ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MDifferentiableAt.continuousAt
theorem MDifferentiableAt.continuousAt {f : M → M'} {x : M} (hf : MDifferentiableAt I I' f x) :
    ContinuousAt f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} (hf : MDifferentiableAt I I' f x)
Goal: ContinuousAt f x
-- MDifferentiableAt.differentiableWithinAt_writtenInExtChartAt
theorem MDifferentiableAt.differentiableWithinAt_writtenInExtChartAt {f : M → M'} {x : M}
    (hf : MDifferentiableAt I I' f x) :
    DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f) (range I) ((extChartAt I x) x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} (hf : MDifferentiableAt I I' f x)
Goal: DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f) (Set.range ↑I) (↑(extChartAt I x) x)
-- Complex.norm_eventually_eq_of_mdifferentiableAt_of_isLocalMax
theorem Complex.norm_eventually_eq_of_mdifferentiableAt_of_isLocalMax {f : M → F} {c : M}
    (hd : ∀ᶠ z in 𝓝 c, MDifferentiableAt I 𝓘(ℂ, F) f z) (hc : IsLocalMax (norm ∘ f) c) :
    ∀ᶠ y in 𝓝 c, ‖f y‖ = ‖f c‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℂ F] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners ℂ E H} [I.Boundaryless] {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {f : M → F} {c : M} (hd : ∀ᶠ (z : M) in 𝓝 c, MDifferentiableAt I 𝓘(ℂ, F) f z) (hc : IsLocalMax (norm ∘ f) c)
Goal: ∀ᶠ (y : M) in 𝓝 c, ‖f y‖ = ‖f c‖
-- MDifferentiableOn.norm_eqOn_of_isPreconnected_of_isMaxOn
theorem norm_eqOn_of_isPreconnected_of_isMaxOn {f : M → F} {U : Set M} {c : M}
    (hd : MDifferentiableOn I 𝓘(ℂ, F) f U) (hc : IsPreconnected U) (ho : IsOpen U)
    (hcU : c ∈ U) (hm : IsMaxOn (norm ∘ f) U c) : EqOn (norm ∘ f) (const M ‖f c‖) U 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℂ F] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners ℂ E H} [I.Boundaryless] {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {f : M → F} {U : Set M} {c : M} (hd : MDifferentiableOn I 𝓘(ℂ, F) f U) (hc : IsPreconnected U) (ho : IsOpen U) (hcU : c ∈ U) (hm : IsMaxOn (norm ∘ f) U c)
Goal: Set.EqOn (norm ∘ f) (Function.const M ‖f c‖) U
-- MDifferentiableOn.eqOn_of_isPreconnected_of_isMaxOn_norm
theorem eqOn_of_isPreconnected_of_isMaxOn_norm [StrictConvexSpace ℝ F] {f : M → F} {U : Set M}
    {c : M} (hd : MDifferentiableOn I 𝓘(ℂ, F) f U) (hc : IsPreconnected U) (ho : IsOpen U)
    (hcU : c ∈ U) (hm : IsMaxOn (norm ∘ f) U c) : EqOn f (const M (f c)) U 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℂ F] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners ℂ E H} [I.Boundaryless] {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] [StrictConvexSpace ℝ F] {f : M → F} {U : Set M} {c : M} (hd : MDifferentiableOn I 𝓘(ℂ, F) f U) (hc : IsPreconnected U) (ho : IsOpen U) (hcU : c ∈ U) (hm : IsMaxOn (norm ∘ f) U c)
Goal: Set.EqOn f (Function.const M (f c)) U
-- MDifferentiableOn.apply_eq_of_isPreconnected_isCompact_isOpen
theorem apply_eq_of_isPreconnected_isCompact_isOpen {f : M → F} {U : Set M} {a b : M}
    (hd : MDifferentiableOn I 𝓘(ℂ, F) f U) (hpc : IsPreconnected U) (hc : IsCompact U)
    (ho : IsOpen U) (ha : a ∈ U) (hb : b ∈ U) : f a = f b 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℂ F] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners ℂ E H} [I.Boundaryless] {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {f : M → F} {U : Set M} {a b : M} (hd : MDifferentiableOn I 𝓘(ℂ, F) f U) (hpc : IsPreconnected U) (hc : IsCompact U) (ho : IsOpen U) (ha : a ∈ U) (hb : b ∈ U)
