mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedRing A
inst✝² : StarRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p
a : A
ha : p a
this : CompactSpace ↑(σₙ 𝕜 a)
⊢ Isometry
    ⇑((↑(cfcHom ha)).comp
        ((↑(ContinuousMap.compStarAlgHom' 𝕜 𝕜 { toFun := Set.inclusion ⋯, continuous_toFun := ⋯ })).comp
          ContinuousMapZero.toContinuousMapHom))

Tactic:
]

Goal state after:
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedRing A
inst✝² : StarRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p
a : A
ha : p a
this : CompactSpace ↑(σₙ 𝕜 a)
⊢ Isometry
    ⇑((↑(cfcHom ha)).comp
        ((↑(ContinuousMap.compStarAlgHom' 𝕜 𝕜 { toFun := Set.inclusion ⋯, continuous_toFun := ⋯ })).comp
          ContinuousMapZero.toContinuousMapHom))
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
U : Set (α × β)
hU : U ∈ {V | IsOpen[instTopologicalSpaceProd] V}
⊢ ∀ x ∈ (SetLike.coe ∘ fun p ↦ p.1 ×ˢ p.2) ⁻¹' (fun V ↦ {s | (s ∩ V).Nonempty}) U,
    ∃ t ⊆ (SetLike.coe ∘ fun p ↦ p.1 ×ˢ p.2) ⁻¹' (fun V ↦ {s | (s ∩ V).Nonempty}) U, IsOpen t ∧ x ∈ t

Tactic:
]

Goal state after:
case inr
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
U : Set (α × β)
hU : U ∈ {V | IsOpen[instTopologicalSpaceProd] V}
⊢ ∀ x ∈ (SetLike.coe ∘ fun p ↦ p.1 ×ˢ p.2) ⁻¹' (fun V ↦ {s | (s ∩ V).Nonempty}) U,
    ∃ t ⊆ (SetLike.coe ∘ fun p ↦ p.1 ×ˢ p.2) ⁻¹' (fun V ↦ {s | (s ∩ V).Nonempty}) U, IsOpen t ∧ x ∈ t
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
S : StarSubalgebra ℂ A
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
a : ↥S
ha : IsUnit ↑a
ha₁ : IsUnit (a⋆ * a)
ha₂ : IsUnit (a * a⋆)
spec_eq : ∀ {x : ↥S}, IsSelfAdjoint x → σ ℂ x = σ ℂ ↑x
⊢ IsUnit a

pp: A : Type u_1
inst✝ : CStarAlgebra A
S : StarSubalgebra ℂ A
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
a : ↥S
ha : IsUnit ↑a
ha₁ : 0 ∉ σ ℂ (a⋆ * a)
ha₂ : 0 ∉ σ ℂ ↑(a * a⋆)
spec_eq : ∀ {x : ↥S}, IsSelfAdjoint x → σ ℂ x = σ ℂ ↑x
⊢ IsSelfAdjoint (a * a⋆)

pp: A : Type u_1
inst✝ : CStarAlgebra A
S : StarSubalgebra ℂ A
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
a : ↥S
ha : IsUnit ↑a
ha₁ : 0 ∉ σ ℂ ↑(a⋆ * a)
ha₂ : 0 ∉ σ ℂ ↑(a * a⋆)
spec_eq : ∀ {x : ↥S}, IsSelfAdjoint x → σ ℂ x = σ ℂ ↑x
⊢ IsSelfAdjoint (a⋆ * a)

Tactic:
]

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
S : StarSubalgebra ℂ A
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
a : ↥S
ha : IsUnit ↑a
ha₁ : IsUnit (a⋆ * a)
ha₂ : IsUnit (a * a⋆)
spec_eq : ∀ {x : ↥S}, IsSelfAdjoint x → σ ℂ x = σ ℂ ↑x
⊢ IsUnit a

