' : β ht' : t' ∈ connectedComponent t htv : f ⁻¹' {t'} ⊆ v ⊢ t' ∈ T₂ exact ⟨ht', htv⟩ case inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ t' : β ht' : t' ∈ connectedComponent t htu : f ⁻¹' {t'} ⊆ u ⊢ t' ∈ T₁ ∨ t' ∈ T₂ left case inl.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ t' : β ht' : t' ∈ connectedComponent t htu : f ⁻¹' {t'} ⊆ u ⊢ t' ∈ T₁ exact ⟨ht', htu⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ ⊢ Disjoint T₁ T₂ refine' Disjoint.of_preimage hf _ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ ⊢ Disjoint (f ⁻¹' T₁) (f ⁻¹' T₂) rw [T₁_u, T₂_v, disjoint_iff_inter_eq_empty, ← inter_inter_distrib_left, uv_disj.inter_eq, inter_empty] case inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ T_disjoint : Disjoint T₁ T₂ h : connectedComponent t ⊆ T₁ ⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v left case inl.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ T_disjoint : Disjoint T₁ T₂ h : connectedComponent t ⊆ T₁ ⊢ f ⁻¹' connectedComponent t ⊆ u rw [Subset.antisymm_iff] at T₁_u case inl.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ ⊆ f ⁻¹' connectedComponent t ∩ u ∧ f ⁻¹' connectedComponent t ∩ u ⊆ f ⁻¹' T₁ T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ T_disjoint : Disjoint T₁ T₂ h : connectedComponent t ⊆ T₁ ⊢ f ⁻¹' connectedComponent t ⊆ u suffices f ⁻¹' connectedComponent t ⊆ f ⁻¹' T₁ from (this.trans T₁_u.1).trans (inter_subset_right _ _) case inl.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ ⊆ f ⁻¹' connectedComponent t ∩ u ∧ f ⁻¹' connectedComponent t ∩ u ⊆ f ⁻¹' T₁ T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ T_disjoint : Disjoint T₁ T₂ h : connectedComponent t ⊆ T₁ ⊢ f ⁻¹' connectedComponent t ⊆ f ⁻¹' T₁ exact preimage_mono h case inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ T_disjoint : Disjoint T₁ T₂ h : connectedComponent t ⊆ T₂ ⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v right case inr.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ T_disjoint : Disjoint T₁ T₂ h : connectedComponent t ⊆ T₂ ⊢ f ⁻¹' connectedComponent t ⊆ v rw [Subset.antisymm_iff] at T₂_v case inr.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ ⊆ f ⁻¹' connectedComponent t ∩ v ∧ f ⁻¹' connectedComponent t ∩ v ⊆ f ⁻¹' T₂ hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ T_disjoint : Disjoint T₁ T₂ h : connectedComponent t ⊆ T₂ ⊢ f ⁻¹' connectedComponent t ⊆ v suffices f ⁻¹' connectedComponent t ⊆ f ⁻¹' T₂ from (this.trans T₂_v.1).trans (inter_subset_right _ _) case inr.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ ⊆ f ⁻¹' connectedComponent t ∩ v ∧ f ⁻¹' connectedComponent t ∩ v ⊆ f ⁻¹' T₂ hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ T_disjoint : Disjoint T₁ T₂ h : connectedComponent t ⊆ T₂ ⊢ f ⁻¹' connectedComponent t ⊆ f ⁻¹' T₂ exact preimage_mono h ** Qed
QuotientMap.image_connectedComponent α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t u v : Set α inst✝ : TopologicalSpace β f : α → β hf : QuotientMap f h_fibers : ∀ (y : β), IsConnected (f ⁻¹' {y}) a : α ⊢ f '' connectedComponent a = connectedComponent (f a) rw [← hf.preimage_connectedComponent h_fibers, image_preimage_eq _ hf.surjective] ** Qed
connectedComponents_preimage_singleton α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α ⊢ ConnectedComponents.mk ⁻¹' {ConnectedComponents.mk x} = connectedComponent x ext y case h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x y : α ⊢ y ∈ ConnectedComponents.mk ⁻¹' {ConnectedComponents.mk x} ↔ y ∈ connectedComponent x rw [mem_preimage, mem_singleton_iff, ConnectedComponents.coe_eq_coe'] ** Qed
connectedComponents_preimage_image α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v U : Set α ⊢ ConnectedComponents.mk ⁻¹' (ConnectedComponents.mk '' U) = ⋃ x ∈ U, connectedComponent x simp only [connectedComponents_preimage_singleton, preimage_iUnion₂, image_eq_iUnion] ** Qed
isPreconnected_of_forall_constant α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y ⊢ IsPreconnected s unfold IsPreconnected α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y ⊢ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) by_contra' α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y this : ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∪ v ∧ Set.Nonempty (s ∩ u) ∧ Set.Nonempty (s ∩ v) ∧ ¬Set.Nonempty (s ∩ (u ∩ v)) ⊢ False rcases this with ⟨u, v, u_op, v_op, hsuv, ⟨x, x_in_s, x_in_u⟩, ⟨y, y_in_s, y_in_v⟩, H⟩ case intro.intro.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✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y u v : Set α u_op : IsOpen u v_op : IsOpen v hsuv : s ⊆ u ∪ v x : α x_in_s : x ∈ s x_in_u : x ∈ u H : ¬Set.Nonempty (s ∩ (u ∩ v)) y : α y_in_s : y ∈ s y_in_v : y ∈ v ⊢ False rw [not_nonempty_iff_eq_empty] at H case intro.intro.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✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y u v : Set α u_op : IsOpen u v_op : IsOpen v hsuv : s ⊆ u ∪ v x : α x_in_s : x ∈ s x_in_u : x ∈ u H : s ∩ (u ∩ v) = ∅ y : α y_in_s : y ∈ s y_in_v : y ∈ v ⊢ False have hy : y ∉ u := fun y_in_u => eq_empty_iff_forall_not_mem.mp H y ⟨y_in_s, ⟨y_in_u, y_in_v⟩⟩ case intro.intro.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✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y u v : Set α u_op : IsOpen u v_op : IsOpen v hsuv : s ⊆ u ∪ v x : α x_in_s : x ∈ s x_in_u : x ∈ u H : s ∩ (u ∩ v) = ∅ y : α y_in_s : y ∈ s y_in_v : y ∈ v hy : ¬y ∈ u this : ContinuousOn (boolIndicator u) s ⊢ False simpa [(u.mem_iff_boolIndicator _).mp x_in_u, (u.not_mem_iff_boolIndicator _).mp hy] using hs _ this x x_in_s y y_in_s α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y u v : Set α u_op : IsOpen u v_op : IsOpen v hsuv : s ⊆ u ∪ v x : α x_in_s : x ∈ s x_in_u : x ∈ u H : s ∩ (u ∩ v) = ∅ y : α y_in_s : y ∈ s y_in_v : y ∈ v hy : ¬y ∈ u ⊢ ContinuousOn (boolIndicator u) s apply (continuousOn_boolIndicator_iff_clopen _ _).mpr ⟨_, _⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y u v : Set α u_op : IsOpen u v_op : IsOpen v hsuv : s ⊆ u ∪ v x : α x_in_s : x ∈ s x_in_u : x ∈ u H : s ∩ (u ∩ v) = ∅ y : α y_in_s : y ∈ s y_in_v : y ∈ v hy : ¬y ∈ u ⊢ IsOpen (Subtype.val ⁻¹' u) exact u_op.preimage continuous_subtype_val α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y u v : Set α u_op : IsOpen u v_op : IsOpen v hsuv : s ⊆ u ∪ v x : α x_in_s : x ∈ s x_in_u : x ∈ u H : s ∩ (u ∩ v) = ∅ y : α y_in_s : y ∈ s y_in_v : y ∈ v hy : ¬y ∈ u ⊢ IsClosed (Subtype.val ⁻¹' u) rw [preimage_subtype_coe_eq_compl hsuv H] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y u v : Set α u_op : IsOpen u v_op : IsOpen v hsuv : s ⊆ u ∪ v x : α x_in_s : x ∈ s x_in_u : x ∈ u H : s ∩ (u ∩ v) = ∅ y : α y_in_s : y ∈ s y_in_v : y ∈ v hy : ¬y ∈ u ⊢ IsClosed (Subtype.val ⁻¹' v)ᶜ exact (v_op.preimage continuous_subtype_val).isClosed_compl ** Qed
polynomialFunctions_closure_eq_top' ⊢ Subalgebra.topologicalClosure (polynomialFunctions I) = ⊤ apply eq_top_iff.mpr ⊢ ⊤ ≤ Subalgebra.topologicalClosure (polynomialFunctions I) rintro f - f : C(↑I, ℝ) ⊢ f ∈ Subalgebra.topologicalClosure (polynomialFunctions I) refine' Filter.Frequently.mem_closure _ f : C(↑I, ℝ) ⊢ ∃ᶠ (x : C(↑I, ℝ)) in nhds f, x ∈ ↑(polynomialFunctions I) refine' Filter.Tendsto.frequently (bernsteinApproximation_uniform f) _ f : C(↑I, ℝ) ⊢ ∃ᶠ (x : ℕ) in atTop, bernsteinApproximation x f ∈ ↑(polynomialFunctions I) apply frequently_of_forall case h f : C(↑I, ℝ) ⊢ ∀ (x : ℕ), bernsteinApproximation x f ∈ ↑(polynomialFunctions I) intro n case h f : C(↑I, ℝ) n : ℕ ⊢ bernsteinApproximation n f ∈ ↑(polynomialFunctions I) simp only [SetLike.mem_coe] case h f : C(↑I, ℝ) n : ℕ ⊢ bernsteinApproximation n f ∈ polynomialFunctions I apply Subalgebra.sum_mem case h.h f : C(↑I, ℝ) n : ℕ ⊢ ∀ (x : Fin (n + 1)), x ∈ Finset.univ → ↑f (bernstein.z x) • bernstein n ↑x ∈ polynomialFunctions I rintro n - case h.h f : C(↑I, ℝ) n✝ : ℕ n : Fin (n✝ + 1) ⊢ ↑f (bernstein.z n) • bernstein n✝ ↑n ∈ polynomialFunctions I apply Subalgebra.smul_mem case h.h.hx f : C(↑I, ℝ) n✝ : ℕ n : Fin (n✝ + 1) ⊢ bernstein n✝ ↑n ∈ polynomialFunctions I dsimp [bernstein, polynomialFunctions] case h.h.hx f : C(↑I, ℝ) n✝ : ℕ n : Fin (n✝ + 1) ⊢ Polynomial.toContinuousMapOn (bernsteinPolynomial ℝ n✝ ↑n) I ∈ Subalgebra.map (Polynomial.toContinuousMapOnAlgHom I) ⊤ simp ** Qed
polynomialFunctions_closure_eq_top a b : ℝ ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ cases' lt_or_le a b with h h case inl a b : ℝ h : a < b ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ let W : C(Set.Icc a b, ℝ) →ₐ[ℝ] C(I, ℝ) := compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap case inl a b : ℝ h : a < b W : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) := compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h))) ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ let W' : C(Set.Icc a b, ℝ) ≃ₜ C(I, ℝ) := compRightHomeomorph ℝ (iccHomeoI a b h).symm case inl a b : ℝ h : a < b W : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) := compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h))) W' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h)) ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ have w : (W : C(Set.Icc a b, ℝ) → C(I, ℝ)) = W' := rfl case inl a b : ℝ h : a < b W : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) := compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h))) W' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h)) w : ↑W = ↑W' ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ have p := polynomialFunctions_closure_eq_top' case inl a b : ℝ h : a < b W : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) := compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h))) W' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h)) w : ↑W = ↑W' p : Subalgebra.topologicalClosure (polynomialFunctions I) = ⊤ ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ apply_fun fun s => s.comap W at p case inl a b : ℝ h : a < b W : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) := compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h))) W' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h)) w : ↑W = ↑W' p : Subalgebra.comap W (Subalgebra.topologicalClosure (polynomialFunctions I)) = Subalgebra.comap W ⊤ ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ simp only [Algebra.comap_top] at p case inl a b : ℝ h : a < b W : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) := compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h))) W' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h)) w : ↑W = ↑W' p : Subalgebra.comap (compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) (Subalgebra.topologicalClosure (polynomialFunctions I)) = ⊤ ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ rw [Subalgebra.topologicalClosure_comap_homeomorph _ W W' w] at p case inl a b : ℝ h : a < b W : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) := compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h))) W' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h)) w : ↑W = ↑W' p : Subalgebra.topologicalClosure (Subalgebra.comap W (polynomialFunctions I)) = ⊤ ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ rw [polynomialFunctions.comap_compRightAlgHom_iccHomeoI] at p case inl a b : ℝ h : a < b W : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) := compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h))) W' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h)) w : ↑W = ↑W' p : Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ exact p case inr a b : ℝ h : b ≤ a ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ have : Subsingleton (Set.Icc a b) := (Set.subsingleton_coe _).mpr $ Set.subsingleton_Icc_of_ge h case inr a b : ℝ h : b ≤ a this : Subsingleton ↑(Set.Icc a b) ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ apply Subsingleton.elim ** Qed
continuousMap_mem_polynomialFunctions_closure a b : ℝ f : C(↑(Set.Icc a b), ℝ) ⊢ f ∈ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) rw [polynomialFunctions_closure_eq_top _ _] a b : ℝ f : C(↑(Set.Icc a b), ℝ) ⊢ f ∈ ⊤ simp ** Qed
exists_polynomial_near_continuousMap a b : ℝ f : C(↑(Set.Icc a b), ℝ) ε : ℝ pos : 0 < ε ⊢ ∃ p, ‖Polynomial.toContinuousMapOn p (Set.Icc a b) - f‖ < ε have w := mem_closure_iff_frequently.mp (continuousMap_mem_polynomialFunctions_closure _ _ f) a b : ℝ f : C(↑(Set.Icc a b), ℝ) ε : ℝ pos : 0 < ε w : ∃ᶠ (x : C(↑(Set.Icc a b), ℝ)) in nhds f, x ∈ ↑(polynomialFunctions (Set.Icc a b)) ⊢ ∃ p, ‖Polynomial.toContinuousMapOn p (Set.Icc a b) - f‖ < ε rw [Metric.nhds_basis_ball.frequently_iff] at w a b : ℝ f : C(↑(Set.Icc a b), ℝ) ε : ℝ pos : 0 < ε w : ∀ (i : ℝ), 0 < i → ∃ x, x ∈ Metric.ball f i ∧ x ∈ ↑(polynomialFunctions (Set.Icc a b)) ⊢ ∃ p, ‖Polynomial.toContinuousMapOn p (Set.Icc a b) - f‖ < ε obtain ⟨-, H, ⟨m, ⟨-, rfl⟩⟩⟩ := w ε pos case intro.intro.intro.intro a b : ℝ f : C(↑(Set.Icc a b), ℝ) ε : ℝ pos : 0 < ε w : ∀ (i : ℝ), 0 < i → ∃ x, x ∈ Metric.ball f i ∧ x ∈ ↑(polynomialFunctions (Set.Icc a b)) m : ℝ[X] H : ↑↑(Polynomial.toContinuousMapOnAlgHom (Set.Icc a b)) m ∈ Metric.ball f ε ⊢ ∃ p, ‖Polynomial.toContinuousMapOn p (Set.Icc a b) - f‖ < ε rw [Metric.mem_ball, dist_eq_norm] at H case intro.intro.intro.intro a b : ℝ f : C(↑(Set.Icc a b), ℝ) ε : ℝ pos : 0 < ε w : ∀ (i : ℝ), 0 < i → ∃ x, x ∈ Metric.ball f i ∧ x ∈ ↑(polynomialFunctions (Set.Icc a b)) m : ℝ[X] H : ‖↑↑(Polynomial.toContinuousMapOnAlgHom (Set.Icc a b)) m - f‖ < ε ⊢ ∃ p, ‖Polynomial.toContinuousMapOn p (Set.Icc a b) - f‖ < ε exact ⟨m, H⟩ ** Qed

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Path.trans_continuous_family X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b c : ι → X γ₁ : (t : ι) → Path (a t) (b t) h₁ : Continuous ↿γ₁ γ₂ : (t : ι) → Path (b t) (c t) h₂ : Continuous ↿γ₂ ⊢ Continuous ↿fun t => trans (γ₁ t) (γ₂ t) have h₁' := Path.continuous_uncurry_extend_of_continuous_family γ₁ h₁ X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b c : ι → X γ₁ : (t : ι) → Path (a t) (b t) h₁ : Continuous ↿γ₁ γ₂ : (t : ι) → Path (b t) (c t) h₂ : Continuous ↿γ₂ h₁' : Continuous ↿fun t => extend (γ₁ t) ⊢ Continuous ↿fun t => trans (γ₁ t) (γ₂ t) have h₂' := Path.continuous_uncurry_extend_of_continuous_family γ₂ h₂ ** X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b c : ι → X γ₁ : (t : ι) → Path (a t) (b t) h₁ : Continuous ↿γ₁ γ₂ : (t : ι) → Path (b t) (c t) h₂ : Continuous ↿γ₂ h₁' : Continuous ↿fun t => extend (γ₁ t) h₂' : Continuous ↿fun t => extend (γ₂ t) ⊢ Continuous fun p => ↑{ toContinuousMap := ContinuousMap.mk fun x => if ↑x ≤ 1 / 2 then extend (γ₁ p.1) (2 * ↑x) else extend (γ₂ p.1) (2 * ↑x - 1), source' := (_ : ContinuousMap.toFun (ContinuousMap.mk fun x => if ↑x ≤ 1 / 2 then extend (γ₁ p.1) (2 * ↑x) else extend (γ₂ p.1) (2 * ↑x - 1)) 0 = a p.1), target' := (_ : ContinuousMap.toFun (ContinuousMap.mk fun x => if ↑x ≤ 1 / 2 then extend (γ₁ p.1) (2 * ↑x) else extend (γ₂ p.1) (2 * ↑x - 1)) 1 = c p.1) } p.2 refine' Continuous.if_le _ _ (continuous_subtype_val.comp continuous_snd) continuous_const _ case refine'_1 X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b c : ι → X γ₁ : (t : ι) → Path (a t) (b t) h₁ : Continuous ↿γ₁ γ₂ : (t : ι) → Path (b t) (c t) h₂ : Continuous ↿γ₂ h₁' : Continuous ↿fun t => extend (γ₁ t) h₂' : Continuous ↿fun t => extend (γ₂ t) ⊢ Continuous fun p => extend (γ₁ p.1) (2 * ↑p.2) ** change Continuous ((fun p : ι × ℝ => (γ₁ p.1).extend p.2) ∘ Prod.map id (fun x => 2 * x : I → ℝ)) ** case refine'_1 X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b c : ι → X γ₁ : (t : ι) → Path (a t) (b t) h₁ : Continuous ↿γ₁ γ₂ : (t : ι) → Path (b t) (c t) h₂ : Continuous ↿γ₂ h₁' : Continuous ↿fun t => extend (γ₁ t) h₂' : Continuous ↿fun t => extend (γ₂ t) ⊢ Continuous ((fun p => extend (γ₁ p.1) p.2) ∘ Prod.map id fun x => 2 * ↑x) exact h₁'.comp (continuous_id.prod_map <\| continuous_const.mul continuous_subtype_val) case refine'_2 X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b c : ι → X γ₁ : (t : ι) → Path (a t) (b t) h₁ : Continuous ↿γ₁ γ₂ : (t : ι) → Path (b t) (c t) h₂ : Continuous ↿γ₂ h₁' : Continuous ↿fun t => extend (γ₁ t) h₂' : Continuous ↿fun t => extend (γ₂ t) ⊢ Continuous fun p => extend (γ₂ p.1) (2 * ↑p.2 - 1) ** change Continuous ((fun p : ι × ℝ => (γ₂ p.1).extend p.2) ∘ Prod.map id (fun x => 2 * x - 1 : I → ℝ)) ** case refine'_2 X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b c : ι → X γ₁ : (t : ι) → Path (a t) (b t) h₁ : Continuous ↿γ₁ γ₂ : (t : ι) → Path (b t) (c t) h₂ : Continuous ↿γ₂ h₁' : Continuous ↿fun t => extend (γ₁ t) h₂' : Continuous ↿fun t => extend (γ₂ t) ⊢ Continuous ((fun p => extend (γ₂ p.1) p.2) ∘ Prod.map id fun x => 2 * ↑x - 1) exact h₂'.comp (continuous_id.prod_map <\| (continuous_const.mul continuous_subtype_val).sub continuous_const) case refine'_3 X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b c : ι → X γ₁ : (t : ι) → Path (a t) (b t) h₁ : Continuous ↿γ₁ γ₂ : (t : ι) → Path (b t) (c t) h₂ : Continuous ↿γ₂ h₁' : Continuous ↿fun t => extend (γ₁ t) h₂' : Continuous ↿fun t => extend (γ₂ t) ⊢ ∀ (x : ι × ↑I), ↑x.2 = 1 / 2 → extend (γ₁ x.1) (2 * ↑x.2) = extend (γ₂ x.1) (2 * ↑x.2 - 1) rintro st hst case refine'_3 X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b c : ι → X γ₁ : (t : ι) → Path (a t) (b t) h₁ : Continuous ↿γ₁ γ₂ : (t : ι) → Path (b t) (c t) h₂ : Continuous ↿γ₂ h₁' : Continuous ↿fun t => extend (γ₁ t) h₂' : Continuous ↿fun t => extend (γ₂ t) st : ι × ↑I hst : ↑st.2 = 1 / 2 ⊢ extend (γ₁ st.1) (2 * ↑st.2) = extend (γ₂ st.1) (2 * ↑st.2 - 1) simp [hst, mul_inv_cancel (two_ne_zero' ℝ)] Qed
Continuous.path_trans X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y f : Y → Path x y g : Y → Path y z ⊢ Continuous f → Continuous g → Continuous fun t => trans (f t) (g t) intro hf hg X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y f : Y → Path x y g : Y → Path y z hf : Continuous f hg : Continuous g ⊢ Continuous fun t => trans (f t) (g t) apply continuous_uncurry_iff.mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y f : Y → Path x y g : Y → Path y z hf : Continuous f hg : Continuous g ⊢ Continuous ↿fun t => trans (f t) (g t) exact trans_continuous_family _ (continuous_uncurry_iff.mpr hf) _ (continuous_uncurry_iff.mpr hg) ** Qed
Path.trans_prod_eq_prod_trans X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y a₁ a₂ a₃ : X b₁ b₂ b₃ : Y γ₁ : Path a₁ a₂ δ₁ : Path a₂ a₃ γ₂ : Path b₁ b₂ δ₂ : Path b₂ b₃ ⊢ trans (Path.prod γ₁ γ₂) (Path.prod δ₁ δ₂) = Path.prod (trans γ₁ δ₁) (trans γ₂ δ₂) ext t <;> unfold Path.trans <;> simp only [Path.coe_mk_mk, Path.prod_coe, Function.comp_apply] <;> split_ifs <;> rfl ** Qed
Path.trans_pi_eq_pi_trans X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y χ : ι → Type u_4 inst✝ : (i : ι) → TopologicalSpace (χ i) as bs cs : (i : ι) → χ i γ₀ : (i : ι) → Path (as i) (bs i) γ₁ : (i : ι) → Path (bs i) (cs i) ⊢ trans (Path.pi γ₀) (Path.pi γ₁) = Path.pi fun i => trans (γ₀ i) (γ₁ i) ext t i case a.h.h X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y χ : ι → Type u_4 inst✝ : (i : ι) → TopologicalSpace (χ i) as bs cs : (i : ι) → χ i γ₀ : (i : ι) → Path (as i) (bs i) γ₁ : (i : ι) → Path (bs i) (cs i) t : ↑I i : ι ⊢ ↑(trans (Path.pi γ₀) (Path.pi γ₁)) t i = ↑(Path.pi fun i => trans (γ₀ i) (γ₁ i)) t i unfold Path.trans ** case a.h.h X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y χ : ι → Type u_4 inst✝ : (i : ι) → TopologicalSpace (χ i) as bs cs : (i : ι) → χ i γ₀ : (i : ι) → Path (as i) (bs i) γ₁ : (i : ι) → Path (bs i) (cs i) t : ↑I i : ι ⊢ ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend (Path.pi γ₀) (2 * t) else extend (Path.pi γ₁) (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend (Path.pi γ₀) (2 * x) else extend (Path.pi γ₁) (2 * x - 1)) ∘ Subtype.val) 0 = fun i => as i), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend (Path.pi γ₀) (2 * x) else extend (Path.pi γ₁) (2 * x - 1)) ∘ Subtype.val) 1 = fun i => cs i) } t i = ↑(Path.pi fun i => { toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend (γ₀ i) (2 * t) else extend (γ₁ i) (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend (γ₀ i) (2 * x) else extend (γ₁ i) (2 * x - 1)) ∘ Subtype.val) 0 = as i), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend (γ₀ i) (2 * x) else extend (γ₁ i) (2 * x - 1)) ∘ Subtype.val) 1 = cs i) }) t i simp only [Path.coe_mk_mk, Function.comp_apply, pi_coe] case a.h.h X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y χ : ι → Type u_4 inst✝ : (i : ι) → TopologicalSpace (χ i) as bs cs : (i : ι) → χ i γ₀ : (i : ι) → Path (as i) (bs i) γ₁ : (i : ι) → Path (bs i) (cs i) t : ↑I i : ι ⊢ ite (↑t ≤ 1 / 2) (extend (Path.pi γ₀) (2 * ↑t)) (extend (Path.pi γ₁) (2 * ↑t - 1)) i = if ↑t ≤ 1 / 2 then extend (γ₀ i) (2 * ↑t) else extend (γ₁ i) (2 * ↑t - 1) split_ifs <;> rfl Qed
Path.truncate_range X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t₀ t₁ : ℝ ⊢ range ↑(truncate γ t₀ t₁) ⊆ range ↑γ rw [← γ.extend_range] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t₀ t₁ : ℝ ⊢ range ↑(truncate γ t₀ t₁) ⊆ range (extend γ) simp only [range_subset_iff, SetCoe.exists, SetCoe.forall] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t₀ t₁ : ℝ ⊢ ∀ (x : ℝ) (h : x ∈ I), ↑(truncate γ t₀ t₁) { val := x, property := h } ∈ range (extend γ) intro x _hx X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ✝ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t₀ t₁ x : ℝ _hx : x ∈ I ⊢ ↑(truncate γ t₀ t₁) { val := x, property := _hx } ∈ range (extend γ) simp only [FunLike.coe, Path.truncate, mem_range_self] ** Qed
Path.truncate_const_continuous_family X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t : ℝ ⊢ Continuous ↿(truncate γ t) have key : Continuous (fun x => (t, x) : ℝ × I → ℝ × ℝ × I) := by continuity X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t : ℝ key : Continuous fun x => (t, x) ⊢ Continuous ↿(truncate γ t) exact γ.truncate_continuous_family.comp key X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t : ℝ ⊢ Continuous fun x => (t, x) continuity ** Qed
Path.truncate_self X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t : ℝ ⊢ extend γ (min t t) = extend γ t rw [min_self] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t : ℝ ⊢ truncate γ t t = cast (refl (extend γ t)) (_ : extend γ (min t t) = extend γ t) (_ : extend γ t = extend γ t) ext x case a.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ✝ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t : ℝ x : ↑I ⊢ ↑(truncate γ t t) x = ↑(cast (refl (extend γ t)) (_ : extend γ (min t t) = extend γ t) (_ : extend γ t = extend γ t)) x rw [cast_coe] case a.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ✝ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t : ℝ x : ↑I ⊢ ↑(truncate γ t t) x = ↑(refl (extend γ t)) x simp only [truncate, FunLike.coe, refl, min_def, max_def] case a.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ✝ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b t : ℝ x : ↑I ⊢ extend γ (if (if ↑x ≤ t then t else ↑x) ≤ t then if ↑x ≤ t then t else ↑x else t) = extend γ t split_ifs with h₁ h₂ <;> congr ** Qed
Path.truncate_zero_zero X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b ⊢ extend γ (min 0 0) = a rw [min_self, γ.extend_zero] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b ⊢ truncate γ 0 0 = cast (refl a) (_ : extend γ (min 0 0) = a) (_ : extend γ 0 = a) convert γ.truncate_self 0 ** Qed
Path.truncate_one_one X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b ⊢ extend γ (min 1 1) = b rw [min_self, γ.extend_one] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b ⊢ truncate γ 1 1 = cast (refl b) (_ : extend γ (min 1 1) = b) (_ : extend γ 1 = b) convert γ.truncate_self 1 ** Qed
Path.truncate_zero_one X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b ⊢ extend γ (min 0 1) = a simp [zero_le_one, extend_zero] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b ⊢ extend γ 1 = b simp X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ✝ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b ⊢ truncate γ 0 1 = cast γ (_ : extend γ (min 0 1) = a) (_ : extend γ 1 = b) ext x case a.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ✝ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b x : ↑I ⊢ ↑(truncate γ 0 1) x = ↑(cast γ (_ : extend γ (min 0 1) = a) (_ : extend γ 1 = b)) x rw [cast_coe] case a.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ✝ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b x : ↑I ⊢ ↑(truncate γ 0 1) x = ↑γ x have : ↑x ∈ (Icc 0 1 : Set ℝ) := x.2 case a.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ✝ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b : X γ : Path a b x : ↑I this : ↑x ∈ Icc 0 1 ⊢ ↑(truncate γ 0 1) x = ↑γ x rw [truncate, coe_mk_mk, max_eq_left this.1, min_eq_left this.2, extend_extends'] ** Qed
Path.reparam_id X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y ⊢ reparam γ id (_ : Continuous id) (_ : id 0 = id 0) (_ : id 1 = id 1) = γ ext case a.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y x✝ : ↑I ⊢ ↑(reparam γ id (_ : Continuous id) (_ : id 0 = id 0) (_ : id 1 = id 1)) x✝ = ↑γ x✝ rfl ** Qed
Path.range_reparam X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 ⊢ range ↑(reparam γ f hfcont hf₀ hf₁) = range ↑γ change range (γ ∘ f) = range γ X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 this : range f = univ ⊢ range (↑γ ∘ f) = range ↑γ rw [range_comp, this, image_univ] X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 ⊢ range f = univ rw [range_iff_surjective] X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 ⊢ Surjective f intro t X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 t : ↑I ⊢ ∃ a, f a = t have h₁ : Continuous (Set.IccExtend (zero_le_one' ℝ) f) := by continuity X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 t : ↑I h₁ : Continuous (IccExtend (_ : 0 ≤ 1) f) ⊢ ∃ a, f a = t have := intermediate_value_Icc (zero_le_one' ℝ) h₁.continuousOn X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 t : ↑I ⊢ Continuous (IccExtend (_ : 0 ≤ 1) f) continuity X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 t : ↑I h₁ : Continuous (IccExtend (_ : 0 ≤ 1) f) this : Icc (IccExtend (_ : 0 ≤ 1) f 0) (IccExtend (_ : 0 ≤ 1) f 1) ⊆ IccExtend (_ : 0 ≤ 1) f '' Icc 0 1 ⊢ ∃ a, f a = t rw [IccExtend_left, IccExtend_right, Icc.mk_zero, Icc.mk_one, hf₀, hf₁] at this X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 t : ↑I h₁ : Continuous (IccExtend (_ : 0 ≤ 1) f) this : Icc 0 1 ⊆ IccExtend (_ : 0 ≤ 1) f '' Icc 0 1 ⊢ ∃ a, f a = t rcases this t.2 with ⟨w, hw₁, hw₂⟩ case intro.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 t : ↑I h₁ : Continuous (IccExtend (_ : 0 ≤ 1) f) this : Icc 0 1 ⊆ IccExtend (_ : 0 ≤ 1) f '' Icc 0 1 w : ℝ hw₁ : w ∈ Icc 0 1 hw₂ : IccExtend (_ : 0 ≤ 1) f w = t ⊢ ∃ a, f a = t rw [IccExtend_of_mem _ _ hw₁] at hw₂ case intro.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 t : ↑I h₁ : Continuous (IccExtend (_ : 0 ≤ 1) f) this : Icc 0 1 ⊆ IccExtend (_ : 0 ≤ 1) f '' Icc 0 1 w : ℝ hw₁ : w ∈ Icc 0 1 hw₂ : f { val := w, property := hw₁ } = t ⊢ ∃ a, f a = t exact ⟨_, hw₂⟩ ** Qed
Path.refl_reparam X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 ⊢ reparam (refl x) f hfcont hf₀ hf₁ = refl x ext case a.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y f : ↑I → ↑I hfcont : Continuous f hf₀ : f 0 = 0 hf₁ : f 1 = 1 x✝ : ↑I ⊢ ↑(reparam (refl x) f hfcont hf₀ hf₁) x✝ = ↑(refl x) x✝ simp ** Qed
JoinedIn.mem X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : JoinedIn F x y ⊢ x ∈ F ∧ y ∈ F rcases h with ⟨γ, γ_in⟩ case intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X γ : Path x y γ_in : ∀ (t : ↑I), ↑γ t ∈ F ⊢ x ∈ F ∧ y ∈ F have : γ 0 ∈ F ∧ γ 1 ∈ F := by constructor <;> apply γ_in case intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X γ : Path x y γ_in : ∀ (t : ↑I), ↑γ t ∈ F this : ↑γ 0 ∈ F ∧ ↑γ 1 ∈ F ⊢ x ∈ F ∧ y ∈ F simpa using this X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X γ : Path x y γ_in : ∀ (t : ↑I), ↑γ t ∈ F ⊢ ↑γ 0 ∈ F ∧ ↑γ 1 ∈ F constructor <;> apply γ_in ** Qed
JoinedIn.joined_subtype X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : JoinedIn F x y ⊢ Continuous fun t => { val := ↑(somePath h) t, property := (_ : ↑(somePath h) t ∈ F) } continuity X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : JoinedIn F x y ⊢ ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(somePath h) t, property := (_ : ↑(somePath h) t ∈ F) }) 0 = { val := x, property := (_ : x ∈ F) } simp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : JoinedIn F x y ⊢ ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(somePath h) t, property := (_ : ↑(somePath h) t ∈ F) }) 1 = { val := y, property := (_ : y ∈ F) } simp ** Qed
joinedIn_iff_joined X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X x_in : x ∈ F y_in : y ∈ F h : Joined { val := x, property := x_in } { val := y, property := y_in } ⊢ ∀ (t : ↑I), ↑(Path.map (Joined.somePath h) (_ : Continuous Subtype.val)) t ∈ F simp ** Qed
joinedIn_univ X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ JoinedIn univ x y ↔ Joined x y simp [JoinedIn, Joined, exists_true_iff_nonempty] ** Qed
JoinedIn.symm X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : JoinedIn F x y ⊢ JoinedIn F y x cases' h.mem with hx hy case intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : JoinedIn F x y hx : x ∈ F hy : y ∈ F ⊢ JoinedIn F y x simp_all [joinedIn_iff_joined] case intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X hx✝ : x ∈ F hy✝ : y ∈ F h : Joined { val := x, property := (_ : x ∈ F) } { val := y, property := (_ : y ∈ F) } hx : x ∈ F hy : y ∈ F ⊢ Joined { val := y, property := (_ : y ∈ F) } { val := x, property := (_ : x ∈ F) } exact h.symm ** Qed
JoinedIn.trans X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X hxy : JoinedIn F x y hyz : JoinedIn F y z ⊢ JoinedIn F x z cases' hxy.mem with hx hy case intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X hxy : JoinedIn F x y hyz : JoinedIn F y z hx : x ∈ F hy : y ∈ F ⊢ JoinedIn F x z cases' hyz.mem with hx hy case intro.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X hxy : JoinedIn F x y hyz : JoinedIn F y z hx✝ : x ∈ F hy✝ hx : y ∈ F hy : z ∈ F ⊢ JoinedIn F x z simp_all [joinedIn_iff_joined] case intro.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X hx✝¹ : x ∈ F hy✝¹ : y ∈ F hy✝ : z ∈ F hxy : Joined { val := x, property := (_ : x ∈ F) } { val := y, property := (_ : y ∈ F) } hyz : Joined { val := y, property := (_ : y ∈ F) } { val := z, property := (_ : z ∈ F) } hx✝ : x ∈ F hx : y ∈ F hy : z ∈ F ⊢ Joined { val := x, property := (_ : x ∈ F) } { val := z, property := (_ : z ∈ F) } exact hxy.trans hyz ** Qed
pathComponent_congr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : x ∈ pathComponent y ⊢ pathComponent x = pathComponent y ext z case h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z✝ : X ι : Type u_3 F : Set X h : x ∈ pathComponent y z : X ⊢ z ∈ pathComponent x ↔ z ∈ pathComponent y constructor case h.mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z✝ : X ι : Type u_3 F : Set X h : x ∈ pathComponent y z : X ⊢ z ∈ pathComponent x → z ∈ pathComponent y intro h' case h.mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z✝ : X ι : Type u_3 F : Set X h : x ∈ pathComponent y z : X h' : z ∈ pathComponent x ⊢ z ∈ pathComponent y rw [pathComponent_symm] case h.mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z✝ : X ι : Type u_3 F : Set X h : x ∈ pathComponent y z : X h' : z ∈ pathComponent x ⊢ y ∈ pathComponent z exact (h.trans h').symm case h.mpr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z✝ : X ι : Type u_3 F : Set X h : x ∈ pathComponent y z : X ⊢ z ∈ pathComponent y → z ∈ pathComponent x intro h' case h.mpr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z✝ : X ι : Type u_3 F : Set X h : x ∈ pathComponent y z : X h' : z ∈ pathComponent y ⊢ z ∈ pathComponent x rw [pathComponent_symm] at h' ⊢ case h.mpr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z✝ : X ι : Type u_3 F : Set X h : x ∈ pathComponent y z : X h' : y ∈ pathComponent z ⊢ x ∈ pathComponent z exact h'.trans h ** Qed
pathComponent_subset_component X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z : X ι : Type u_3 F : Set X x y : X h : y ∈ pathComponent x ⊢ ↑(Joined.somePath h) 0 = x simp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z : X ι : Type u_3 F : Set X x y : X h : y ∈ pathComponent x ⊢ ↑(Joined.somePath h) 1 = y simp ** Qed
pathComponentIn_univ X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X x : X ⊢ pathComponentIn x univ = pathComponent x simp [pathComponentIn, pathComponent, JoinedIn, Joined, exists_true_iff_nonempty] ** Qed
isPathConnected_iff_eq X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ IsPathConnected F ↔ ∃ x, x ∈ F ∧ pathComponentIn x F = F constructor <;> rintro ⟨x, x_in, h⟩ <;> use x, x_in case right X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X x : X x_in : x ∈ F h : ∀ {y : X}, y ∈ F → JoinedIn F x y ⊢ pathComponentIn x F = F ext y case right.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z : X ι : Type u_3 F : Set X x : X x_in : x ∈ F h : ∀ {y : X}, y ∈ F → JoinedIn F x y y : X ⊢ y ∈ pathComponentIn x F ↔ y ∈ F exact ⟨fun hy => hy.mem.2, h⟩ case right X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X x : X x_in : x ∈ F h : pathComponentIn x F = F ⊢ ∀ {y : X}, y ∈ F → JoinedIn F x y intro y y_in case right X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z : X ι : Type u_3 F : Set X x : X x_in : x ∈ F h : pathComponentIn x F = F y : X y_in : y ∈ F ⊢ JoinedIn F x y rwa [← h] at y_in ** Qed
IsPathConnected.image' X : Type u_1 Y✝ : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y✝ x y z : X ι : Type u_3 F : Set X Y : Type u_4 inst✝ : TopologicalSpace Y hF : IsPathConnected F f : X → Y hf : ContinuousOn f F ⊢ IsPathConnected (f '' F) rcases hF with ⟨x, x_in, hx⟩ case intro.intro X : Type u_1 Y✝ : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y✝ x✝ y z : X ι : Type u_3 F : Set X Y : Type u_4 inst✝ : TopologicalSpace Y f : X → Y hf : ContinuousOn f F x : X x_in : x ∈ F hx : ∀ {y : X}, y ∈ F → JoinedIn F x y ⊢ IsPathConnected (f '' F) use f x, mem_image_of_mem f x_in case right X : Type u_1 Y✝ : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y✝ x✝ y z : X ι : Type u_3 F : Set X Y : Type u_4 inst✝ : TopologicalSpace Y f : X → Y hf : ContinuousOn f F x : X x_in : x ∈ F hx : ∀ {y : X}, y ∈ F → JoinedIn F x y ⊢ ∀ {y : Y}, y ∈ f '' F → JoinedIn (f '' F) (f x) y rintro _ ⟨y, y_in, rfl⟩ case right.intro.intro X : Type u_1 Y✝ : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y✝ x✝ y✝ z : X ι : Type u_3 F : Set X Y : Type u_4 inst✝ : TopologicalSpace Y f : X → Y hf : ContinuousOn f F x : X x_in : x ∈ F hx : ∀ {y : X}, y ∈ F → JoinedIn F x y y : X y_in : y ∈ F ⊢ JoinedIn (f '' F) (f x) (f y) refine ⟨(hx y_in).somePath.map' ?_, fun t ↦ ⟨_, (hx y_in).somePath_mem t, rfl⟩⟩ case right.intro.intro X : Type u_1 Y✝ : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y✝ x✝ y✝ z : X ι : Type u_3 F : Set X Y : Type u_4 inst✝ : TopologicalSpace Y f : X → Y hf : ContinuousOn f F x : X x_in : x ∈ F hx : ∀ {y : X}, y ∈ F → JoinedIn F x y y : X y_in : y ∈ F ⊢ ContinuousOn f (range ↑(JoinedIn.somePath (_ : JoinedIn F x y))) exact hf.mono (range_subset_iff.2 (hx y_in).somePath_mem) ** Qed
isPathConnected_singleton X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X x : X ⊢ IsPathConnected {x} refine ⟨x, rfl, ?_⟩ X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X x : X ⊢ ∀ {y : X}, y ∈ {x} → JoinedIn {x} x y rintro y rfl X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y✝ z : X ι : Type u_3 F : Set X y : X ⊢ JoinedIn {y} y y exact JoinedIn.refl rfl ** Qed
IsPathConnected.union X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F U V : Set X hU : IsPathConnected U hV : IsPathConnected V hUV : Set.Nonempty (U ∩ V) ⊢ IsPathConnected (U ∪ V) rcases hUV with ⟨x, xU, xV⟩ case intro.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F U V : Set X hU : IsPathConnected U hV : IsPathConnected V x : X xU : x ∈ U xV : x ∈ V ⊢ IsPathConnected (U ∪ V) use x, Or.inl xU case right X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F U V : Set X hU : IsPathConnected U hV : IsPathConnected V x : X xU : x ∈ U xV : x ∈ V ⊢ ∀ {y : X}, y ∈ U ∪ V → JoinedIn (U ∪ V) x y rintro y (yU \| yV) case right.inl X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z : X ι : Type u_3 F U V : Set X hU : IsPathConnected U hV : IsPathConnected V x : X xU : x ∈ U xV : x ∈ V y : X yU : y ∈ U ⊢ JoinedIn (U ∪ V) x y exact (hU.joinedIn x xU y yU).mono (subset_union_left U V) case right.inr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z : X ι : Type u_3 F U V : Set X hU : IsPathConnected U hV : IsPathConnected V x : X xU : x ∈ U xV : x ∈ V y : X yV : y ∈ V ⊢ JoinedIn (U ∪ V) x y exact (hV.joinedIn x xV y yV).mono (subset_union_right U V) ** Qed
IsPathConnected.preimage_coe X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F U W : Set X hW : IsPathConnected W hWU : W ⊆ U ⊢ IsPathConnected (Subtype.val ⁻¹' W) rcases hW with ⟨x, x_in, hx⟩ case intro.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F U W : Set X hWU : W ⊆ U x : X x_in : x ∈ W hx : ∀ {y : X}, y ∈ W → JoinedIn W x y ⊢ IsPathConnected (Subtype.val ⁻¹' W) use ⟨x, hWU x_in⟩, by simp [x_in] case right X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F U W : Set X hWU : W ⊆ U x : X x_in : x ∈ W hx : ∀ {y : X}, y ∈ W → JoinedIn W x y ⊢ ∀ {y : { x // x ∈ U }}, y ∈ Subtype.val ⁻¹' W → JoinedIn (Subtype.val ⁻¹' W) { val := x, property := (_ : x ∈ U) } y rintro ⟨y, hyU⟩ hyW case right.mk X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z : X ι : Type u_3 F U W : Set X hWU : W ⊆ U x : X x_in : x ∈ W hx : ∀ {y : X}, y ∈ W → JoinedIn W x y y : X hyU : y ∈ U hyW : { val := y, property := hyU } ∈ Subtype.val ⁻¹' W ⊢ JoinedIn (Subtype.val ⁻¹' W) { val := x, property := (_ : x ∈ U) } { val := y, property := hyU } exact ⟨(hx hyW).joined_subtype.somePath.map (continuous_inclusion hWU), by simp⟩ X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F U W : Set X hWU : W ⊆ U x : X x_in : x ∈ W hx : ∀ {y : X}, y ∈ W → JoinedIn W x y ⊢ { val := x, property := (_ : x ∈ U) } ∈ Subtype.val ⁻¹' W simp [x_in] X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z : X ι : Type u_3 F U W : Set X hWU : W ⊆ U x : X x_in : x ∈ W hx : ∀ {y : X}, y ∈ W → JoinedIn W x y y : X hyU : y ∈ U hyW : { val := y, property := hyU } ∈ Subtype.val ⁻¹' W ⊢ ∀ (t : ↑I), ↑(Path.map (Joined.somePath (_ : Joined { val := x, property := (_ : x ∈ W) } { val := ↑{ val := y, property := hyU }, property := (_ : ↑{ val := y, property := hyU } ∈ W) })) (_ : Continuous (inclusion hWU))) t ∈ Subtype.val ⁻¹' W simp ** Qed
IsPathConnected.exists_path_through_family X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s ⊢ ∃ γ, range ↑γ ⊆ s ∧ ∀ (i : Fin (n + 1)), p i ∈ range ↑γ let p' : ℕ → X := fun k => if h : k < n + 1 then p ⟨k, h⟩ else p ⟨0, n.zero_lt_succ⟩ case intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s ⊢ ∃ γ, range ↑γ ⊆ s ∧ ∀ (i : Fin (n + 1)), p i ∈ range ↑γ have hpp' : ∀ k < n + 1, p k = p' k := by intro k hk simp only [hk, dif_pos] congr ext rw [Fin.val_cast_of_lt hk] case intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hpp' : ∀ (k : ℕ), k < n + 1 → p ↑k = p' k ⊢ ∃ γ, range ↑γ ⊆ s ∧ ∀ (i : Fin (n + 1)), p i ∈ range ↑γ use γ.cast (hpp' 0 n.zero_lt_succ) (hpp' n n.lt_succ_self) case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hpp' : ∀ (k : ℕ), k < n + 1 → p ↑k = p' k ⊢ range ↑(Path.cast γ (_ : p ↑0 = p' 0) (_ : p ↑n = p' n)) ⊆ s ∧ ∀ (i : Fin (n + 1)), p i ∈ range ↑(Path.cast γ (_ : p ↑0 = p' 0) (_ : p ↑n = p' n)) simp only [γ.cast_coe] case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hpp' : ∀ (k : ℕ), k < n + 1 → p ↑k = p' k ⊢ range ↑γ ⊆ s ∧ ∀ (i : Fin (n + 1)), p i ∈ range ↑γ refine' And.intro hγ.2 _ case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hpp' : ∀ (k : ℕ), k < n + 1 → p ↑k = p' k ⊢ ∀ (i : Fin (n + 1)), p i ∈ range ↑γ rintro ⟨i, hi⟩ case h.mk X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hpp' : ∀ (k : ℕ), k < n + 1 → p ↑k = p' k i : ℕ hi : i < n + 1 ⊢ p { val := i, isLt := hi } ∈ range ↑γ suffices p ⟨i, hi⟩ = p' i by convert hγ.1 i (Nat.le_of_lt_succ hi) case h.mk X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hpp' : ∀ (k : ℕ), k < n + 1 → p ↑k = p' k i : ℕ hi : i < n + 1 ⊢ p { val := i, isLt := hi } = p' i rw [← hpp' i hi] case h.mk X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hpp' : ∀ (k : ℕ), k < n + 1 → p ↑k = p' k i : ℕ hi : i < n + 1 ⊢ p { val := i, isLt := hi } = p ↑i suffices i = i % n.succ by congr case h.mk X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hpp' : ∀ (k : ℕ), k < n + 1 → p ↑k = p' k i : ℕ hi : i < n + 1 ⊢ i = i % Nat.succ n rw [Nat.mod_eq_of_lt hi] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } ⊢ ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s have hp' : ∀ i ≤ n, p' i ∈ s := by intro i hi simp [Nat.lt_succ_of_le hi, hp] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } hp' : ∀ (i : ℕ), i ≤ n → p' i ∈ s ⊢ ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s clear_value p' X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X hp' : ∀ (i : ℕ), i ≤ n → p' i ∈ s ⊢ ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s clear hp p X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp' : ∀ (i : ℕ), i ≤ n → p' i ∈ s ⊢ ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s induction' n with n hn X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } ⊢ ∀ (i : ℕ), i ≤ n → p' i ∈ s intro i hi X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } i : ℕ hi : i ≤ n ⊢ p' i ∈ s simp [Nat.lt_succ_of_le hi, hp] case zero X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n → p' i ∈ s hp' : ∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ s ⊢ ∃ γ, (∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s use Path.refl (p' 0) case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n → p' i ∈ s hp' : ∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ s ⊢ (∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ range ↑(Path.refl (p' 0))) ∧ range ↑(Path.refl (p' 0)) ⊆ s constructor case h.left X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n → p' i ∈ s hp' : ∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ s ⊢ ∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ range ↑(Path.refl (p' 0)) rintro i hi case h.left X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n → p' i ∈ s hp' : ∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ s i : ℕ hi : i ≤ Nat.zero ⊢ p' i ∈ range ↑(Path.refl (p' 0)) rw [le_zero_iff.mp hi] case h.left X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n → p' i ∈ s hp' : ∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ s i : ℕ hi : i ≤ Nat.zero ⊢ p' 0 ∈ range ↑(Path.refl (p' 0)) exact ⟨0, rfl⟩ case h.right X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n → p' i ∈ s hp' : ∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ s ⊢ range ↑(Path.refl (p' 0)) ⊆ s rw [range_subset_iff] case h.right X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n → p' i ∈ s hp' : ∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ s ⊢ ∀ (y : ↑I), ↑(Path.refl (p' 0)) y ∈ s rintro _x case h.right X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n → p' i ∈ s hp' : ∀ (i : ℕ), i ≤ Nat.zero → p' i ∈ s _x : ↑I ⊢ ↑(Path.refl (p' 0)) _x ∈ s exact hp' 0 le_rfl case succ X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s ⊢ ∃ γ, (∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s rcases hn fun i hi => hp' i <\| Nat.le_succ_of_le hi with ⟨γ₀, hγ₀⟩ case succ.intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s ⊢ ∃ γ, (∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s rcases h.joinedIn (p' n) (hp' n n.le_succ) (p' <\| n + 1) (hp' (n + 1) <\| le_rfl) with ⟨γ₁, hγ₁⟩ case succ.intro.intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s ⊢ ∃ γ, (∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s let γ : Path (p' 0) (p' <\| n + 1) := γ₀.trans γ₁ case succ.intro.intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ ⊢ ∃ γ, (∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s use γ case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ ⊢ (∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s have range_eq : range γ = range γ₀ ∪ range γ₁ := γ₀.trans_range γ₁ case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ ⊢ (∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s constructor case h.left X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ ⊢ ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ range ↑γ rintro i hi case h.left X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ i : ℕ hi : i ≤ Nat.succ n ⊢ p' i ∈ range ↑γ by_cases hi' : i ≤ n case pos X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ i : ℕ hi : i ≤ Nat.succ n hi' : i ≤ n ⊢ p' i ∈ range ↑γ rw [range_eq] case pos X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ i : ℕ hi : i ≤ Nat.succ n hi' : i ≤ n ⊢ p' i ∈ range ↑γ₀ ∪ range ↑γ₁ left case pos.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ i : ℕ hi : i ≤ Nat.succ n hi' : i ≤ n ⊢ p' i ∈ range ↑γ₀ exact hγ₀.1 i hi' case neg X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ i : ℕ hi : i ≤ Nat.succ n hi' : ¬i ≤ n ⊢ p' i ∈ range ↑γ rw [not_le, ← Nat.succ_le_iff] at hi' case neg X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ i : ℕ hi : i ≤ Nat.succ n hi' : Nat.succ n ≤ i ⊢ p' i ∈ range ↑γ have : i = n.succ := le_antisymm hi hi' case neg X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ i : ℕ hi : i ≤ Nat.succ n hi' : Nat.succ n ≤ i this : i = Nat.succ n ⊢ p' i ∈ range ↑γ rw [this] case neg X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ i : ℕ hi : i ≤ Nat.succ n hi' : Nat.succ n ≤ i this : i = Nat.succ n ⊢ p' (Nat.succ n) ∈ range ↑γ use 1 case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ i : ℕ hi : i ≤ Nat.succ n hi' : Nat.succ n ≤ i this : i = Nat.succ n ⊢ ↑γ 1 = p' (Nat.succ n) exact γ.target case h.right X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ ⊢ range ↑γ ⊆ s rw [range_eq] case h.right X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ ⊢ range ↑γ₀ ∪ range ↑γ₁ ⊆ s apply union_subset hγ₀.2 case h.right X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ ⊢ range ↑γ₁ ⊆ s rw [range_subset_iff] case h.right X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n✝ : ℕ s : Set X h : IsPathConnected s p' : ℕ → X hp'✝ : ∀ (i : ℕ), i ≤ n✝ → p' i ∈ s n : ℕ hn : (∀ (i : ℕ), i ≤ n → p' i ∈ s) → ∃ γ, (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hp' : ∀ (i : ℕ), i ≤ Nat.succ n → p' i ∈ s γ₀ : Path (p' 0) (p' n) hγ₀ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ₀) ∧ range ↑γ₀ ⊆ s γ₁ : Path (p' n) (p' (n + 1)) hγ₁ : ∀ (t : ↑I), ↑γ₁ t ∈ s γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁ range_eq : range ↑γ = range ↑γ₀ ∪ range ↑γ₁ ⊢ ∀ (y : ↑I), ↑γ₁ y ∈ s exact hγ₁ X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s ⊢ ∀ (k : ℕ), k < n + 1 → p ↑k = p' k intro k hk X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s k : ℕ hk : k < n + 1 ⊢ p ↑k = p' k simp only [hk, dif_pos] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s k : ℕ hk : k < n + 1 ⊢ p ↑k = p { val := k, isLt := (_ : k < n + 1) } congr case e_a X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s k : ℕ hk : k < n + 1 ⊢ ↑k = { val := k, isLt := (_ : k < n + 1) } ext case e_a.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s k : ℕ hk : k < n + 1 ⊢ ↑↑k = ↑{ val := k, isLt := (_ : k < n + 1) } rw [Fin.val_cast_of_lt hk] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hpp' : ∀ (k : ℕ), k < n + 1 → p ↑k = p' k i : ℕ hi : i < n + 1 this : p { val := i, isLt := hi } = p' i ⊢ p { val := i, isLt := hi } ∈ range ↑γ convert hγ.1 i (Nat.le_of_lt_succ hi) X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) } γ : Path (p' 0) (p' n) hγ : (∀ (i : ℕ), i ≤ n → p' i ∈ range ↑γ) ∧ range ↑γ ⊆ s hpp' : ∀ (k : ℕ), k < n + 1 → p ↑k = p' k i : ℕ hi : i < n + 1 this : i = i % Nat.succ n ⊢ p { val := i, isLt := hi } = p ↑i congr ** Qed
IsPathConnected.exists_path_through_family' X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s ⊢ ∃ γ t, (∀ (t : ↑I), ↑γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), ↑γ (t i) = p i rcases h.exists_path_through_family p hp with ⟨γ, hγ⟩ case intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s γ : Path (p 0) (p ↑n) hγ : range ↑γ ⊆ s ∧ ∀ (i : Fin (n + 1)), p i ∈ range ↑γ ⊢ ∃ γ t, (∀ (t : ↑I), ↑γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), ↑γ (t i) = p i rcases hγ with ⟨h₁, h₂⟩ case intro.intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s γ : Path (p 0) (p ↑n) h₁ : range ↑γ ⊆ s h₂ : ∀ (i : Fin (n + 1)), p i ∈ range ↑γ ⊢ ∃ γ t, (∀ (t : ↑I), ↑γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), ↑γ (t i) = p i simp only [range, mem_setOf_eq] at h₂ case intro.intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s γ : Path (p 0) (p ↑n) h₁ : range ↑γ ⊆ s h₂ : ∀ (i : Fin (n + 1)), ∃ y, ↑γ y = p i ⊢ ∃ γ t, (∀ (t : ↑I), ↑γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), ↑γ (t i) = p i rw [range_subset_iff] at h₁ case intro.intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s γ : Path (p 0) (p ↑n) h₁ : ∀ (y : ↑I), ↑γ y ∈ s h₂ : ∀ (i : Fin (n + 1)), ∃ y, ↑γ y = p i ⊢ ∃ γ t, (∀ (t : ↑I), ↑γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), ↑γ (t i) = p i choose! t ht using h₂ case intro.intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 F : Set X✝ X : Type u_4 inst✝ : TopologicalSpace X n : ℕ s : Set X h : IsPathConnected s p : Fin (n + 1) → X hp : ∀ (i : Fin (n + 1)), p i ∈ s γ : Path (p 0) (p ↑n) h₁ : ∀ (y : ↑I), ↑γ y ∈ s t : Fin (n + 1) → ↑I ht : ∀ (i : Fin (n + 1)), ↑γ (t i) = p i ⊢ ∃ γ t, (∀ (t : ↑I), ↑γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), ↑γ (t i) = p i exact ⟨γ, t, h₁, ht⟩ ** Qed
pathConnectedSpace_iff_zerothHomotopy X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ PathConnectedSpace X ↔ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) letI := pathSetoid X X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X this : Setoid X := pathSetoid X ⊢ PathConnectedSpace X ↔ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) constructor case mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X this : Setoid X := pathSetoid X ⊢ PathConnectedSpace X → Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) intro h case mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X this : Setoid X := pathSetoid X h : PathConnectedSpace X ⊢ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) refine' ⟨(nonempty_quotient_iff _).mpr h.1, ⟨_⟩⟩ case mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X this : Setoid X := pathSetoid X h : PathConnectedSpace X ⊢ ∀ (a b : ZerothHomotopy X), a = b rintro ⟨x⟩ ⟨y⟩ case mp.mk.mk X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z : X ι : Type u_3 F : Set X this : Setoid X := pathSetoid X h : PathConnectedSpace X a✝ : ZerothHomotopy X x : X b✝ : ZerothHomotopy X y : X ⊢ Quot.mk Setoid.r x = Quot.mk Setoid.r y exact Quotient.sound (PathConnectedSpace.Joined x y) case mpr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X this : Setoid X := pathSetoid X ⊢ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) → PathConnectedSpace X unfold ZerothHomotopy case mpr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X this : Setoid X := pathSetoid X ⊢ Nonempty (Quotient (pathSetoid X)) ∧ Subsingleton (Quotient (pathSetoid X)) → PathConnectedSpace X rintro ⟨h, h'⟩ case mpr.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X this : Setoid X := pathSetoid X h : Nonempty (Quotient (pathSetoid X)) h' : Subsingleton (Quotient (pathSetoid X)) ⊢ PathConnectedSpace X exact ⟨(nonempty_quotient_iff _).mp h, fun x y => Quotient.exact <\| Subsingleton.elim ⟦x⟧ ⟦y⟧⟩ ** Qed
isPathConnected_iff_pathConnectedSpace X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ IsPathConnected F ↔ PathConnectedSpace ↑F rw [isPathConnected_iff] X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ (Set.Nonempty F ∧ ∀ (x : X), x ∈ F → ∀ (y : X), y ∈ F → JoinedIn F x y) ↔ PathConnectedSpace ↑F constructor case mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ (Set.Nonempty F ∧ ∀ (x : X), x ∈ F → ∀ (y : X), y ∈ F → JoinedIn F x y) → PathConnectedSpace ↑F rintro ⟨⟨x, x_in⟩, h⟩ case mp.intro.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X h : ∀ (x : X), x ∈ F → ∀ (y : X), y ∈ F → JoinedIn F x y x : X x_in : x ∈ F ⊢ PathConnectedSpace ↑F refine' ⟨⟨⟨x, x_in⟩⟩, _⟩ case mp.intro.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X h : ∀ (x : X), x ∈ F → ∀ (y : X), y ∈ F → JoinedIn F x y x : X x_in : x ∈ F ⊢ ∀ (x y : ↑F), Joined x y rintro ⟨y, y_in⟩ ⟨z, z_in⟩ case mp.intro.intro.mk.mk X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z✝ : X ι : Type u_3 F : Set X h : ∀ (x : X), x ∈ F → ∀ (y : X), y ∈ F → JoinedIn F x y x : X x_in : x ∈ F y : X y_in : y ∈ F z : X z_in : z ∈ F ⊢ Joined { val := y, property := y_in } { val := z, property := z_in } have H := h y y_in z z_in case mp.intro.intro.mk.mk X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z✝ : X ι : Type u_3 F : Set X h : ∀ (x : X), x ∈ F → ∀ (y : X), y ∈ F → JoinedIn F x y x : X x_in : x ∈ F y : X y_in : y ∈ F z : X z_in : z ∈ F H : JoinedIn F y z ⊢ Joined { val := y, property := y_in } { val := z, property := z_in } rwa [joinedIn_iff_joined y_in z_in] at H case mpr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ PathConnectedSpace ↑F → Set.Nonempty F ∧ ∀ (x : X), x ∈ F → ∀ (y : X), y ∈ F → JoinedIn F x y rintro ⟨⟨x, x_in⟩, H⟩ case mpr.mk.intro.mk X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X H : ∀ (x y : ↑F), Joined x y x : X x_in : x ∈ F ⊢ Set.Nonempty F ∧ ∀ (x : X), x ∈ F → ∀ (y : X), y ∈ F → JoinedIn F x y refine' ⟨⟨x, x_in⟩, fun y y_in z z_in => _⟩ case mpr.mk.intro.mk X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z✝ : X ι : Type u_3 F : Set X H : ∀ (x y : ↑F), Joined x y x : X x_in : x ∈ F y : X y_in : y ∈ F z : X z_in : z ∈ F ⊢ JoinedIn F y z rw [joinedIn_iff_joined y_in z_in] case mpr.mk.intro.mk X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z✝ : X ι : Type u_3 F : Set X H : ∀ (x y : ↑F), Joined x y x : X x_in : x ∈ F y : X y_in : y ∈ F z : X z_in : z ∈ F ⊢ Joined { val := y, property := y_in } { val := z, property := z_in } apply H ** Qed
pathConnectedSpace_iff_univ X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ PathConnectedSpace X ↔ IsPathConnected univ constructor case mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ PathConnectedSpace X → IsPathConnected univ intro h case mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : PathConnectedSpace X ⊢ IsPathConnected univ haveI := @PathConnectedSpace.Nonempty X _ _ case mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : PathConnectedSpace X this : Nonempty X ⊢ IsPathConnected univ inhabit X case mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : PathConnectedSpace X this : Nonempty X inhabited_h : Inhabited X ⊢ IsPathConnected univ refine' ⟨default, mem_univ _, _⟩ case mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : PathConnectedSpace X this : Nonempty X inhabited_h : Inhabited X ⊢ ∀ {y : X}, y ∈ univ → JoinedIn univ default y intros y _hy case mp X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y✝ z : X ι : Type u_3 F : Set X h : PathConnectedSpace X this : Nonempty X inhabited_h : Inhabited X y : X _hy : y ∈ univ ⊢ JoinedIn univ default y simpa using PathConnectedSpace.Joined default y case mpr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ IsPathConnected univ → PathConnectedSpace X intro h case mpr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : IsPathConnected univ ⊢ PathConnectedSpace X have h' := h.joinedIn case mpr X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X h : IsPathConnected univ h' : ∀ (x : X), x ∈ univ → ∀ (y : X), y ∈ univ → JoinedIn univ x y ⊢ PathConnectedSpace X cases' h with x h case mpr.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X h' : ∀ (x : X), x ∈ univ → ∀ (y : X), y ∈ univ → JoinedIn univ x y x : X h : x ∈ univ ∧ ∀ {y : X}, y ∈ univ → JoinedIn univ x y ⊢ PathConnectedSpace X exact ⟨⟨x⟩, by simpa using h'⟩ X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X h' : ∀ (x : X), x ∈ univ → ∀ (y : X), y ∈ univ → JoinedIn univ x y x : X h : x ∈ univ ∧ ∀ {y : X}, y ∈ univ → JoinedIn univ x y ⊢ ∀ (x y : X), Joined x y simpa using h' ** Qed
isPathConnected_range X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X f : X → Y hf : Continuous f ⊢ IsPathConnected (range f) rw [← image_univ] X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X f : X → Y hf : Continuous f ⊢ IsPathConnected (f '' univ) exact isPathConnected_univ.image hf ** Qed
Function.Surjective.pathConnectedSpace X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X f : X → Y hf : Surjective f hf' : Continuous f ⊢ PathConnectedSpace Y rw [pathConnectedSpace_iff_univ, ← hf.range_eq] X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X f : X → Y hf : Surjective f hf' : Continuous f ⊢ IsPathConnected (range f) exact isPathConnected_range hf' ** Qed
pathConnectedSpace_iff_eq X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X ⊢ PathConnectedSpace X ↔ ∃ x, pathComponent x = univ simp [pathConnectedSpace_iff_univ, isPathConnected_iff_eq] ** Qed
IsPathConnected.isConnected X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X hF : IsPathConnected F ⊢ IsConnected F rw [isConnected_iff_connectedSpace] X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X hF : IsPathConnected F ⊢ ConnectedSpace ↑F rw [isPathConnected_iff_pathConnectedSpace] at hF X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X hF : PathConnectedSpace ↑F ⊢ ConnectedSpace ↑F exact @PathConnectedSpace.connectedSpace _ _ hF ** Qed
PathConnectedSpace.exists_path_through_family X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X n : ℕ p : Fin (n + 1) → X ⊢ ∃ γ, ∀ (i : Fin (n + 1)), p i ∈ range ↑γ have : IsPathConnected (univ : Set X) := pathConnectedSpace_iff_univ.mp (by infer_instance) X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X n : ℕ p : Fin (n + 1) → X this : IsPathConnected univ ⊢ ∃ γ, ∀ (i : Fin (n + 1)), p i ∈ range ↑γ rcases this.exists_path_through_family p fun _i => True.intro with ⟨γ, -, h⟩ case intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X n : ℕ p : Fin (n + 1) → X this : IsPathConnected univ γ : Path (p 0) (p ↑n) h : ∀ (i : Fin (n + 1)), p i ∈ range ↑γ ⊢ ∃ γ, ∀ (i : Fin (n + 1)), p i ∈ range ↑γ exact ⟨γ, h⟩ X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X n : ℕ p : Fin (n + 1) → X ⊢ PathConnectedSpace X infer_instance ** Qed
PathConnectedSpace.exists_path_through_family' X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X n : ℕ p : Fin (n + 1) → X ⊢ ∃ γ t, ∀ (i : Fin (n + 1)), ↑γ (t i) = p i have : IsPathConnected (univ : Set X) := pathConnectedSpace_iff_univ.mp (by infer_instance) X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X n : ℕ p : Fin (n + 1) → X this : IsPathConnected univ ⊢ ∃ γ t, ∀ (i : Fin (n + 1)), ↑γ (t i) = p i rcases this.exists_path_through_family' p fun _i => True.intro with ⟨γ, t, -, h⟩ case intro.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X n : ℕ p : Fin (n + 1) → X this : IsPathConnected univ γ : Path (p 0) (p ↑n) t : Fin (n + 1) → ↑I h : ∀ (i : Fin (n + 1)), ↑γ (t i) = p i ⊢ ∃ γ t, ∀ (i : Fin (n + 1)), ↑γ (t i) = p i exact ⟨γ, t, h⟩ X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : PathConnectedSpace X n : ℕ p : Fin (n + 1) → X ⊢ PathConnectedSpace X infer_instance ** Qed
locPathConnected_of_bases X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X p : ι → Prop s : X → ι → Set X h : ∀ (x : X), HasBasis (𝓝 x) p (s x) h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i) ⊢ LocPathConnectedSpace X constructor case path_connected_basis X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X p : ι → Prop s : X → ι → Set X h : ∀ (x : X), HasBasis (𝓝 x) p (s x) h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i) ⊢ ∀ (x : X), HasBasis (𝓝 x) (fun s => s ∈ 𝓝 x ∧ IsPathConnected s) id intro x case path_connected_basis X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X p : ι → Prop s : X → ι → Set X h : ∀ (x : X), HasBasis (𝓝 x) p (s x) h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i) x : X ⊢ HasBasis (𝓝 x) (fun s => s ∈ 𝓝 x ∧ IsPathConnected s) id apply (h x).to_hasBasis case path_connected_basis.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X p : ι → Prop s : X → ι → Set X h : ∀ (x : X), HasBasis (𝓝 x) p (s x) h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i) x : X ⊢ ∀ (i : ι), p i → ∃ i', (i' ∈ 𝓝 x ∧ IsPathConnected i') ∧ id i' ⊆ s x i intro i pi case path_connected_basis.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X p : ι → Prop s : X → ι → Set X h : ∀ (x : X), HasBasis (𝓝 x) p (s x) h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i) x : X i : ι pi : p i ⊢ ∃ i', (i' ∈ 𝓝 x ∧ IsPathConnected i') ∧ id i' ⊆ s x i exact ⟨s x i, ⟨(h x).mem_of_mem pi, h' x i pi⟩, by rfl⟩ X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X p : ι → Prop s : X → ι → Set X h : ∀ (x : X), HasBasis (𝓝 x) p (s x) h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i) x : X i : ι pi : p i ⊢ id (s x i) ⊆ s x i rfl case path_connected_basis.h' X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X p : ι → Prop s : X → ι → Set X h : ∀ (x : X), HasBasis (𝓝 x) p (s x) h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i) x : X ⊢ ∀ (i' : Set X), i' ∈ 𝓝 x ∧ IsPathConnected i' → ∃ i, p i ∧ s x i ⊆ id i' rintro U ⟨U_in, _hU⟩ case path_connected_basis.h'.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X p : ι → Prop s : X → ι → Set X h : ∀ (x : X), HasBasis (𝓝 x) p (s x) h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i) x : X U : Set X U_in : U ∈ 𝓝 x _hU : IsPathConnected U ⊢ ∃ i, p i ∧ s x i ⊆ id U rcases (h x).mem_iff.mp U_in with ⟨i, pi, hi⟩ case path_connected_basis.h'.intro.intro.intro X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X p : ι → Prop s : X → ι → Set X h : ∀ (x : X), HasBasis (𝓝 x) p (s x) h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i) x : X U : Set X U_in : U ∈ 𝓝 x _hU : IsPathConnected U i : ι pi : p i hi : s x i ⊆ U ⊢ ∃ i, p i ∧ s x i ⊆ id U tauto ** Qed
pathConnectedSpace_iff_connectedSpace X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X ⊢ PathConnectedSpace X ↔ ConnectedSpace X constructor case mp X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X ⊢ PathConnectedSpace X → ConnectedSpace X intro h case mp X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X h : PathConnectedSpace X ⊢ ConnectedSpace X infer_instance case mpr X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X ⊢ ConnectedSpace X → PathConnectedSpace X intro hX case mpr X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X ⊢ PathConnectedSpace X rw [pathConnectedSpace_iff_eq] case mpr X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X ⊢ ∃ x, pathComponent x = univ use Classical.arbitrary X case h X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X ⊢ pathComponent (Classical.arbitrary X) = univ refine' IsClopen.eq_univ ⟨_, _⟩ (by simp) X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X ⊢ Set.Nonempty (pathComponent (Classical.arbitrary X)) simp case h.refine'_1 X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X ⊢ IsOpen (pathComponent (Classical.arbitrary X)) rw [isOpen_iff_mem_nhds] case h.refine'_1 X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X ⊢ ∀ (a : X), a ∈ pathComponent (Classical.arbitrary X) → pathComponent (Classical.arbitrary X) ∈ 𝓝 a intro y y_in case h.refine'_1 X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y✝ z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X y : X y_in : y ∈ pathComponent (Classical.arbitrary X) ⊢ pathComponent (Classical.arbitrary X) ∈ 𝓝 y rcases (path_connected_basis y).ex_mem with ⟨U, ⟨U_in, hU⟩⟩ case h.refine'_1.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y✝ z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X y : X y_in : y ∈ pathComponent (Classical.arbitrary X) U : Set X U_in : U ∈ 𝓝 y hU : IsPathConnected U ⊢ pathComponent (Classical.arbitrary X) ∈ 𝓝 y apply mem_of_superset U_in case h.refine'_1.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y✝ z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X y : X y_in : y ∈ pathComponent (Classical.arbitrary X) U : Set X U_in : U ∈ 𝓝 y hU : IsPathConnected U ⊢ U ⊆ pathComponent (Classical.arbitrary X) rw [← pathComponent_congr y_in] case h.refine'_1.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y✝ z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X y : X y_in : y ∈ pathComponent (Classical.arbitrary X) U : Set X U_in : U ∈ 𝓝 y hU : IsPathConnected U ⊢ U ⊆ pathComponent y exact hU.subset_pathComponent (mem_of_mem_nhds U_in) case h.refine'_2 X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X ⊢ IsClosed (pathComponent (Classical.arbitrary X)) rw [isClosed_iff_nhds] case h.refine'_2 X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X ⊢ ∀ (x : X), (∀ (U : Set X), U ∈ 𝓝 x → Set.Nonempty (U ∩ pathComponent (Classical.arbitrary X))) → x ∈ pathComponent (Classical.arbitrary X) intro y H case h.refine'_2 X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y✝ z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X y : X H : ∀ (U : Set X), U ∈ 𝓝 y → Set.Nonempty (U ∩ pathComponent (Classical.arbitrary X)) ⊢ y ∈ pathComponent (Classical.arbitrary X) rcases (path_connected_basis y).ex_mem with ⟨U, ⟨U_in, hU⟩⟩ case h.refine'_2.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y✝ z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X y : X H : ∀ (U : Set X), U ∈ 𝓝 y → Set.Nonempty (U ∩ pathComponent (Classical.arbitrary X)) U : Set X U_in : U ∈ 𝓝 y hU : IsPathConnected U ⊢ y ∈ pathComponent (Classical.arbitrary X) rcases H U U_in with ⟨z, hz, hz'⟩ case h.refine'_2.intro.intro.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y✝ z✝ : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X hX : ConnectedSpace X y : X H : ∀ (U : Set X), U ∈ 𝓝 y → Set.Nonempty (U ∩ pathComponent (Classical.arbitrary X)) U : Set X U_in : U ∈ 𝓝 y hU : IsPathConnected U z : X hz : z ∈ U hz' : z ∈ pathComponent (Classical.arbitrary X) ⊢ y ∈ pathComponent (Classical.arbitrary X) exact (hU.joinedIn z hz y <\| mem_of_mem_nhds U_in).joined.mem_pathComponent hz' ** Qed
locPathConnected_of_isOpen X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U ⊢ ∀ (x : ↑U), HasBasis (𝓝 x) (fun s => s ∈ 𝓝 x ∧ IsPathConnected s) id rintro ⟨x, x_in⟩ case mk X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x : X x_in : x ∈ U ⊢ HasBasis (𝓝 { val := x, property := x_in }) (fun s => s ∈ 𝓝 { val := x, property := x_in } ∧ IsPathConnected s) id rw [nhds_subtype_eq_comap] case mk X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x : X x_in : x ∈ U ⊢ HasBasis (comap Subtype.val (𝓝 x)) (fun s => s ∈ comap Subtype.val (𝓝 x) ∧ IsPathConnected s) id constructor case mk.mem_iff' X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x : X x_in : x ∈ U ⊢ ∀ (t : Set ↑U), t ∈ comap Subtype.val (𝓝 x) ↔ ∃ i, (i ∈ comap Subtype.val (𝓝 x) ∧ IsPathConnected i) ∧ id i ⊆ t intro V case mk.mem_iff' X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x : X x_in : x ∈ U V : Set ↑U ⊢ V ∈ comap Subtype.val (𝓝 x) ↔ ∃ i, (i ∈ comap Subtype.val (𝓝 x) ∧ IsPathConnected i) ∧ id i ⊆ V rw [(HasBasis.comap ((↑) : U → X) (pathConnected_subset_basis h x_in)).mem_iff] case mk.mem_iff' X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x : X x_in : x ∈ U V : Set ↑U ⊢ (∃ i, (i ∈ 𝓝 x ∧ IsPathConnected i ∧ i ⊆ U) ∧ Subtype.val ⁻¹' id i ⊆ V) ↔ ∃ i, (i ∈ comap Subtype.val (𝓝 x) ∧ IsPathConnected i) ∧ id i ⊆ V constructor case mk.mem_iff'.mp X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x : X x_in : x ∈ U V : Set ↑U ⊢ (∃ i, (i ∈ 𝓝 x ∧ IsPathConnected i ∧ i ⊆ U) ∧ Subtype.val ⁻¹' id i ⊆ V) → ∃ i, (i ∈ comap Subtype.val (𝓝 x) ∧ IsPathConnected i) ∧ id i ⊆ V rintro ⟨W, ⟨W_in, hW, hWU⟩, hWV⟩ case mk.mem_iff'.mp.intro.intro.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x : X x_in : x ∈ U V : Set ↑U W : Set X hWV : Subtype.val ⁻¹' id W ⊆ V W_in : W ∈ 𝓝 x hW : IsPathConnected W hWU : W ⊆ U ⊢ ∃ i, (i ∈ comap Subtype.val (𝓝 x) ∧ IsPathConnected i) ∧ id i ⊆ V exact ⟨Subtype.val ⁻¹' W, ⟨⟨preimage_mem_comap W_in, hW.preimage_coe hWU⟩, hWV⟩⟩ case mk.mem_iff'.mpr X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x : X x_in : x ∈ U V : Set ↑U ⊢ (∃ i, (i ∈ comap Subtype.val (𝓝 x) ∧ IsPathConnected i) ∧ id i ⊆ V) → ∃ i, (i ∈ 𝓝 x ∧ IsPathConnected i ∧ i ⊆ U) ∧ Subtype.val ⁻¹' id i ⊆ V rintro ⟨W, ⟨W_in, hW⟩, hWV⟩ case mk.mem_iff'.mpr.intro.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x : X x_in : x ∈ U V W : Set ↑U hWV : id W ⊆ V W_in : W ∈ comap Subtype.val (𝓝 x) hW : IsPathConnected W ⊢ ∃ i, (i ∈ 𝓝 x ∧ IsPathConnected i ∧ i ⊆ U) ∧ Subtype.val ⁻¹' id i ⊆ V refine' ⟨(↑) '' W, ⟨Filter.image_coe_mem_of_mem_comap (IsOpen.mem_nhds h x_in) W_in, hW.image continuous_subtype_val, Subtype.coe_image_subset U W⟩, _⟩ case mk.mem_iff'.mpr.intro.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝ y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x : X x_in : x ∈ U V W : Set ↑U hWV : id W ⊆ V W_in : W ∈ comap Subtype.val (𝓝 x) hW : IsPathConnected W ⊢ Subtype.val ⁻¹' id (Subtype.val '' W) ⊆ V rintro x ⟨y, ⟨y_in, hy⟩⟩ case mk.mem_iff'.mpr.intro.intro.intro.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝¹ y✝ z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x✝ : X x_in : x✝ ∈ U V W : Set ↑U hWV : id W ⊆ V W_in : W ∈ comap Subtype.val (𝓝 x✝) hW : IsPathConnected W x : { x // x ∈ U } y : ↑U y_in : y ∈ W hy : ↑y = ↑x ⊢ x ∈ V rw [← Subtype.coe_injective hy] case mk.mem_iff'.mpr.intro.intro.intro.intro.intro X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x✝¹ y✝ z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X h : IsOpen U x✝ : X x_in : x✝ ∈ U V W : Set ↑U hWV : id W ⊆ V W_in : W ∈ comap Subtype.val (𝓝 x✝) hW : IsPathConnected W x : { x // x ∈ U } y : ↑U y_in : y ∈ W hy : ↑y = ↑x ⊢ y ∈ V tauto ** Qed
IsOpen.isConnected_iff_isPathConnected X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X U_op : IsOpen U ⊢ IsPathConnected U ↔ IsConnected U rw [isConnected_iff_connectedSpace, isPathConnected_iff_pathConnectedSpace] X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X U_op : IsOpen U ⊢ PathConnectedSpace ↑U ↔ ConnectedSpace ↑U haveI := locPathConnected_of_isOpen U_op X : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x y z : X ι : Type u_3 F : Set X inst✝ : LocPathConnectedSpace X U : Set X U_op : IsOpen U this : LocPathConnectedSpace ↑U ⊢ PathConnectedSpace ↑U ↔ ConnectedSpace ↑U exact pathConnectedSpace_iff_connectedSpace ** Qed
isPreconnected_of_forall α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t ⊢ IsPreconnected s rintro u v hu hv hs ⟨z, zs, zu⟩ ⟨y, ys, yv⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) have xs : x ∈ s := by rcases H y ys with ⟨t, ts, xt, -, -⟩ exact ts xt case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s ⊢ Set.Nonempty (s ∩ (u ∩ v)) cases hs xs case intro.intro.intro.intro.inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s h✝ : x ∈ u ⊢ Set.Nonempty (s ∩ (u ∩ v)) case intro.intro.intro.intro.inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s h✝ : x ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) case inl xu => rcases H y ys with ⟨t, ts, xt, yt, ht⟩ have := ht u v hu hv (ts.trans hs) ⟨x, xt, xu⟩ ⟨y, yt, yv⟩ exact this.imp fun z hz => ⟨ts hz.1, hz.2⟩ case intro.intro.intro.intro.inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s h✝ : x ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) case inr xv => rcases H z zs with ⟨t, ts, xt, zt, ht⟩ have := ht v u hv hu (ts.trans <\| by rwa [union_comm]) ⟨x, xt, xv⟩ ⟨z, zt, zu⟩ exact this.imp fun _ h => ⟨ts h.1, h.2.2, h.2.1⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v ⊢ x ∈ s rcases H y ys with ⟨t, ts, xt, -, -⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v t : Set α ts : t ⊆ s xt : x ∈ t ⊢ x ∈ s exact ts xt α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s xu : x ∈ u ⊢ Set.Nonempty (s ∩ (u ∩ v)) rcases H y ys with ⟨t, ts, xt, yt, ht⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s xu : x ∈ u t : Set α ts : t ⊆ s xt : x ∈ t yt : y ∈ t ht : IsPreconnected t ⊢ Set.Nonempty (s ∩ (u ∩ v)) have := ht u v hu hv (ts.trans hs) ⟨x, xt, xu⟩ ⟨y, yt, yv⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s xu : x ∈ u t : Set α ts : t ⊆ s xt : x ∈ t yt : y ∈ t ht : IsPreconnected t this : Set.Nonempty (t ∩ (u ∩ v)) ⊢ Set.Nonempty (s ∩ (u ∩ v)) exact this.imp fun z hz => ⟨ts hz.1, hz.2⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s xv : x ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) rcases H z zs with ⟨t, ts, xt, zt, ht⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s xv : x ∈ v t : Set α ts : t ⊆ s xt : x ∈ t zt : z ∈ t ht : IsPreconnected t ⊢ Set.Nonempty (s ∩ (u ∩ v)) have := ht v u hv hu (ts.trans <\| by rwa [union_comm]) ⟨x, xt, xv⟩ ⟨z, zt, zu⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s xv : x ∈ v t : Set α ts : t ⊆ s xt : x ∈ t zt : z ∈ t ht : IsPreconnected t this : Set.Nonempty (t ∩ (v ∩ u)) ⊢ Set.Nonempty (s ∩ (u ∩ v)) exact this.imp fun _ h => ⟨ts h.1, h.2.2, h.2.1⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u✝ v✝ s : Set α x : α H : ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v z : α zs : z ∈ s zu : z ∈ u y : α ys : y ∈ s yv : y ∈ v xs : x ∈ s xv : x ∈ v t : Set α ts : t ⊆ s xt : x ∈ t zt : z ∈ t ht : IsPreconnected t ⊢ s ⊆ v ∪ u rwa [union_comm] ** Qed
isPreconnected_of_forall_pair α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α H : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t ⊢ IsPreconnected s rcases eq_empty_or_nonempty s with (rfl \| ⟨x, hx⟩) case inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α H : ∀ (x : α), x ∈ ∅ → ∀ (y : α), y ∈ ∅ → ∃ t, t ⊆ ∅ ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t ⊢ IsPreconnected ∅ case inr.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α H : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t x : α hx : x ∈ s ⊢ IsPreconnected s exacts [isPreconnected_empty, isPreconnected_of_forall x fun y => H x hx y] ** Qed
isPreconnected_sUnion α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α c : Set (Set α) H1 : ∀ (s : Set α), s ∈ c → x ∈ s H2 : ∀ (s : Set α), s ∈ c → IsPreconnected s ⊢ IsPreconnected (⋃₀ c) apply isPreconnected_of_forall x α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α c : Set (Set α) H1 : ∀ (s : Set α), s ∈ c → x ∈ s H2 : ∀ (s : Set α), s ∈ c → IsPreconnected s ⊢ ∀ (y : α), y ∈ ⋃₀ c → ∃ t, t ⊆ ⋃₀ c ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t rintro y ⟨s, sc, ys⟩ case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v : Set α x : α c : Set (Set α) H1 : ∀ (s : Set α), s ∈ c → x ∈ s H2 : ∀ (s : Set α), s ∈ c → IsPreconnected s y : α s : Set α sc : s ∈ c ys : y ∈ s ⊢ ∃ t, t ⊆ ⋃₀ c ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t exact ⟨s, subset_sUnion_of_mem sc, H1 s sc, ys, H2 s sc⟩ ** Qed
IsPreconnected.union α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α x : α s t : Set α H1 : x ∈ s H2 : x ∈ t H3 : IsPreconnected s H4 : IsPreconnected t ⊢ ∀ (s_1 : Set α), s_1 ∈ {s, t} → x ∈ s_1 rintro r (rfl \| rfl \| h) <;> assumption α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α x : α s t : Set α H1 : x ∈ s H2 : x ∈ t H3 : IsPreconnected s H4 : IsPreconnected t ⊢ ∀ (s_1 : Set α), s_1 ∈ {s, t} → IsPreconnected s_1 rintro r (rfl \| rfl \| h) <;> assumption ** Qed
IsPreconnected.union' α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s t : Set α H : Set.Nonempty (s ∩ t) hs : IsPreconnected s ht : IsPreconnected t ⊢ IsPreconnected (s ∪ t) rcases H with ⟨x, hxs, hxt⟩ case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s t : Set α hs : IsPreconnected s ht : IsPreconnected t x : α hxs : x ∈ s hxt : x ∈ t ⊢ IsPreconnected (s ∪ t) exact hs.union x hxs hxt ht ** Qed
IsConnected.union α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s t : Set α H : Set.Nonempty (s ∩ t) Hs : IsConnected s Ht : IsConnected t ⊢ IsConnected (s ∪ t) rcases H with ⟨x, hx⟩ case intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s t : Set α Hs : IsConnected s Ht : IsConnected t x : α hx : x ∈ s ∩ t ⊢ IsConnected (s ∪ t) refine' ⟨⟨x, mem_union_left t (mem_of_mem_inter_left hx)⟩, _⟩ case intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s t : Set α Hs : IsConnected s Ht : IsConnected t x : α hx : x ∈ s ∩ t ⊢ IsPreconnected (s ∪ t) exact Hs.isPreconnected.union x (mem_of_mem_inter_left hx) (mem_of_mem_inter_right hx) Ht.isPreconnected ** Qed
IsPreconnected.sUnion_directed α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α S : Set (Set α) K : DirectedOn (fun x x_1 => x ⊆ x_1) S H : ∀ (s : Set α), s ∈ S → IsPreconnected s ⊢ IsPreconnected (⋃₀ S) rintro u v hu hv Huv ⟨a, ⟨s, hsS, has⟩, hau⟩ ⟨b, ⟨t, htS, hbt⟩, hbv⟩ 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✝ : TopologicalSpace α s✝ t✝ u✝ v✝ : Set α S : Set (Set α) K : DirectedOn (fun x x_1 => x ⊆ x_1) S H : ∀ (s : Set α), s ∈ S → IsPreconnected s u v : Set α hu : IsOpen u hv : IsOpen v Huv : ⋃₀ S ⊆ u ∪ v a : α hau : a ∈ u s : Set α hsS : s ∈ S has : a ∈ s b : α hbv : b ∈ v t : Set α htS : t ∈ S hbt : b ∈ t ⊢ Set.Nonempty (⋃₀ S ∩ (u ∩ v)) obtain ⟨r, hrS, hsr, htr⟩ : ∃ r ∈ S, s ⊆ r ∧ t ⊆ r := K s hsS t htS case intro.intro.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✝ : TopologicalSpace α s✝ t✝ u✝ v✝ : Set α S : Set (Set α) K : DirectedOn (fun x x_1 => x ⊆ x_1) S H : ∀ (s : Set α), s ∈ S → IsPreconnected s u v : Set α hu : IsOpen u hv : IsOpen v Huv : ⋃₀ S ⊆ u ∪ v a : α hau : a ∈ u s : Set α hsS : s ∈ S has : a ∈ s b : α hbv : b ∈ v t : Set α htS : t ∈ S hbt : b ∈ t r : Set α hrS : r ∈ S hsr : s ⊆ r htr : t ⊆ r ⊢ Set.Nonempty (⋃₀ S ∩ (u ∩ v)) have Hnuv : (r ∩ (u ∩ v)).Nonempty := H _ hrS u v hu hv ((subset_sUnion_of_mem hrS).trans Huv) ⟨a, hsr has, hau⟩ ⟨b, htr hbt, hbv⟩ case intro.intro.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✝ : TopologicalSpace α s✝ t✝ u✝ v✝ : Set α S : Set (Set α) K : DirectedOn (fun x x_1 => x ⊆ x_1) S H : ∀ (s : Set α), s ∈ S → IsPreconnected s u v : Set α hu : IsOpen u hv : IsOpen v Huv : ⋃₀ S ⊆ u ∪ v a : α hau : a ∈ u s : Set α hsS : s ∈ S has : a ∈ s b : α hbv : b ∈ v t : Set α htS : t ∈ S hbt : b ∈ t r : Set α hrS : r ∈ S hsr : s ⊆ r htr : t ⊆ r Hnuv : Set.Nonempty (r ∩ (u ∩ v)) ⊢ Set.Nonempty (⋃₀ S ∩ (u ∩ v)) have Kruv : r ∩ (u ∩ v) ⊆ ⋃₀ S ∩ (u ∩ v) := inter_subset_inter_left _ (subset_sUnion_of_mem hrS) case intro.intro.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✝ : TopologicalSpace α s✝ t✝ u✝ v✝ : Set α S : Set (Set α) K : DirectedOn (fun x x_1 => x ⊆ x_1) S H : ∀ (s : Set α), s ∈ S → IsPreconnected s u v : Set α hu : IsOpen u hv : IsOpen v Huv : ⋃₀ S ⊆ u ∪ v a : α hau : a ∈ u s : Set α hsS : s ∈ S has : a ∈ s b : α hbv : b ∈ v t : Set α htS : t ∈ S hbt : b ∈ t r : Set α hrS : r ∈ S hsr : s ⊆ r htr : t ⊆ r Hnuv : Set.Nonempty (r ∩ (u ∩ v)) Kruv : r ∩ (u ∩ v) ⊆ ⋃₀ S ∩ (u ∩ v) ⊢ Set.Nonempty (⋃₀ S ∩ (u ∩ v)) exact Hnuv.mono Kruv ** Qed
IsPreconnected.biUnion_of_reflTransGen α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j ⊢ IsPreconnected (⋃ n ∈ t, s n) let R := fun i j : ι => (s i ∩ s j).Nonempty ∧ i ∈ t α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t ⊢ IsPreconnected (⋃ n ∈ t, s n) have P : ∀ i, i ∈ t → ∀ j, j ∈ t → ReflTransGen R i j → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) := fun i hi j hj h => by induction h case refl => refine ⟨{i}, singleton_subset_iff.mpr hi, mem_singleton i, mem_singleton i, ?_⟩ rw [biUnion_singleton] exact H i hi case tail j k _ hjk ih => obtain ⟨p, hpt, hip, hjp, hp⟩ := ih hjk.2 refine ⟨insert k p, insert_subset_iff.mpr ⟨hj, hpt⟩, mem_insert_of_mem k hip, mem_insert k p, ?_⟩ rw [biUnion_insert] refine (H k hj).union' (hjk.1.mono ?_) hp rw [inter_comm] exact inter_subset_inter_right _ (subset_biUnion_of_mem hjp) α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t P : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen R i j → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) ⊢ IsPreconnected (⋃ n ∈ t, s n) refine' isPreconnected_of_forall_pair _ α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t P : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen R i j → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) ⊢ ∀ (x : α), x ∈ ⋃ n ∈ t, s n → ∀ (y : α), y ∈ ⋃ n ∈ t, s n → ∃ t_1, t_1 ⊆ ⋃ n ∈ t, s n ∧ x ∈ t_1 ∧ y ∈ t_1 ∧ IsPreconnected t_1 intro x hx y hy α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t P : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen R i j → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) x : α hx : x ∈ ⋃ n ∈ t, s n y : α hy : y ∈ ⋃ n ∈ t, s n ⊢ ∃ t_1, t_1 ⊆ ⋃ n ∈ t, s n ∧ x ∈ t_1 ∧ y ∈ t_1 ∧ IsPreconnected t_1 obtain ⟨i : ι, hi : i ∈ t, hxi : x ∈ s i⟩ := mem_iUnion₂.1 hx case intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t P : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen R i j → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) x : α hx : x ∈ ⋃ n ∈ t, s n y : α hy : y ∈ ⋃ n ∈ t, s n i : ι hi : i ∈ t hxi : x ∈ s i ⊢ ∃ t_1, t_1 ⊆ ⋃ n ∈ t, s n ∧ x ∈ t_1 ∧ y ∈ t_1 ∧ IsPreconnected t_1 obtain ⟨j : ι, hj : j ∈ t, hyj : y ∈ s j⟩ := mem_iUnion₂.1 hy case intro.intro.intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t P : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen R i j → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) x : α hx : x ∈ ⋃ n ∈ t, s n y : α hy : y ∈ ⋃ n ∈ t, s n i : ι hi : i ∈ t hxi : x ∈ s i j : ι hj : j ∈ t hyj : y ∈ s j ⊢ ∃ t_1, t_1 ⊆ ⋃ n ∈ t, s n ∧ x ∈ t_1 ∧ y ∈ t_1 ∧ IsPreconnected t_1 obtain ⟨p, hpt, hip, hjp, hp⟩ := P i hi j hj (K i hi j hj) 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✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t P : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen R i j → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) x : α hx : x ∈ ⋃ n ∈ t, s n y : α hy : y ∈ ⋃ n ∈ t, s n i : ι hi : i ∈ t hxi : x ∈ s i j : ι hj : j ∈ t hyj : y ∈ s j p : Set ι hpt : p ⊆ t hip : i ∈ p hjp : j ∈ p hp : IsPreconnected (⋃ j ∈ p, s j) ⊢ ∃ t_1, t_1 ⊆ ⋃ n ∈ t, s n ∧ x ∈ t_1 ∧ y ∈ t_1 ∧ IsPreconnected t_1 exact ⟨⋃ j ∈ p, s j, biUnion_subset_biUnion_left hpt, mem_biUnion hip hxi, mem_biUnion hjp hyj, hp⟩ α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j : ι hj : j ∈ t h : ReflTransGen R i j ⊢ ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) induction h case refl α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j : ι hj : i ∈ t ⊢ ∃ p, p ⊆ t ∧ i ∈ p ∧ i ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) case tail α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j b✝ c✝ : ι a✝¹ : ReflTransGen R i b✝ a✝ : R b✝ c✝ a_ih✝ : b✝ ∈ t → ∃ p, p ⊆ t ∧ i ∈ p ∧ b✝ ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) hj : c✝ ∈ t ⊢ ∃ p, p ⊆ t ∧ i ∈ p ∧ c✝ ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) case refl => refine ⟨{i}, singleton_subset_iff.mpr hi, mem_singleton i, mem_singleton i, ?_⟩ rw [biUnion_singleton] exact H i hi case tail α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j b✝ c✝ : ι a✝¹ : ReflTransGen R i b✝ a✝ : R b✝ c✝ a_ih✝ : b✝ ∈ t → ∃ p, p ⊆ t ∧ i ∈ p ∧ b✝ ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) hj : c✝ ∈ t ⊢ ∃ p, p ⊆ t ∧ i ∈ p ∧ c✝ ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) case tail j k _ hjk ih => obtain ⟨p, hpt, hip, hjp, hp⟩ := ih hjk.2 refine ⟨insert k p, insert_subset_iff.mpr ⟨hj, hpt⟩, mem_insert_of_mem k hip, mem_insert k p, ?_⟩ rw [biUnion_insert] refine (H k hj).union' (hjk.1.mono ?_) hp rw [inter_comm] exact inter_subset_inter_right _ (subset_biUnion_of_mem hjp) α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j : ι hj : i ∈ t ⊢ ∃ p, p ⊆ t ∧ i ∈ p ∧ i ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) refine ⟨{i}, singleton_subset_iff.mpr hi, mem_singleton i, mem_singleton i, ?_⟩ α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j : ι hj : i ∈ t ⊢ IsPreconnected (⋃ j ∈ {i}, s j) rw [biUnion_singleton] α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j : ι hj : i ∈ t ⊢ IsPreconnected (s i) exact H i hi α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j✝ j k : ι a✝ : ReflTransGen R i j hjk : R j k ih : j ∈ t → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) hj : k ∈ t ⊢ ∃ p, p ⊆ t ∧ i ∈ p ∧ k ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) obtain ⟨p, hpt, hip, hjp, hp⟩ := ih hjk.2 case intro.intro.intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j✝ j k : ι a✝ : ReflTransGen R i j hjk : R j k ih : j ∈ t → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) hj : k ∈ t p : Set ι hpt : p ⊆ t hip : i ∈ p hjp : j ∈ p hp : IsPreconnected (⋃ j ∈ p, s j) ⊢ ∃ p, p ⊆ t ∧ i ∈ p ∧ k ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) refine ⟨insert k p, insert_subset_iff.mpr ⟨hj, hpt⟩, mem_insert_of_mem k hip, mem_insert k p, ?_⟩ case intro.intro.intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j✝ j k : ι a✝ : ReflTransGen R i j hjk : R j k ih : j ∈ t → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) hj : k ∈ t p : Set ι hpt : p ⊆ t hip : i ∈ p hjp : j ∈ p hp : IsPreconnected (⋃ j ∈ p, s j) ⊢ IsPreconnected (⋃ j ∈ insert k p, s j) rw [biUnion_insert] case intro.intro.intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j✝ j k : ι a✝ : ReflTransGen R i j hjk : R j k ih : j ∈ t → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) hj : k ∈ t p : Set ι hpt : p ⊆ t hip : i ∈ p hjp : j ∈ p hp : IsPreconnected (⋃ j ∈ p, s j) ⊢ IsPreconnected (s k ∪ ⋃ x ∈ p, s x) refine (H k hj).union' (hjk.1.mono ?_) hp case intro.intro.intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j✝ j k : ι a✝ : ReflTransGen R i j hjk : R j k ih : j ∈ t → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) hj : k ∈ t p : Set ι hpt : p ⊆ t hip : i ∈ p hjp : j ∈ p hp : IsPreconnected (⋃ j ∈ p, s j) ⊢ s j ∩ s k ⊆ s k ∩ ⋃ x ∈ p, s x rw [inter_comm] case intro.intro.intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v : Set α ι : Type u_3 t : Set ι s : ι → Set α H : ∀ (i : ι), i ∈ t → IsPreconnected (s i) K : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j R : ι → ι → Prop := fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t i : ι hi : i ∈ t j✝ j k : ι a✝ : ReflTransGen R i j hjk : R j k ih : j ∈ t → ∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) hj : k ∈ t p : Set ι hpt : p ⊆ t hip : i ∈ p hjp : j ∈ p hp : IsPreconnected (⋃ j ∈ p, s j) ⊢ s k ∩ s j ⊆ s k ∩ ⋃ x ∈ p, s x exact inter_subset_inter_right _ (subset_biUnion_of_mem hjp) ** Qed
IsPreconnected.iUnion_of_reflTransGen α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t u v : Set α ι : Type u_3 s : ι → Set α H : ∀ (i : ι), IsPreconnected (s i) K : ∀ (i j : ι), ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j)) i j ⊢ IsPreconnected (⋃ n, s n) rw [← biUnion_univ] α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t u v : Set α ι : Type u_3 s : ι → Set α H : ∀ (i : ι), IsPreconnected (s i) K : ∀ (i j : ι), ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j)) i j ⊢ IsPreconnected (⋃ x ∈ univ, s x) exact IsPreconnected.biUnion_of_reflTransGen (fun i _ => H i) fun i _ j _ => by simpa [mem_univ] using K i j α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝ : TopologicalSpace α s✝ t u v : Set α ι : Type u_3 s : ι → Set α H : ∀ (i : ι), IsPreconnected (s i) K : ∀ (i j : ι), ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j)) i j i : ι x✝¹ : i ∈ univ j : ι x✝ : j ∈ univ ⊢ ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ univ) i j simpa [mem_univ] using K i j ** Qed
IsPreconnected.iUnion_of_chain α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α s✝ t u v : Set α inst✝² : LinearOrder β inst✝¹ : SuccOrder β inst✝ : IsSuccArchimedean β s : β → Set α H : ∀ (n : β), IsPreconnected (s n) K : ∀ (n : β), Set.Nonempty (s n ∩ s (succ n)) i✝ j i : β x✝ : i ∈ Ico j i✝ ⊢ Set.Nonempty (s (succ i) ∩ s i) rw [inter_comm] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α s✝ t u v : Set α inst✝² : LinearOrder β inst✝¹ : SuccOrder β inst✝ : IsSuccArchimedean β s : β → Set α H : ∀ (n : β), IsPreconnected (s n) K : ∀ (n : β), Set.Nonempty (s n ∩ s (succ n)) i✝ j i : β x✝ : i ∈ Ico j i✝ ⊢ Set.Nonempty (s i ∩ s (succ i)) exact K i ** Qed
IsConnected.iUnion_of_chain α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝⁴ : TopologicalSpace α s✝ t u v : Set α inst✝³ : LinearOrder β inst✝² : SuccOrder β inst✝¹ : IsSuccArchimedean β inst✝ : Nonempty β s : β → Set α H : ∀ (n : β), IsConnected (s n) K : ∀ (n : β), Set.Nonempty (s n ∩ s (succ n)) i✝ j i : β x✝ : i ∈ Ico j i✝ ⊢ Set.Nonempty (s (succ i) ∩ s i) rw [inter_comm] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝⁴ : TopologicalSpace α s✝ t u v : Set α inst✝³ : LinearOrder β inst✝² : SuccOrder β inst✝¹ : IsSuccArchimedean β inst✝ : Nonempty β s : β → Set α H : ∀ (n : β), IsConnected (s n) K : ∀ (n : β), Set.Nonempty (s n ∩ s (succ n)) i✝ j i : β x✝ : i ∈ Ico j i✝ ⊢ Set.Nonempty (s i ∩ s (succ i)) exact K i ** Qed
IsPreconnected.biUnion_of_chain α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α s✝ t✝ u v : Set α inst✝² : LinearOrder β inst✝¹ : SuccOrder β inst✝ : IsSuccArchimedean β s : β → Set α t : Set β ht : OrdConnected t H : ∀ (n : β), n ∈ t → IsPreconnected (s n) K : ∀ (n : β), n ∈ t → succ n ∈ t → Set.Nonempty (s n ∩ s (succ n)) ⊢ IsPreconnected (⋃ n ∈ t, s n) have h1 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → k ∈ t := fun hi hj hk => ht.out hi hj (Ico_subset_Icc_self hk) α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α s✝ t✝ u v : Set α inst✝² : LinearOrder β inst✝¹ : SuccOrder β inst✝ : IsSuccArchimedean β s : β → Set α t : Set β ht : OrdConnected t H : ∀ (n : β), n ∈ t → IsPreconnected (s n) K : ∀ (n : β), n ∈ t → succ n ∈ t → Set.Nonempty (s n ∩ s (succ n)) h1 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → k ∈ t ⊢ IsPreconnected (⋃ n ∈ t, s n) have h2 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → succ k ∈ t := fun hi hj hk => ht.out hi hj ⟨hk.1.trans <\| le_succ _, succ_le_of_lt hk.2⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α s✝ t✝ u v : Set α inst✝² : LinearOrder β inst✝¹ : SuccOrder β inst✝ : IsSuccArchimedean β s : β → Set α t : Set β ht : OrdConnected t H : ∀ (n : β), n ∈ t → IsPreconnected (s n) K : ∀ (n : β), n ∈ t → succ n ∈ t → Set.Nonempty (s n ∩ s (succ n)) h1 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → k ∈ t h2 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → succ k ∈ t ⊢ IsPreconnected (⋃ n ∈ t, s n) have h3 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → (s k ∩ s (succ k)).Nonempty := fun hi hj hk => K _ (h1 hi hj hk) (h2 hi hj hk) α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α s✝ t✝ u v : Set α inst✝² : LinearOrder β inst✝¹ : SuccOrder β inst✝ : IsSuccArchimedean β s : β → Set α t : Set β ht : OrdConnected t H : ∀ (n : β), n ∈ t → IsPreconnected (s n) K : ∀ (n : β), n ∈ t → succ n ∈ t → Set.Nonempty (s n ∩ s (succ n)) h1 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → k ∈ t h2 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → succ k ∈ t h3 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → Set.Nonempty (s k ∩ s (succ k)) ⊢ IsPreconnected (⋃ n ∈ t, s n) refine' IsPreconnected.biUnion_of_reflTransGen H fun i hi j hj => _ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α s✝ t✝ u v : Set α inst✝² : LinearOrder β inst✝¹ : SuccOrder β inst✝ : IsSuccArchimedean β s : β → Set α t : Set β ht : OrdConnected t H : ∀ (n : β), n ∈ t → IsPreconnected (s n) K : ∀ (n : β), n ∈ t → succ n ∈ t → Set.Nonempty (s n ∩ s (succ n)) h1 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → k ∈ t h2 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → succ k ∈ t h3 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → Set.Nonempty (s k ∩ s (succ k)) i : β hi : i ∈ t j : β hj : j ∈ t ⊢ ReflTransGen (fun i j => Set.Nonempty (s i ∩ s j) ∧ i ∈ t) i j exact reflTransGen_of_succ _ (fun k hk => ⟨h3 hi hj hk, h1 hi hj hk⟩) fun k hk => ⟨by rw [inter_comm]; exact h3 hj hi hk, h2 hj hi hk⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α s✝ t✝ u v : Set α inst✝² : LinearOrder β inst✝¹ : SuccOrder β inst✝ : IsSuccArchimedean β s : β → Set α t : Set β ht : OrdConnected t H : ∀ (n : β), n ∈ t → IsPreconnected (s n) K : ∀ (n : β), n ∈ t → succ n ∈ t → Set.Nonempty (s n ∩ s (succ n)) h1 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → k ∈ t h2 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → succ k ∈ t h3 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → Set.Nonempty (s k ∩ s (succ k)) i : β hi : i ∈ t j : β hj : j ∈ t k : β hk : k ∈ Ico j i ⊢ Set.Nonempty (s (succ k) ∩ s k) rw [inter_comm] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝³ : TopologicalSpace α s✝ t✝ u v : Set α inst✝² : LinearOrder β inst✝¹ : SuccOrder β inst✝ : IsSuccArchimedean β s : β → Set α t : Set β ht : OrdConnected t H : ∀ (n : β), n ∈ t → IsPreconnected (s n) K : ∀ (n : β), n ∈ t → succ n ∈ t → Set.Nonempty (s n ∩ s (succ n)) h1 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → k ∈ t h2 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → succ k ∈ t h3 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → Set.Nonempty (s k ∩ s (succ k)) i : β hi : i ∈ t j : β hj : j ∈ t k : β hk : k ∈ Ico j i ⊢ Set.Nonempty (s k ∩ s (succ k)) exact h3 hj hi hk ** Qed
IsPreconnected.image α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s ⊢ IsPreconnected (f '' s) rintro u v hu hv huv ⟨_, ⟨x, xs, rfl⟩, xu⟩ ⟨_, ⟨y, ys, rfl⟩, yv⟩ 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✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v huv : f '' s ⊆ u ∪ v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v ⊢ Set.Nonempty (f '' s ∩ (u ∩ v)) rcases continuousOn_iff'.1 hf u hu with ⟨u', hu', u'_eq⟩ case intro.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 s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v huv : f '' s ⊆ u ∪ v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s ⊢ Set.Nonempty (f '' s ∩ (u ∩ v)) rcases continuousOn_iff'.1 hf v hv with ⟨v', hv', v'_eq⟩ case intro.intro.intro.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 s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v huv : f '' s ⊆ u ∪ v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s ⊢ Set.Nonempty (f '' s ∩ (u ∩ v)) replace huv : s ⊆ u' ∪ v' case intro.intro.intro.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 s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s huv : s ⊆ u' ∪ v' ⊢ Set.Nonempty (f '' s ∩ (u ∩ v)) obtain ⟨z, hz⟩ : (s ∩ (u' ∩ v')).Nonempty := by refine H u' v' hu' hv' huv ⟨x, ?_⟩ ⟨y, ?_⟩ <;> rw [inter_comm] exacts [u'_eq ▸ ⟨xu, xs⟩, v'_eq ▸ ⟨yv, ys⟩] case intro.intro.intro.intro.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 s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s huv : s ⊆ u' ∪ v' z : α hz : z ∈ s ∩ (u' ∩ v') ⊢ Set.Nonempty (f '' s ∩ (u ∩ v)) rw [← inter_self s, inter_assoc, inter_left_comm s u', ← inter_assoc, inter_comm s, inter_comm s, ← u'_eq, ← v'_eq] at hz case intro.intro.intro.intro.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 s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s huv : s ⊆ u' ∪ v' z : α hz : z ∈ f ⁻¹' u ∩ s ∩ (f ⁻¹' v ∩ s) ⊢ Set.Nonempty (f '' s ∩ (u ∩ v)) exact ⟨f z, ⟨z, hz.1.2, rfl⟩, hz.1.1, hz.2.1⟩ case huv α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v huv : f '' s ⊆ u ∪ v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s ⊢ s ⊆ u' ∪ v' rw [image_subset_iff, preimage_union] at huv case huv α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v huv : s ⊆ f ⁻¹' u ∪ f ⁻¹' v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s ⊢ s ⊆ u' ∪ v' replace huv := subset_inter huv Subset.rfl case huv α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s huv : s ⊆ (f ⁻¹' u ∪ f ⁻¹' v) ∩ s ⊢ s ⊆ u' ∪ v' rw [inter_distrib_right, u'_eq, v'_eq, ← inter_distrib_right] at huv case huv α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s huv : s ⊆ (u' ∪ v') ∩ s ⊢ s ⊆ u' ∪ v' exact (subset_inter_iff.1 huv).1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s huv : s ⊆ u' ∪ v' ⊢ Set.Nonempty (s ∩ (u' ∩ v')) refine H u' v' hu' hv' huv ⟨x, ?_⟩ ⟨y, ?_⟩ <;> rw [inter_comm] case refine_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s huv : s ⊆ u' ∪ v' ⊢ x ∈ u' ∩ s case refine_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α H : IsPreconnected s f : α → β hf : ContinuousOn f s u v : Set β hu : IsOpen u hv : IsOpen v x : α xs : x ∈ s xu : f x ∈ u y : α ys : y ∈ s yv : f y ∈ v u' : Set α hu' : IsOpen u' u'_eq : f ⁻¹' u ∩ s = u' ∩ s v' : Set α hv' : IsOpen v' v'_eq : f ⁻¹' v ∩ s = v' ∩ s huv : s ⊆ u' ∪ v' ⊢ y ∈ v' ∩ s exacts [u'_eq ▸ ⟨xu, xs⟩, v'_eq ▸ ⟨yv, ys⟩] ** Qed
isPreconnected_closed_iff α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α ⊢ IsPreconnected s → ∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t')) rintro h t t' ht ht' htt' ⟨x, xs, xt⟩ ⟨y, ys, yt'⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s : Set α h : IsPreconnected s t t' : Set α ht : IsClosed t ht' : IsClosed t' htt' : s ⊆ t ∪ t' x : α xs : x ∈ s xt : x ∈ t y : α ys : y ∈ s yt' : y ∈ t' ⊢ Set.Nonempty (s ∩ (t ∩ t')) rw [← not_disjoint_iff_nonempty_inter, ← subset_compl_iff_disjoint_right, compl_inter] case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s : Set α h : IsPreconnected s t t' : Set α ht : IsClosed t ht' : IsClosed t' htt' : s ⊆ t ∪ t' x : α xs : x ∈ s xt : x ∈ t y : α ys : y ∈ s yt' : y ∈ t' ⊢ ¬s ⊆ tᶜ ∪ t'ᶜ intro h' case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s : Set α h : IsPreconnected s t t' : Set α ht : IsClosed t ht' : IsClosed t' htt' : s ⊆ t ∪ t' x : α xs : x ∈ s xt : x ∈ t y : α ys : y ∈ s yt' : y ∈ t' h' : s ⊆ tᶜ ∪ t'ᶜ ⊢ False have xt' : x ∉ t' := (h' xs).resolve_left (absurd xt) case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s : Set α h : IsPreconnected s t t' : Set α ht : IsClosed t ht' : IsClosed t' htt' : s ⊆ t ∪ t' x : α xs : x ∈ s xt : x ∈ t y : α ys : y ∈ s yt' : y ∈ t' h' : s ⊆ tᶜ ∪ t'ᶜ xt' : ¬x ∈ t' ⊢ False have yt : y ∉ t := (h' ys).resolve_right (absurd yt') case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s : Set α h : IsPreconnected s t t' : Set α ht : IsClosed t ht' : IsClosed t' htt' : s ⊆ t ∪ t' x : α xs : x ∈ s xt : x ∈ t y : α ys : y ∈ s yt' : y ∈ t' h' : s ⊆ tᶜ ∪ t'ᶜ xt' : ¬x ∈ t' yt : ¬y ∈ t ⊢ False have := h _ _ ht.isOpen_compl ht'.isOpen_compl h' ⟨y, ys, yt⟩ ⟨x, xs, xt'⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s : Set α h : IsPreconnected s t t' : Set α ht : IsClosed t ht' : IsClosed t' htt' : s ⊆ t ∪ t' x : α xs : x ∈ s xt : x ∈ t y : α ys : y ∈ s yt' : y ∈ t' h' : s ⊆ tᶜ ∪ t'ᶜ xt' : ¬x ∈ t' yt : ¬y ∈ t this : Set.Nonempty (s ∩ (tᶜ ∩ t'ᶜ)) ⊢ False rw [← compl_union] at this case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s : Set α h : IsPreconnected s t t' : Set α ht : IsClosed t ht' : IsClosed t' htt' : s ⊆ t ∪ t' x : α xs : x ∈ s xt : x ∈ t y : α ys : y ∈ s yt' : y ∈ t' h' : s ⊆ tᶜ ∪ t'ᶜ xt' : ¬x ∈ t' yt : ¬y ∈ t this : Set.Nonempty (s ∩ (t ∪ t')ᶜ) ⊢ False exact this.ne_empty htt'.disjoint_compl_right.inter_eq α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α ⊢ (∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t'))) → IsPreconnected s rintro h u v hu hv huv ⟨x, xs, xu⟩ ⟨y, ys, yv⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t')) u v : Set α hu : IsOpen u hv : IsOpen v huv : s ⊆ u ∪ v x : α xs : x ∈ s xu : x ∈ u y : α ys : y ∈ s yv : y ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) rw [← not_disjoint_iff_nonempty_inter, ← subset_compl_iff_disjoint_right, compl_inter] case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t')) u v : Set α hu : IsOpen u hv : IsOpen v huv : s ⊆ u ∪ v x : α xs : x ∈ s xu : x ∈ u y : α ys : y ∈ s yv : y ∈ v ⊢ ¬s ⊆ uᶜ ∪ vᶜ intro h' case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t')) u v : Set α hu : IsOpen u hv : IsOpen v huv : s ⊆ u ∪ v x : α xs : x ∈ s xu : x ∈ u y : α ys : y ∈ s yv : y ∈ v h' : s ⊆ uᶜ ∪ vᶜ ⊢ False have xv : x ∉ v := (h' xs).elim (absurd xu) id case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t')) u v : Set α hu : IsOpen u hv : IsOpen v huv : s ⊆ u ∪ v x : α xs : x ∈ s xu : x ∈ u y : α ys : y ∈ s yv : y ∈ v h' : s ⊆ uᶜ ∪ vᶜ xv : ¬x ∈ v ⊢ False have yu : y ∉ u := (h' ys).elim id (absurd yv) case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t')) u v : Set α hu : IsOpen u hv : IsOpen v huv : s ⊆ u ∪ v x : α xs : x ∈ s xu : x ∈ u y : α ys : y ∈ s yv : y ∈ v h' : s ⊆ uᶜ ∪ vᶜ xv : ¬x ∈ v yu : ¬y ∈ u ⊢ False have := h _ _ hu.isClosed_compl hv.isClosed_compl h' ⟨y, ys, yu⟩ ⟨x, xs, xv⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t')) u v : Set α hu : IsOpen u hv : IsOpen v huv : s ⊆ u ∪ v x : α xs : x ∈ s xu : x ∈ u y : α ys : y ∈ s yv : y ∈ v h' : s ⊆ uᶜ ∪ vᶜ xv : ¬x ∈ v yu : ¬y ∈ u this : Set.Nonempty (s ∩ (uᶜ ∩ vᶜ)) ⊢ False rw [← compl_union] at this case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t')) u v : Set α hu : IsOpen u hv : IsOpen v huv : s ⊆ u ∪ v x : α xs : x ∈ s xu : x ∈ u y : α ys : y ∈ s yv : y ∈ v h' : s ⊆ uᶜ ∪ vᶜ xv : ¬x ∈ v yu : ¬y ∈ u this : Set.Nonempty (s ∩ (u ∪ v)ᶜ) ⊢ False exact this.ne_empty huv.disjoint_compl_right.inter_eq ** Qed
Inducing.isPreconnected_image α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set α f : α → β hf : Inducing f ⊢ IsPreconnected (f '' s) ↔ IsPreconnected s refine' ⟨fun h => _, fun h => h.image _ hf.continuous.continuousOn⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set α f : α → β hf : Inducing f h : IsPreconnected (f '' s) ⊢ IsPreconnected s rintro u v hu' hv' huv ⟨x, hxs, hxu⟩ ⟨y, hys, hyv⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α f : α → β hf : Inducing f h : IsPreconnected (f '' s) u v : Set α hu' : IsOpen u hv' : IsOpen v huv : s ⊆ u ∪ v x : α hxs : x ∈ s hxu : x ∈ u y : α hys : y ∈ s hyv : y ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) rcases hf.isOpen_iff.1 hu' with ⟨u, hu, rfl⟩ case 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 s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α f : α → β hf : Inducing f h : IsPreconnected (f '' s) v : Set α hv' : IsOpen v x : α hxs : x ∈ s y : α hys : y ∈ s hyv : y ∈ v u : Set β hu : IsOpen u hu' : IsOpen (f ⁻¹' u) huv : s ⊆ f ⁻¹' u ∪ v hxu : x ∈ f ⁻¹' u ⊢ Set.Nonempty (s ∩ (f ⁻¹' u ∩ v)) rcases hf.isOpen_iff.1 hv' with ⟨v, hv, 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✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α f : α → β hf : Inducing f h : IsPreconnected (f '' s) x : α hxs : x ∈ s y : α hys : y ∈ s u : Set β hu : IsOpen u hu' : IsOpen (f ⁻¹' u) hxu : x ∈ f ⁻¹' u v : Set β hv : IsOpen v hv' : IsOpen (f ⁻¹' v) hyv : y ∈ f ⁻¹' v huv : s ⊆ f ⁻¹' u ∪ f ⁻¹' v ⊢ Set.Nonempty (s ∩ (f ⁻¹' u ∩ f ⁻¹' v)) replace huv : f '' s ⊆ u ∪ v 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✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α f : α → β hf : Inducing f h : IsPreconnected (f '' s) x : α hxs : x ∈ s y : α hys : y ∈ s u : Set β hu : IsOpen u hu' : IsOpen (f ⁻¹' u) hxu : x ∈ f ⁻¹' u v : Set β hv : IsOpen v hv' : IsOpen (f ⁻¹' v) hyv : y ∈ f ⁻¹' v huv : f '' s ⊆ u ∪ v ⊢ Set.Nonempty (s ∩ (f ⁻¹' u ∩ f ⁻¹' v)) rcases h u v hu hv huv ⟨f x, mem_image_of_mem _ hxs, hxu⟩ ⟨f y, mem_image_of_mem _ hys, hyv⟩ with ⟨_, ⟨z, hzs, rfl⟩, hzuv⟩ case intro.intro.intro.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 s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α f : α → β hf : Inducing f h : IsPreconnected (f '' s) x : α hxs : x ∈ s y : α hys : y ∈ s u : Set β hu : IsOpen u hu' : IsOpen (f ⁻¹' u) hxu : x ∈ f ⁻¹' u v : Set β hv : IsOpen v hv' : IsOpen (f ⁻¹' v) hyv : y ∈ f ⁻¹' v huv : f '' s ⊆ u ∪ v z : α hzs : z ∈ s hzuv : f z ∈ u ∩ v ⊢ Set.Nonempty (s ∩ (f ⁻¹' u ∩ f ⁻¹' v)) exact ⟨z, hzs, hzuv⟩ case huv α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set α f : α → β hf : Inducing f h : IsPreconnected (f '' s) x : α hxs : x ∈ s y : α hys : y ∈ s u : Set β hu : IsOpen u hu' : IsOpen (f ⁻¹' u) hxu : x ∈ f ⁻¹' u v : Set β hv : IsOpen v hv' : IsOpen (f ⁻¹' v) hyv : y ∈ f ⁻¹' v huv : s ⊆ f ⁻¹' u ∪ f ⁻¹' v ⊢ f '' s ⊆ u ∪ v rwa [image_subset_iff] ** Qed
IsPreconnected.preimage_of_open_map α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsOpenMap f hsf : s ⊆ range f u v : Set α hu : IsOpen u hv : IsOpen v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) ⊢ Set.Nonempty (f ⁻¹' s ∩ (u ∩ v)) replace hsf : f '' (f ⁻¹' s) = s := image_preimage_eq_of_subset hsf α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsOpenMap f u v : Set α hu : IsOpen u hv : IsOpen v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s ⊢ Set.Nonempty (f ⁻¹' s ∩ (u ∩ v)) obtain ⟨_, has, ⟨a, hau, rfl⟩, hav⟩ : (s ∩ (f '' u ∩ f '' v)).Nonempty α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsOpenMap f u v : Set α hu : IsOpen u hv : IsOpen v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s ⊢ Set.Nonempty (s ∩ (f '' u ∩ f '' v)) refine hs (f '' u) (f '' v) (hf u hu) (hf v hv) ?_ ?_ ?_ case refine_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsOpenMap f u v : Set α hu : IsOpen u hv : IsOpen v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s ⊢ s ⊆ f '' u ∪ f '' v simpa only [hsf, image_union] using image_subset f hsuv case refine_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsOpenMap f u v : Set α hu : IsOpen u hv : IsOpen v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s ⊢ Set.Nonempty (s ∩ f '' u) simpa only [image_preimage_inter] using hsu.image f case refine_3 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsOpenMap f u v : Set α hu : IsOpen u hv : IsOpen v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s ⊢ Set.Nonempty (s ∩ f '' v) simpa only [image_preimage_inter] using hsv.image f 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 s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsOpenMap f u v : Set α hu : IsOpen u hv : IsOpen v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s a : α hau : a ∈ u has : f a ∈ s hav : f a ∈ f '' v ⊢ Set.Nonempty (f ⁻¹' s ∩ (u ∩ v)) exact ⟨a, has, hau, hinj.mem_set_image.1 hav⟩ ** Qed
IsPreconnected.preimage_of_closed_map α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsClosedMap f hsf : s ⊆ range f u v : Set α hu : IsClosed u hv : IsClosed v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) ⊢ Set.Nonempty (f ⁻¹' s ∩ (u ∩ v)) replace hsf : f '' (f ⁻¹' s) = s := image_preimage_eq_of_subset hsf α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsClosedMap f u v : Set α hu : IsClosed u hv : IsClosed v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s ⊢ Set.Nonempty (f ⁻¹' s ∩ (u ∩ v)) obtain ⟨_, has, ⟨a, hau, rfl⟩, hav⟩ : (s ∩ (f '' u ∩ f '' v)).Nonempty α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsClosedMap f u v : Set α hu : IsClosed u hv : IsClosed v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s ⊢ Set.Nonempty (s ∩ (f '' u ∩ f '' v)) refine isPreconnected_closed_iff.1 hs (f '' u) (f '' v) (hf u hu) (hf v hv) ?_ ?_ ?_ case refine_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsClosedMap f u v : Set α hu : IsClosed u hv : IsClosed v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s ⊢ s ⊆ f '' u ∪ f '' v simpa only [hsf, image_union] using image_subset f hsuv case refine_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsClosedMap f u v : Set α hu : IsClosed u hv : IsClosed v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s ⊢ Set.Nonempty (s ∩ f '' u) simpa only [image_preimage_inter] using hsu.image f case refine_3 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsClosedMap f u v : Set α hu : IsClosed u hv : IsClosed v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s ⊢ Set.Nonempty (s ∩ f '' v) simpa only [image_preimage_inter] using hsv.image f 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 s✝ t u✝ v✝ : Set α inst✝ : TopologicalSpace β s : Set β hs : IsPreconnected s f : α → β hinj : Injective f hf : IsClosedMap f u v : Set α hu : IsClosed u hv : IsClosed v hsuv : f ⁻¹' s ⊆ u ∪ v hsu : Set.Nonempty (f ⁻¹' s ∩ u) hsv : Set.Nonempty (f ⁻¹' s ∩ v) hsf : f '' (f ⁻¹' s) = s a : α hau : a ∈ u has : f a ∈ s hav : f a ∈ f '' v ⊢ Set.Nonempty (f ⁻¹' s ∩ (u ∩ v)) exact ⟨a, has, hau, hinj.mem_set_image.1 hav⟩ ** Qed
IsPreconnected.subset_or_subset α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hu : IsOpen u hv : IsOpen v huv : Disjoint u v hsuv : s ⊆ u ∪ v hs : IsPreconnected s ⊢ s ⊆ u ∨ s ⊆ v specialize hs u v hu hv hsuv α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hu : IsOpen u hv : IsOpen v huv : Disjoint u v hsuv : s ⊆ u ∪ v hs : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) ⊢ s ⊆ u ∨ s ⊆ v obtain hsu \| hsu := (s ∩ u).eq_empty_or_nonempty case inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hu : IsOpen u hv : IsOpen v huv : Disjoint u v hsuv : s ⊆ u ∪ v hs : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) hsu : s ∩ u = ∅ ⊢ s ⊆ u ∨ s ⊆ v exact Or.inr ((Set.disjoint_iff_inter_eq_empty.2 hsu).subset_right_of_subset_union hsuv) case inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hu : IsOpen u hv : IsOpen v huv : Disjoint u v hsuv : s ⊆ u ∪ v hs : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) hsu : Set.Nonempty (s ∩ u) ⊢ s ⊆ u ∨ s ⊆ v replace hs := mt (hs hsu) case inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hu : IsOpen u hv : IsOpen v huv : Disjoint u v hsuv : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hs : ¬Set.Nonempty (s ∩ (u ∩ v)) → ¬Set.Nonempty (s ∩ v) ⊢ s ⊆ u ∨ s ⊆ v simp_rw [Set.not_nonempty_iff_eq_empty, ← Set.disjoint_iff_inter_eq_empty, disjoint_iff_inter_eq_empty.1 huv] at hs case inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hu : IsOpen u hv : IsOpen v huv : Disjoint u v hsuv : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hs : Disjoint s ∅ → Disjoint s v ⊢ s ⊆ u ∨ s ⊆ v exact Or.inl ((hs s.disjoint_empty).subset_left_of_subset_union hsuv) ** Qed
IsPreconnected.subset_left_of_subset_union α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hu : IsOpen u hv : IsOpen v huv : Disjoint u v hsuv : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hs : IsPreconnected s ⊢ Disjoint s v by_contra hsv α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hu : IsOpen u hv : IsOpen v huv : Disjoint u v hsuv : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hs : IsPreconnected s hsv : ¬Disjoint s v ⊢ False rw [not_disjoint_iff_nonempty_inter] at hsv α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hu : IsOpen u hv : IsOpen v huv : Disjoint u v hsuv : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hs : IsPreconnected s hsv : Set.Nonempty (s ∩ v) ⊢ False obtain ⟨x, _, hx⟩ := hs u v hu hv hsuv hsu hsv case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hu : IsOpen u hv : IsOpen v huv : Disjoint u v hsuv : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hs : IsPreconnected s hsv : Set.Nonempty (s ∩ v) x : α left✝ : x ∈ s hx : x ∈ u ∩ v ⊢ False exact Set.disjoint_iff.1 huv hx ** Qed
IsPreconnected.subset_clopen α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t✝ u v s t : Set α hs : IsPreconnected s ht : IsClopen t hne : Set.Nonempty (s ∩ t) ⊢ s ⊆ t ∪ tᶜ simp ** Qed
IsPreconnected.subset_of_closure_inter_subset α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hs : IsPreconnected s hu : IsOpen u h'u : Set.Nonempty (s ∩ u) h : closure u ∩ s ⊆ u A : s ⊆ u ∪ (closure u)ᶜ ⊢ s ⊆ u apply hs.subset_left_of_subset_union hu isClosed_closure.isOpen_compl _ A h'u α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hs : IsPreconnected s hu : IsOpen u h'u : Set.Nonempty (s ∩ u) h : closure u ∩ s ⊆ u A : s ⊆ u ∪ (closure u)ᶜ ⊢ Disjoint u (closure u)ᶜ exact disjoint_compl_right.mono_right (compl_subset_compl.2 subset_closure) α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hs : IsPreconnected s hu : IsOpen u h'u : Set.Nonempty (s ∩ u) h : closure u ∩ s ⊆ u ⊢ s ⊆ u ∪ (closure u)ᶜ intro x hx α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hs : IsPreconnected s hu : IsOpen u h'u : Set.Nonempty (s ∩ u) h : closure u ∩ s ⊆ u x : α hx : x ∈ s ⊢ x ∈ u ∪ (closure u)ᶜ by_cases xu : x ∈ u case pos α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hs : IsPreconnected s hu : IsOpen u h'u : Set.Nonempty (s ∩ u) h : closure u ∩ s ⊆ u x : α hx : x ∈ s xu : x ∈ u ⊢ x ∈ u ∪ (closure u)ᶜ exact Or.inl xu case neg α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hs : IsPreconnected s hu : IsOpen u h'u : Set.Nonempty (s ∩ u) h : closure u ∩ s ⊆ u x : α hx : x ∈ s xu : ¬x ∈ u ⊢ x ∈ u ∪ (closure u)ᶜ right case neg.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hs : IsPreconnected s hu : IsOpen u h'u : Set.Nonempty (s ∩ u) h : closure u ∩ s ⊆ u x : α hx : x ∈ s xu : ¬x ∈ u ⊢ x ∈ (closure u)ᶜ intro h'x case neg.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α hs : IsPreconnected s hu : IsOpen u h'u : Set.Nonempty (s ∩ u) h : closure u ∩ s ⊆ u x : α hx : x ∈ s xu : ¬x ∈ u h'x : x ∈ closure u ⊢ False exact xu (h (mem_inter h'x hx)) ** Qed
IsPreconnected.prod α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t✝ u v : Set α inst✝ : TopologicalSpace β s : Set α t : Set β hs : IsPreconnected s ht : IsPreconnected t ⊢ IsPreconnected (s ×ˢ t) apply isPreconnected_of_forall_pair case H α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t✝ u v : Set α inst✝ : TopologicalSpace β s : Set α t : Set β hs : IsPreconnected s ht : IsPreconnected t ⊢ ∀ (x : α × β), x ∈ s ×ˢ t → ∀ (y : α × β), y ∈ s ×ˢ t → ∃ t_1, t_1 ⊆ s ×ˢ t ∧ x ∈ t_1 ∧ y ∈ t_1 ∧ IsPreconnected t_1 rintro ⟨a₁, b₁⟩ ⟨ha₁, hb₁⟩ ⟨a₂, b₂⟩ ⟨ha₂, hb₂⟩ case H.mk.intro.mk.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t✝ u v : Set α inst✝ : TopologicalSpace β s : Set α t : Set β hs : IsPreconnected s ht : IsPreconnected t a₁ : α b₁ : β ha₁ : (a₁, b₁).1 ∈ s hb₁ : (a₁, b₁).2 ∈ t a₂ : α b₂ : β ha₂ : (a₂, b₂).1 ∈ s hb₂ : (a₂, b₂).2 ∈ t ⊢ ∃ t_1, t_1 ⊆ s ×ˢ t ∧ (a₁, b₁) ∈ t_1 ∧ (a₂, b₂) ∈ t_1 ∧ IsPreconnected t_1 refine' ⟨Prod.mk a₁ '' t ∪ flip Prod.mk b₂ '' s, _, .inl ⟨b₁, hb₁, rfl⟩, .inr ⟨a₂, ha₂, rfl⟩, _⟩ case H.mk.intro.mk.intro.refine'_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t✝ u v : Set α inst✝ : TopologicalSpace β s : Set α t : Set β hs : IsPreconnected s ht : IsPreconnected t a₁ : α b₁ : β ha₁ : (a₁, b₁).1 ∈ s hb₁ : (a₁, b₁).2 ∈ t a₂ : α b₂ : β ha₂ : (a₂, b₂).1 ∈ s hb₂ : (a₂, b₂).2 ∈ t ⊢ Prod.mk a₁ '' t ∪ flip Prod.mk b₂ '' s ⊆ s ×ˢ t rintro _ (⟨y, hy, rfl⟩ \| ⟨x, hx, rfl⟩) case H.mk.intro.mk.intro.refine'_1.inl.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t✝ u v : Set α inst✝ : TopologicalSpace β s : Set α t : Set β hs : IsPreconnected s ht : IsPreconnected t a₁ : α b₁ : β ha₁ : (a₁, b₁).1 ∈ s hb₁ : (a₁, b₁).2 ∈ t a₂ : α b₂ : β ha₂ : (a₂, b₂).1 ∈ s hb₂ : (a₂, b₂).2 ∈ t y : β hy : y ∈ t ⊢ (a₁, y) ∈ s ×ˢ t case H.mk.intro.mk.intro.refine'_1.inr.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t✝ u v : Set α inst✝ : TopologicalSpace β s : Set α t : Set β hs : IsPreconnected s ht : IsPreconnected t a₁ : α b₁ : β ha₁ : (a₁, b₁).1 ∈ s hb₁ : (a₁, b₁).2 ∈ t a₂ : α b₂ : β ha₂ : (a₂, b₂).1 ∈ s hb₂ : (a₂, b₂).2 ∈ t x : α hx : x ∈ s ⊢ flip Prod.mk b₂ x ∈ s ×ˢ t exacts [⟨ha₁, hy⟩, ⟨hx, hb₂⟩] case H.mk.intro.mk.intro.refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t✝ u v : Set α inst✝ : TopologicalSpace β s : Set α t : Set β hs : IsPreconnected s ht : IsPreconnected t a₁ : α b₁ : β ha₁ : (a₁, b₁).1 ∈ s hb₁ : (a₁, b₁).2 ∈ t a₂ : α b₂ : β ha₂ : (a₂, b₂).1 ∈ s hb₂ : (a₂, b₂).2 ∈ t ⊢ IsPreconnected (Prod.mk a₁ '' t ∪ flip Prod.mk b₂ '' s) exact (ht.image _ (Continuous.Prod.mk _).continuousOn).union (a₁, b₂) ⟨b₂, hb₂, rfl⟩ ⟨a₁, ha₁, rfl⟩ (hs.image _ (continuous_id.prod_mk continuous_const).continuousOn) ** Qed
isPreconnected_univ_pi α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) ⊢ IsPreconnected (pi univ s) rintro u v uo vo hsuv ⟨f, hfs, hfu⟩ ⟨g, hgs, hgv⟩ case intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) rcases exists_finset_piecewise_mem_of_mem_nhds (uo.mem_nhds hfu) g with ⟨I, hI⟩ 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 s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I : Finset ι hI : Finset.piecewise I f g ∈ u ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) induction' I using Finset.induction_on with i I _ ihI case intro.intro.intro.intro.intro.empty α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I : Finset ι hI✝ : Finset.piecewise I f g ∈ u hI : Finset.piecewise ∅ f g ∈ u ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) refine' ⟨g, hgs, ⟨_, hgv⟩⟩ case intro.intro.intro.intro.intro.empty α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I : Finset ι hI✝ : Finset.piecewise I f g ∈ u hI : Finset.piecewise ∅ f g ∈ u ⊢ g ∈ u simpa using hI case intro.intro.intro.intro.intro.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : Finset.piecewise (insert i I) f g ∈ u ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) rw [Finset.piecewise_insert] at hI case intro.intro.intro.intro.intro.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) have := I.piecewise_mem_set_pi hfs hgs case intro.intro.intro.intro.intro.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) refine' (hsuv this).elim ihI fun h => _ case intro.intro.intro.intro.intro.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s h : Finset.piecewise I f g ∈ v ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) set S := update (I.piecewise f g) i '' s i case intro.intro.intro.intro.intro.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s h : Finset.piecewise I f g ∈ v S : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) have hsub : S ⊆ pi univ s := by refine' image_subset_iff.2 fun z hz => _ rwa [update_preimage_univ_pi] exact fun j _ => this j trivial case intro.intro.intro.intro.intro.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s h : Finset.piecewise I f g ∈ v S : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i hsub : S ⊆ pi univ s ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) have hconn : IsPreconnected S := (hs i).image _ (continuous_const.update i continuous_id).continuousOn case intro.intro.intro.intro.intro.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s h : Finset.piecewise I f g ∈ v S : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i hsub : S ⊆ pi univ s hconn : IsPreconnected S ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) have hSu : (S ∩ u).Nonempty := ⟨_, mem_image_of_mem _ (hfs _ trivial), hI⟩ case intro.intro.intro.intro.intro.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s h : Finset.piecewise I f g ∈ v S : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i hsub : S ⊆ pi univ s hconn : IsPreconnected S hSu : Set.Nonempty (S ∩ u) ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) have hSv : (S ∩ v).Nonempty := ⟨_, ⟨_, this _ trivial, update_eq_self _ _⟩, h⟩ case intro.intro.intro.intro.intro.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s h : Finset.piecewise I f g ∈ v S : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i hsub : S ⊆ pi univ s hconn : IsPreconnected S hSu : Set.Nonempty (S ∩ u) hSv : Set.Nonempty (S ∩ v) ⊢ Set.Nonempty (pi univ s ∩ (u ∩ v)) refine' (hconn u v uo vo (hsub.trans hsuv) hSu hSv).mono _ case intro.intro.intro.intro.intro.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s h : Finset.piecewise I f g ∈ v S : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i hsub : S ⊆ pi univ s hconn : IsPreconnected S hSu : Set.Nonempty (S ∩ u) hSv : Set.Nonempty (S ∩ v) ⊢ S ∩ (u ∩ v) ⊆ pi univ s ∩ (u ∩ v) exact inter_subset_inter_left _ hsub α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s h : Finset.piecewise I f g ∈ v S : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i ⊢ S ⊆ pi univ s refine' image_subset_iff.2 fun z hz => _ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s h : Finset.piecewise I f g ∈ v S : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i z : π i hz : z ∈ s i ⊢ z ∈ update (Finset.piecewise I f g) i ⁻¹' pi univ s rwa [update_preimage_univ_pi] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u✝ v✝ : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hs : ∀ (i : ι), IsPreconnected (s i) u v : Set ((i : ι) → π i) uo : IsOpen u vo : IsOpen v hsuv : pi univ s ⊆ u ∪ v f : (i : ι) → π i hfs : f ∈ pi univ s hfu : f ∈ u g : (i : ι) → π i hgs : g ∈ pi univ s hgv : g ∈ v I✝ : Finset ι hI✝ : Finset.piecewise I✝ f g ∈ u i : ι I : Finset ι a✝ : ¬i ∈ I ihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v)) hI : update (Finset.piecewise I f g) i (f i) ∈ u this : Finset.piecewise I f g ∈ pi univ s h : Finset.piecewise I f g ∈ v S : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i z : π i hz : z ∈ s i ⊢ ∀ (j : ι), j ≠ i → Finset.piecewise I f g j ∈ s j exact fun j _ => this j trivial ** Qed
isConnected_univ_pi α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) ⊢ IsConnected (pi univ s) ↔ ∀ (i : ι), IsConnected (s i) simp only [IsConnected, ← univ_pi_nonempty_iff, forall_and, and_congr_right_iff] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) ⊢ Set.Nonempty (pi univ s) → (IsPreconnected (pi univ s) ↔ ∀ (x : ι), IsPreconnected (s x)) refine' fun hne => ⟨fun hc i => _, isPreconnected_univ_pi⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hne : Set.Nonempty (pi univ s) hc : IsPreconnected (pi univ s) i : ι ⊢ IsPreconnected (s i) rw [← eval_image_univ_pi hne] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : (i : ι) → Set (π i) hne : Set.Nonempty (pi univ s) hc : IsPreconnected (pi univ s) i : ι ⊢ IsPreconnected ((fun f => f i) '' pi univ s) exact hc.image _ (continuous_apply _).continuousOn ** Qed
Sigma.isConnected_iff α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : Set ((i : ι) × π i) ⊢ IsConnected s ↔ ∃ i t, IsConnected t ∧ s = mk i '' t refine' ⟨fun hs => _, _⟩ case refine'_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : Set ((i : ι) × π i) hs : IsConnected s ⊢ ∃ i t, IsConnected t ∧ s = mk i '' t obtain ⟨⟨i, x⟩, hx⟩ := hs.nonempty case refine'_1.intro.mk α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : Set ((i : ι) × π i) hs : IsConnected s i : ι x : π i hx : { fst := i, snd := x } ∈ s ⊢ ∃ i t, IsConnected t ∧ s = mk i '' t have : s ⊆ range (Sigma.mk i) := hs.isPreconnected.subset_clopen isClopen_range_sigmaMk ⟨⟨i, x⟩, hx, x, rfl⟩ case refine'_1.intro.mk α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : Set ((i : ι) × π i) hs : IsConnected s i : ι x : π i hx : { fst := i, snd := x } ∈ s this : s ⊆ range (mk i) ⊢ ∃ i t, IsConnected t ∧ s = mk i '' t exact ⟨i, Sigma.mk i ⁻¹' s, hs.preimage_of_openMap sigma_mk_injective isOpenMap_sigmaMk this, (Set.image_preimage_eq_of_subset this).symm⟩ case refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) s : Set ((i : ι) × π i) ⊢ (∃ i t, IsConnected t ∧ s = mk i '' t) → IsConnected s rintro ⟨i, t, ht, rfl⟩ case refine'_2.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : (i : ι) → TopologicalSpace (π i) i : ι t : Set (π i) ht : IsConnected t ⊢ IsConnected (mk i '' t) exact ht.image _ continuous_sigmaMk.continuousOn ** Qed
Sigma.isPreconnected_iff α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α hι : Nonempty ι inst✝ : (i : ι) → TopologicalSpace (π i) s : Set ((i : ι) × π i) ⊢ IsPreconnected s ↔ ∃ i t, IsPreconnected t ∧ s = mk i '' t refine' ⟨fun hs => _, _⟩ case refine'_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α hι : Nonempty ι inst✝ : (i : ι) → TopologicalSpace (π i) s : Set ((i : ι) × π i) hs : IsPreconnected s ⊢ ∃ i t, IsPreconnected t ∧ s = mk i '' t obtain rfl \| h := s.eq_empty_or_nonempty case refine'_1.inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t u v : Set α hι : Nonempty ι inst✝ : (i : ι) → TopologicalSpace (π i) hs : IsPreconnected ∅ ⊢ ∃ i t, IsPreconnected t ∧ ∅ = mk i '' t exact ⟨Classical.choice hι, ∅, isPreconnected_empty, (Set.image_empty _).symm⟩ case refine'_1.inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α hι : Nonempty ι inst✝ : (i : ι) → TopologicalSpace (π i) s : Set ((i : ι) × π i) hs : IsPreconnected s h : Set.Nonempty s ⊢ ∃ i t, IsPreconnected t ∧ s = mk i '' t obtain ⟨a, t, ht, rfl⟩ := Sigma.isConnected_iff.1 ⟨h, hs⟩ case refine'_1.inr.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α hι : Nonempty ι inst✝ : (i : ι) → TopologicalSpace (π i) a : ι t : Set (π a) ht : IsConnected t hs : IsPreconnected (mk a '' t) h : Set.Nonempty (mk a '' t) ⊢ ∃ i t_1, IsPreconnected t_1 ∧ mk a '' t = mk i '' t_1 refine' ⟨a, t, ht.isPreconnected, rfl⟩ case refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α hι : Nonempty ι inst✝ : (i : ι) → TopologicalSpace (π i) s : Set ((i : ι) × π i) ⊢ (∃ i t, IsPreconnected t ∧ s = mk i '' t) → IsPreconnected s rintro ⟨a, t, ht, rfl⟩ case refine'_2.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α hι : Nonempty ι inst✝ : (i : ι) → TopologicalSpace (π i) a : ι t : Set (π a) ht : IsPreconnected t ⊢ IsPreconnected (mk a '' t) exact ht.image _ continuous_sigmaMk.continuousOn ** Qed
Sum.isConnected_iff α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) ⊢ IsConnected s ↔ (∃ t, IsConnected t ∧ s = inl '' t) ∨ ∃ t, IsConnected t ∧ s = inr '' t refine' ⟨fun hs => _, _⟩ case refine'_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsConnected s ⊢ (∃ t, IsConnected t ∧ s = inl '' t) ∨ ∃ t, IsConnected t ∧ s = inr '' t obtain ⟨x \| x, hx⟩ := hs.nonempty case refine'_1.intro.inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsConnected s x : α hx : inl x ∈ s ⊢ (∃ t, IsConnected t ∧ s = inl '' t) ∨ ∃ t, IsConnected t ∧ s = inr '' t have h : s ⊆ range Sum.inl := hs.isPreconnected.subset_clopen isClopen_range_inl ⟨.inl x, hx, x, rfl⟩ case refine'_1.intro.inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsConnected s x : α hx : inl x ∈ s h : s ⊆ range inl ⊢ (∃ t, IsConnected t ∧ s = inl '' t) ∨ ∃ t, IsConnected t ∧ s = inr '' t refine' Or.inl ⟨Sum.inl ⁻¹' s, _, _⟩ case refine'_1.intro.inl.refine'_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsConnected s x : α hx : inl x ∈ s h : s ⊆ range inl ⊢ IsConnected (inl ⁻¹' s) exact hs.preimage_of_openMap Sum.inl_injective isOpenMap_inl h case refine'_1.intro.inl.refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsConnected s x : α hx : inl x ∈ s h : s ⊆ range inl ⊢ s = inl '' (inl ⁻¹' s) exact (image_preimage_eq_of_subset h).symm case refine'_1.intro.inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsConnected s x : β hx : inr x ∈ s ⊢ (∃ t, IsConnected t ∧ s = inl '' t) ∨ ∃ t, IsConnected t ∧ s = inr '' t have h : s ⊆ range Sum.inr := hs.isPreconnected.subset_clopen isClopen_range_inr ⟨.inr x, hx, x, rfl⟩ case refine'_1.intro.inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsConnected s x : β hx : inr x ∈ s h : s ⊆ range inr ⊢ (∃ t, IsConnected t ∧ s = inl '' t) ∨ ∃ t, IsConnected t ∧ s = inr '' t refine' Or.inr ⟨Sum.inr ⁻¹' s, _, _⟩ case refine'_1.intro.inr.refine'_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsConnected s x : β hx : inr x ∈ s h : s ⊆ range inr ⊢ IsConnected (inr ⁻¹' s) exact hs.preimage_of_openMap Sum.inr_injective isOpenMap_inr h case refine'_1.intro.inr.refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsConnected s x : β hx : inr x ∈ s h : s ⊆ range inr ⊢ s = inr '' (inr ⁻¹' s) exact (image_preimage_eq_of_subset h).symm case refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) ⊢ ((∃ t, IsConnected t ∧ s = inl '' t) ∨ ∃ t, IsConnected t ∧ s = inr '' t) → IsConnected s rintro (⟨t, ht, rfl⟩ \| ⟨t, ht, rfl⟩) case refine'_2.inl.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β t : Set α ht : IsConnected t ⊢ IsConnected (inl '' t) exact ht.image _ continuous_inl.continuousOn case refine'_2.inr.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β t : Set β ht : IsConnected t ⊢ IsConnected (inr '' t) exact ht.image _ continuous_inr.continuousOn ** Qed
Sum.isPreconnected_iff α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) ⊢ IsPreconnected s ↔ (∃ t, IsPreconnected t ∧ s = inl '' t) ∨ ∃ t, IsPreconnected t ∧ s = inr '' t refine' ⟨fun hs => _, _⟩ case refine'_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsPreconnected s ⊢ (∃ t, IsPreconnected t ∧ s = inl '' t) ∨ ∃ t, IsPreconnected t ∧ s = inr '' t obtain rfl \| h := s.eq_empty_or_nonempty case refine'_1.inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) hs : IsPreconnected s h : Set.Nonempty s ⊢ (∃ t, IsPreconnected t ∧ s = inl '' t) ∨ ∃ t, IsPreconnected t ∧ s = inr '' t obtain ⟨t, ht, rfl⟩ \| ⟨t, ht, rfl⟩ := Sum.isConnected_iff.1 ⟨h, hs⟩ case refine'_1.inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t u v : Set α inst✝ : TopologicalSpace β hs : IsPreconnected ∅ ⊢ (∃ t, IsPreconnected t ∧ ∅ = inl '' t) ∨ ∃ t, IsPreconnected t ∧ ∅ = inr '' t exact Or.inl ⟨∅, isPreconnected_empty, (Set.image_empty _).symm⟩ case refine'_1.inr.inl.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β t : Set α ht : IsConnected t hs : IsPreconnected (inl '' t) h : Set.Nonempty (inl '' t) ⊢ (∃ t_1, IsPreconnected t_1 ∧ inl '' t = inl '' t_1) ∨ ∃ t_1, IsPreconnected t_1 ∧ inl '' t = inr '' t_1 exact Or.inl ⟨t, ht.isPreconnected, rfl⟩ case refine'_1.inr.inr.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β t : Set β ht : IsConnected t hs : IsPreconnected (inr '' t) h : Set.Nonempty (inr '' t) ⊢ (∃ t_1, IsPreconnected t_1 ∧ inr '' t = inl '' t_1) ∨ ∃ t_1, IsPreconnected t_1 ∧ inr '' t = inr '' t_1 exact Or.inr ⟨t, ht.isPreconnected, rfl⟩ case refine'_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : TopologicalSpace β s : Set (α ⊕ β) ⊢ ((∃ t, IsPreconnected t ∧ s = inl '' t) ∨ ∃ t, IsPreconnected t ∧ s = inr '' t) → IsPreconnected s rintro (⟨t, ht, rfl⟩ \| ⟨t, ht, rfl⟩) case refine'_2.inl.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β t : Set α ht : IsPreconnected t ⊢ IsPreconnected (inl '' t) exact ht.image _ continuous_inl.continuousOn case refine'_2.inr.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β t : Set β ht : IsPreconnected t ⊢ IsPreconnected (inr '' t) exact ht.image _ continuous_inr.continuousOn ** Qed
mem_connectedComponentIn α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hx : x ∈ F ⊢ x ∈ connectedComponentIn F x simp [connectedComponentIn_eq_image hx, mem_connectedComponent, hx] ** Qed
connectedComponentIn_nonempty_iff α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α ⊢ Set.Nonempty (connectedComponentIn F x) ↔ x ∈ F rw [connectedComponentIn] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α ⊢ Set.Nonempty (if h : x ∈ F then Subtype.val '' connectedComponent { val := x, property := h } else ∅) ↔ x ∈ F split_ifs <;> simp [connectedComponent_nonempty, ] * Qed
connectedComponentIn_subset α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v F : Set α x : α ⊢ connectedComponentIn F x ⊆ F rw [connectedComponentIn] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v F : Set α x : α ⊢ (if h : x ∈ F then Subtype.val '' connectedComponent { val := x, property := h } else ∅) ⊆ F split_ifs <;> simp ** Qed
isPreconnected_connectedComponentIn α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α ⊢ IsPreconnected (connectedComponentIn F x) rw [connectedComponentIn] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α ⊢ IsPreconnected (if h : x ∈ F then Subtype.val '' connectedComponent { val := x, property := h } else ∅) split_ifs case pos α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α h✝ : x ∈ F ⊢ IsPreconnected (Subtype.val '' connectedComponent { val := x, property := h✝ }) exact inducing_subtype_val.isPreconnected_image.mpr isPreconnected_connectedComponent case neg α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α h✝ : ¬x ∈ F ⊢ IsPreconnected ∅ exact isPreconnected_empty ** Qed
isConnected_connectedComponentIn_iff α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α ⊢ IsConnected (connectedComponentIn F x) ↔ x ∈ F simp_rw [← connectedComponentIn_nonempty_iff, IsConnected, isPreconnected_connectedComponentIn, and_true_iff] ** Qed
IsPreconnected.subset_connectedComponentIn α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hs : IsPreconnected s hxs : x ∈ s hsF : s ⊆ F ⊢ s ⊆ connectedComponentIn F x have : IsPreconnected (((↑) : F → α) ⁻¹' s) := by refine' inducing_subtype_val.isPreconnected_image.mp _ rwa [Subtype.image_preimage_coe, inter_eq_left.mpr hsF] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hs : IsPreconnected s hxs : x ∈ s hsF : s ⊆ F this : IsPreconnected (Subtype.val ⁻¹' s) ⊢ s ⊆ connectedComponentIn F x have h2xs : (⟨x, hsF hxs⟩ : F) ∈ (↑) ⁻¹' s := by rw [mem_preimage] exact hxs α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hs : IsPreconnected s hxs : x ∈ s hsF : s ⊆ F this : IsPreconnected (Subtype.val ⁻¹' s) h2xs : { val := x, property := (_ : x ∈ F) } ∈ Subtype.val ⁻¹' s ⊢ s ⊆ connectedComponentIn F x have := this.subset_connectedComponent h2xs α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hs : IsPreconnected s hxs : x ∈ s hsF : s ⊆ F this✝ : IsPreconnected (Subtype.val ⁻¹' s) h2xs : { val := x, property := (_ : x ∈ F) } ∈ Subtype.val ⁻¹' s this : Subtype.val ⁻¹' s ⊆ connectedComponent { val := x, property := (_ : x ∈ F) } ⊢ s ⊆ connectedComponentIn F x rw [connectedComponentIn_eq_image (hsF hxs)] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hs : IsPreconnected s hxs : x ∈ s hsF : s ⊆ F this✝ : IsPreconnected (Subtype.val ⁻¹' s) h2xs : { val := x, property := (_ : x ∈ F) } ∈ Subtype.val ⁻¹' s this : Subtype.val ⁻¹' s ⊆ connectedComponent { val := x, property := (_ : x ∈ F) } ⊢ s ⊆ Subtype.val '' connectedComponent { val := x, property := (_ : x ∈ F) } refine' Subset.trans _ (image_subset _ this) α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hs : IsPreconnected s hxs : x ∈ s hsF : s ⊆ F this✝ : IsPreconnected (Subtype.val ⁻¹' s) h2xs : { val := x, property := (_ : x ∈ F) } ∈ Subtype.val ⁻¹' s this : Subtype.val ⁻¹' s ⊆ connectedComponent { val := x, property := (_ : x ∈ F) } ⊢ s ⊆ Subtype.val '' (Subtype.val ⁻¹' s) rw [Subtype.image_preimage_coe, inter_eq_left.mpr hsF] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hs : IsPreconnected s hxs : x ∈ s hsF : s ⊆ F ⊢ IsPreconnected (Subtype.val ⁻¹' s) refine' inducing_subtype_val.isPreconnected_image.mp _ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hs : IsPreconnected s hxs : x ∈ s hsF : s ⊆ F ⊢ IsPreconnected (Subtype.val '' (Subtype.val ⁻¹' s)) rwa [Subtype.image_preimage_coe, inter_eq_left.mpr hsF] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hs : IsPreconnected s hxs : x ∈ s hsF : s ⊆ F this : IsPreconnected (Subtype.val ⁻¹' s) ⊢ { val := x, property := (_ : x ∈ F) } ∈ Subtype.val ⁻¹' s rw [mem_preimage] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hs : IsPreconnected s hxs : x ∈ s hsF : s ⊆ F this : IsPreconnected (Subtype.val ⁻¹' s) ⊢ ↑{ val := x, property := (_ : x ∈ F) } ∈ s exact hxs ** Qed
connectedComponentIn_eq α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x y : α F : Set α h : y ∈ connectedComponentIn F x ⊢ connectedComponentIn F x = connectedComponentIn F y have hx : x ∈ F := connectedComponentIn_nonempty_iff.mp ⟨y, h⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x y : α F : Set α h : y ∈ connectedComponentIn F x hx : x ∈ F ⊢ connectedComponentIn F x = connectedComponentIn F y simp_rw [connectedComponentIn_eq_image hx] at h ⊢ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x y : α F : Set α hx : x ∈ F h : y ∈ Subtype.val '' connectedComponent { val := x, property := hx } ⊢ Subtype.val '' connectedComponent { val := x, property := hx } = connectedComponentIn F y obtain ⟨⟨y, hy⟩, h2y, rfl⟩ := h case intro.mk.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F : Set α hx : x ∈ F y : α hy : y ∈ F h2y : { val := y, property := hy } ∈ connectedComponent { val := x, property := hx } ⊢ Subtype.val '' connectedComponent { val := x, property := hx } = connectedComponentIn F ↑{ val := y, property := hy } simp_rw [connectedComponentIn_eq_image hy, connectedComponent_eq h2y] ** Qed
connectedComponentIn_mono α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F G : Set α h : F ⊆ G ⊢ connectedComponentIn F x ⊆ connectedComponentIn G x by_cases hx : x ∈ F case pos α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F G : Set α h : F ⊆ G hx : x ∈ F ⊢ connectedComponentIn F x ⊆ connectedComponentIn G x rw [connectedComponentIn_eq_image hx, connectedComponentIn_eq_image (h hx), ← show ((↑) : G → α) ∘ inclusion h = (↑) from rfl, image_comp] case pos α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F G : Set α h : F ⊆ G hx : x ∈ F ⊢ Subtype.val '' (inclusion h '' connectedComponent { val := x, property := hx }) ⊆ Subtype.val '' connectedComponent { val := x, property := (_ : x ∈ G) } exact image_subset _ ((continuous_inclusion h).image_connectedComponent_subset ⟨x, hx⟩) case neg α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F G : Set α h : F ⊆ G hx : ¬x ∈ F ⊢ connectedComponentIn F x ⊆ connectedComponentIn G x rw [connectedComponentIn_eq_empty hx] case neg α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α F G : Set α h : F ⊆ G hx : ¬x ∈ F ⊢ ∅ ⊆ connectedComponentIn G x exact Set.empty_subset _ ** Qed
Function.Surjective.connectedSpace α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α s t u v : Set α inst✝¹ : ConnectedSpace α inst✝ : TopologicalSpace β f : α → β hf : Surjective f hf' : Continuous f ⊢ ConnectedSpace β rw [connectedSpace_iff_univ, ← hf.range_eq] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α s t u v : Set α inst✝¹ : ConnectedSpace α inst✝ : TopologicalSpace β f : α → β hf : Surjective f hf' : Continuous f ⊢ IsConnected (range f) exact isConnected_range hf' ** Qed
connectedSpace_iff_connectedComponent α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α ⊢ ConnectedSpace α ↔ ∃ x, connectedComponent x = univ constructor case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α ⊢ ConnectedSpace α → ∃ x, connectedComponent x = univ rintro ⟨⟨x⟩⟩ case mp.mk.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α toPreconnectedSpace✝ : PreconnectedSpace α x : α ⊢ ∃ x, connectedComponent x = univ exact ⟨x, eq_univ_of_univ_subset <\| isPreconnected_univ.subset_connectedComponent (mem_univ x)⟩ case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α ⊢ (∃ x, connectedComponent x = univ) → ConnectedSpace α rintro ⟨x, h⟩ case mpr.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α h : connectedComponent x = univ ⊢ ConnectedSpace α haveI : PreconnectedSpace α := ⟨by rw [← h]; exact isPreconnected_connectedComponent⟩ case mpr.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α h : connectedComponent x = univ this : PreconnectedSpace α ⊢ ConnectedSpace α exact ⟨⟨x⟩⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α h : connectedComponent x = univ ⊢ IsPreconnected univ rw [← h] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α h : connectedComponent x = univ ⊢ IsPreconnected (connectedComponent x) exact isPreconnected_connectedComponent ** Qed
preconnectedSpace_iff_connectedComponent α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α ⊢ PreconnectedSpace α ↔ ∀ (x : α), connectedComponent x = univ constructor case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α ⊢ PreconnectedSpace α → ∀ (x : α), connectedComponent x = univ intro h x case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : PreconnectedSpace α x : α ⊢ connectedComponent x = univ exact eq_univ_of_univ_subset <\| isPreconnected_univ.subset_connectedComponent (mem_univ x) case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α ⊢ (∀ (x : α), connectedComponent x = univ) → PreconnectedSpace α intro h case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : ∀ (x : α), connectedComponent x = univ ⊢ PreconnectedSpace α cases' isEmpty_or_nonempty α with hα hα case mpr.inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : ∀ (x : α), connectedComponent x = univ hα : IsEmpty α ⊢ PreconnectedSpace α exact ⟨by rw [univ_eq_empty_iff.mpr hα]; exact isPreconnected_empty⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : ∀ (x : α), connectedComponent x = univ hα : IsEmpty α ⊢ IsPreconnected univ rw [univ_eq_empty_iff.mpr hα] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : ∀ (x : α), connectedComponent x = univ hα : IsEmpty α ⊢ IsPreconnected ∅ exact isPreconnected_empty case mpr.inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : ∀ (x : α), connectedComponent x = univ hα : Nonempty α ⊢ PreconnectedSpace α exact ⟨by rw [← h (Classical.choice hα)]; exact isPreconnected_connectedComponent⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : ∀ (x : α), connectedComponent x = univ hα : Nonempty α ⊢ IsPreconnected univ rw [← h (Classical.choice hα)] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : ∀ (x : α), connectedComponent x = univ hα : Nonempty α ⊢ IsPreconnected (connectedComponent (Classical.choice hα)) exact isPreconnected_connectedComponent ** Qed
Continuous.exists_lift_sigma α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α s t u v : Set α inst✝¹ : ConnectedSpace α inst✝ : (i : ι) → TopologicalSpace (π i) f : α → (i : ι) × π i hf : Continuous f ⊢ ∃ i g, Continuous g ∧ f = Sigma.mk i ∘ g obtain ⟨i, hi⟩ : ∃ i, range f ⊆ range (.mk i) case intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α s t u v : Set α inst✝¹ : ConnectedSpace α inst✝ : (i : ι) → TopologicalSpace (π i) f : α → (i : ι) × π i hf : Continuous f i : ι hi : range f ⊆ range (Sigma.mk i) ⊢ ∃ i g, Continuous g ∧ f = Sigma.mk i ∘ g rcases range_subset_range_iff_exists_comp.1 hi with ⟨g, rfl⟩ case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α s t u v : Set α inst✝¹ : ConnectedSpace α inst✝ : (i : ι) → TopologicalSpace (π i) i : ι g : α → π i hf : Continuous (Sigma.mk i ∘ g) hi : range (Sigma.mk i ∘ g) ⊆ range (Sigma.mk i) ⊢ ∃ i_1 g_1, Continuous g_1 ∧ Sigma.mk i ∘ g = Sigma.mk i_1 ∘ g_1 refine ⟨i, g, ?_, rfl⟩ case intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α s t u v : Set α inst✝¹ : ConnectedSpace α inst✝ : (i : ι) → TopologicalSpace (π i) i : ι g : α → π i hf : Continuous (Sigma.mk i ∘ g) hi : range (Sigma.mk i ∘ g) ⊆ range (Sigma.mk i) ⊢ Continuous g rwa [← embedding_sigmaMk.continuous_iff] at hf α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α s t u v : Set α inst✝¹ : ConnectedSpace α inst✝ : (i : ι) → TopologicalSpace (π i) f : α → (i : ι) × π i hf : Continuous f ⊢ ∃ i, range f ⊆ range (Sigma.mk i) rcases Sigma.isConnected_iff.1 (isConnected_range hf) with ⟨i, s, -, hs⟩ case intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α s✝ t u v : Set α inst✝¹ : ConnectedSpace α inst✝ : (i : ι) → TopologicalSpace (π i) f : α → (i : ι) × π i hf : Continuous f i : ι s : Set (π i) hs : range f = Sigma.mk i '' s ⊢ ∃ i, range f ⊆ range (Sigma.mk i) exact ⟨i, hs.trans_subset (image_subset_range _ _)⟩ ** Qed
nonempty_inter α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t✝ u v : Set α inst✝ : PreconnectedSpace α s t : Set α ⊢ IsOpen s → IsOpen t → s ∪ t = univ → Set.Nonempty s → Set.Nonempty t → Set.Nonempty (s ∩ t) simpa only [univ_inter, univ_subset_iff] using @PreconnectedSpace.isPreconnected_univ α _ _ s t ** Qed
isClopen_iff α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : PreconnectedSpace α s : Set α hs : IsClopen s h : ¬(s = ∅ ∨ s = univ) h2 : sᶜ = ∅ ⊢ s = univ rw [← compl_compl s, h2, compl_empty] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : PreconnectedSpace α s : Set α ⊢ s = ∅ ∨ s = univ → IsClopen s rintro (rfl \| rfl) <;> [exact isClopen_empty; exact isClopen_univ] ** Qed
nonempty_frontier_iff α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s✝ t u v : Set α inst✝ : PreconnectedSpace α s : Set α ⊢ Set.Nonempty (frontier s) ↔ Set.Nonempty s ∧ s ≠ univ simp only [nonempty_iff_ne_empty, Ne.def, frontier_eq_empty_iff, not_or] ** Qed
Subtype.preconnectedSpace α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α h : IsPreconnected s ⊢ IsPreconnected univ rwa [← inducing_subtype_val.isPreconnected_image, image_univ, Subtype.range_val] ** Qed
isPreconnected_iff_preconnectedSpace α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α h : PreconnectedSpace ↑s ⊢ IsPreconnected s simpa using isPreconnected_univ.image ((↑) : s → α) continuous_subtype_val.continuousOn ** Qed
isPreconnected_iff_subset_of_disjoint α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α ⊢ IsPreconnected s ↔ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v constructor <;> intro h case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α h : IsPreconnected s ⊢ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v intro u v hu hv hs huv case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : IsPreconnected s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ ⊢ s ⊆ u ∨ s ⊆ v specialize h u v hu hv hs case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) ⊢ s ⊆ u ∨ s ⊆ v contrapose! huv case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) huv : ¬s ⊆ u ∧ ¬s ⊆ v ⊢ s ∩ (u ∩ v) ≠ ∅ rw [← nonempty_iff_ne_empty] case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) huv : ¬s ⊆ u ∧ ¬s ⊆ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) simp [not_subset] at huv case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) huv : (∃ a, a ∈ s ∧ ¬a ∈ u) ∧ ∃ a, a ∈ s ∧ ¬a ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) rcases huv with ⟨⟨x, hxs, hxu⟩, ⟨y, hys, hyv⟩⟩ case mp.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) x : α hxs : x ∈ s hxu : ¬x ∈ u y : α hys : y ∈ s hyv : ¬y ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) have hxv : x ∈ v := or_iff_not_imp_left.mp (hs hxs) hxu case mp.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) x : α hxs : x ∈ s hxu : ¬x ∈ u y : α hys : y ∈ s hyv : ¬y ∈ v hxv : x ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) have hyu : y ∈ u := or_iff_not_imp_right.mp (hs hys) hyv case mp.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) x : α hxs : x ∈ s hxu : ¬x ∈ u y : α hys : y ∈ s hyv : ¬y ∈ v hxv : x ∈ v hyu : y ∈ u ⊢ Set.Nonempty (s ∩ (u ∩ v)) exact h ⟨y, hys, hyu⟩ ⟨x, hxs, hxv⟩ case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v ⊢ IsPreconnected s intro u v hu hv hs hsu hsv case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) ⊢ Set.Nonempty (s ∩ (u ∩ v)) rw [nonempty_iff_ne_empty] case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) ⊢ s ∩ (u ∩ v) ≠ ∅ intro H case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) H : s ∩ (u ∩ v) = ∅ ⊢ False specialize h u v hu hv hs H case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) H : s ∩ (u ∩ v) = ∅ h : s ⊆ u ∨ s ⊆ v ⊢ False contrapose H case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) h : s ⊆ u ∨ s ⊆ v H : ¬False ⊢ ¬s ∩ (u ∩ v) = ∅ apply Nonempty.ne_empty case mpr.a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) h : s ⊆ u ∨ s ⊆ v H : ¬False ⊢ Set.Nonempty (s ∩ (u ∩ v)) cases' h with h h case mpr.a.inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) H : ¬False h : s ⊆ u ⊢ Set.Nonempty (s ∩ (u ∩ v)) rcases hsv with ⟨x, hxs, hxv⟩ case mpr.a.inl.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) H : ¬False h : s ⊆ u x : α hxs : x ∈ s hxv : x ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) exact ⟨x, hxs, ⟨h hxs, hxv⟩⟩ case mpr.a.inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) H : ¬False h : s ⊆ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) rcases hsu with ⟨x, hxs, hxu⟩ case mpr.a.inr.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsv : Set.Nonempty (s ∩ v) H : ¬False h : s ⊆ v x : α hxs : x ∈ s hxu : x ∈ u ⊢ Set.Nonempty (s ∩ (u ∩ v)) exact ⟨x, hxs, ⟨hxu, h hxs⟩⟩ ** Qed
isConnected_iff_sUnion_disjoint_open α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α ⊢ IsConnected s ↔ ∀ (U : Finset (Set α)), (∀ (u v : Set α), u ∈ U → v ∈ U → Set.Nonempty (s ∩ (u ∩ v)) → u = v) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u rw [IsConnected, isPreconnected_iff_subset_of_disjoint] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α ⊢ (Set.Nonempty s ∧ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v) ↔ ∀ (U : Finset (Set α)), (∀ (u v : Set α), u ∈ U → v ∈ U → Set.Nonempty (s ∩ (u ∩ v)) → u = v) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u refine ⟨fun ⟨hne, h⟩ U hU hUo hsU => ?_, fun h => ⟨?_, fun u v hu hv hs hsuv => ?_⟩⟩ case refine_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α x✝ : Set.Nonempty s ∧ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v U : Finset (Set α) hU : ∀ (u v : Set α), u ∈ U → v ∈ U → Set.Nonempty (s ∩ (u ∩ v)) → u = v hUo : ∀ (u : Set α), u ∈ U → IsOpen u hsU : s ⊆ ⋃₀ ↑U hne : Set.Nonempty s h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v ⊢ ∃ u, u ∈ U ∧ s ⊆ u induction U using Finset.induction_on case refine_1.empty α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α x✝ : Set.Nonempty s ∧ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v hne : Set.Nonempty s h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v hU : ∀ (u v : Set α), u ∈ ∅ → v ∈ ∅ → Set.Nonempty (s ∩ (u ∩ v)) → u = v hUo : ∀ (u : Set α), u ∈ ∅ → IsOpen u hsU : s ⊆ ⋃₀ ↑∅ ⊢ ∃ u, u ∈ ∅ ∧ s ⊆ u case refine_1.insert α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝¹ t u v s : Set α x✝ : Set.Nonempty s ∧ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v hne : Set.Nonempty s h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v a✝² : Set α s✝ : Finset (Set α) a✝¹ : ¬a✝² ∈ s✝ a✝ : (∀ (u v : Set α), u ∈ s✝ → v ∈ s✝ → Set.Nonempty (s ∩ (u ∩ v)) → u = v) → (∀ (u : Set α), u ∈ s✝ → IsOpen u) → s ⊆ ⋃₀ ↑s✝ → ∃ u, u ∈ s✝ ∧ s ⊆ u hU : ∀ (u v : Set α), u ∈ insert a✝² s✝ → v ∈ insert a✝² s✝ → Set.Nonempty (s ∩ (u ∩ v)) → u = v hUo : ∀ (u : Set α), u ∈ insert a✝² s✝ → IsOpen u hsU : s ⊆ ⋃₀ ↑(insert a✝² s✝) ⊢ ∃ u, u ∈ insert a✝² s✝ ∧ s ⊆ u case empty => exact absurd (by simpa using hsU) hne.not_subset_empty α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α x✝ : Set.Nonempty s ∧ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v hne : Set.Nonempty s h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v hU : ∀ (u v : Set α), u ∈ ∅ → v ∈ ∅ → Set.Nonempty (s ∩ (u ∩ v)) → u = v hUo : ∀ (u : Set α), u ∈ ∅ → IsOpen u hsU : s ⊆ ⋃₀ ↑∅ ⊢ ∃ u, u ∈ ∅ ∧ s ⊆ u exact absurd (by simpa using hsU) hne.not_subset_empty α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α x✝ : Set.Nonempty s ∧ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v hne : Set.Nonempty s h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v hU : ∀ (u v : Set α), u ∈ ∅ → v ∈ ∅ → Set.Nonempty (s ∩ (u ∩ v)) → u = v hUo : ∀ (u : Set α), u ∈ ∅ → IsOpen u hsU : s ⊆ ⋃₀ ↑∅ ⊢ s ⊆ ∅ simpa using hsU α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v s : Set α x✝ : Set.Nonempty s ∧ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v hne : Set.Nonempty s h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v u : Set α U : Finset (Set α) uU : ¬u ∈ U IH : (∀ (u v : Set α), u ∈ U → v ∈ U → Set.Nonempty (s ∩ (u ∩ v)) → u = v) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u hU : ∀ (u_1 v : Set α), u_1 ∈ insert u U → v ∈ insert u U → Set.Nonempty (s ∩ (u_1 ∩ v)) → u_1 = v hUo : ∀ (u_1 : Set α), u_1 ∈ insert u U → IsOpen u_1 hsU : s ⊆ ⋃₀ ↑(insert u U) ⊢ ∃ u_1, u_1 ∈ insert u U ∧ s ⊆ u_1 simp only [← ball_cond_comm, Finset.forall_mem_insert, Finset.exists_mem_insert, Finset.coe_insert, sUnion_insert, implies_true, true_and] at * α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v s : Set α hne : Set.Nonempty s u : Set α U : Finset (Set α) uU : ¬u ∈ U x✝ : Set.Nonempty s ∧ ∀ (a : Set α), IsOpen a → ∀ (b : Set α), IsOpen b → s ⊆ a ∪ b → s ∩ (a ∩ b) = ∅ → s ⊆ a ∨ s ⊆ b h : ∀ (a : Set α), IsOpen a → ∀ (b : Set α), IsOpen b → s ⊆ a ∪ b → s ∩ (a ∩ b) = ∅ → s ⊆ a ∨ s ⊆ b IH : (∀ (a : Set α), a ∈ U → ∀ (b : Set α), b ∈ U → Set.Nonempty (s ∩ (a ∩ b)) → a = b) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u hU : (∀ (x : Set α), x ∈ U → Set.Nonempty (s ∩ (u ∩ x)) → u = x) ∧ ∀ (x : Set α), x ∈ U → (Set.Nonempty (s ∩ (x ∩ u)) → x = u) ∧ ∀ (x_1 : Set α), x_1 ∈ U → Set.Nonempty (s ∩ (x ∩ x_1)) → x = x_1 hUo : IsOpen u ∧ ∀ (x : Set α), x ∈ U → IsOpen x hsU : s ⊆ u ∪ ⋃₀ ↑U ⊢ s ⊆ u ∨ ∃ x, x ∈ U ∧ s ⊆ x refine (h _ hUo.1 (⋃₀ ↑U) (isOpen_sUnion hUo.2) hsU ?_).imp_right ?_ case refine_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v s : Set α hne : Set.Nonempty s u : Set α U : Finset (Set α) uU : ¬u ∈ U x✝ : Set.Nonempty s ∧ ∀ (a : Set α), IsOpen a → ∀ (b : Set α), IsOpen b → s ⊆ a ∪ b → s ∩ (a ∩ b) = ∅ → s ⊆ a ∨ s ⊆ b h : ∀ (a : Set α), IsOpen a → ∀ (b : Set α), IsOpen b → s ⊆ a ∪ b → s ∩ (a ∩ b) = ∅ → s ⊆ a ∨ s ⊆ b IH : (∀ (a : Set α), a ∈ U → ∀ (b : Set α), b ∈ U → Set.Nonempty (s ∩ (a ∩ b)) → a = b) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u hU : (∀ (x : Set α), x ∈ U → Set.Nonempty (s ∩ (u ∩ x)) → u = x) ∧ ∀ (x : Set α), x ∈ U → (Set.Nonempty (s ∩ (x ∩ u)) → x = u) ∧ ∀ (x_1 : Set α), x_1 ∈ U → Set.Nonempty (s ∩ (x ∩ x_1)) → x = x_1 hUo : IsOpen u ∧ ∀ (x : Set α), x ∈ U → IsOpen x hsU : s ⊆ u ∪ ⋃₀ ↑U ⊢ s ∩ (u ∩ ⋃₀ ↑U) = ∅ refine subset_empty_iff.1 fun x ⟨hxs, hxu, v, hvU, hxv⟩ => ?_ case refine_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hne : Set.Nonempty s u : Set α U : Finset (Set α) uU : ¬u ∈ U x✝¹ : Set.Nonempty s ∧ ∀ (a : Set α), IsOpen a → ∀ (b : Set α), IsOpen b → s ⊆ a ∪ b → s ∩ (a ∩ b) = ∅ → s ⊆ a ∨ s ⊆ b h : ∀ (a : Set α), IsOpen a → ∀ (b : Set α), IsOpen b → s ⊆ a ∪ b → s ∩ (a ∩ b) = ∅ → s ⊆ a ∨ s ⊆ b IH : (∀ (a : Set α), a ∈ U → ∀ (b : Set α), b ∈ U → Set.Nonempty (s ∩ (a ∩ b)) → a = b) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u hU : (∀ (x : Set α), x ∈ U → Set.Nonempty (s ∩ (u ∩ x)) → u = x) ∧ ∀ (x : Set α), x ∈ U → (Set.Nonempty (s ∩ (x ∩ u)) → x = u) ∧ ∀ (x_1 : Set α), x_1 ∈ U → Set.Nonempty (s ∩ (x ∩ x_1)) → x = x_1 hUo : IsOpen u ∧ ∀ (x : Set α), x ∈ U → IsOpen x hsU : s ⊆ u ∪ ⋃₀ ↑U x : α x✝ : x ∈ s ∩ (u ∩ ⋃₀ ↑U) hxs : x ∈ s hxu : x ∈ u v : Set α hvU : v ∈ ↑U hxv : x ∈ v ⊢ x ∈ ∅ exact ne_of_mem_of_not_mem hvU uU (hU.1 v hvU ⟨x, hxs, hxu, hxv⟩).symm case refine_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v s : Set α hne : Set.Nonempty s u : Set α U : Finset (Set α) uU : ¬u ∈ U x✝ : Set.Nonempty s ∧ ∀ (a : Set α), IsOpen a → ∀ (b : Set α), IsOpen b → s ⊆ a ∪ b → s ∩ (a ∩ b) = ∅ → s ⊆ a ∨ s ⊆ b h : ∀ (a : Set α), IsOpen a → ∀ (b : Set α), IsOpen b → s ⊆ a ∪ b → s ∩ (a ∩ b) = ∅ → s ⊆ a ∨ s ⊆ b IH : (∀ (a : Set α), a ∈ U → ∀ (b : Set α), b ∈ U → Set.Nonempty (s ∩ (a ∩ b)) → a = b) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u hU : (∀ (x : Set α), x ∈ U → Set.Nonempty (s ∩ (u ∩ x)) → u = x) ∧ ∀ (x : Set α), x ∈ U → (Set.Nonempty (s ∩ (x ∩ u)) → x = u) ∧ ∀ (x_1 : Set α), x_1 ∈ U → Set.Nonempty (s ∩ (x ∩ x_1)) → x = x_1 hUo : IsOpen u ∧ ∀ (x : Set α), x ∈ U → IsOpen x hsU : s ⊆ u ∪ ⋃₀ ↑U ⊢ s ⊆ ⋃₀ ↑U → ∃ x, x ∈ U ∧ s ⊆ x exact IH (fun u hu => (hU.2 u hu).2) hUo.2 case refine_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α h : ∀ (U : Finset (Set α)), (∀ (u v : Set α), u ∈ U → v ∈ U → Set.Nonempty (s ∩ (u ∩ v)) → u = v) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u ⊢ Set.Nonempty s simpa [subset_empty_iff, nonempty_iff_ne_empty] using h ∅ case refine_3 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (U : Finset (Set α)), (∀ (u v : Set α), u ∈ U → v ∈ U → Set.Nonempty (s ∩ (u ∩ v)) → u = v) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsuv : s ∩ (u ∩ v) = ∅ ⊢ s ⊆ u ∨ s ⊆ v rw [← not_nonempty_iff_eq_empty] at hsuv case refine_3 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (U : Finset (Set α)), (∀ (u v : Set α), u ∈ U → v ∈ U → Set.Nonempty (s ∩ (u ∩ v)) → u = v) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsuv : ¬Set.Nonempty (s ∩ (u ∩ v)) ⊢ s ⊆ u ∨ s ⊆ v have := hsuv case refine_3 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (U : Finset (Set α)), (∀ (u v : Set α), u ∈ U → v ∈ U → Set.Nonempty (s ∩ (u ∩ v)) → u = v) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsuv this : ¬Set.Nonempty (s ∩ (u ∩ v)) ⊢ s ⊆ u ∨ s ⊆ v rw [inter_comm u] at this case refine_3 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α h : ∀ (U : Finset (Set α)), (∀ (u v : Set α), u ∈ U → v ∈ U → Set.Nonempty (s ∩ (u ∩ v)) → u = v) → (∀ (u : Set α), u ∈ U → IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u, u ∈ U ∧ s ⊆ u u v : Set α hu : IsOpen u hv : IsOpen v hs : s ⊆ u ∪ v hsuv : ¬Set.Nonempty (s ∩ (u ∩ v)) this : ¬Set.Nonempty (s ∩ (v ∩ u)) ⊢ s ⊆ u ∨ s ⊆ v simpa [, or_imp, forall_and] using h {u, v} * Qed
isPreconnected_iff_subset_of_disjoint_closed α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α ⊢ IsPreconnected s ↔ ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v constructor <;> intro h case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : IsPreconnected s ⊢ ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v intro u v hu hv hs huv case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ : Set α h : IsPreconnected s u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ ⊢ s ⊆ u ∨ s ⊆ v rw [isPreconnected_closed_iff] at h case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ : Set α h : ∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t')) u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ ⊢ s ⊆ u ∨ s ⊆ v specialize h u v hu hv hs case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) ⊢ s ⊆ u ∨ s ⊆ v contrapose! huv case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) huv : ¬s ⊆ u ∧ ¬s ⊆ v ⊢ s ∩ (u ∩ v) ≠ ∅ rw [← nonempty_iff_ne_empty] case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) huv : ¬s ⊆ u ∧ ¬s ⊆ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) simp [not_subset] at huv case mp α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) huv : (∃ a, a ∈ s ∧ ¬a ∈ u) ∧ ∃ a, a ∈ s ∧ ¬a ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) rcases huv with ⟨⟨x, hxs, hxu⟩, ⟨y, hys, hyv⟩⟩ case mp.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) x : α hxs : x ∈ s hxu : ¬x ∈ u y : α hys : y ∈ s hyv : ¬y ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) have hxv : x ∈ v := or_iff_not_imp_left.mp (hs hxs) hxu case mp.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) x : α hxs : x ∈ s hxu : ¬x ∈ u y : α hys : y ∈ s hyv : ¬y ∈ v hxv : x ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) have hyu : y ∈ u := or_iff_not_imp_right.mp (hs hys) hyv case mp.intro.intro.intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v h : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) x : α hxs : x ∈ s hxu : ¬x ∈ u y : α hys : y ∈ s hyv : ¬y ∈ v hxv : x ∈ v hyu : y ∈ u ⊢ Set.Nonempty (s ∩ (u ∩ v)) exact h ⟨y, hys, hyu⟩ ⟨x, hxs, hxv⟩ case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v ⊢ IsPreconnected s rw [isPreconnected_closed_iff] case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α h : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v ⊢ ∀ (t t' : Set α), IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t')) intro u v hu hv hs hsu hsv case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ : Set α h : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) ⊢ Set.Nonempty (s ∩ (u ∩ v)) rw [nonempty_iff_ne_empty] case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ : Set α h : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) ⊢ s ∩ (u ∩ v) ≠ ∅ intro H case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ : Set α h : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) H : s ∩ (u ∩ v) = ∅ ⊢ False specialize h u v hu hv hs H case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) H : s ∩ (u ∩ v) = ∅ h : s ⊆ u ∨ s ⊆ v ⊢ False contrapose H case mpr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) h : s ⊆ u ∨ s ⊆ v H : ¬False ⊢ ¬s ∩ (u ∩ v) = ∅ apply Nonempty.ne_empty case mpr.a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) h : s ⊆ u ∨ s ⊆ v H : ¬False ⊢ Set.Nonempty (s ∩ (u ∩ v)) cases' h with h h case mpr.a.inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) H : ¬False h : s ⊆ u ⊢ Set.Nonempty (s ∩ (u ∩ v)) rcases hsv with ⟨x, hxs, hxv⟩ case mpr.a.inl.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) H : ¬False h : s ⊆ u x : α hxs : x ∈ s hxv : x ∈ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) exact ⟨x, hxs, ⟨h hxs, hxv⟩⟩ case mpr.a.inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v hsu : Set.Nonempty (s ∩ u) hsv : Set.Nonempty (s ∩ v) H : ¬False h : s ⊆ v ⊢ Set.Nonempty (s ∩ (u ∩ v)) rcases hsu with ⟨x, hxs, hxu⟩ case mpr.a.inr.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u✝ v✝ u v : Set α hu : IsClosed u hv : IsClosed v hs : s ⊆ u ∪ v hsv : Set.Nonempty (s ∩ v) H : ¬False h : s ⊆ v x : α hxs : x ∈ s hxu : x ∈ u ⊢ Set.Nonempty (s ∩ (u ∩ v)) exact ⟨x, hxs, ⟨hxu, h hxs⟩⟩ ** Qed
isPreconnected_iff_subset_of_fully_disjoint_closed α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α hs : IsClosed s ⊢ IsPreconnected s ↔ ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v refine isPreconnected_iff_subset_of_disjoint_closed.trans ⟨?_, ?_⟩ <;> intro H u v hu hv hss huv case refine_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : IsClosed s H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hss : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ ⊢ s ⊆ u ∨ s ⊆ v have H1 := H (u ∩ s) (v ∩ s) case refine_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : IsClosed s H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hss : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ H1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∩ s ∨ s ⊆ v ∩ s ⊢ s ⊆ u ∨ s ⊆ v rw [subset_inter_iff, subset_inter_iff] at H1 case refine_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : IsClosed s H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hss : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ H1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∧ s ⊆ s ∨ s ⊆ v ∧ s ⊆ s ⊢ s ⊆ u ∨ s ⊆ v simp only [Subset.refl, and_true] at H1 case refine_2 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : IsClosed s H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hss : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ H1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∨ s ⊆ v ⊢ s ⊆ u ∨ s ⊆ v apply H1 (hu.inter hs) (hv.inter hs) case refine_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : IsClosed s H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hss : s ⊆ u ∪ v huv : Disjoint u v ⊢ s ⊆ u ∨ s ⊆ v apply H u v hu hv hss case refine_1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : IsClosed s H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hss : s ⊆ u ∪ v huv : Disjoint u v ⊢ s ∩ (u ∩ v) = ∅ rw [huv.inter_eq, inter_empty] case refine_2.a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : IsClosed s H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hss : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ H1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∨ s ⊆ v ⊢ s ⊆ u ∩ s ∪ v ∩ s rw [← inter_distrib_right] case refine_2.a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : IsClosed s H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hss : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ H1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∨ s ⊆ v ⊢ s ⊆ (u ∪ v) ∩ s exact subset_inter hss Subset.rfl case refine_2.a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : IsClosed s H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v u v : Set α hu : IsClosed u hv : IsClosed v hss : s ⊆ u ∪ v huv : s ∩ (u ∩ v) = ∅ H1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∨ s ⊆ v ⊢ Disjoint (u ∩ s) (v ∩ s) rwa [disjoint_iff_inter_eq_empty, ← inter_inter_distrib_right, inter_comm] ** Qed
preimage_connectedComponent_connected α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β ⊢ IsConnected (f ⁻¹' connectedComponent t) have hf : Surjective f := Surjective.of_comp fun t : β => (connected_fibers t).1 α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f ⊢ IsConnected (f ⁻¹' connectedComponent t) refine ⟨Nonempty.preimage connectedComponent_nonempty hf, ?_⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f ⊢ IsPreconnected (f ⁻¹' connectedComponent t) have hT : IsClosed (f ⁻¹' connectedComponent t) := (hcl (connectedComponent t)).1 isClosed_connectedComponent α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) ⊢ IsPreconnected (f ⁻¹' connectedComponent t) rw [isPreconnected_iff_subset_of_fully_disjoint_closed hT] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u v : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) ⊢ ∀ (u v : Set α), IsClosed u → IsClosed v → f ⁻¹' connectedComponent t ⊆ u ∪ v → Disjoint u v → f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v intro u v hu hv huv uv_disj α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v ⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v let T₁ := { t' ∈ connectedComponent t \| f ⁻¹' {t'} ⊆ u } α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} ⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v let T₂ := { t' ∈ connectedComponent t \| f ⁻¹' {t'} ⊆ v } α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v ⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v have hT₁ : IsClosed T₁ := (hcl T₁).2 (T₁_u.symm ▸ IsClosed.inter hT hu) α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ ⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v have hT₂ : IsClosed T₂ := (hcl T₂).2 (T₂_v.symm ▸ IsClosed.inter hT hv) α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ ⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v have T_disjoint : Disjoint T₁ T₂ := by refine' Disjoint.of_preimage hf _ rw [T₁_u, T₂_v, disjoint_iff_inter_eq_empty, ← inter_inter_distrib_left, uv_disj.inter_eq, inter_empty] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ T_disjoint : Disjoint T₁ T₂ ⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v cases' (isPreconnected_iff_subset_of_fully_disjoint_closed isClosed_connectedComponent).1 isPreconnected_connectedComponent T₁ T₂ hT₁ hT₂ T_decomp T_disjoint with h h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} ⊢ ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v intro t' ht' α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} t' : β ht' : t' ∈ connectedComponent t ⊢ f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v apply isPreconnected_iff_subset_of_disjoint_closed.1 (connected_fibers t').2 u v hu hv case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} t' : β ht' : t' ∈ connectedComponent t ⊢ f ⁻¹' {t'} ∩ (u ∩ v) = ∅ rw [uv_disj.inter_eq, inter_empty] case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} t' : β ht' : t' ∈ connectedComponent t ⊢ f ⁻¹' {t'} ⊆ u ∪ v exact Subset.trans (preimage_mono (singleton_subset_iff.2 ht')) huv α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v ⊢ f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u apply eq_of_subset_of_subset case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v ⊢ f ⁻¹' connectedComponent t ∩ u ⊆ f ⁻¹' T₁ rintro a ⟨hat, hau⟩ case a.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v a : α hat : a ∈ f ⁻¹' connectedComponent t hau : a ∈ u ⊢ a ∈ f ⁻¹' T₁ constructor case a.intro.right α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v a : α hat : a ∈ f ⁻¹' connectedComponent t hau : a ∈ u ⊢ f ⁻¹' {f a} ⊆ u refine (fiber_decomp (f a) (mem_preimage.1 hat)).resolve_right fun h => ?_ case a.intro.right α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v a : α hat : a ∈ f ⁻¹' connectedComponent t hau : a ∈ u h : f ⁻¹' {f a} ⊆ v ⊢ False exact uv_disj.subset_compl_right hau (h rfl) case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v ⊢ f ⁻¹' T₁ ⊆ f ⁻¹' connectedComponent t ∩ u rw [← biUnion_preimage_singleton] case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v ⊢ ⋃ y ∈ T₁, f ⁻¹' {y} ⊆ f ⁻¹' connectedComponent t ∩ u refine' iUnion₂_subset fun t' ht' => subset_inter _ ht'.2 case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v t' : β ht' : t' ∈ T₁ ⊢ f ⁻¹' {t'} ⊆ f ⁻¹' connectedComponent t rw [hf.preimage_subset_preimage_iff, singleton_subset_iff] case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v t' : β ht' : t' ∈ T₁ ⊢ t' ∈ connectedComponent t exact ht'.1 case a.intro.left α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v a : α hat : a ∈ f ⁻¹' connectedComponent t hau : a ∈ u ⊢ f a ∈ connectedComponent t exact mem_preimage.1 hat α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u ⊢ f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v apply eq_of_subset_of_subset case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u ⊢ f ⁻¹' connectedComponent t ∩ v ⊆ f ⁻¹' T₂ rintro a ⟨hat, hav⟩ case a.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u a : α hat : a ∈ f ⁻¹' connectedComponent t hav : a ∈ v ⊢ a ∈ f ⁻¹' T₂ constructor case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u ⊢ f ⁻¹' T₂ ⊆ f ⁻¹' connectedComponent t ∩ v rw [← biUnion_preimage_singleton] case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u ⊢ ⋃ y ∈ T₂, f ⁻¹' {y} ⊆ f ⁻¹' connectedComponent t ∩ v refine' iUnion₂_subset fun t' ht' => subset_inter _ ht'.2 case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u t' : β ht' : t' ∈ T₂ ⊢ f ⁻¹' {t'} ⊆ f ⁻¹' connectedComponent t rw [hf.preimage_subset_preimage_iff, singleton_subset_iff] case a α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u t' : β ht' : t' ∈ T₂ ⊢ t' ∈ connectedComponent t exact ht'.1 case a.intro.left α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u a : α hat : a ∈ f ⁻¹' connectedComponent t hav : a ∈ v ⊢ f a ∈ connectedComponent t exact mem_preimage.1 hat case a.intro.right α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u a : α hat : a ∈ f ⁻¹' connectedComponent t hav : a ∈ v ⊢ f ⁻¹' {f a} ⊆ v refine (fiber_decomp (f a) (mem_preimage.1 hat)).resolve_left fun h => ?_ case a.intro.right α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u a : α hat : a ∈ f ⁻¹' connectedComponent t hav : a ∈ v h : f ⁻¹' {f a} ⊆ u ⊢ False exact uv_disj.subset_compl_left hav (h rfl) α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ t' : β ht' : t' ∈ connectedComponent t ⊢ t' ∈ T₁ ∪ T₂ rw [mem_union t' T₁ T₂] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ t' : β ht' : t' ∈ connectedComponent t ⊢ t' ∈ T₁ ∨ t' ∈ T₂ cases' fiber_decomp t' ht' with htu htv case inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ t' : β ht' : t' ∈ connectedComponent t htv : f ⁻¹' {t'} ⊆ v ⊢ t' ∈ T₁ ∨ t' ∈ T₂ right case inr.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ t' : β ht' : t' ∈ connectedComponent t htv : f ⁻¹' {t'} ⊆ v ⊢ t' ∈ T₂ exact ⟨ht', htv⟩ case inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ t' : β ht' : t' ∈ connectedComponent t htu : f ⁻¹' {t'} ⊆ u ⊢ t' ∈ T₁ ∨ t' ∈ T₂ left case inl.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ t' : β ht' : t' ∈ connectedComponent t htu : f ⁻¹' {t'} ⊆ u ⊢ t' ∈ T₁ exact ⟨ht', htu⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ ⊢ Disjoint T₁ T₂ refine' Disjoint.of_preimage hf _ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ ⊢ Disjoint (f ⁻¹' T₁) (f ⁻¹' T₂) rw [T₁_u, T₂_v, disjoint_iff_inter_eq_empty, ← inter_inter_distrib_left, uv_disj.inter_eq, inter_empty] case inl α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ T_disjoint : Disjoint T₁ T₂ h : connectedComponent t ⊆ T₁ ⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v left case inl.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ T_disjoint : Disjoint T₁ T₂ h : connectedComponent t ⊆ T₁ ⊢ f ⁻¹' connectedComponent t ⊆ u rw [Subset.antisymm_iff] at T₁_u case inl.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ ⊆ f ⁻¹' connectedComponent t ∩ u ∧ f ⁻¹' connectedComponent t ∩ u ⊆ f ⁻¹' T₁ T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ T_disjoint : Disjoint T₁ T₂ h : connectedComponent t ⊆ T₁ ⊢ f ⁻¹' connectedComponent t ⊆ u suffices f ⁻¹' connectedComponent t ⊆ f ⁻¹' T₁ from (this.trans T₁_u.1).trans (inter_subset_right _ _) case inl.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ ⊆ f ⁻¹' connectedComponent t ∩ u ∧ f ⁻¹' connectedComponent t ∩ u ⊆ f ⁻¹' T₁ T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ T_disjoint : Disjoint T₁ T₂ h : connectedComponent t ⊆ T₁ ⊢ f ⁻¹' connectedComponent t ⊆ f ⁻¹' T₁ exact preimage_mono h case inr α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ T_disjoint : Disjoint T₁ T₂ h : connectedComponent t ⊆ T₂ ⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v right case inr.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ T_disjoint : Disjoint T₁ T₂ h : connectedComponent t ⊆ T₂ ⊢ f ⁻¹' connectedComponent t ⊆ v rw [Subset.antisymm_iff] at T₂_v case inr.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ ⊆ f ⁻¹' connectedComponent t ∩ v ∧ f ⁻¹' connectedComponent t ∩ v ⊆ f ⁻¹' T₂ hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ T_disjoint : Disjoint T₁ T₂ h : connectedComponent t ⊆ T₂ ⊢ f ⁻¹' connectedComponent t ⊆ v suffices f ⁻¹' connectedComponent t ⊆ f ⁻¹' T₂ from (this.trans T₂_v.1).trans (inter_subset_right _ _) case inr.h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t✝ u✝ v✝ : Set α inst✝ : TopologicalSpace β f : α → β connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t}) hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T) t : β hf : Surjective f hT : IsClosed (f ⁻¹' connectedComponent t) u v : Set α hu : IsClosed u hv : IsClosed v huv : f ⁻¹' connectedComponent t ⊆ u ∪ v uv_disj : Disjoint u v T₁ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u} T₂ : Set β := {t' \| t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v} fiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u T₂_v : f ⁻¹' T₂ ⊆ f ⁻¹' connectedComponent t ∩ v ∧ f ⁻¹' connectedComponent t ∩ v ⊆ f ⁻¹' T₂ hT₁ : IsClosed T₁ hT₂ : IsClosed T₂ T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ T_disjoint : Disjoint T₁ T₂ h : connectedComponent t ⊆ T₂ ⊢ f ⁻¹' connectedComponent t ⊆ f ⁻¹' T₂ exact preimage_mono h ** Qed
QuotientMap.image_connectedComponent α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝¹ : TopologicalSpace α s t u v : Set α inst✝ : TopologicalSpace β f : α → β hf : QuotientMap f h_fibers : ∀ (y : β), IsConnected (f ⁻¹' {y}) a : α ⊢ f '' connectedComponent a = connectedComponent (f a) rw [← hf.preimage_connectedComponent h_fibers, image_preimage_eq _ hf.surjective] ** Qed
connectedComponents_preimage_singleton α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x : α ⊢ ConnectedComponents.mk ⁻¹' {ConnectedComponents.mk x} = connectedComponent x ext y case h α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v : Set α x y : α ⊢ y ∈ ConnectedComponents.mk ⁻¹' {ConnectedComponents.mk x} ↔ y ∈ connectedComponent x rw [mem_preimage, mem_singleton_iff, ConnectedComponents.coe_eq_coe'] ** Qed
connectedComponents_preimage_image α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s t u v U : Set α ⊢ ConnectedComponents.mk ⁻¹' (ConnectedComponents.mk '' U) = ⋃ x ∈ U, connectedComponent x simp only [connectedComponents_preimage_singleton, preimage_iUnion₂, image_eq_iUnion] ** Qed
isPreconnected_of_forall_constant α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y ⊢ IsPreconnected s unfold IsPreconnected α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y ⊢ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v)) by_contra' α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u v s : Set α hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y this : ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∪ v ∧ Set.Nonempty (s ∩ u) ∧ Set.Nonempty (s ∩ v) ∧ ¬Set.Nonempty (s ∩ (u ∩ v)) ⊢ False rcases this with ⟨u, v, u_op, v_op, hsuv, ⟨x, x_in_s, x_in_u⟩, ⟨y, y_in_s, y_in_v⟩, H⟩ case intro.intro.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✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y u v : Set α u_op : IsOpen u v_op : IsOpen v hsuv : s ⊆ u ∪ v x : α x_in_s : x ∈ s x_in_u : x ∈ u H : ¬Set.Nonempty (s ∩ (u ∩ v)) y : α y_in_s : y ∈ s y_in_v : y ∈ v ⊢ False rw [not_nonempty_iff_eq_empty] at H case intro.intro.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✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y u v : Set α u_op : IsOpen u v_op : IsOpen v hsuv : s ⊆ u ∪ v x : α x_in_s : x ∈ s x_in_u : x ∈ u H : s ∩ (u ∩ v) = ∅ y : α y_in_s : y ∈ s y_in_v : y ∈ v ⊢ False have hy : y ∉ u := fun y_in_u => eq_empty_iff_forall_not_mem.mp H y ⟨y_in_s, ⟨y_in_u, y_in_v⟩⟩ case intro.intro.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✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y u v : Set α u_op : IsOpen u v_op : IsOpen v hsuv : s ⊆ u ∪ v x : α x_in_s : x ∈ s x_in_u : x ∈ u H : s ∩ (u ∩ v) = ∅ y : α y_in_s : y ∈ s y_in_v : y ∈ v hy : ¬y ∈ u this : ContinuousOn (boolIndicator u) s ⊢ False simpa [(u.mem_iff_boolIndicator _).mp x_in_u, (u.not_mem_iff_boolIndicator _).mp hy] using hs _ this x x_in_s y y_in_s α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y u v : Set α u_op : IsOpen u v_op : IsOpen v hsuv : s ⊆ u ∪ v x : α x_in_s : x ∈ s x_in_u : x ∈ u H : s ∩ (u ∩ v) = ∅ y : α y_in_s : y ∈ s y_in_v : y ∈ v hy : ¬y ∈ u ⊢ ContinuousOn (boolIndicator u) s apply (continuousOn_boolIndicator_iff_clopen _ _).mpr ⟨_, _⟩ α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y u v : Set α u_op : IsOpen u v_op : IsOpen v hsuv : s ⊆ u ∪ v x : α x_in_s : x ∈ s x_in_u : x ∈ u H : s ∩ (u ∩ v) = ∅ y : α y_in_s : y ∈ s y_in_v : y ∈ v hy : ¬y ∈ u ⊢ IsOpen (Subtype.val ⁻¹' u) exact u_op.preimage continuous_subtype_val α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y u v : Set α u_op : IsOpen u v_op : IsOpen v hsuv : s ⊆ u ∪ v x : α x_in_s : x ∈ s x_in_u : x ∈ u H : s ∩ (u ∩ v) = ∅ y : α y_in_s : y ∈ s y_in_v : y ∈ v hy : ¬y ∈ u ⊢ IsClosed (Subtype.val ⁻¹' u) rw [preimage_subtype_coe_eq_compl hsuv H] α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝ : TopologicalSpace α s✝ t u✝ v✝ s : Set α hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y u v : Set α u_op : IsOpen u v_op : IsOpen v hsuv : s ⊆ u ∪ v x : α x_in_s : x ∈ s x_in_u : x ∈ u H : s ∩ (u ∩ v) = ∅ y : α y_in_s : y ∈ s y_in_v : y ∈ v hy : ¬y ∈ u ⊢ IsClosed (Subtype.val ⁻¹' v)ᶜ exact (v_op.preimage continuous_subtype_val).isClosed_compl ** Qed
polynomialFunctions_closure_eq_top' ⊢ Subalgebra.topologicalClosure (polynomialFunctions I) = ⊤ apply eq_top_iff.mpr ⊢ ⊤ ≤ Subalgebra.topologicalClosure (polynomialFunctions I) rintro f - f : C(↑I, ℝ) ⊢ f ∈ Subalgebra.topologicalClosure (polynomialFunctions I) refine' Filter.Frequently.mem_closure _ f : C(↑I, ℝ) ⊢ ∃ᶠ (x : C(↑I, ℝ)) in nhds f, x ∈ ↑(polynomialFunctions I) refine' Filter.Tendsto.frequently (bernsteinApproximation_uniform f) _ f : C(↑I, ℝ) ⊢ ∃ᶠ (x : ℕ) in atTop, bernsteinApproximation x f ∈ ↑(polynomialFunctions I) apply frequently_of_forall case h f : C(↑I, ℝ) ⊢ ∀ (x : ℕ), bernsteinApproximation x f ∈ ↑(polynomialFunctions I) intro n case h f : C(↑I, ℝ) n : ℕ ⊢ bernsteinApproximation n f ∈ ↑(polynomialFunctions I) simp only [SetLike.mem_coe] case h f : C(↑I, ℝ) n : ℕ ⊢ bernsteinApproximation n f ∈ polynomialFunctions I apply Subalgebra.sum_mem case h.h f : C(↑I, ℝ) n : ℕ ⊢ ∀ (x : Fin (n + 1)), x ∈ Finset.univ → ↑f (bernstein.z x) • bernstein n ↑x ∈ polynomialFunctions I rintro n - case h.h f : C(↑I, ℝ) n✝ : ℕ n : Fin (n✝ + 1) ⊢ ↑f (bernstein.z n) • bernstein n✝ ↑n ∈ polynomialFunctions I apply Subalgebra.smul_mem case h.h.hx f : C(↑I, ℝ) n✝ : ℕ n : Fin (n✝ + 1) ⊢ bernstein n✝ ↑n ∈ polynomialFunctions I dsimp [bernstein, polynomialFunctions] case h.h.hx f : C(↑I, ℝ) n✝ : ℕ n : Fin (n✝ + 1) ⊢ Polynomial.toContinuousMapOn (bernsteinPolynomial ℝ n✝ ↑n) I ∈ Subalgebra.map (Polynomial.toContinuousMapOnAlgHom I) ⊤ simp ** Qed
polynomialFunctions_closure_eq_top a b : ℝ ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ cases' lt_or_le a b with h h case inl a b : ℝ h : a < b ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ let W : C(Set.Icc a b, ℝ) →ₐ[ℝ] C(I, ℝ) := compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap case inl a b : ℝ h : a < b W : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) := compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h))) ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ let W' : C(Set.Icc a b, ℝ) ≃ₜ C(I, ℝ) := compRightHomeomorph ℝ (iccHomeoI a b h).symm case inl a b : ℝ h : a < b W : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) := compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h))) W' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h)) ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ have w : (W : C(Set.Icc a b, ℝ) → C(I, ℝ)) = W' := rfl case inl a b : ℝ h : a < b W : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) := compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h))) W' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h)) w : ↑W = ↑W' ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ have p := polynomialFunctions_closure_eq_top' case inl a b : ℝ h : a < b W : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) := compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h))) W' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h)) w : ↑W = ↑W' p : Subalgebra.topologicalClosure (polynomialFunctions I) = ⊤ ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ apply_fun fun s => s.comap W at p case inl a b : ℝ h : a < b W : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) := compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h))) W' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h)) w : ↑W = ↑W' p : Subalgebra.comap W (Subalgebra.topologicalClosure (polynomialFunctions I)) = Subalgebra.comap W ⊤ ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ simp only [Algebra.comap_top] at p case inl a b : ℝ h : a < b W : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) := compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h))) W' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h)) w : ↑W = ↑W' p : Subalgebra.comap (compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))) (Subalgebra.topologicalClosure (polynomialFunctions I)) = ⊤ ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ rw [Subalgebra.topologicalClosure_comap_homeomorph _ W W' w] at p case inl a b : ℝ h : a < b W : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) := compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h))) W' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h)) w : ↑W = ↑W' p : Subalgebra.topologicalClosure (Subalgebra.comap W (polynomialFunctions I)) = ⊤ ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ rw [polynomialFunctions.comap_compRightAlgHom_iccHomeoI] at p case inl a b : ℝ h : a < b W : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) := compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h))) W' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h)) w : ↑W = ↑W' p : Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ exact p case inr a b : ℝ h : b ≤ a ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ have : Subsingleton (Set.Icc a b) := (Set.subsingleton_coe _).mpr $ Set.subsingleton_Icc_of_ge h case inr a b : ℝ h : b ≤ a this : Subsingleton ↑(Set.Icc a b) ⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤ apply Subsingleton.elim ** Qed
continuousMap_mem_polynomialFunctions_closure a b : ℝ f : C(↑(Set.Icc a b), ℝ) ⊢ f ∈ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) rw [polynomialFunctions_closure_eq_top _ _] a b : ℝ f : C(↑(Set.Icc a b), ℝ) ⊢ f ∈ ⊤ simp ** Qed
exists_polynomial_near_continuousMap a b : ℝ f : C(↑(Set.Icc a b), ℝ) ε : ℝ pos : 0 < ε ⊢ ∃ p, ‖Polynomial.toContinuousMapOn p (Set.Icc a b) - f‖ < ε have w := mem_closure_iff_frequently.mp (continuousMap_mem_polynomialFunctions_closure _ _ f) a b : ℝ f : C(↑(Set.Icc a b), ℝ) ε : ℝ pos : 0 < ε w : ∃ᶠ (x : C(↑(Set.Icc a b), ℝ)) in nhds f, x ∈ ↑(polynomialFunctions (Set.Icc a b)) ⊢ ∃ p, ‖Polynomial.toContinuousMapOn p (Set.Icc a b) - f‖ < ε rw [Metric.nhds_basis_ball.frequently_iff] at w a b : ℝ f : C(↑(Set.Icc a b), ℝ) ε : ℝ pos : 0 < ε w : ∀ (i : ℝ), 0 < i → ∃ x, x ∈ Metric.ball f i ∧ x ∈ ↑(polynomialFunctions (Set.Icc a b)) ⊢ ∃ p, ‖Polynomial.toContinuousMapOn p (Set.Icc a b) - f‖ < ε obtain ⟨-, H, ⟨m, ⟨-, rfl⟩⟩⟩ := w ε pos case intro.intro.intro.intro a b : ℝ f : C(↑(Set.Icc a b), ℝ) ε : ℝ pos : 0 < ε w : ∀ (i : ℝ), 0 < i → ∃ x, x ∈ Metric.ball f i ∧ x ∈ ↑(polynomialFunctions (Set.Icc a b)) m : ℝ[X] H : ↑↑(Polynomial.toContinuousMapOnAlgHom (Set.Icc a b)) m ∈ Metric.ball f ε ⊢ ∃ p, ‖Polynomial.toContinuousMapOn p (Set.Icc a b) - f‖ < ε rw [Metric.mem_ball, dist_eq_norm] at H case intro.intro.intro.intro a b : ℝ f : C(↑(Set.Icc a b), ℝ) ε : ℝ pos : 0 < ε w : ∀ (i : ℝ), 0 < i → ∃ x, x ∈ Metric.ball f i ∧ x ∈ ↑(polynomialFunctions (Set.Icc a b)) m : ℝ[X] H : ‖↑↑(Polynomial.toContinuousMapOnAlgHom (Set.Icc a b)) m - f‖ < ε ⊢ ∃ p, ‖Polynomial.toContinuousMapOn p (Set.Icc a b) - f‖ < ε exact ⟨m, H⟩ ** Qed