mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr
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).Nonempty}) U
⊢ ∃ s ∈ 𝓝ˢ ((fun a ↦ {a_1 | (↑a_1 ∩ a).Nonempty}) U)ᶜ, ∃ t ∈ 𝓝 K, Disjoint s t

Tactic:
·

Goal state after:
case inr
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).Nonempty}) U
⊢ ∃ s ∈ 𝓝ˢ ((fun a ↦ {a_1 | (↑a_1 ∩ a).Nonempty}) U)ᶜ, ∃ t ∈ 𝓝 K, Disjoint s t
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr.refine_1
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 ⊆ W} ∈ 𝓝ˢ ((fun a ↦ {a_1 | (↑a_1 ∩ a).Nonempty}) U)ᶜ

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 ∩ V).Nonempty} ∈ 𝓝 K

Tactic:
·

Goal state after:
case inr.refine_1
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 ⊆ W} ∈ 𝓝ˢ ((fun a ↦ {a_1 | (↑a_1 ∩ a).Nonempty}) U)ᶜ
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 ∩ V).Nonempty} ∈ 𝓝 K

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 ∩ V).Nonempty} ∈ 𝓝 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:
intro ι U hU h

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 : ⋃ K ∈ S, ↑K ⊆ ⋃ i, U i
⊢ ∃ t, ⋃ K ∈ S, ↑K ⊆ ⋃ i ∈ t, U i
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_1
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
⊢ {K' | ↑K' ⊆ L ∧ ∀ U ∈ u, (↑K' ∩ M U).Nonempty} ∈ 𝓝 K

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 {K' | ↑K' ⊆ L ∧ ∀ U ∈ u, (↑K' ∩ M U).Nonempty}

Tactic:
·

Goal state after:
case refine_1
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
⊢ {K' | ↑K' ⊆ L ∧ ∀ U ∈ u, (↑K' ∩ M U).Nonempty} ∈ 𝓝 K
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 {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 {K' | ↑K' ⊆ L ∧ ∀ U ∈ u, (↑K' ∩ M U).Nonempty}
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inl
α : Type u_1
inst✝ : TopologicalSpace α
x✝ : SeparableSpace (Compacts α)
h✝ : IsEmpty α
⊢ SeparableSpace α

pp: case inr
α : Type u_1
inst✝ : TopologicalSpace α
x✝ : SeparableSpace (Compacts α)
h✝ : Nonempty α
⊢ SeparableSpace α

Tactic:
·

Goal state after:
case inl
α : Type u_1
inst✝ : TopologicalSpace α
x✝ : SeparableSpace (Compacts α)
h✝ : IsEmpty α
⊢ SeparableSpace α
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr
α : Type u_1
inst✝ : TopologicalSpace α
K L : Compacts α
hK : K ≠ ⊥
hL : IsPreconnected ↑L
this :
  ∀ {α : Type u_1} [inst : TopologicalSpace α] {K L : Compacts α},
    K ≠ ⊥ → IsPreconnected ↑L → K ≤ L → IsPreconnected (Icc K L)
hKL : ¬K ≤ L
⊢ IsPreconnected (Icc K L)

pp: α✝ : Type u_1
inst✝¹ : TopologicalSpace α✝
α : Type u_1
inst✝ : TopologicalSpace α
K L : Compacts α
hK : K ≠ ⊥
hL : IsPreconnected ↑L
hKL : K ≤ L
⊢ IsPreconnected (Icc K L)

Tactic:
·

Goal state after:
case inr
α : Type u_1
inst✝ : TopologicalSpace α
K L : Compacts α
hK : K ≠ ⊥
hL : IsPreconnected ↑L
this :
  ∀ {α : Type u_1} [inst : TopologicalSpace α] {K L : Compacts α},
    K ≠ ⊥ → IsPreconnected ↑L → K ≤ L → IsPreconnected (Icc K L)
hKL : ¬K ≤ L
⊢ IsPreconnected (Icc K L)
Module: Mathlib.Topology.ContinuousMap.ZeroAtInfty

Goal state before:
pp: case pos
F : Type u_1
α : Type u
β : Type v
inst✝⁴ : TopologicalSpace α
inst✝³ : PseudoMetricSpace β
inst✝² : Zero β
inst✝¹ : FunLike F α β
inst✝ : ZeroAtInftyContinuousMapClass F α β
f : F
K : Set α
hK₁ : IsCompact K
hK₂ : Kᶜ ⊆ ⇑f ⁻¹' closedBall 0 1
C : ℝ
hC : ⇑f '' K ⊆ closedBall 0 C
x✝ y x : α
hx : x ∈ K
⊢ f x ∈ closedBall 0 (max C 1)

pp: case neg
F : Type u_1
α : Type u
β : Type v
inst✝⁴ : TopologicalSpace α
inst✝³ : PseudoMetricSpace β
inst✝² : Zero β
inst✝¹ : FunLike F α β
inst✝ : ZeroAtInftyContinuousMapClass F α β
f : F
K : Set α
hK₁ : IsCompact K
hK₂ : Kᶜ ⊆ ⇑f ⁻¹' closedBall 0 1
C : ℝ
hC : ⇑f '' K ⊆ closedBall 0 C
x✝ y x : α
hx : x ∉ K
⊢ f x ∈ closedBall 0 (max C 1)