A : Type u_1
inst✝ : CStarAlgebra A
S : StarSubalgebra ℂ A
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
a : ↥S
ha : IsUnit ↑a
ha₁ : 0 ∉ σ ℂ (a⋆ * a)
ha₂ : 0 ∉ σ ℂ ↑(a * a⋆)
spec_eq : ∀ {x : ↥S}, IsSelfAdjoint x → σ ℂ x = σ ℂ ↑x
⊢ IsSelfAdjoint (a * a⋆)

A : Type u_1
inst✝ : CStarAlgebra A
S : StarSubalgebra ℂ A
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
a : ↥S
ha : IsUnit ↑a
ha₁ : 0 ∉ σ ℂ ↑(a⋆ * a)
ha₂ : 0 ∉ σ ℂ ↑(a * a⋆)
spec_eq : ∀ {x : ↥S}, IsSelfAdjoint x → σ ℂ x = σ ℂ ↑x
⊢ IsSelfAdjoint (a⋆ * a)
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
U : Set (α × β)
hU : U ∈ {V | IsOpen[instTopologicalSpaceProd] V}
K : Compacts α
L : Compacts β
x : α
y : β
hx : x ∈ K
hy : y ∈ L
hxy : (x, y) ∈ U
V : Set α
W : Set β
hV : IsOpen[inst✝¹] V
hW : IsOpen[inst✝] W
hxV : x ∈ V
hyW : y ∈ W
hVW : V ×ˢ W ⊆ U
⊢ ∃ t ⊆ {a | ((SetLike.coe ∘ fun p ↦ p.1 ×ˢ p.2) a ∩ V ×ˢ W).Nonempty}, IsOpen t ∧ (K, L) ∈ t

Tactic:
]

Goal state after:
case inr
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
U : Set (α × β)
hU : U ∈ {V | IsOpen[instTopologicalSpaceProd] V}
K : Compacts α
L : Compacts β
x : α
y : β
hx : x ∈ K
hy : y ∈ L
hxy : (x, y) ∈ U
V : Set α
W : Set β
hV : IsOpen[inst✝¹] V
hW : IsOpen[inst✝] W
hxV : x ∈ V
hyW : y ∈ W
hVW : V ×ˢ W ⊆ U
⊢ ∃ t ⊆ {a | ((SetLike.coe ∘ fun p ↦ p.1 ×ˢ p.2) a ∩ V ×ˢ W).Nonempty}, IsOpen t ∧ (K, L) ∈ t
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: F : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : NonUnitalCStarAlgebra B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F ℂ A B
inst✝ : StarHomClass F A B
φ : F
a : A
ψ : Unitization ℂ A →⋆ₐ[ℂ] Unitization ℂ B
x s : Unitization ℂ A
hs : IsSelfAdjoint s
this : ‖ψ s‖₊ ≤ ‖s‖₊
⊢ ‖ψ s‖₊ ≤ ‖s‖₊

Tactic:
]

Goal state after:
F : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : NonUnitalCStarAlgebra B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F ℂ A B
inst✝ : StarHomClass F A B
φ : F
a : A
ψ : Unitization ℂ A →⋆ₐ[ℂ] Unitization ℂ B
x s : Unitization ℂ A
hs : IsSelfAdjoint s
this : ‖ψ s‖₊ ≤ ‖s‖₊
⊢ ‖ψ s‖₊ ≤ ‖s‖₊
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : DiscreteTopology α
⊢ ∀ (a : Compacts α), IsOpen {a}

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : DiscreteTopology α
⊢ ∀ (a : Compacts α), IsOpen {a}
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: F : Type u_1
A : Type u_2
inst✝¹ : CStarAlgebra A
inst✝ : FunLike F A ℂ
hF : AlgHomClass F ℂ A ℂ
φ : F
a : A
hsa : ∀ (s : ↥(selfAdjoint A)), (φ ↑s)⋆ = φ ↑s
⊢ φ (↑(ℜ a) + I • ↑(ℑ a))⋆ = (φ (↑(ℜ a) + I • ↑(ℑ a)))⋆

