mathlib-initiative/mathlib-tactics huggingface dataset:
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
  have hexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z) := by
    rw [algebraMap_exp_comm z, ← exp_add_of_commute (Algebra.commutes z (a - ↑ₐ z)).symm,
      sub_add_cancel]
  let b := ∑' n : ℕ, ((n + 1).factorial⁻¹ : 𝕜) • (a - ↑ₐ z) ^ n
  have hb : Summable fun n : ℕ => ((n + 1).factorial⁻¹ : 𝕜) • (a - ↑ₐ z) ^ n := by
    refine .of_norm_bounded_eventually (Real.summable_pow_div_factorial ‖a - ↑ₐ z‖) ?_
    filter_upwards [Filter.eventually_cofinite_ne 0] with n hn
    rw [norm_smul, mul_comm, norm_inv, RCLike.norm_natCast, ← div_eq_mul_inv]
    gcongr
    · exact norm_pow_le' _ (pos_iff_ne_zero.mpr hn)
    · exact n.le_succ
  have h₀ : (∑' n : ℕ, ((n + 1).factorial⁻¹ : 𝕜) • (a - ↑ₐ z) ^ (n + 1)) = (a - ↑ₐ z) * b := by
    simpa only [mul_smul_comm, pow_succ'] using hb.tsum_mul_left (a - ↑ₐ z)
  have h₁ : (∑' n : ℕ, ((n + 1).factorial⁻¹ : 𝕜) • (a - ↑ₐ z) ^ (n + 1)) = b * (a - ↑ₐ z) := by
    simpa only [pow_succ, Algebra.smul_mul_assoc] using hb.tsum_mul_right (a - ↑ₐ z)
  have h₃ : exp (a - ↑ₐ z) = 1 + (a - ↑ₐ z) * b := by
    rw [exp_eq_tsum 𝕜]
    convert! (expSeries_summable' (𝕂 := 𝕜) (a - ↑ₐ z)).tsum_eq_zero_add
    · simp only [Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, one_smul]
    · exact h₀.symm
  rw [spectrum.mem_iff, IsUnit.sub_iff, ← one_mul (↑ₐ (exp z)), hexpmul, ← _root_.sub_mul,
    Commute.isUnit_mul_iff (Algebra.commutes (exp z) (exp (a - ↑ₐ z) - 1)).symm,
    sub_eq_iff_eq_add'.mpr h₃, Commute.isUnit_mul_iff (h₀ ▸ h₁ : (a - ↑ₐ z) * b = b * (a - ↑ₐ z))]
  exact not_and_of_not_left _ (not_and_of_not_left _ ((not_iff_not.mpr IsUnit.sub_iff).mp hz))

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 : CompleteSpace ↥S
μ : 𝕜
hμ : μ ∈ frontier (σ 𝕜 x)
h : μ ∉ σ 𝕜 ↑x
⊢ False

Tactic:
spectrum.notMem_iff

Goal state after:
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 : CompleteSpace ↥S
μ : 𝕜
hμ : μ ∈ frontier (σ 𝕜 x)
h : IsUnit ((algebraMap 𝕜 A) μ - ↑x)
⊢ False
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 : CompleteSpace ↥S
μ : 𝕜
h : IsUnit ((algebraMap 𝕜 A) μ - ↑x)
hμ₁ : ClusterPt μ (𝓟 (σ 𝕜 x)ᶜ)
hμ₂ : μ ∉ (σ 𝕜 x)ᶜ
⊢ μ ∉ σ 𝕜 x

Tactic:
spectrum.notMem_iff

Goal state after:
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 : CompleteSpace ↥S
μ : 𝕜
h : IsUnit ((algebraMap 𝕜 A) μ - ↑x)
hμ₁ : ClusterPt μ (𝓟 (σ 𝕜 x)ᶜ)
hμ₂ : μ ∉ (σ 𝕜 x)ᶜ
⊢ IsUnit ((algebraMap 𝕜 ↥S) μ - x)
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:
apply Set.singleton_subset_iff.mpr

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)}))
⊢ ContinuousMapZero.id (σₙ 𝕜 a) ∈
    ↑(NonUnitalStarSubalgebra.comap (cfcₙAux ⋯ a ha)
        (NonUnitalStarAlgHom.range (Unitization.inrNonUnitalStarAlgHom 𝕜 A)))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case e'_4
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
x : 𝕜
hx' : x ∈ σₙ 𝕜 a
hx : IsGreatest ((fun x ↦ ‖f x‖) '' σₙ 𝕜 a) ((fun x ↦ ‖f x‖) x)
⊢ ‖(cfcₙHom ha) { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ }‖ = (fun x ↦ ‖f x‖) x

Tactic:
norm_cfcₙHom a _

Goal state after:
case e'_4
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
x : 𝕜
hx' : x ∈ σₙ 𝕜 a
hx : IsGreatest ((fun x ↦ ‖f x‖) '' σₙ 𝕜 a) ((fun x ↦ ‖f x‖) x)
⊢ ‖{ toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ }‖ = (fun x ↦ ‖f x‖) x
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: A : Type u_2
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
inst✝ : CompleteSpace A
⊢ UpperHemicontinuous (spectrum ℝ≥0)

Tactic:
by
  obtain ⟨⟨h₁, -⟩, h₂⟩ : IsClosedEmbedding ((↑) : ℝ≥0 → ℝ) := NNReal.isClosedEmbedding_coe
  exact upperHemicontinuous_spectrum ℝ A |>.isInducing_comp h₁ h₂

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NormedField 𝕜
E : Type u_2
inst✝⁴ : NormedRing E
inst✝³ : StarRing E
inst✝² : CStarRing E
inst✝¹ : NormedAlgebra 𝕜 E
inst✝ : CompleteSpace E
u : ↥(unitary E)
a✝ : Nontrivial E
⊢ σ 𝕜 ↑u ⊆ Metric.sphere 0 1

Tactic:
refine fun k hk => mem_sphere_zero_iff_norm.mpr (le_antisymm ?_ ?_)

Goal state after:
case refine_1
𝕜 : Type u_1
inst✝⁵ : NormedField 𝕜
E : Type u_2
inst✝⁴ : NormedRing E
inst✝³ : StarRing E
inst✝² : CStarRing E
inst✝¹ : NormedAlgebra 𝕜 E
inst✝ : CompleteSpace E
u : ↥(unitary E)
a✝ : Nontrivial E
k : 𝕜
hk : k ∈ σ 𝕜 ↑u
⊢ ‖k‖ ≤ 1

case refine_2
𝕜 : Type u_1
inst✝⁵ : NormedField 𝕜
E : Type u_2
inst✝⁴ : NormedRing E
inst✝³ : StarRing E
inst✝² : CStarRing E
inst✝¹ : NormedAlgebra 𝕜 E
inst✝ : CompleteSpace E
u : ↥(unitary E)
a✝ : Nontrivial E
k : 𝕜
hk : k ∈ σ 𝕜 ↑u
⊢ 1 ≤ ‖k‖
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
z : ℂ
hz : z ∈ σ ℂ a
⊢ z = ↑z.re

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

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
z : ℂ
hz : z ∈ σ ℂ a
this : NormedAlgebra ℚ A
⊢ z = ↑z.re
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → β
⊢ ContinuousSup (Compacts α)

Tactic:
constructor
  simp_rw [isEmbedding_coe.continuous_iff, Function.comp_def, coe_sup]
  fun_prop

Goal state after:
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
⊢ expUnitary (a + b) = expUnitary a * expUnitary b

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

Goal state after:
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
this : NormedAlgebra ℚ A
⊢ expUnitary (a + b) = expUnitary a * expUnitary b
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:
by
    rw [h]
    exact f.continuous_toFun

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✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → NonemptyCompacts β
g : α → β → γ
hf : Continuous[inst✝², _] f
hg : Continuous[instTopologicalSpaceProd, inst✝] (Function.uncurry g)
⊢ Continuous[inst✝², _] fun x ↦ NonemptyCompacts.map (g x) ⋯ (f x)

Tactic:
simp_rw [isEmbedding_toCompacts.continuous_iff, Function.comp_def, toCompacts_map]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → NonemptyCompacts β
g : α → β → γ
hf : Continuous[inst✝², _] f
hg : Continuous[instTopologicalSpaceProd, inst✝] (Function.uncurry g)
⊢ Continuous[inst✝², _] fun x ↦ Compacts.map (g x) ⋯ (f x).toCompacts
Module: Mathlib.Analysis.Normed.Operator.Completeness

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
F : Type u_4
Fₗ : Type u_5
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup Fₗ
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NontriviallyNormedField 𝕜₂
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜₂ F
inst✝³ : NormedSpace 𝕜 Fₗ
σ₁₂ : 𝕜 →+* 𝕜₂
f✝ g : E →SL[σ₁₂] F
E' : Type u_6
inst✝² : SeminormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : RingHomIsometric σ₁₂
f : E' → F
s : Set (E' →SL[σ₁₂] F)
hs : Bornology.IsBounded s
hf : f ∈ closure[Pi.topologicalSpace] (DFunLike.coe '' s)
⊢ E' →SL[σ₁₂] F

Tactic:
refine (linearMapOfMemClosureRangeCoe f ?_).mkContinuousOfExistsBound ?_

Goal state after:
case refine_1
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
F : Type u_4
Fₗ : Type u_5
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup Fₗ
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NontriviallyNormedField 𝕜₂
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜₂ F
inst✝³ : NormedSpace 𝕜 Fₗ
σ₁₂ : 𝕜 →+* 𝕜₂
f✝ g : E →SL[σ₁₂] F
E' : Type u_6
inst✝² : SeminormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : RingHomIsometric σ₁₂
f : E' → F
s : Set (E' →SL[σ₁₂] F)
hs : Bornology.IsBounded s
hf : f ∈ closure[Pi.topologicalSpace] (DFunLike.coe '' s)
⊢ f ∈ closure[Pi.topologicalSpace] (range DFunLike.coe)

case refine_2
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
F : Type u_4
Fₗ : Type u_5
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup Fₗ
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NontriviallyNormedField 𝕜₂
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜₂ F
inst✝³ : NormedSpace 𝕜 Fₗ
σ₁₂ : 𝕜 →+* 𝕜₂
f✝ g : E →SL[σ₁₂] F
E' : Type u_6
inst✝² : SeminormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : RingHomIsometric σ₁₂
f : E' → F
s : Set (E' →SL[σ₁₂] F)
hs : Bornology.IsBounded s
hf : f ∈ closure[Pi.topologicalSpace] (DFunLike.coe '' s)
⊢ ∃ C, ∀ (x : E'), ‖(linearMapOfMemClosureRangeCoe f ?refine_1) x‖ ≤ C * ‖x‖
Module: Mathlib.Analysis.Normed.Operator.Completeness

Goal state before:
pp: case refine_1
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
F : Type u_4
Fₗ : Type u_5
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup Fₗ
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NontriviallyNormedField 𝕜₂
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜₂ F
inst✝³ : NormedSpace 𝕜 Fₗ
σ₁₂ : 𝕜 →+* 𝕜₂
f✝ g : E →SL[σ₁₂] F
E' : Type u_6
inst✝² : SeminormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : RingHomIsometric σ₁₂
f : E' → F
s : Set (E' →SL[σ₁₂] F)
hs : Bornology.IsBounded s
hf : f ∈ closure[Pi.topologicalSpace] (DFunLike.coe '' s)
⊢ f ∈ closure[Pi.topologicalSpace] (range DFunLike.coe)

Tactic:
refine closure_mono (image_subset_iff.2 fun g _ => ?_) hf
    exact ⟨g, rfl⟩

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

Goal state before:
pp: case pos
ι : Type u_1
R : Type u_2
S : Type u_3
A : ι → Type u_4
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : MetricSpace R
inst✝¹⁴ : IsTopologicalSemiring R
inst✝¹³ : ContinuousStar R
inst✝¹² : CommRing S
inst✝¹¹ : Algebra R S
inst✝¹⁰ : (i : ι) → NonUnitalRing (A i)
inst✝⁹ : (i : ι) → Module S (A i)
inst✝⁸ : (i : ι) → Module R (A i)
inst✝⁷ : ∀ (i : ι), IsScalarTower R S (A i)
inst✝⁶ : ∀ (i : ι), SMulCommClass R (A i) (A i)
inst✝⁵ : ∀ (i : ι), IsScalarTower R (A i) (A i)
inst✝⁴ : (i : ι) → StarRing (A i)
inst✝³ : (i : ι) → TopologicalSpace (A i)
p : ((i : ι) → A i) → Prop
q : (i : ι) → A i → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R ((i : ι) → A i) p
inst✝¹ : ∀ (i : ι), NonUnitalContinuousFunctionalCalculus R (A i) (q i)
inst✝ : ∀ (i : ι), ContinuousMapZero.UniqueHom R (A i)
f : R → R
a : (i : ι) → A i
hf : ContinuousOn f (⋃ i, quasispectrum R (a i))
ha : p a
ha' : ∀ (i : ι), q i (a i)
hf₀ : f 0 = 0
i : ι
⊢ cfcₙ f a i = cfcₙ f (a i)

Tactic:
have : Nonempty ι := ⟨i⟩

Goal state after:
case pos
ι : Type u_1
R : Type u_2
S : Type u_3
A : ι → Type u_4
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : MetricSpace R
inst✝¹⁴ : IsTopologicalSemiring R
inst✝¹³ : ContinuousStar R
inst✝¹² : CommRing S
inst✝¹¹ : Algebra R S
inst✝¹⁰ : (i : ι) → NonUnitalRing (A i)
inst✝⁹ : (i : ι) → Module S (A i)
inst✝⁸ : (i : ι) → Module R (A i)
inst✝⁷ : ∀ (i : ι), IsScalarTower R S (A i)
inst✝⁶ : ∀ (i : ι), SMulCommClass R (A i) (A i)
inst✝⁵ : ∀ (i : ι), IsScalarTower R (A i) (A i)
inst✝⁴ : (i : ι) → StarRing (A i)
inst✝³ : (i : ι) → TopologicalSpace (A i)
p : ((i : ι) → A i) → Prop
q : (i : ι) → A i → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R ((i : ι) → A i) p
inst✝¹ : ∀ (i : ι), NonUnitalContinuousFunctionalCalculus R (A i) (q i)
inst✝ : ∀ (i : ι), ContinuousMapZero.UniqueHom R (A i)
f : R → R
a : (i : ι) → A i
hf : ContinuousOn f (⋃ i, quasispectrum R (a i))
ha : p a
ha' : ∀ (i : ι), q i (a i)
hf₀ : f 0 = 0
i : ι
this : Nonempty ι
⊢ cfcₙ f a i = cfcₙ f (a i)
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
s t : Set X
hs : IsOpen[inst✝¹] s
hscp : IsCompact (closure[inst✝¹] sᶜᶜ)
ht : IsClosed[inst✝¹] t
hst : t ⊆ s
u v : Set X
huIsOpen : IsOpen[inst✝¹] u
hvIsOpen : IsOpen[inst✝¹] v
hscompl_subset_u : sᶜ ⊆ u
ht_subset_v : t ⊆ v
hDisjointuv : u ⊆ vᶜ
huvc : closure[inst✝¹] u ⊆ vᶜ
P : Set X → Set X → Prop := fun C x ↦ sᶜ ⊆ C
c : Urysohns.CU P := { C := closure[inst✝¹] u, U := tᶜ, P_C_U := ⋯, closed_C := ⋯, open_U := ⋯, subset := ⋯, hP := ⋯ }
⊢ tsupport c.lim ⊆ closure[inst✝¹] uᶜ

Tactic:
apply closure_mono

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
s t : Set X
hs : IsOpen[inst✝¹] s
hscp : IsCompact (closure[inst✝¹] sᶜᶜ)
ht : IsClosed[inst✝¹] t
hst : t ⊆ s
u v : Set X
huIsOpen : IsOpen[inst✝¹] u
hvIsOpen : IsOpen[inst✝¹] v
hscompl_subset_u : sᶜ ⊆ u
ht_subset_v : t ⊆ v
hDisjointuv : u ⊆ vᶜ
huvc : closure[inst✝¹] u ⊆ vᶜ
P : Set X → Set X → Prop := ⋯
c : Urysohns.CU P := ⋯
⊢ Function.support c.lim ⊆ uᶜ
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a b c : A
habc : a = b - c
hbc : b * c = 0
hb : 0 ≤ b
hc : 0 ≤ c
hb' : IsSelfAdjoint b
hc' : IsSelfAdjoint (-c)
⊢ a⁺ = b ∧ a⁻ = c

Tactic:
have ha : IsSelfAdjoint a := habc ▸ hb'.sub <| .of_nonneg hc

Goal state after:
A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a b c : A
habc : a = b - c
hbc : b * c = 0
hb : 0 ≤ b
hc : 0 ≤ c
hb' : IsSelfAdjoint b
hc' : IsSelfAdjoint (-c)
ha : IsSelfAdjoint a
⊢ a⁺ = b ∧ a⁻ = c
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a b c : A
habc : a = b - c
hbc : b * c = 0
hb : 0 ≤ b
hc : 0 ≤ c
hb' : IsSelfAdjoint b
hc' : IsSelfAdjoint (-c)
ha : IsSelfAdjoint a
⊢ a⁺ = b ∧ a⁻ = c

Tactic:
rw [and_iff_left_of_imp ?of_b_eq]

Goal state after:
A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a b c : A
habc : a = b - c
hbc : b * c = 0
hb : 0 ≤ b
hc : 0 ≤ c
hb' : IsSelfAdjoint b
hc' : IsSelfAdjoint (-c)
ha : IsSelfAdjoint a
⊢ a⁺ = b

case of_b_eq
A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a b c : A
habc : a = b - c
hbc : b * c = 0
hb : 0 ≤ b
hc : 0 ≤ c
hb' : IsSelfAdjoint b
hc' : IsSelfAdjoint (-c)
ha : IsSelfAdjoint a
⊢ a⁺ = b → a⁻ = c
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
t : ℝ≥0
ha : IsSelfAdjoint a
ht : spectralRadius ℝ a ≤ ↑t
this : IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)
⊢ SpectrumRestricts ((algebraMap ℝ A) ↑t - a) ⇑Complex.reCLM

Tactic:
try apply IsSelfAdjoint.spectrumRestricts

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
t : ℝ≥0
ha : IsSelfAdjoint a
ht : spectralRadius ℝ a ≤ ↑t
this : IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)
⊢ IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
t : ℝ≥0
ha : IsSelfAdjoint a
ht : spectralRadius ℂ a ≤ ↑t
this : IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)
⊢ SpectrumRestricts a ⇑Complex.reCLM

Tactic:
try apply IsSelfAdjoint.spectrumRestricts

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
t : ℝ≥0
ha : IsSelfAdjoint a
ht : spectralRadius ℂ a ≤ ↑t
this : IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)
⊢ IsSelfAdjoint a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
t : ℝ≥0
ha : IsSelfAdjoint a
ht : spectralRadius ℂ a ≤ ↑t
this : IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)
⊢ IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)

Tactic:
try apply IsSelfAdjoint.spectrumRestricts

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
t : ℝ≥0
ha : IsSelfAdjoint a
ht : spectralRadius ℂ a ≤ ↑t
this : IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)
⊢ IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
t : ℝ≥0
ha : IsSelfAdjoint a
ht : ↑‖a‖₊ ≤ ↑t
this : IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)
⊢ IsSelfAdjoint a

Tactic:
try apply IsSelfAdjoint.spectrumRestricts

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
t : ℝ≥0
ha : IsSelfAdjoint a
ht : ↑‖a‖₊ ≤ ↑t
this : IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)
⊢ IsSelfAdjoint a
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case h.left.«0».left
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ 0 ≤ ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨0, ⋯⟩)

Tactic:
simp
        cfc_tac

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case h.left.«1».left
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ 0 ≤ ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨1, ⋯⟩)

Tactic:
simp
        cfc_tac

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case h.left.«2».left
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ 0 ≤ ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨2, ⋯⟩)

Tactic:
simp
        cfc_tac

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case h.left.«3».left
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ 0 ≤ ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨3, ⋯⟩)

Tactic:
simp
        cfc_tac

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

Goal state before:
pp: R : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : MetricSpace A
inst✝¹ : Algebra R A
inst✝ : IsometricContinuousFunctionalCalculus R A p
a : A
s : Set R
hs : spectrum R a ⊆ s
h₁ :
  LipschitzOnWith 1 (fun f ↦ cfc ((toFun {spectrum R a}) f) a)
    {f | ContinuousOn ((toFun {spectrum R a}) f) (spectrum R a)}
h₂ : LipschitzWith 1 (⇑(ofFun {spectrum R a}) ∘ ⇑(toFun {s}))
h₃ : LipschitzOnWith 1 (⇑(ofFun {spectrum R a}) ∘ ⇑(toFun {s})) {f | ContinuousOn ((toFun {s}) f) s}
⊢ LipschitzOnWith 1 (fun f ↦ cfc ((toFun {s}) f) a) {f | ContinuousOn ((toFun {s}) f) s}

Tactic:
simpa using! h₁.comp h₃ (fun f hf ↦ hf.mono hs)

Goal state after: