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

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
f g : C(X, ℝ)
x : X
⊢ max (f x + g x) 0 + (-f x + max (f x) 0) + (-g x + max (g x) 0) =
    -(f x + g x) + max (f x + g x) 0 + max (f x) 0 + max (g x) 0

Tactic:
abel

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
f g : C(X, ℝ)
x : X
⊢ max (f x * g x) 0 + (-(f x * max (g x) 0) + max (f x) 0 * max (g x) 0) +
      (-(max (f x) 0 * g x) + max (f x) 0 * max (g x) 0) =
    -(f x * g x) + max (f x * g x) 0 + max (f x) 0 * max (g x) 0 +
      (-(max (f x) 0 * g x) + f x * g x + (-(f x * max (g x) 0) + max (f x) 0 * max (g x) 0))

Tactic:
abel

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f g : C(X, ℝ)
this :
  φ (f * g).toNNReal + φ (-f).toNNReal * φ g.toNNReal + φ f.toNNReal * φ (-g).toNNReal -
      (φ (-(f * g)).toNNReal + φ f.toNNReal * φ g.toNNReal + φ (-f).toNNReal * φ (-g).toNNReal) =
    0
⊢ φ (f * g).toNNReal - φ (-(f * g)).toNNReal -
      (φ f.toNNReal * φ g.toNNReal - φ (-f).toNNReal * φ g.toNNReal -
        (φ f.toNNReal * φ (-g).toNNReal - φ (-f).toNNReal * φ (-g).toNNReal)) =
    φ (f * g).toNNReal + φ (-f).toNNReal * φ g.toNNReal + φ f.toNNReal * φ (-g).toNNReal -
      (φ (-(f * g)).toNNReal + φ f.toNNReal * φ g.toNNReal + φ (-f).toNNReal * φ (-g).toNNReal)

Tactic:
abel

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f g : C(X, ℝ)
this :
  φ (f + g).toNNReal + φ (-f).toNNReal + φ (-g).toNNReal - (φ (-(f + g)).toNNReal + φ f.toNNReal + φ g.toNNReal) = 0
⊢ φ (f + g).toNNReal - φ (-(f + g)).toNNReal - (φ f.toNNReal - φ (-f).toNNReal + (φ g.toNNReal - φ (-g).toNNReal)) =
    φ (f + g).toNNReal + φ (-f).toNNReal + φ (-g).toNNReal - (φ (-(f + g)).toNNReal + φ f.toNNReal + φ g.toNNReal)

Tactic:
abel

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f g : C(X, ℝ)₀
this :
  φ (f + g).toNNReal + φ (-f).toNNReal + φ (-g).toNNReal - (φ (-(f + g)).toNNReal + φ f.toNNReal + φ g.toNNReal) = 0
⊢ φ (f + g).toNNReal - φ (-(f + g)).toNNReal - (φ f.toNNReal - φ (-f).toNNReal + (φ g.toNNReal - φ (-g).toNNReal)) =
    φ (f + g).toNNReal + φ (-f).toNNReal + φ (-g).toNNReal - (φ (-(f + g)).toNNReal + φ f.toNNReal + φ g.toNNReal)

Tactic:
abel

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f g : C(X, ℝ)₀
this :
  φ (f * g).toNNReal + φ (-f).toNNReal * φ g.toNNReal + φ f.toNNReal * φ (-g).toNNReal -
      (φ (-(f * g)).toNNReal + φ f.toNNReal * φ g.toNNReal + φ (-f).toNNReal * φ (-g).toNNReal) =
    0
⊢ φ (f * g).toNNReal - φ (-(f * g)).toNNReal -
      (φ f.toNNReal * φ g.toNNReal - φ (-f).toNNReal * φ g.toNNReal -
        (φ f.toNNReal * φ (-g).toNNReal - φ (-f).toNNReal * φ (-g).toNNReal)) =
    φ (f * g).toNNReal + φ (-f).toNNReal * φ g.toNNReal + φ f.toNNReal * φ (-g).toNNReal -
      (φ (-(f * g)).toNNReal + φ f.toNNReal * φ g.toNNReal + φ (-f).toNNReal * φ (-g).toNNReal)