Tactic:
]

Goal state after:
F : Type u_1
A : Type u_2
inst✝¹ : CStarAlgebra A
inst✝ : FunLike F A ℂ
hF : AlgHomClass F ℂ A ℂ
φ : F
a : A
hsa : ∀ (s : ↥(selfAdjoint A)), (φ ↑s)⋆ = φ ↑s
⊢ φ (↑(ℜ a) + I • ↑(ℑ a))⋆ = (φ (↑(ℜ a) + I • ↑(ℑ a)))⋆
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: F : Type u_1
A : Type u_2
inst✝¹ : CStarAlgebra A
inst✝ : FunLike F A ℂ
hF : AlgHomClass F ℂ A ℂ
φ : F
a : A
s : ↥(selfAdjoint A)
⊢ (↑(φ ↑s).re)⋆ = ↑(φ ↑s).re

Tactic:
]

Goal state after:
F : Type u_1
A : Type u_2
inst✝¹ : CStarAlgebra A
inst✝ : FunLike F A ℂ
hF : AlgHomClass F ℂ A ℂ
φ : F
a : A
s : ↥(selfAdjoint A)
⊢ (↑(φ ↑s).re)⋆ = ↑(φ ↑s).re
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : T2Space α
K₁ K₂ : Compacts α
h : K₁ ≠ K₂
h' : ∃ x ∈ K₁, x ∉ K₂
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ K₁ ∈ u ∧ K₂ ∈ v ∧ Disjoint u v

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : T2Space α
K₁ K₂ : Compacts α
h : K₁ ≠ K₂
h' : ∃ x ∈ K₁, x ∉ K₂
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ K₁ ∈ u ∧ K₂ ∈ v ∧ Disjoint u v
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : RegularSpace α
U : Set α
hU : U ∈ {U | IsOpen[inst✝³] U}
K : Compacts α
hK : K ∈ (fun a ↦ {a_1 | ↑a_1 ⊆ a}) U
V W : Set α
hV : IsOpen[inst✝³] V
hW : IsOpen[inst✝³] W
hKV : ↑K ⊆ V
hUW : Uᶜ ⊆ W
hVW : Disjoint V W
⊢ {a | (↑a ∩ W).Nonempty} ⊆ {K | (↑K ∩ W).Nonempty}

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr.refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : RegularSpace α
U : Set α
hU : U ∈ {U | IsOpen[inst✝³] U}
K : Compacts α
x : α
hx₁ : x ∈ ↑K
hx₂ : x ∈ U
V W : Set α
hV : IsOpen[inst✝³] V
hW : IsOpen[inst✝³] W
hxV : {x} ⊆ V
hUW : Uᶜ ⊆ W
hVW : Disjoint V W
⊢ K ∈ {K | (↑K ∩ V).Nonempty}

Tactic:
]

Goal state after:
case inr.refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : RegularSpace α
U : Set α
hU : U ∈ {U | IsOpen[inst✝³] U}
K : Compacts α
x : α
hx₁ : x ∈ ↑K
hx₂ : x ∈ U
V W : Set α
hV : IsOpen[inst✝³] V
hW : IsOpen[inst✝³] W
hxV : {x} ⊆ V
hUW : Uᶜ ⊆ W
hVW : Disjoint V W
⊢ K ∈ {K | (↑K ∩ V).Nonempty}
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
⊢ IsCompact (SetLike.coe '' {L | ↑L ⊆ K})

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
⊢ IsCompact (SetLike.coe '' {L | ↑L ⊆ K})
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
S : Set (Compacts α)
hS : IsCompact S
⊢ ∀ {ι : Type u_1} (U : ι → Set α),
    (∀ (i : ι), IsOpen[inst✝] (U i)) → ⋃ K ∈ S, ↑K ⊆ ⋃ i, U i → ∃ t, ⋃ K ∈ S, ↑K ⊆ ⋃ i ∈ t, U i