Tactic:
·

Goal state after:
case pos
F : Type u_1
α : Type u
β : Type v
inst✝⁴ : TopologicalSpace α
inst✝³ : PseudoMetricSpace β
inst✝² : Zero β
inst✝¹ : FunLike F α β
inst✝ : ZeroAtInftyContinuousMapClass F α β
f : F
K : Set α
hK₁ : IsCompact K
hK₂ : Kᶜ ⊆ ⇑f ⁻¹' closedBall 0 1
C : ℝ
hC : ⇑f '' K ⊆ closedBall 0 C
x✝ y x : α
hx : x ∈ K
⊢ f x ∈ closedBall 0 (max C 1)
Module: Mathlib.Topology.ContinuousMap.ZeroAtInfty

Goal state before:
pp: case neg
F : Type u_1
α : Type u
β : Type v
inst✝⁴ : TopologicalSpace α
inst✝³ : PseudoMetricSpace β
inst✝² : Zero β
inst✝¹ : FunLike F α β
inst✝ : ZeroAtInftyContinuousMapClass F α β
f : F
K : Set α
hK₁ : IsCompact K
hK₂ : Kᶜ ⊆ ⇑f ⁻¹' closedBall 0 1
C : ℝ
hC : ⇑f '' K ⊆ closedBall 0 C
x✝ y x : α
hx : x ∉ K
⊢ f x ∈ closedBall 0 (max C 1)

Tactic:
·

Goal state after:
case neg
F : Type u_1
α : Type u
β : Type v
inst✝⁴ : TopologicalSpace α
inst✝³ : PseudoMetricSpace β
inst✝² : Zero β
inst✝¹ : FunLike F α β
inst✝ : ZeroAtInftyContinuousMapClass F α β
f : F
K : Set α
hK₁ : IsCompact K
hK₂ : Kᶜ ⊆ ⇑f ⁻¹' closedBall 0 1
C : ℝ
hC : ⇑f '' K ⊆ closedBall 0 C
x✝ y x : α
hx : x ∉ K
⊢ f x ∈ closedBall 0 (max C 1)
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case a
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 := ⋯
⊢ {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}

pp: case a
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 := ⋯
⊢ (Finset.univ.sup '' univ.pi fun U ↦ {K | (↑K).Nonempty ∧ (↑K).Finite ∧ ↑K ⊆ ↑U}) ⊆
    {K | ↑K ⊆ ⋃₀ u ∧ (∀ U ∈ u, (↑K ∩ U).Nonempty) ∧ (↑K).Finite}

Tactic:
·

Goal state after:
case a
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 := ⋯
⊢ {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}
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
⊢ ‖expMulMulExp a b x z‖ = ‖x‖

Tactic:
expMulMulExp_eq_expUnitary_mul_mul_expUnitary h z,

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
hf : Differentiable ℂ (expMulMulExp a b x)
z : ℂ
⊢ ‖↑(expUnitary (2 • ℑ (z • star b))) * x * ↑(expUnitary (2 • ℑ (star z • a)))‖ = ‖x‖
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case a
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 := ⋯
⊢ (Finset.univ.sup '' univ.pi fun U ↦ {K | (↑K).Nonempty ∧ (↑K).Finite ∧ ↑K ⊆ ↑U}) ⊆
    {K | ↑K ⊆ ⋃₀ u ∧ (∀ U ∈ u, (↑K ∩ U).Nonempty) ∧ (↑K).Finite}

Tactic:
·

Goal state after:
case a
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 := ⋯
⊢ (Finset.univ.sup '' univ.pi fun U ↦ {K | (↑K).Nonempty ∧ (↑K).Finite ∧ ↑K ⊆ ↑U}) ⊆
    {K | ↑K ⊆ ⋃₀ u ∧ (∀ U ∈ u, (↑K ∩ U).Nonempty) ∧ (↑K).Finite}
Module: Mathlib.Analysis.CStarAlgebra.Fuglede

Goal state before:
pp: case refine_1
A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : IsStarNormal a
x : ℝ
⊢ NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A

pp: case refine_2
A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : ∀ (x : ℝ), NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A
⊢ IsStarNormal a

Tactic:
·

Goal state after:
case refine_1
A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : IsStarNormal a
x : ℝ
⊢ NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: case mp
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
S : Type u_4
inst✝¹ : SetLike S E
inst✝ : SMulMemClass S 𝕜 E
m : S
y : F
⊢ y ∈ B.polar ↑m → y ∈ {y | ∀ x ∈ m, (B x) y = 0}

pp: case mpr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
S : Type u_4
inst✝¹ : SetLike S E
inst✝ : SMulMemClass S 𝕜 E
m : S
y : F
⊢ y ∈ {y | ∀ x ∈ m, (B x) y = 0} → y ∈ B.polar ↑m

Tactic:
·

Goal state after:
case mp
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
S : Type u_4
inst✝¹ : SetLike S E
inst✝ : SMulMemClass S 𝕜 E
m : S
y : F
⊢ y ∈ B.polar ↑m → y ∈ {y | ∀ x ∈ m, (B x) y = 0}
Module: Mathlib.Analysis.CStarAlgebra.Fuglede

Goal state before:
pp: case refine_2
A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : ∀ (x : ℝ), NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A
⊢ IsStarNormal a

Tactic:
·

Goal state after:
case refine_2
A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : ∀ (x : ℝ), NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A
⊢ IsStarNormal a
Module: Mathlib.Analysis.LocallyConvex.AbsConvex

Goal state before:
pp: case mp
𝕜 : Type u_1
E : Type u_2
inst✝³ : SeminormedRing 𝕜
inst✝² : SMul 𝕜 E
inst✝¹ : AddCommMonoid E
inst✝ : PartialOrder 𝕜
s : Set E
⊢ (absConvexHull 𝕜) s = ∅ → s = ∅

pp: case mpr
𝕜 : Type u_1
E : Type u_2
inst✝³ : SeminormedRing 𝕜
inst✝² : SMul 𝕜 E
inst✝¹ : AddCommMonoid E
inst✝ : PartialOrder 𝕜
s : Set E
⊢ s = ∅ → (absConvexHull 𝕜) s = ∅

Tactic:
·

Goal state after:
case mp
𝕜 : Type u_1
E : Type u_2
inst✝³ : SeminormedRing 𝕜
inst✝² : SMul 𝕜 E
inst✝¹ : AddCommMonoid E
inst✝ : PartialOrder 𝕜
s : Set E
⊢ (absConvexHull 𝕜) s = ∅ → s = ∅
Module: Mathlib.Analysis.LocallyConvex.AbsConvex

Goal state before:
pp: case mpr
𝕜 : Type u_1
E : Type u_2
inst✝³ : SeminormedRing 𝕜
inst✝² : SMul 𝕜 E
inst✝¹ : AddCommMonoid E
inst✝ : PartialOrder 𝕜
s : Set E
⊢ s = ∅ → (absConvexHull 𝕜) s = ∅

Tactic:
·

Goal state after:
case mpr
𝕜 : Type u_1
E : Type u_2
inst✝³ : SeminormedRing 𝕜
inst✝² : SMul 𝕜 E
inst✝¹ : AddCommMonoid E
inst✝ : PartialOrder 𝕜
s : Set E
⊢ s = ∅ → (absConvexHull 𝕜) s = ∅
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: case mpr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
S : Type u_4
inst✝¹ : SetLike S E
inst✝ : SMulMemClass S 𝕜 E
m : S
y : F
⊢ y ∈ {y | ∀ x ∈ m, (B x) y = 0} → y ∈ B.polar ↑m

Tactic:
·

Goal state after:
case mpr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
S : Type u_4
inst✝¹ : SetLike S E
inst✝ : SMulMemClass S 𝕜 E
m : S
y : F
⊢ y ∈ {y | ∀ x ∈ m, (B x) y = 0} → y ∈ B.polar ↑m
Module: Mathlib.Analysis.CStarAlgebra.Fuglede

Goal state before:
pp: case e'_2
A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : ∀ (x : ℝ), NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A
x : ℝ
u : ↥(unitary A) := ⟨NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a), ⋯⟩
⊢ NormedSpace.exp (-x • a) * NormedSpace.exp (x • star a) = ↑(star u)

pp: case e'_3
A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : ∀ (x : ℝ), NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A
x : ℝ
u : ↥(unitary A) := ⟨NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a), ⋯⟩
⊢ NormedSpace.exp (x • star a) * NormedSpace.exp (-x • a) = ↑u⁻¹

Tactic:
·

Goal state after:
case e'_2
A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : ∀ (x : ℝ), NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A
x : ℝ
u : ↥(unitary A) := ⟨NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a), ⋯⟩
⊢ NormedSpace.exp (-x • a) * NormedSpace.exp (x • star a) = ↑(star u)
Module: Mathlib.Analysis.CStarAlgebra.Fuglede

Goal state before:
pp: case e'_3
A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : ∀ (x : ℝ), NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A
x : ℝ
u : ↥(unitary A) := ⟨NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a), ⋯⟩
⊢ NormedSpace.exp (x • star a) * NormedSpace.exp (-x • a) = ↑u⁻¹

Tactic:
·

Goal state after:
case e'_3
A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : ∀ (x : ℝ), NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A
x : ℝ
u : ↥(unitary A) := ⟨NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a), ⋯⟩
⊢ NormedSpace.exp (x • star a) * NormedSpace.exp (-x • a) = ↑u⁻¹
Module: Mathlib.Analysis.LocallyConvex.AbsConvexOpen

Goal state before:
pp: case refine_1
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : AddCommGroup E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : Module ℝ E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : ContinuousSMul ℝ E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s : Set E
hs : 0 ∈ s ∧ IsOpen[inst✝⁷] s ∧ AbsConvex 𝕜 s
⊢ ∃ i' ∈ (gaugeSeminormFamily 𝕜 E).basisSets, id i' ⊆ id s

pp: case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : AddCommGroup E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : Module ℝ E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : ContinuousSMul ℝ E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s : Set E
hs : s ∈ (gaugeSeminormFamily 𝕜 E).basisSets
⊢ ∃ i, (0 ∈ i ∧ IsOpen[inst✝⁷] i ∧ AbsConvex 𝕜 i) ∧ id i ⊆ id s

Tactic:
·

Goal state after:
case refine_1
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : AddCommGroup E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : Module ℝ E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : ContinuousSMul ℝ E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s : Set E
hs : 0 ∈ s ∧ IsOpen[inst✝⁷] s ∧ AbsConvex 𝕜 s
⊢ ∃ i' ∈ (gaugeSeminormFamily 𝕜 E).basisSets, id i' ⊆ id s
Module: Mathlib.Analysis.LocallyConvex.AbsConvex

Goal state before:
pp: case refine_1
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : PartialOrder 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : LocallyConvexSpace 𝕜 E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ZeroLEOneClass 𝕜
⊢ ∀ (i : Set E), i ∈ 𝓝 0 ∧ AbsConvex 𝕜 i → ∃ i', (0 ∈ i' ∧ IsOpen[inst✝⁴] i' ∧ AbsConvex 𝕜 i') ∧ id i' ⊆ id i

pp: case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : PartialOrder 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : LocallyConvexSpace 𝕜 E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ZeroLEOneClass 𝕜
⊢ ∀ (i' : Set E), 0 ∈ i' ∧ IsOpen[inst✝⁴] i' ∧ AbsConvex 𝕜 i' → ∃ i, (i ∈ 𝓝 0 ∧ AbsConvex 𝕜 i) ∧ id i ⊆ id i'

Tactic:
·

Goal state after:
case refine_1
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : PartialOrder 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : LocallyConvexSpace 𝕜 E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ZeroLEOneClass 𝕜
⊢ ∀ (i : Set E), i ∈ 𝓝 0 ∧ AbsConvex 𝕜 i → ∃ i', (0 ∈ i' ∧ IsOpen[inst✝⁴] i' ∧ AbsConvex 𝕜 i') ∧ id i' ⊆ id i
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
⊢ ∀ ⦃x : Set (Set α)⦄, x ∈ {u | u.Finite ∧ u.Nonempty ∧ u ⊆ B} → IsOpen {K | ↑K ⊆ ⋃₀ x ∧ ∀ U ∈ x, (↑K ∩ U).Nonempty}

pp: case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
⊢ ∀ ⦃x : Set (Set α)⦄,
    x ∈ {u | u.Finite ∧ u ⊆ B} →
      ∀ x_1 ∈ toCompacts ⁻¹' {K | ↑K ⊆ ⋃₀ x ∧ ∀ U ∈ x, (↑K ∩ U).Nonempty},
        ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u.Nonempty ∧ u ⊆ B},
          x_1 ∈ v ∧ v ⊆ toCompacts ⁻¹' {K | ↑K ⊆ ⋃₀ x ∧ ∀ U ∈ x, (↑K ∩ U).Nonempty}

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
⊢ ∀ ⦃x : Set (Set α)⦄, x ∈ {u | u.Finite ∧ u.Nonempty ∧ u ⊆ B} → IsOpen {K | ↑K ⊆ ⋃₀ x ∧ ∀ U ∈ x, (↑K ∩ U).Nonempty}