Tactic:
abel

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: case h₁
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing E
inst✝⁶ : StarRing E
inst✝⁵ : CStarRing E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : IsScalarTower 𝕜 E E
inst✝² : SMulCommClass 𝕜 E E
inst✝¹ : StarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
b : E
hb : b ∈ Metric.closedBall 0 1
y : E
hy : y ∈ Metric.closedBall 0 1
⊢ ‖↑((algebraMap 𝕜 (E →L[𝕜] E)) (x⋆ * x).toProd.1) y‖ ≤ ‖(x⋆ * x).toProd.1‖

Tactic:
Algebra.algebraMap_eq_smul_one

Goal state after:
case h₁
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing E
inst✝⁶ : StarRing E
inst✝⁵ : CStarRing E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : IsScalarTower 𝕜 E E
inst✝² : SMulCommClass 𝕜 E E
inst✝¹ : StarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
b : E
hb : b ∈ Metric.closedBall 0 1
y : E
hy : y ∈ Metric.closedBall 0 1
⊢ ‖↑((x⋆ * x).toProd.1 • 1) y‖ ≤ ‖(x⋆ * x).toProd.1‖
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: 𝕜 : Type u_1
X : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
I : C(X, ℝ) →L[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealCLM
A₀ : Submodule ℝ C(X, ℝ) := Submodule.comap (↑I) (Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra))
SW : A₀.topologicalClosure = ⊤
h₁ : Submodule.map (↑I) A₀.topologicalClosure ≤ (Submodule.map (↑I) A₀).topologicalClosure
h₂ :
  Submodule.map (↑I) (Submodule.comap (↑I) (Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra))) ≤
    Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra)
⊢ Submodule.map (↑I) A₀.topologicalClosure ≤
    (Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra)).topologicalClosure

Tactic:
exact h₁.trans (Submodule.topologicalClosure_mono h₂)

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: case pos
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : DenselyNormedField 𝕜
inst✝⁸ : NonUnitalNormedRing E
inst✝⁷ : StarRing E
inst✝⁶ : CStarRing E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : IsScalarTower 𝕜 E E
inst✝³ : SMulCommClass 𝕜 E E
inst✝² : StarRing 𝕜
inst✝¹ : StarModule 𝕜 E
inst✝ : CStarRing 𝕜
x : Unitization 𝕜 E
h₁ : ∀ (x : Unitization 𝕜 E), ‖((splitMul 𝕜 E) x).2‖ ≤ ‖((splitMul 𝕜 E) x⋆).2‖
h₂ : ‖((splitMul 𝕜 E) (x⋆ * x)).2‖ = ‖((splitMul 𝕜 E) x).2‖ ^ 2
h₃ : ‖((splitMul 𝕜 E) (x⋆ * x)).1‖ = ‖((splitMul 𝕜 E) x).1‖ ^ 2
h : ‖((splitMul 𝕜 E) x).1‖ ≤ ‖((splitMul 𝕜 E) x).2‖
⊢ max ‖((splitMul 𝕜 E) x).1‖ ‖((splitMul 𝕜 E) x).2‖ * max ‖((splitMul 𝕜 E) x).1‖ ‖((splitMul 𝕜 E) x).2‖ ≤
    max (‖((splitMul 𝕜 E) x).1‖ ^ 2) (‖((splitMul 𝕜 E) x).2‖ ^ 2)

Tactic:
rw [sq, sq, sup_eq_right.mpr h, sup_eq_right.mpr (mul_self_le_mul_self (norm_nonneg _) h)]

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

Goal state before:
pp: case mpr
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
f : C(↑s, 𝕜)
⊢ f ∈ ↑(adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}) →
    f ∈ ↑(StarAlgebra.adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}) ∩ ↑(RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩))

Tactic:
simp only [Set.mem_inter_iff, SetLike.mem_coe]
    refine fun hf ↦ ⟨?_, nonUnitalStarAlgebraAdjoin_id_subset_ker_evalStarAlgHom h0 hf⟩
    exact adjoin_le_starAlgebra_adjoin _ _ hf

Goal state after:
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
f : α → Set β
inst✝ : TopologicalSpace β
⊢ LowerHemicontinuous f ↔ ∀ (u : Set β), IsOpen[inst✝] u → IsOpen[inst✝¹] (f ⁻¹' Iic uᶜ)ᶜ