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
S : Set (Compacts α)
hS : IsCompact S
⊢ ∀ {ι : Type u_1} (U : ι → Set α),
    (∀ (i : ι), IsOpen[inst✝] (U i)) → ⋃ K ∈ S, ↑K ⊆ ⋃ i, U i → ∃ t, ⋃ K ∈ S, ↑K ⊆ ⋃ i ∈ t, U i
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
S : Set (Compacts α)
hS : IsCompact S
ι : Type u_1
U : ι → Set α
hU : ∀ (i : ι), IsOpen[inst✝] (U i)
h : ∀ i ∈ S, ↑i ⊆ ⋃ i, U i
⊢ ∃ t, ⋃ K ∈ S, ↑K ⊆ ⋃ i ∈ t, U i

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
S : Set (Compacts α)
hS : IsCompact S
ι : Type u_1
U : ι → Set α
hU : ∀ (i : ι), IsOpen[inst✝] (U i)
h : ∀ i ∈ S, ↑i ⊆ ⋃ i, U i
⊢ ∃ t, ⋃ K ∈ S, ↑K ⊆ ⋃ i ∈ t, U i
Module: Mathlib.Analysis.CStarAlgebra.Exponential

Goal state before:
pp: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra ℂ A
inst✝³ : StarRing A
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : StarModule ℂ A
a b : ↥(selfAdjoint A)
h : Commute ↑a ↑b
⊢ selfAdjoint.expUnitary (b + a) = selfAdjoint.expUnitary (b + a)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyCompactSpace α
K : Compacts α
U : Set (Compacts α)
hU :
  ∃ x ∈ {u | u.Finite ∧ ∀ U ∈ u, IsOpen[inst✝³] U},
    K ∈ {K | ↑K ⊆ ⋃₀ x ∧ ∀ U ∈ x, (↑K ∩ U).Nonempty} ∧ {K | ↑K ⊆ ⋃₀ x ∧ ∀ U ∈ x, (↑K ∩ U).Nonempty} ⊆ U
⊢ ∃ s ∈ 𝓝 K, s ⊆ U ∧ IsCompact s

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyCompactSpace α
K : Compacts α
U : Set (Compacts α)
hU :
  ∃ x ∈ {u | u.Finite ∧ ∀ U ∈ u, IsOpen[inst✝³] U},
    K ∈ {K | ↑K ⊆ ⋃₀ x ∧ ∀ U ∈ x, (↑K ∩ U).Nonempty} ∧ {K | ↑K ⊆ ⋃₀ x ∧ ∀ U ∈ x, (↑K ∩ U).Nonempty} ⊆ U
⊢ ∃ s ∈ 𝓝 K, s ⊆ U ∧ IsCompact s
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyCompactSpace α
K : Compacts α
U : Set (Compacts α)
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝³] U
huU : {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty} ⊆ U
hKu₁ : ↑K ⊆ ⋃₀ u
hKu₂ : ∀ U ∈ u, (↑K ∩ U).Nonempty
⊢ ∃ s ∈ 𝓝 K, s ⊆ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty} ∧ IsCompact s

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyCompactSpace α
K : Compacts α
U : Set (Compacts α)
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝³] U
huU : {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty} ⊆ U
hKu₁ : ↑K ⊆ ⋃₀ u
hKu₂ : ∀ U ∈ u, (↑K ∩ U).Nonempty
⊢ ∃ s ∈ 𝓝 K, s ⊆ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty} ∧ IsCompact s
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyCompactSpace α
K : Compacts α
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝³] U
hKu₁ : ↑K ⊆ ⋃₀ u
hKu₂ : ∀ U ∈ u, (↑K ∩ U).Nonempty
L : Set α
hL : IsCompact L
hLK : ↑K ⊆ interior L
hLu : L ⊆ ⋃₀ u
M : Set α → Set α
hM : ∀ U ∈ u, IsCompact (M U)
hML : ∀ U ∈ u, M U ⊆ L
hMU : ∀ U ∈ u, M U ⊆ U
hMK : ∀ U ∈ u, (↑K ∩ interior (M U)).Nonempty
⊢ IsCompact (SetLike.coe '' {K' | ↑K' ⊆ L ∧ ∀ U ∈ u, (↑K' ∩ M U).Nonempty})