Goal: f a = f b
-- MDifferentiable.isLocallyConstant
protected theorem isLocallyConstant {f : M → F} (hf : MDifferentiable I 𝓘(ℂ, F) f) :
    IsLocallyConstant f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℂ F] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners ℂ E H} [I.Boundaryless] {M : Type u_4} [TopologicalSpace M] [CompactSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {f : M → F} (hf : MDifferentiable I 𝓘(ℂ, F) f)
Goal: IsLocallyConstant f
-- MDifferentiable.apply_eq_of_compactSpace
theorem apply_eq_of_compactSpace [PreconnectedSpace M] {f : M → F}
    (hf : MDifferentiable I 𝓘(ℂ, F) f) (a b : M) : f a = f b 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℂ F] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners ℂ E H} [I.Boundaryless] {M : Type u_4} [TopologicalSpace M] [CompactSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] [PreconnectedSpace M] {f : M → F} (hf : MDifferentiable I 𝓘(ℂ, F) f) (a b : M)
Goal: f a = f b
-- MDifferentiable.exists_eq_const_of_compactSpace
theorem exists_eq_const_of_compactSpace [PreconnectedSpace M] {f : M → F}
    (hf : MDifferentiable I 𝓘(ℂ, F) f) : ∃ v : F, f = Function.const M v 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℂ F] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners ℂ E H} [I.Boundaryless] {M : Type u_4} [TopologicalSpace M] [CompactSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] [PreconnectedSpace M] {f : M → F} (hf : MDifferentiable I 𝓘(ℂ, F) f)
Goal: ∃ v, f = Function.const M v
-- IsPreconnected.subset_isClopen
theorem IsPreconnected.subset_isClopen {s t : Set α} (hs : IsPreconnected s) (ht : IsClopen t)
    (hne : (s ∩ t).Nonempty) : s ⊆ t 
Arguments: {α : Type u} [TopologicalSpace α] {s t : Set α} (hs : IsPreconnected s) (ht : IsClopen t) (hne : (s ∩ t).Nonempty)
Goal: s ⊆ t
-- Sigma.isConnected_iff
theorem Sigma.isConnected_iff [∀ i, TopologicalSpace (π i)] {s : Set (Σi, π i)} :
    IsConnected s ↔ ∃ i t, IsConnected t ∧ s = Sigma.mk i '' t 
Arguments: {ι : Type u_1} {π : ι → Type u_2} [(i : ι) → TopologicalSpace (π i)] {s : Set ((i : ι) × π i)}
Goal: IsConnected s ↔ ∃ i t, IsConnected t ∧ s = Sigma.mk i '' t
-- Sigma.isPreconnected_iff
theorem Sigma.isPreconnected_iff [hι : Nonempty ι] [∀ i, TopologicalSpace (π i)]
    {s : Set (Σi, π i)} : IsPreconnected s ↔ ∃ i t, IsPreconnected t ∧ s = Sigma.mk i '' t 
Arguments: {ι : Type u_1} {π : ι → Type u_2} [hι : Nonempty ι] [(i : ι) → TopologicalSpace (π i)] {s : Set ((i : ι) × π i)}
Goal: IsPreconnected s ↔ ∃ i t, IsPreconnected t ∧ s = Sigma.mk i '' t
-- Sum.isConnected_iff
theorem Sum.isConnected_iff [TopologicalSpace β] {s : Set (α ⊕ β)} :
    IsConnected s ↔
      (∃ t, IsConnected t ∧ s = Sum.inl '' t) ∨ ∃ t, IsConnected t ∧ s = Sum.inr '' t 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {s : Set (α ⊕ β)}
Goal: IsConnected s ↔ (∃ t, IsConnected t ∧ s = Sum.inl '' t) ∨ ∃ t, IsConnected t ∧ s = Sum.inr '' t
-- Sum.isPreconnected_iff
theorem Sum.isPreconnected_iff [TopologicalSpace β] {s : Set (α ⊕ β)} :
    IsPreconnected s ↔
      (∃ t, IsPreconnected t ∧ s = Sum.inl '' t) ∨ ∃ t, IsPreconnected t ∧ s = Sum.inr '' t 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {s : Set (α ⊕ β)}
Goal: IsPreconnected s ↔ (∃ t, IsPreconnected t ∧ s = Sum.inl '' t) ∨ ∃ t, IsPreconnected t ∧ s = Sum.inr '' t