Tactic:
have (u : Set β) : (f ⁻¹' (Iic uᶜ))ᶜ = {x | (f x ∩ u).Nonempty} := by
    simp [Set.ext_iff, Iic, Set.mem_compl_iff, Set.not_subset, Set.Nonempty]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
f : α → Set β
inst✝ : TopologicalSpace β
this : ∀ (u : Set β), (f ⁻¹' Iic uᶜ)ᶜ = {x | (f x ∩ u).Nonempty}
⊢ LowerHemicontinuous f ↔ ∀ (u : Set β), IsOpen[inst✝] u → IsOpen[inst✝¹] (f ⁻¹' Iic uᶜ)ᶜ
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
U V : SetRel α α
h : U ⊆ V
⊢ hausdorffEntourage U ⊆ hausdorffEntourage V

Tactic:
by
  unfold hausdorffEntourage
  gcongr

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

Goal state before:
pp: case refine_2.inr
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
h✝ : s.Nonempty
u : Set (Set α)
hu₁ : u ⊆ {U | IsOpen[inst✝] U}
hu₂ : u.Finite
hu : powerset '' u ⊆ powerset '' {U | IsOpen[inst✝] U}
v : Set (Set α)
hv₁ : v ⊆ {U | IsOpen[inst✝] U}
hv₂ : v.Finite
hv : (fun V ↦ {s | (s ∩ V).Nonempty}) '' v ⊆ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {U | IsOpen[inst✝] U}
ht₁ : (powerset '' u).Finite ∧ ((fun V ↦ {s | (s ∩ V).Nonempty}) '' v).Finite
ht₂ :
  powerset '' u ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' v ⊆
    powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
U : Set α := ⋂₀ u
hU : IsOpen[inst✝] U
hs : s ⊆ U ∧ ∀ i ∈ v, s ∈ {s | (s ∩ i).Nonempty}
⊢ (((fun x ↦ U ∩ x) '' v).Finite ∧ IsOpen[inst✝] U ∧ ∀ ⦃x : Set α⦄, x ∈ v → IsOpen[inst✝] (U ∩ x)) ∧
    (s ⊆ U ∧ s.Nonempty ∧ ∀ ⦃x : Set α⦄, x ∈ v → (s ∩ x).Nonempty) ∧
      {s | s ⊆ U ∧ (s ∩ U).Nonempty ∧ ∀ ⦃x : Set α⦄, x ∈ v → (s ∩ U ∩ x).Nonempty} ⊆
        𝒫 U ∩ ⋂ a ∈ v, {s | (s ∩ a).Nonempty}

Tactic:
refine ⟨⟨hv₂.image _, hU, fun V hV => hU.inter (hv₁ hV)⟩, by grind,
      fun t ⟨htU, _, ht⟩ => ⟨htU, mem_iInter₂_of_mem ?_⟩⟩

Goal state after:
case refine_2.inr
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
h✝ : s.Nonempty
u : Set (Set α)
hu₁ : u ⊆ {U | IsOpen[inst✝] U}
hu₂ : u.Finite
hu : powerset '' u ⊆ powerset '' {U | IsOpen[inst✝] U}
v : Set (Set α)
hv₁ : v ⊆ {U | IsOpen[inst✝] U}
hv₂ : v.Finite
hv : (fun V ↦ {s | (s ∩ V).Nonempty}) '' v ⊆ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {U | IsOpen[inst✝] U}
ht₁ : (powerset '' u).Finite ∧ ((fun V ↦ {s | (s ∩ V).Nonempty}) '' v).Finite
ht₂ :
  powerset '' u ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' v ⊆
    powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
U : Set α := ⋂₀ u
hU : IsOpen[inst✝] U
hs : s ⊆ U ∧ ∀ i ∈ v, s ∈ {s | (s ∩ i).Nonempty}
t : Set α
x✝ : t ∈ {s | s ⊆ U ∧ (s ∩ U).Nonempty ∧ ∀ ⦃x : Set α⦄, x ∈ v → (s ∩ U ∩ x).Nonempty}
htU : t ⊆ U
left✝ : (t ∩ U).Nonempty
ht : ∀ ⦃x : Set α⦄, x ∈ v → (t ∩ U ∩ x).Nonempty
⊢ ∀ i ∈ v, t ∈ {s | (s ∩ i).Nonempty}
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
U : SetRel α α
x y : α
⊢ ({x}, {y}) ∈ hausdorffEntourage U ↔ (x, y) ∈ U

Tactic:
simp [hausdorffEntourage]

Goal state after:
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
f : α → Set β
s : Set α
ι : Type u_3
F : ι → α → Set β
l : Filter ι
inst✝¹ : l.NeBot
inst✝ : UniformSpace β
htendsto : TendstoUniformlyOn F f l s
hF : ∀ (n : ι), LowerHemicontinuousOn (F n) s
x₀ : α
hx₀s : x₀ ∈ s
V : Set β
hV : IsOpen[inst✝.toTopologicalSpace] V
y₀ : β
hy₀f : y₀ ∈ f x₀
hy₀V : y₀ ∈ V
W : Set (β × β)
hW : W ∈ uniformity β
hWsub : ball y₀ W ⊆ V
U₁ : Set (β × β)
hU₁ : U₁ ∈ uniformity β
hU₁sym : SetRel.IsSymm U₁
hU₁comp : SetRel.comp U₁ U₁ ⊆ W
U : Set (β × β)
hU : U ∈ uniformity β
hUsym : SetRel.IsSymm U
hUcomp : SetRel.comp U U ⊆ U₁
hU_le_U₁ : U ⊆ U₁
hHU : hausdorffEntourage U ∈ uniformity (Set β)
N : ι
hN : ∀ x ∈ s, (f x, F N x) ∈ hausdorffEntourage U
⊢ ∀ᶠ (x' : α) in 𝓝[s] x₀, (f x' ∩ V).Nonempty

Tactic:
obtain ⟨z₀, hz₀FN, hz₀y₀⟩ :=
    ((mem_hausdorffEntourage U (f x₀) (F N x₀)).mp (hN x₀ hx₀s)).1 hy₀f

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
f : α → Set β
s : Set α
ι : Type u_3
F : ι → α → Set β
l : Filter ι
inst✝¹ : l.NeBot
inst✝ : UniformSpace β
htendsto : TendstoUniformlyOn F f l s
hF : ∀ (n : ι), LowerHemicontinuousOn (F n) s
x₀ : α
hx₀s : x₀ ∈ s
V : Set β
hV : IsOpen[inst✝.toTopologicalSpace] V
y₀ : β
hy₀f : y₀ ∈ f x₀
hy₀V : y₀ ∈ V
W : Set (β × β)
hW : W ∈ uniformity β
hWsub : ball y₀ W ⊆ V
U₁ : Set (β × β)
hU₁ : U₁ ∈ uniformity β
hU₁sym : SetRel.IsSymm U₁
hU₁comp : SetRel.comp U₁ U₁ ⊆ W
U : Set (β × β)
hU : U ∈ uniformity β
hUsym : SetRel.IsSymm U
hUcomp : SetRel.comp U U ⊆ U₁
hU_le_U₁ : U ⊆ U₁
hHU : hausdorffEntourage U ∈ uniformity (Set β)
N : ι
hN : ∀ x ∈ s, (f x, F N x) ∈ hausdorffEntourage U
z₀ : β
hz₀FN : z₀ ∈ F N x₀
hz₀y₀ : (y₀, z₀) ∈ U
⊢ ∀ᶠ (x' : α) in 𝓝[s] x₀, (f x' ∩ V).Nonempty
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
k : 𝕜
h : ‖a‖ * ‖1‖ < ‖k‖
⊢ IsUnit (↑ₐ k - a)

Tactic:
Algebra.algebraMap_eq_smul_one

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
k : 𝕜
h : ‖a‖ * ‖1‖ < ‖k‖
⊢ IsUnit (k • 1 - a)
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A✝ : Type u_2
inst✝⁶ : NormedField 𝕜
inst✝⁵ : NormedRing A✝
inst✝⁴ : NormedAlgebra 𝕜 A✝
inst✝³ : CompleteSpace A✝
A : Type u_3
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a : A
inst✝ : CompactSpace ↑(spectrum ℝ a)
⊢ CompactSpace ↑(spectrum ℝ≥0 a)

Tactic:
rw [← isCompact_iff_compactSpace] at *
  rw [← preimage_algebraMap ℝ]
  exact isClosed_nonneg.isClosedEmbedding_subtypeVal.isCompact_preimage <| by assumption

Goal state after:
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case left
α : Type u_1
inst✝ : UniformSpace α
U : Set (α × α)
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
x✝¹ : Set α × Set α
s t : Set α
x✝ : (s, t) ∈ hausdorffEntourage V
hst : s ⊆ V.preimage t
hts : t ⊆ V.image s
⊢ closure[inst✝.toTopologicalSpace] s ⊆ SetRel.preimage U (closure[inst✝.toTopologicalSpace] t)

Tactic:
grw [closure_subset_preimage hV s, hst, ← subset_closure, ← hVU, SetRel.preimage_comp]

Goal state after:
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
f : α → Set β
x✝ :
  (∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)) ∧
    ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)
h₁ : ∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)
h₂ : ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)
U : Set β
hU : U ∈ {V | IsOpen[inst✝] V}
⊢ IsOpen[inst✝¹] {a | Disjoint (f a) U}ᶜ

Tactic:
isOpen_compl_iff,

Goal state after:
case inr
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → Set β
x✝ :
  (∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)) ∧
    ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)
h₁ : ∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)
h₂ : ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)
U : Set β
hU : U ∈ {V | IsOpen[inst✝] V}
⊢ IsClosed[inst✝¹] {a | Disjoint (f a) U}
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case neg
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
a : A
y : 𝕜
hy : y ∈ Metric.eball 0 (↑‖a‖₊)⁻¹
h : ¬‖a‖₊ = 0
nnnorm_lt : ‖y‖₊ < ‖a‖₊⁻¹
⊢ ↑‖y • a‖₊ < 1

Tactic:
← Real.lt_toNNReal_iff_coe_lt,

Goal state after:
case neg
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
a : A
y : 𝕜
hy : y ∈ Metric.eball 0 (↑‖a‖₊)⁻¹
h : ¬‖a‖₊ = 0
nnnorm_lt : ‖y‖₊ < ‖a‖₊⁻¹
⊢ ‖y • a‖₊ < Real.toNNReal 1
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
a : A
z : 𝕜
h : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹
hz : ¬z = 0
u : 𝕜ˣ := Units.mk0 z hz
hu : IsUnit (u⁻¹ • 1 - a)
⊢ IsUnit (1 - z • a)

Tactic:
IsUnit.smul_sub_iff_sub_inv_smul,

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
a : A
z : 𝕜
h : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹
hz : ¬z = 0
u : 𝕜ˣ := Units.mk0 z hz
hu : IsUnit (1 - u⁻¹⁻¹ • a)
⊢ IsUnit (1 - z • a)
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ spectrum 𝕜 a
⊢ exp z ∈ spectrum 𝕜 (exp a)

Tactic:
let +nondep : NormedAlgebra ℚ A := .restrictScalars ℚ 𝕜 A

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ spectrum 𝕜 a
this : NormedAlgebra ℚ A
⊢ exp z ∈ spectrum 𝕜 (exp a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: case refine_1
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
a : A
ha : p a
inst✝¹ : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝ : CompleteSpace A
f : C(↑(σₙ 𝕜 a), 𝕜)₀
h₁ :
  (cfcₙAux ⋯ a ha) f ∈
    closure[Unitization.instUniformSpace.toTopologicalSpace]
      (⇑(cfcₙAux ⋯ a ha) '' ↑(NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMapZero.id (σₙ 𝕜 a)}))
⊢ ContinuousMapZero.id (σₙ 𝕜 a) ∈
    ↑(NonUnitalStarSubalgebra.comap (cfcₙAux ⋯ a ha)
        (NonUnitalStarAlgHom.range (Unitization.inrNonUnitalStarAlgHom 𝕜 A)))

Tactic:
rw [SetLike.mem_coe, NonUnitalStarSubalgebra.mem_comap, cfcₙAux_id hp₁ a ha]

Goal state after:
case refine_1
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
a : A
ha : p a
inst✝¹ : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝ : CompleteSpace A
f : C(↑(σₙ 𝕜 a), 𝕜)₀
h₁ :
  (cfcₙAux ⋯ a ha) f ∈
    closure[Unitization.instUniformSpace.toTopologicalSpace]
      (⇑(cfcₙAux ⋯ a ha) '' ↑(NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMapZero.id (σₙ 𝕜 a)}))
⊢ ↑a ∈ NonUnitalStarAlgHom.range (Unitization.inrNonUnitalStarAlgHom 𝕜 A)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: case refine_1
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
a : A
ha : p a
inst✝¹ : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝ : CompleteSpace A
f : C(↑(σₙ 𝕜 a), 𝕜)₀
h₁ :
  (cfcₙAux ⋯ a ha) f ∈
    closure[Unitization.instUniformSpace.toTopologicalSpace]
      (⇑(cfcₙAux ⋯ a ha) '' ↑(NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMapZero.id (σₙ 𝕜 a)}))
⊢ ⇑(cfcₙAux ⋯ a ha) '' ↑(NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMapZero.id (σₙ 𝕜 a)}) ⊆
    ↑(NonUnitalStarAlgHom.range (Unitization.inrNonUnitalStarAlgHom 𝕜 A))

Tactic:
rw [← NonUnitalStarSubalgebra.coe_map, SetLike.coe_subset_coe, NonUnitalStarSubalgebra.map_le]
    apply NonUnitalStarAlgebra.adjoin_le
    apply Set.singleton_subset_iff.mpr
    rw [SetLike.mem_coe, NonUnitalStarSubalgebra.mem_comap, cfcₙAux_id hp₁ a ha]
    exact ⟨a, rfl⟩

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_3
A : Type u_4
SA : Type u_5
inst✝⁵ : NormedRing A
inst✝⁴ : CompleteSpace A
inst✝³ : SetLike SA A
inst✝² : SubringClass SA A
inst✝¹ : NormedField 𝕜
inst✝ : NormedAlgebra 𝕜 A
instSMulMem : SMulMemClass SA 𝕜 A
S : SA
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
x : ↥S
this : IsClopen ((σ 𝕜 ↑x)ᶜ ↓∩ σ 𝕜 x \ σ 𝕜 ↑x)
⊢ σ 𝕜 x = σ 𝕜 ↑x ∪ ⋃ z ∈ σ 𝕜 x \ σ 𝕜 ↑x, connectedComponentIn (σ 𝕜 ↑x)ᶜ z

Tactic:
by
    rw [← this.biUnion_connectedComponentIn (sdiff_subset_compl _ _),
      union_sdiff_cancel (spectrum.subset_subalgebra x)]

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
⊢ (inrNonUnitalStarAlgHom 𝕜 A).comp (cfcₙHom ha) = cfcₙAux ⋯ a ha

Tactic:
by
  let _ := RCLike.nonUnitalContinuousFunctionalCalculus hp₁
  apply ContinuousMapZero.UniqueHom.eq_of_continuous_of_map_id _ _ _
    (Unitization.continuous_inr.comp <| cfcₙHom_continuous ha)
    (continuous_cfcₙAux hp₁ a ha)
    (by simp [cfcₙHom_id ha, cfcₙAux_id hp₁ a ha])
  all_goals infer_instance

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha₁ : IsSelfAdjoint a
ha₂ : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
x : ℝ
hx : x ∈ σₙ ℝ a
⊢ √x * √x = id x

Tactic:
QuasispectrumRestricts.nnreal_iff

Goal state after:
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha₁ : IsSelfAdjoint a
ha₂ : ∀ x ∈ σₙ ℝ a, 0 ≤ x
x : ℝ
hx : x ∈ σₙ ℝ a
⊢ √x * √x = id x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : SMulCommClass R A A
inst✝³ : MetricSpace A
inst✝² : NonUnitalIsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMapZero.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : QuasispectrumRestricts a ⇑f
⊢ Isometry ⇑(cfcₙHom ha)

Tactic:
have := QuasispectrumRestricts.cfc f halg.isClosedEmbedding h0 h

Goal state after:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : SMulCommClass R A A
inst✝³ : MetricSpace A
inst✝² : NonUnitalIsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMapZero.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : QuasispectrumRestricts a ⇑f
this : NonUnitalContinuousFunctionalCalculus R A p
⊢ Isometry ⇑(cfcₙHom ha)