Tactic:
]

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyCompactSpace α
K : Compacts α
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝³] U
hKu₁ : ↑K ⊆ ⋃₀ u
hKu₂ : ∀ U ∈ u, (↑K ∩ U).Nonempty
L : Set α
hL : IsCompact L
hLK : ↑K ⊆ interior L
hLu : L ⊆ ⋃₀ u
M : Set α → Set α
hM : ∀ U ∈ u, IsCompact (M U)
hML : ∀ U ∈ u, M U ⊆ L
hMU : ∀ U ∈ u, M U ⊆ U
hMK : ∀ U ∈ u, (↑K ∩ interior (M U)).Nonempty
⊢ IsCompact (SetLike.coe '' {K' | ↑K' ⊆ L ∧ ∀ U ∈ u, (↑K' ∩ M U).Nonempty})
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyCompactSpace α
K : Compacts α
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝³] U
hKu₁ : ↑K ⊆ ⋃₀ u
hKu₂ : ∀ U ∈ u, (↑K ∩ U).Nonempty
L : Set α
hL : IsCompact L
hLK : ↑K ⊆ interior L
hLu : L ⊆ ⋃₀ u
M : Set α → Set α
hM : ∀ U ∈ u, IsCompact (M U)
hML : ∀ U ∈ u, M U ⊆ L
hMU : ∀ U ∈ u, M U ⊆ U
hMK : ∀ U ∈ u, (↑K ∩ interior (M U)).Nonempty
s : Set α
x✝ : s ∈ {t | t ⊆ L ∧ ∀ L ∈ M '' u, (t ∩ L).Nonempty}
hsL : s ⊆ L
hsu : ∀ ⦃x : Set α⦄, x ∈ u → (s ∩ M x).Nonempty
⊢ ∃ y ∈ SetLike.coe '' {K' | ↑K' ⊆ L ∧ ∀ U ∈ u, (↑K' ∩ M U).Nonempty}, s ⤳ y

Tactic:
]

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyCompactSpace α
K : Compacts α
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝³] U
hKu₁ : ↑K ⊆ ⋃₀ u
hKu₂ : ∀ U ∈ u, (↑K ∩ U).Nonempty
L : Set α
hL : IsCompact L
hLK : ↑K ⊆ interior L
hLu : L ⊆ ⋃₀ u
M : Set α → Set α
hM : ∀ U ∈ u, IsCompact (M U)
hML : ∀ U ∈ u, M U ⊆ L
hMU : ∀ U ∈ u, M U ⊆ U
hMK : ∀ U ∈ u, (↑K ∩ interior (M U)).Nonempty
s : Set α
x✝ : s ∈ {t | t ⊆ L ∧ ∀ L ∈ M '' u, (t ∩ L).Nonempty}
hsL : s ⊆ L
hsu : ∀ ⦃x : Set α⦄, x ∈ u → (s ∩ M x).Nonempty
⊢ ∃ y ∈ SetLike.coe '' {K' | ↑K' ⊆ L ∧ ∀ U ∈ u, (↑K' ∩ M U).Nonempty}, s ⤳ y
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
hs : IsPreconnected s
⊢ IsPreconnected (SetLike.coe '' {K | (↑K).Nonempty ∧ (↑K).Finite ∧ ↑K ⊆ s})

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
hs : IsPreconnected s
⊢ IsPreconnected (SetLike.coe '' {K | (↑K).Nonempty ∧ (↑K).Finite ∧ ↑K ⊆ s})
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
hs : IsPreconnected s
⊢ {a | (↑a).Nonempty} ∩ {a | ↑a ⊆ s} ⊆ closure ({a | (↑a).Nonempty} ∩ {a | (↑a).Finite ∧ ↑a ⊆ s})