-- Continuous.exists_lift_sigma
theorem Continuous.exists_lift_sigma [ConnectedSpace α] [∀ i, TopologicalSpace (π i)]
    {f : α → Σ i, π i} (hf : Continuous f) :
    ∃ (i : ι) (g : α → π i), Continuous g ∧ f = Sigma.mk i ∘ g 
Arguments: {α : Type u} {ι : Type u_1} {π : ι → Type u_2} [TopologicalSpace α] [ConnectedSpace α] [(i : ι) → TopologicalSpace (π i)] {f : α → (i : ι) × π i} (hf : Continuous f)
Goal: ∃ i g, Continuous g ∧ f = Sigma.mk i ∘ g
-- nonempty_inter
theorem nonempty_inter [PreconnectedSpace α] {s t : Set α} :
    IsOpen s → IsOpen t → s ∪ t = univ → s.Nonempty → t.Nonempty → (s ∩ t).Nonempty 
Arguments: {α : Type u} [TopologicalSpace α] [PreconnectedSpace α] {s t : Set α}
Goal: IsOpen s → IsOpen t → s ∪ t = Set.univ → s.Nonempty → t.Nonempty → (s ∩ t).Nonempty
-- isClopen_iff
theorem isClopen_iff [PreconnectedSpace α] {s : Set α} : IsClopen s ↔ s = ∅ ∨ s = univ 
Arguments: {α : Type u} [TopologicalSpace α] [PreconnectedSpace α] {s : Set α}
Goal: IsClopen s ↔ s = ∅ ∨ s = Set.univ
-- IsClopen.eq_univ
theorem IsClopen.eq_univ [PreconnectedSpace α] {s : Set α} (h' : IsClopen s) (h : s.Nonempty) :
    s = univ 
Arguments: {α : Type u} [TopologicalSpace α] [PreconnectedSpace α] {s : Set α} (h' : IsClopen s) (h : s.Nonempty)
Goal: s = Set.univ
-- subsingleton_of_disjoint_isClopen
lemma subsingleton_of_disjoint_isClopen
    (h_clopen : ∀ i, IsClopen (s i)) :
    Subsingleton ι 
Arguments: {α : Type u} {ι : Type u_1} [TopologicalSpace α] [PreconnectedSpace α] {s : ι → Set α} (h_nonempty : ∀ (i : ι), (s i).Nonempty) (h_disj : Pairwise (Disjoint on s)) (h_clopen : ∀ (i : ι), IsClopen (s i))
Goal: Subsingleton ι
-- subsingleton_of_disjoint_isOpen_iUnion_eq_univ
lemma subsingleton_of_disjoint_isOpen_iUnion_eq_univ
    (h_open : ∀ i, IsOpen (s i)) (h_Union : ⋃ i, s i = univ) :
    Subsingleton ι 
Arguments: {α : Type u} {ι : Type u_1} [TopologicalSpace α] [PreconnectedSpace α] {s : ι → Set α} (h_nonempty : ∀ (i : ι), (s i).Nonempty) (h_disj : Pairwise (Disjoint on s)) (h_open : ∀ (i : ι), IsOpen (s i)) (h_Union : ⋃ i, s i = Set.univ)
Goal: Subsingleton ι
-- subsingleton_of_disjoint_isClosed_iUnion_eq_univ
lemma subsingleton_of_disjoint_isClosed_iUnion_eq_univ [Finite ι]
    (h_closed : ∀ i, IsClosed (s i)) (h_Union : ⋃ i, s i = univ) :
    Subsingleton ι 
Arguments: {α : Type u} {ι : Type u_1} [TopologicalSpace α] [PreconnectedSpace α] {s : ι → Set α} (h_nonempty : ∀ (i : ι), (s i).Nonempty) (h_disj : Pairwise (Disjoint on s)) [Finite ι] (h_closed : ∀ (i : ι), IsClosed (s i)) (h_Union : ⋃ i, s i = Set.univ)
Goal: Subsingleton ι
-- frontier_eq_empty_iff
theorem frontier_eq_empty_iff [PreconnectedSpace α] {s : Set α} :
    frontier s = ∅ ↔ s = ∅ ∨ s = univ 
Arguments: {α : Type u} [TopologicalSpace α] [PreconnectedSpace α] {s : Set α}
Goal: frontier s = ∅ ↔ s = ∅ ∨ s = Set.univ
-- nonempty_frontier_iff
theorem nonempty_frontier_iff [PreconnectedSpace α] {s : Set α} :
    (frontier s).Nonempty ↔ s.Nonempty ∧ s ≠ univ 
Arguments: {α : Type u} [TopologicalSpace α] [PreconnectedSpace α] {s : Set α}
Goal: (frontier s).Nonempty ↔ s.Nonempty ∧ s ≠ Set.univ
-- PreconnectedSpace.induction₂'
lemma PreconnectedSpace.induction₂' [PreconnectedSpace α] (P : α → α → Prop)
    (h : ∀ x, ∀ᶠ y in 𝓝 x, P x y ∧ P y x) (h' : Transitive P) (x y : α) :
    P x y 
Arguments: {α : Type u} [TopologicalSpace α] [PreconnectedSpace α] (P : α → α → Prop) (h : ∀ (x : α), ∀ᶠ (y : α) in 𝓝 x, P x y ∧ P y x) (h' : Transitive P) (x y : α)
Goal: P x y
-- PreconnectedSpace.induction₂
lemma PreconnectedSpace.induction₂ [PreconnectedSpace α] (P : α → α → Prop)
    (h : ∀ x, ∀ᶠ y in 𝓝 x, P x y) (h' : Transitive P) (h'' : Symmetric P) (x y : α) :
    P x y 
Arguments: {α : Type u} [TopologicalSpace α] [PreconnectedSpace α] (P : α → α → Prop) (h : ∀ (x : α), ∀ᶠ (y : α) in 𝓝 x, P x y) (h' : Transitive P) (h'' : Symmetric P) (x y : α)
Goal: P x y
-- IsPreconnected.induction₂'
lemma IsPreconnected.induction₂' {s : Set α} (hs : IsPreconnected s) (P : α → α → Prop)
    (h : ∀ x ∈ s, ∀ᶠ y in 𝓝[s] x, P x y ∧ P y x)
    (h' : ∀ x y z, x ∈ s → y ∈ s → z ∈ s → P x y → P y z → P x z)
    {x y : α} (hx : x ∈ s) (hy : y ∈ s) : P x y 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α} (hs : IsPreconnected s) (P : α → α → Prop) (h : ∀ x ∈ s, ∀ᶠ (y : α) in 𝓝[s] x, P x y ∧ P y x) (h' : ∀ (x y z : α), x ∈ s → y ∈ s → z ∈ s → P x y → P y z → P x z) {x y : α} (hx : x ∈ s) (hy : y ∈ s)
Goal: P x y
-- IsPreconnected.induction₂
lemma IsPreconnected.induction₂ {s : Set α} (hs : IsPreconnected s) (P : α → α → Prop)
    (h : ∀ x ∈ s, ∀ᶠ y in 𝓝[s] x, P x y)
    (h' : ∀ x y z, x ∈ s → y ∈ s → z ∈ s → P x y → P y z → P x z)
    (h'' : ∀ x y, x ∈ s → y ∈ s → P x y → P y x)
    {x y : α} (hx : x ∈ s) (hy : y ∈ s) : P x y 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α} (hs : IsPreconnected s) (P : α → α → Prop) (h : ∀ x ∈ s, ∀ᶠ (y : α) in 𝓝[s] x, P x y) (h' : ∀ (x y z : α), x ∈ s → y ∈ s → z ∈ s → P x y → P y z → P x z) (h'' : ∀ (x y : α), x ∈ s → y ∈ s → P x y → P y x) {x y : α} (hx : x ∈ s) (hy : y ∈ s)
Goal: P x y
-- isPreconnected_iff_subset_of_disjoint
theorem isPreconnected_iff_subset_of_disjoint {s : Set α} :
    IsPreconnected s ↔
      ∀ u v, IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α}
Goal: IsPreconnected s ↔ ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
-- isConnected_iff_sUnion_disjoint_open
theorem isConnected_iff_sUnion_disjoint_open {s : Set α} :
    IsConnected s ↔
      ∀ U : Finset (Set α), (∀ u v : Set α, u ∈ U → v ∈ U → (s ∩ (u ∩ v)).Nonempty → u = v) →
        (∀ u ∈ U, IsOpen u) → (s ⊆ ⋃₀ ↑U) → ∃ u ∈ U, s ⊆ u 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α}
Goal: IsConnected s ↔
  ∀ (U : Finset (Set α)),
    (∀ (u v : Set α), u ∈ U → v ∈ U → (s ∩ (u ∩ v)).Nonempty → u = v) → (∀ u ∈ U, IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u ∈ U, s ⊆ u
-- disjoint_or_subset_of_isClopen
theorem disjoint_or_subset_of_isClopen {s t : Set α} (hs : IsPreconnected s) (ht : IsClopen t) :
    Disjoint s t ∨ s ⊆ t 
Arguments: {α : Type u} [TopologicalSpace α] {s t : Set α} (hs : IsPreconnected s) (ht : IsClopen t)
Goal: Disjoint s t ∨ s ⊆ t
-- isPreconnected_iff_subset_of_disjoint_closed
theorem isPreconnected_iff_subset_of_disjoint_closed :
    IsPreconnected s ↔
      ∀ u v, IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α}
Goal: IsPreconnected s ↔ ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
-- isPreconnected_iff_subset_of_fully_disjoint_closed
theorem isPreconnected_iff_subset_of_fully_disjoint_closed {s : Set α} (hs : IsClosed s) :
    IsPreconnected s ↔
      ∀ u v, IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α} (hs : IsClosed s)
Goal: IsPreconnected s ↔ ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
-- IsClopen.connectedComponent_subset
theorem IsClopen.connectedComponent_subset {x} (hs : IsClopen s) (hx : x ∈ s) :
    connectedComponent x ⊆ s 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α} {x : α} (hs : IsClopen s) (hx : x ∈ s)
Goal: connectedComponent x ⊆ s
-- connectedComponent_subset_iInter_isClopen
theorem connectedComponent_subset_iInter_isClopen {x : α} :
    connectedComponent x ⊆ ⋂ Z : { Z : Set α // IsClopen Z ∧ x ∈ Z }, Z 
Arguments: {α : Type u} [TopologicalSpace α] {x : α}
Goal: connectedComponent x ⊆ ⋂ Z, ↑Z
-- IsClopen.biUnion_connectedComponent_eq
theorem IsClopen.biUnion_connectedComponent_eq {Z : Set α} (h : IsClopen Z) :
    ⋃ x ∈ Z, connectedComponent x = Z 
Arguments: {α : Type u} [TopologicalSpace α] {Z : Set α} (h : IsClopen Z)
Goal: ⋃ x ∈ Z, connectedComponent x = Z
-- preimage_connectedComponent_connected
theorem preimage_connectedComponent_connected [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) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [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)
Goal: IsConnected (f ⁻¹' connectedComponent t)
-- QuotientMap.preimage_connectedComponent
theorem QuotientMap.preimage_connectedComponent [TopologicalSpace β] {f : α → β}
    (hf : QuotientMap f) (h_fibers : ∀ y : β, IsConnected (f ⁻¹' {y})) (a : α) :
    f ⁻¹' connectedComponent (f a) = connectedComponent a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : QuotientMap f) (h_fibers : ∀ (y : β), IsConnected (f ⁻¹' {y})) (a : α)
Goal: f ⁻¹' connectedComponent (f a) = connectedComponent a
-- QuotientMap.image_connectedComponent
theorem QuotientMap.image_connectedComponent [TopologicalSpace β] {f : α → β} (hf : QuotientMap f)
    (h_fibers : ∀ y : β, IsConnected (f ⁻¹' {y})) (a : α) :
    f '' connectedComponent a = connectedComponent (f a) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : QuotientMap f) (h_fibers : ∀ (y : β), IsConnected (f ⁻¹' {y})) (a : α)
Goal: f '' connectedComponent a = connectedComponent (f a)
-- ConnectedComponents.coe_eq_coe
@[simp]
theorem coe_eq_coe {x y : α} :
    (x : ConnectedComponents α) = y ↔ connectedComponent x = connectedComponent y 
Arguments: {α : Type u} [TopologicalSpace α] {x y : α}
Goal: ConnectedComponents.mk x = ConnectedComponents.mk y ↔ connectedComponent x = connectedComponent y
-- ConnectedComponents.coe_ne_coe
theorem coe_ne_coe {x y : α} :
    (x : ConnectedComponents α) ≠ y ↔ connectedComponent x ≠ connectedComponent y 
Arguments: {α : Type u} [TopologicalSpace α] {x y : α}
Goal: ConnectedComponents.mk x ≠ ConnectedComponents.mk y ↔ connectedComponent x ≠ connectedComponent y
-- ConnectedComponents.coe_eq_coe'
theorem coe_eq_coe' {x y : α} : (x : ConnectedComponents α) = y ↔ x ∈ connectedComponent y 
Arguments: {α : Type u} [TopologicalSpace α] {x y : α}
Goal: ConnectedComponents.mk x = ConnectedComponents.mk y ↔ x ∈ connectedComponent y
-- ConnectedComponents.surjective_coe
theorem surjective_coe : Surjective (mk : α → ConnectedComponents α) 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: Function.Surjective ConnectedComponents.mk
-- ConnectedComponents.quotientMap_coe
theorem quotientMap_coe : QuotientMap (mk : α → ConnectedComponents α) 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: QuotientMap ConnectedComponents.mk
-- ConnectedComponents.continuous_coe
@[continuity]
theorem continuous_coe : Continuous (mk : α → ConnectedComponents α) 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: Continuous ConnectedComponents.mk
-- ConnectedComponents.range_coe
@[simp]
theorem range_coe : range (mk : α → ConnectedComponents α) = univ 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: Set.range ConnectedComponents.mk = Set.univ
-- connectedComponents_preimage_singleton
theorem connectedComponents_preimage_singleton {x : α} :
    (↑) ⁻¹' ({↑x} : Set (ConnectedComponents α)) = connectedComponent x 
Arguments: {α : Type u} [TopologicalSpace α] {x : α}
Goal: ConnectedComponents.mk ⁻¹' {ConnectedComponents.mk x} = connectedComponent x
-- connectedComponents_preimage_image
theorem connectedComponents_preimage_image (U : Set α) :
    (↑) ⁻¹' ((↑) '' U : Set (ConnectedComponents α)) = ⋃ x ∈ U, connectedComponent x 
Arguments: {α : Type u} [TopologicalSpace α] (U : Set α)
Goal: ConnectedComponents.mk ⁻¹' (ConnectedComponents.mk '' U) = ⋃ x ∈ U, connectedComponent x
-- isPreconnected_of_forall_constant
theorem isPreconnected_of_forall_constant {s : Set α}
    (hs : ∀ f : α → Bool, ContinuousOn f s → ∀ x ∈ s, ∀ y ∈ s, f x = f y) : IsPreconnected s 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α} (hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ x ∈ s, ∀ y ∈ s, f x = f y)
Goal: IsPreconnected s
-- preconnectedSpace_of_forall_constant
theorem preconnectedSpace_of_forall_constant
    (hs : ∀ f : α → Bool, Continuous f → ∀ x y, f x = f y) : PreconnectedSpace α 
Arguments: {α : Type u} [TopologicalSpace α] (hs : ∀ (f : α → Bool), Continuous f → ∀ (x y : α), f x = f y)
Goal: PreconnectedSpace α
-- ContMDiffMap.contMDiff
protected theorem contMDiff (f : C^n⟮I, M; I', M'⟯) : ContMDiff I I' n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {H' : Type u_5} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (f : C^n⟮I, M; I', M'⟯)
Goal: ContMDiff I I' n ⇑f
-- ContMDiffMap.smooth
protected theorem smooth (f : C^∞⟮I, M; I', M'⟯) : Smooth I I' f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {H' : Type u_5} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] (f : C^⊤⟮I, M; I', M'⟯)
Goal: Smooth I I' ⇑f
-- ContMDiffMap.coeFn_mk
@[simp]
theorem coeFn_mk (f : M → M') (hf : ContMDiff I I' n f) :
    DFunLike.coe (F := C^n⟮I, M; I', M'⟯) ⟨f, hf⟩ = f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {H' : Type u_5} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (f : M → M') (hf : ContMDiff I I' n f)
Goal: ⇑⟨f, hf⟩ = f
-- ContMDiffMap.coe_injective
theorem coe_injective ⦃f g : C^n⟮I, M; I', M'⟯⦄ (h : (f : M → M') = g) : f = g 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {H' : Type u_5} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} ⦃f g : C^n⟮I, M; I', M'⟯⦄ (h : ⇑f = ⇑g)
Goal: f = g
-- ContMDiffMap.ext
@[ext]
theorem ext (h : ∀ x, f x = g x) : f = g 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {H' : Type u_5} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} {f g : C^n⟮I, M; I', M'⟯} (h : ∀ (x : M), f x = g x)
Goal: f = g
-- ContMDiffMap.comp_apply
@[simp]
theorem comp_apply (f : C^n⟮I', M'; I'', M''⟯) (g : C^n⟮I, M; I', M'⟯) (x : M) :
    f.comp g x = f (g x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {H' : Type u_5} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {n : ℕ∞} (f : C^n⟮I', M'; I'', M''⟯) (g : C^n⟮I, M; I', M'⟯) (x : M)
Goal: (f.comp g) x = f (g x)