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
hs : IsPreconnected s
⊢ {a | (↑a).Nonempty} ∩ {a | ↑a ⊆ s} ⊆ closure ({a | (↑a).Nonempty} ∩ {a | (↑a).Finite ∧ ↑a ⊆ s})
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
hs : IsPreconnected s
⊢ {⊥}ᶜ ∩ {a | ↑a ⊆ s} ⊆ {⊥}ᶜ ∩ {K | ↑K ⊆ s}

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyConnectedSpace α
⊢ IsTopologicalBasis {s | IsOpen s ∧ IsPreconnected s}

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyConnectedSpace α
⊢ IsTopologicalBasis {s | IsOpen s ∧ IsPreconnected s}
Module: Mathlib.Topology.ContinuousMap.ZeroAtInfty

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
inst✝² : Zero β
inst✝¹ : FunLike F α β
inst✝ : ZeroAtInftyContinuousMapClass F α β
f : α →C₀ β
f' : α → β
h : f' = ⇑f
⊢ Continuous[inst✝⁴, inst✝³] ⇑f

Tactic:
]

Goal state after:
F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
inst✝² : Zero β
inst✝¹ : FunLike F α β
inst✝ : ZeroAtInftyContinuousMapClass F α β
f : α →C₀ β
f' : α → β
h : f' = ⇑f
⊢ Continuous[inst✝⁴, inst✝³] ⇑f
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyConnectedSpace α
basis :
  IsTopologicalBasis
    ((fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
      {u | u.Finite ∧ u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}})
U : Set (Compacts α)
hU :
  U ∈
    (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
      {u | u.Finite ∧ u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}}
this : IsPreconnected (U ∩ {K | (↑K).Finite})
⊢ U ⊆ U

Tactic:
]

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ZeroAtInfty

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
x : α
inst✝¹ : MulZeroClass β
inst✝ : ContinuousMul β
f g : α →C₀ β
⊢ Tendsto (↑f * ↑g).toFun (cocompact α) (𝓝 0)

Tactic:
simpa only [mul_zero] using! (zero_at_infty f).mul (zero_at_infty g)

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ZeroAtInfty

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
x : α
R : Type u_2
inst✝⁵ : Semiring R
inst✝⁴ : NonUnitalNonAssocSemiring β
inst✝³ : IsTopologicalSemiring β
inst✝² : Module R β
inst✝¹ : ContinuousConstSMul R β
inst✝ : IsScalarTower R β β
r : R
f g : α →C₀ β
x✝ : α
⊢ r • f x✝ • g x✝ = r • f x✝ • g x✝

Tactic:
]

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ZeroAtInfty

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
x : α
R : Type u_2
inst✝⁵ : Semiring R
inst✝⁴ : NonUnitalNonAssocSemiring β
inst✝³ : IsTopologicalSemiring β
inst✝² : Module R β
inst✝¹ : ContinuousConstSMul R β
inst✝ : SMulCommClass R β β
r : R
f g : α →C₀ β
x✝ : α
⊢ f x✝ • r • g x✝ = f x✝ • r • g x✝

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyConnectedSpace α
basis :
  IsTopologicalBasis
    ((fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
      {u | u.Finite ∧ u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}})
u : Set (Set α)
hu' : u.Finite
hu : u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}
this✝ : Fintype ↑u := hu'.fintype
this :
  {K | ↑K ⊆ ⋃₀ u ∧ (∀ U ∈ u, (↑K ∩ U).Nonempty) ∧ (↑K).Finite} =
    Finset.univ.sup '' univ.pi fun U ↦ {K | (↑K).Nonempty ∧ (↑K).Finite ∧ ↑K ⊆ ↑U}
⊢ IsPreconnected (Finset.univ.sup '' univ.pi fun U ↦ {K | (↑K).Nonempty ∧ (↑K).Finite ∧ ↑K ⊆ ↑U})

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyConnectedSpace α
basis :
  IsTopologicalBasis
    ((fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
      {u | u.Finite ∧ u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}})
u : Set (Set α)
hu' : u.Finite
hu : u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}
this✝ : Fintype ↑u := hu'.fintype
this :
  {K | ↑K ⊆ ⋃₀ u ∧ (∀ U ∈ u, (↑K ∩ U).Nonempty) ∧ (↑K).Finite} =
    Finset.univ.sup '' univ.pi fun U ↦ {K | (↑K).Nonempty ∧ (↑K).Finite ∧ ↑K ⊆ ↑U}
⊢ IsPreconnected (Finset.univ.sup '' univ.pi fun U ↦ {K | (↑K).Nonempty ∧ (↑K).Finite ∧ ↑K ⊆ ↑U})
Module: Mathlib.Analysis.CStarAlgebra.Fuglede

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
a b x : A
inst✝¹ : IsStarNormal a
inst✝ : IsStarNormal b
h : SemiconjBy x a b
z : ℂ
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
⊢ NormedSpace.exp (z • star b) * (NormedSpace.exp (-(star z • b)) * x * NormedSpace.exp (star z • a)) *
      NormedSpace.exp (z • star (-a)) =
    ↑(expUnitary (2 • ℑ (z • star b))) * x * ↑(expUnitary (2 • ℑ (star z • a)))

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
a b x : A
inst✝¹ : IsStarNormal a
inst✝ : IsStarNormal b
h : SemiconjBy x a b
z : ℂ
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
⊢ NormedSpace.exp (z • star b) * (NormedSpace.exp (-(star z • b)) * x * NormedSpace.exp (star z • a)) *
      NormedSpace.exp (z • star (-a)) =
    ↑(expUnitary (2 • ℑ (z • star b))) * x * ↑(expUnitary (2 • ℑ (star z • a)))
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedCommRing 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
s : Set E
⊢ IsClosed[WeakBilin.instTopologicalSpace B.flip] (⋂ x ∈ s, ⇑(B x) ⁻¹' Metric.closedBall 0 1)

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedCommRing 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
s : Set E
⊢ IsClosed[WeakBilin.instTopologicalSpace B.flip] (⋂ x ∈ s, ⇑(B x) ⁻¹' Metric.closedBall 0 1)
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedCommRing 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
a : E
y : F
hy : y ∈ {y | ‖(B a) y‖ ≤ 1}
x✝ : E
hb : x✝ ∈ {a}
⊢ ‖(B a) y‖ ≤ 1

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedCommRing 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
a : E
y : F
hy : y ∈ {y | ‖(B a) y‖ ≤ 1}
x✝ : E
hb : x✝ ∈ {a}
⊢ ‖(B a) y‖ ≤ 1
Module: Mathlib.Topology.ContinuousMap.ZeroAtInfty

Goal state before:
pp: α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : Zero β
f : α →ᵇ β
hf : ∀ U ∈ 𝓝 f, (U ∩ range toBCF).Nonempty
⊢ f ∈ range toBCF

Tactic:
]

Goal state after:
α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : Zero β
f : α →ᵇ β
hf : ∀ U ∈ 𝓝 f, (U ∩ range toBCF).Nonempty
⊢ f ∈ range toBCF
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedCommRing 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
s : Set E
x : E
hx : x ∈ s
y : F
hy : y ∈ B.polar s
⊢ ‖(B x) y‖ ≤ 1

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedCommRing 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
s : Set E
x : E
hx : x ∈ s
y : F
hy : y ∈ B.polar s
⊢ ‖(B x) y‖ ≤ 1
Module: Mathlib.Analysis.CStarAlgebra.Fuglede

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
a b x : A
inst✝¹ : IsStarNormal a
inst✝ : IsStarNormal b
h : SemiconjBy x a b
z✝ : ℂ
hf : Differentiable ℂ (expMulMulExp a b x)
z : ℂ
⊢ ‖x‖ = ‖x‖

Tactic:
]

Goal state after: