mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.LocallyConvex.WeakDual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
f : F →ₗ[𝕜] 𝕜
g : E →ₗ[𝕜] F
⊢ f.toSeminorm.comp g = (f ∘ₗ g).toSeminorm

Tactic:
ext

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
f : F →ₗ[𝕜] 𝕜
g : E →ₗ[𝕜] F
x✝ : E
⊢ (f.toSeminorm.comp g) x✝ = (f ∘ₗ g).toSeminorm x✝
Module: Mathlib.Analysis.LocallyConvex.WeakDual

Goal state before:
pp: ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝⁵ : Finite ι
inst✝⁴ : Field 𝕜
t𝕜 : TopologicalSpace 𝕜
inst✝³ : IsTopologicalRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : T0Space 𝕜
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
x✝ : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜
⊢ φ ∈ Submodule.span 𝕜 (Set.range f) ↔ Continuous[⨅ i, induced (⇑(f i)) t𝕜, t𝕜] ⇑φ

Tactic:
constructor

Goal state after:
case mp
ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝⁵ : Finite ι
inst✝⁴ : Field 𝕜
t𝕜 : TopologicalSpace 𝕜
inst✝³ : IsTopologicalRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : T0Space 𝕜
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
x✝ : TopologicalSpace E := ⋯
⊢ φ ∈ Submodule.span 𝕜 (Set.range f) → Continuous[⨅ i, induced (⇑(f i)) t𝕜, t𝕜] ⇑φ

case mpr
ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝⁵ : Finite ι
inst✝⁴ : Field 𝕜
t𝕜 : TopologicalSpace 𝕜
inst✝³ : IsTopologicalRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : T0Space 𝕜
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
x✝ : TopologicalSpace E := ⋯
⊢ Continuous[⨅ i, induced (⇑(f i)) t𝕜, t𝕜] ⇑φ → φ ∈ Submodule.span 𝕜 (Set.range f)
Module: Mathlib.Analysis.LocallyConvex.HahnBanach

Goal state before:
pp: case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : NormedField 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : Module 𝕜 E
S : Submodule 𝕜 E
f : Dual 𝕜 ↥S
p : Seminorm 𝕜 E
hp : ∀ (x : ↥S), ‖f x‖ ≤ p ↑x
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this✝ : Module ℝ E := restrictScalars ℝ 𝕜 E
this : IsScalarTower ℝ 𝕜 E := IsScalarTower.restrictScalars ℝ 𝕜 E
fr : Dual ℝ ↥S := reLm ∘ₗ ↑ℝ f
g : Dual ℝ E
hg : ∀ (x : ↥S), g ↑x = fr x
hgp : ∀ (x : E), |g x| ≤ (Seminorm.restrictScalars ℝ p) x
x : E
⊢ ‖g.extendRCLike x‖ ≤ p x

Tactic:
apply norm_extendRCLike_le_seminorm
    exact hgp

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Dual

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
⊢ IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    (((topDualPairing 𝕜 E).flip.flip.polar ∘ ⇑OrderDual.ofDual) ((topDualPairing 𝕜 E).flip.polar s))

Tactic:
by
        simpa [LinearMap.flip_flip] using!
          (isClosed_polar _ _).preimage (ContinuousLinearMap.apply 𝕜 𝕜 (E := E)).continuous

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.WeakDual

Goal state before:
pp: ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
t𝕜 : TopologicalSpace 𝕜 := inferInstance
t : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜
this✝ : IsTopologicalAddGroup E
this : WithSeminorms fun i ↦ (f i).toSeminorm
⊢ Continuous[⨅ i, induced (⇑(f i)) inferInstance, inferInstance] ⇑φ ↔
    ∃ s c, φ.toSeminorm ≤ c • s.sup fun i ↦ (f i).toSeminorm

Tactic:
constructor

Goal state after:
case mp
ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
t𝕜 : TopologicalSpace 𝕜 := ⋯
t : TopologicalSpace E := ⋯
this✝ : IsTopologicalAddGroup E
this : WithSeminorms fun i ↦ (f i).toSeminorm
⊢ Continuous[⨅ i, induced (⇑(f i)) inferInstance, inferInstance] ⇑φ →
    ∃ s c, φ.toSeminorm ≤ c • s.sup fun i ↦ (f i).toSeminorm

case mpr
ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
t𝕜 : TopologicalSpace 𝕜 := ⋯
t : TopologicalSpace E := ⋯
this✝ : IsTopologicalAddGroup E
this : WithSeminorms fun i ↦ (f i).toSeminorm
⊢ (∃ s c, φ.toSeminorm ≤ c • s.sup fun i ↦ (f i).toSeminorm) →
    Continuous[⨅ i, induced (⇑(f i)) inferInstance, inferInstance] ⇑φ
Module: Mathlib.Analysis.Normed.Operator.BanachSteinhaus

Goal state before:
pp: E : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : SeminormedAddCommGroup F
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NontriviallyNormedField 𝕜₂
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹ : RingHomIsometric σ₁₂
ι : Type u_5
inst✝ : CompleteSpace E
g : ι → E →SL[σ₁₂] F
h : ∀ (x : E), ∃ C, ∀ (i : ι), ‖(g i) x‖ ≤ C
⊢ ∃ C', ∀ (i : ι), ‖g i‖ ≤ C'

Tactic:
rw [show (∃ C, ∀ i, ‖g i‖ ≤ C) ↔ _ from (NormedSpace.equicontinuous_TFAE g).out 5 2]
  refine (norm_withSeminorms 𝕜₂ F).banach_steinhaus (fun _ x ↦ ?_)
  simpa [bddAbove_def, forall_mem_range] using h x

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

Goal state before:
pp: X : Type u_1
M : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
f : C(X, R)
h : ∀ (x : X), IsUnit (f x)
⊢ { toFun := fun x ↦ ↑((unitsOfForallIsUnit h) x), continuous_toFun := ⋯ } = f

Tactic:
ext

Goal state after:
X : Type u_1
M : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
f : C(X, R)
h : ∀ (x : X), IsUnit (f x)
a✝ : X
⊢ { toFun := fun x ↦ ↑((unitsOfForallIsUnit h) x), continuous_toFun := ⋯ } a✝ = f a✝
Module: Mathlib.Topology.ContinuousMap.Units

Goal state before:
pp: X : Type u_1
R : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
f : C(X, R)
h : ∀ (x : X), IsUnit (f x)
⊢ ↑(unitsLift (unitsOfForallIsUnit h)) = f

Tactic:
ext

Goal state after:
X : Type u_1
R : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
f : C(X, R)
h : ∀ (x : X), IsUnit (f x)
a✝ : X
⊢ ↑(unitsLift (unitsOfForallIsUnit h)) a✝ = f a✝
Module: Mathlib.Analysis.Normed.Module.WeakDual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
s : Set (WeakDual 𝕜 E)
⊢ Bornology.IsBounded s ↔ IsVonNBounded 𝕜 s

Tactic:
constructor

Goal state after:
case mp
𝕜 : Type u_1
E : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
s : Set (WeakDual 𝕜 E)
⊢ Bornology.IsBounded s → IsVonNBounded 𝕜 s

case mpr
𝕜 : Type u_1
E : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
s : Set (WeakDual 𝕜 E)
⊢ IsVonNBounded 𝕜 s → Bornology.IsBounded s
Module: Mathlib.Analysis.Normed.Module.WeakDual

Goal state before:
pp: case h
𝕜 : Type u_1
E : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : SeparableSpace E
⊢ (∀ (n : ℕ), Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ((fun n φ ↦ φ (denseSeq E n)) n)) ∧
    ∀ ⦃x y : WeakDual 𝕜 E⦄, x ≠ y → ∃ n, (fun n φ ↦ φ (denseSeq E n)) n x ≠ (fun n φ ↦ φ (denseSeq E n)) n y

Tactic:
constructor

Goal state after:
case h.left
𝕜 : Type u_1
E : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : SeparableSpace E
⊢ ∀ (n : ℕ), Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun φ ↦ φ (denseSeq E n)

case h.right
𝕜 : Type u_1
E : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : SeparableSpace E
⊢ ∀ ⦃x y : WeakDual 𝕜 E⦄, x ≠ y → ∃ n, x (denseSeq E n) ≠ y (denseSeq E n)
Module: Mathlib.Analysis.Normed.Module.WeakDual

Goal state before:
pp: 𝕜 : Type u_5
F : Type u_7
inst✝⁶ : RCLike 𝕜
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module 𝕜 F
inst✝³ : TopologicalSpace F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : Module ℝ F
inst✝ : IsScalarTower ℝ 𝕜 F
f : WeakDual ℝ F
x : F
⊢ re ((extendRCLikeL f) x) = f x

Tactic:
simp [extendRCLikeL_apply_apply]

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
k : Set X
k_comp : IsCompact k
k_closed : IsClosed[inst✝²] k
sk : s ⊆ interior k
kt : k ⊆ tᶜ
P : Set X → Set X → Prop := fun C x ↦ IsCompact C
⊢ ∀ {c u : Set X},
    IsClosed[inst✝²] c →
      P c u →
        IsOpen[inst✝²] u →
          c ⊆ u → ∃ v, IsOpen[inst✝²] v ∧ c ⊆ v ∧ closure[inst✝²] v ⊆ u ∧ P c v ∧ P (closure[inst✝²] v) u

Tactic:
rintro c u - c_comp u_open cu

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
k : Set X
k_comp : IsCompact k
k_closed : IsClosed[inst✝²] k
sk : s ⊆ interior k
kt : k ⊆ tᶜ
P : Set X → Set X → Prop := fun C x ↦ IsCompact C
c u : Set X
c_comp : P c u
u_open : IsOpen[inst✝²] u
cu : c ⊆ u
⊢ ∃ v, IsOpen[inst✝²] v ∧ c ⊆ v ∧ closure[inst✝²] v ⊆ u ∧ P c v ∧ P (closure[inst✝²] v) u
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
⊢ (↑(algebraMapCLM ℝ≥0 𝕜)).comp 0 = 0

Tactic:
ext

Goal state after:
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
a✝ : X
⊢ ((↑(algebraMapCLM ℝ≥0 𝕜)).comp 0) a✝ = 0 a✝
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
x : X
g : C(X, 𝕜)
hI : g ∈ I
hgx : g x ≠ 0
this : ∀ᶠ (z : X) in nhds x, g z ≠ 0
⊢ (↑(algebraMapCLM ℝ≥0 𝕜)).comp { toFun := fun x ↦ ‖g x‖₊ ^ 2, continuous_toFun := ⋯ } = star g * g

Tactic:
ext

Goal state after:
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
x : X
g : C(X, 𝕜)
hI : g ∈ I
hgx : g x ≠ 0
this : ∀ᶠ (z : X) in nhds x, g z ≠ 0
a✝ : X
⊢ ((↑(algebraMapCLM ℝ≥0 𝕜)).comp { toFun := fun x ↦ ‖g x‖₊ ^ 2, continuous_toFun := ⋯ }) a✝ = (star g * g) a✝
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
g' : C(X, ℝ≥0)
hI' : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I
hgt' : ∀ x ∈ t, 0 < g' x
c : ℝ≥0
hc : c > 0
hgc' : t ⊆ {y | c ≤ g' y}
g : C(X, ℝ≥0)
hg : ∀ (x : X), (g * g') x ≤ 1
hgc : Set.EqOn (⇑(g * g')) 1 {x | c ≤ g' x}
⊢ (↑(algebraMapCLM ℝ≥0 𝕜)).comp (g * g') = (↑(algebraMapCLM ℝ≥0 𝕜)).comp g * (↑(algebraMapCLM ℝ≥0 𝕜)).comp g'

Tactic:
ext

Goal state after:
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
g' : C(X, ℝ≥0)
hI' : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I
hgt' : ∀ x ∈ t, 0 < g' x
c : ℝ≥0
hc : c > 0
hgc' : t ⊆ {y | c ≤ g' y}
g : C(X, ℝ≥0)
hg : ∀ (x : X), (g * g') x ≤ 1
hgc : Set.EqOn (⇑(g * g')) 1 {x | c ≤ g' x}
a✝ : X
⊢ ((↑(algebraMapCLM ℝ≥0 𝕜)).comp (g * g')) a✝ = ((↑(algebraMapCLM ℝ≥0 𝕜)).comp g * (↑(algebraMapCLM ℝ≥0 𝕜)).comp g') a✝
Module: Mathlib.Analysis.CStarAlgebra.GelfandDuality

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
a b : A
ha : IsStarNormal a
hb : IsStarNormal b
hcomm : Commute a b
hab : a * b = 0
S : NonUnitalStarSubalgebra ℂ A := (adjoin ℂ {a, b}).topologicalClosure
hS : IsClosed ↑S
hcomm₁ : Commute (star a) b
hcomm₂ : Commute a (star b)
this : IsMulCommutative ↥(adjoin ℂ {a, b})
x✝¹ : NonUnitalCommRing ↥S := (adjoin ℂ {a, b}).nonUnitalCommRingTopologicalClosure ⋯
x✝ : NonUnitalCommCStarAlgebra ↥S :=
  { toNonUnitalNormedRing := (nonUnitalCStarAlgebra S).toNonUnitalNormedRing, mul_comm := ⋯,
    toStarRing := (nonUnitalCStarAlgebra S).toStarRing, toCompleteSpace := ⋯, toCStarRing := ⋯,
    toNormedSpace := (nonUnitalCStarAlgebra S).toNormedSpace, toIsScalarTower := ⋯, toSMulCommClass := ⋯,
    toStarModule := ⋯ }
⊢ ⟨a, ?m.172⟩ * ⟨b, ?m.175⟩ = 0

Tactic:
ext

Goal state after:
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
a b : A
ha : IsStarNormal a
hb : IsStarNormal b
hcomm : Commute a b
hab : a * b = 0
S : NonUnitalStarSubalgebra ℂ A := ⋯
hS : IsClosed ↑S
hcomm₁ : Commute (star a) b
hcomm₂ : Commute a (star b)
this : IsMulCommutative ↥(adjoin ℂ {a, b})
x✝¹ : NonUnitalCommRing ↥S := ⋯
x✝ : NonUnitalCommCStarAlgebra ↥S := ⋯
⊢ ↑(⟨a, ?m.172⟩ * ⟨b, ?m.175⟩) = ↑0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Pi

Goal state before:
pp: A : Type u_1
B : Type u_2
R : Type u_3
S : Type u_4
inst✝²³ : CommSemiring R
inst✝²² : StarRing R
inst✝²¹ : MetricSpace R
inst✝²⁰ : IsTopologicalSemiring R
inst✝¹⁹ : ContinuousStar R
inst✝¹⁸ : CommRing S
inst✝¹⁷ : Algebra R S
inst✝¹⁶ : Ring A
inst✝¹⁵ : Ring B
inst✝¹⁴ : Algebra S A
inst✝¹³ : Algebra S B
inst✝¹² : Algebra R A
inst✝¹¹ : Algebra R B
inst✝¹⁰ : IsScalarTower R S A
inst✝⁹ : IsScalarTower R S B
inst✝⁸ : StarRing A
inst✝⁷ : StarRing B
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace B
pab : A × B → Prop
pa : A → Prop
pb : B → Prop
inst✝⁴ : ContinuousFunctionalCalculus R (A × B) pab
inst✝³ : ContinuousFunctionalCalculus R A pa
inst✝² : ContinuousFunctionalCalculus R B pb
inst✝¹ : ContinuousMap.UniqueHom R A
inst✝ : ContinuousMap.UniqueHom R B
f : R → R
a : A
b : B
hf : ContinuousOn f (spectrum R a ∪ spectrum R b)
hab : pab (a, b)
ha : pa a
hb : pb b
⊢ cfc f (a, b) = (cfc f a, cfc f b)

Tactic:
ext

Goal state after:
case fst
A : Type u_1
B : Type u_2
R : Type u_3
S : Type u_4
inst✝²³ : CommSemiring R
inst✝²² : StarRing R
inst✝²¹ : MetricSpace R
inst✝²⁰ : IsTopologicalSemiring R
inst✝¹⁹ : ContinuousStar R
inst✝¹⁸ : CommRing S
inst✝¹⁷ : Algebra R S
inst✝¹⁶ : Ring A
inst✝¹⁵ : Ring B
inst✝¹⁴ : Algebra S A
inst✝¹³ : Algebra S B
inst✝¹² : Algebra R A
inst✝¹¹ : Algebra R B
inst✝¹⁰ : IsScalarTower R S A
inst✝⁹ : IsScalarTower R S B
inst✝⁸ : StarRing A
inst✝⁷ : StarRing B
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace B
pab : A × B → Prop
pa : A → Prop
pb : B → Prop
inst✝⁴ : ContinuousFunctionalCalculus R (A × B) pab
inst✝³ : ContinuousFunctionalCalculus R A pa
inst✝² : ContinuousFunctionalCalculus R B pb
inst✝¹ : ContinuousMap.UniqueHom R A
inst✝ : ContinuousMap.UniqueHom R B
f : R → R
a : A
b : B
hf : ContinuousOn f (spectrum R a ∪ spectrum R b)
hab : pab (a, b)
ha : pa a
hb : pb b
⊢ (cfc f (a, b)).1 = (cfc f a, cfc f b).1

case snd
A : Type u_1
B : Type u_2
R : Type u_3
S : Type u_4
inst✝²³ : CommSemiring R
inst✝²² : StarRing R
inst✝²¹ : MetricSpace R
inst✝²⁰ : IsTopologicalSemiring R
inst✝¹⁹ : ContinuousStar R
inst✝¹⁸ : CommRing S
inst✝¹⁷ : Algebra R S
inst✝¹⁶ : Ring A
inst✝¹⁵ : Ring B
inst✝¹⁴ : Algebra S A
inst✝¹³ : Algebra S B
inst✝¹² : Algebra R A
inst✝¹¹ : Algebra R B
inst✝¹⁰ : IsScalarTower R S A
inst✝⁹ : IsScalarTower R S B
inst✝⁸ : StarRing A
inst✝⁷ : StarRing B
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace B
pab : A × B → Prop
pa : A → Prop
pb : B → Prop
inst✝⁴ : ContinuousFunctionalCalculus R (A × B) pab
inst✝³ : ContinuousFunctionalCalculus R A pa
inst✝² : ContinuousFunctionalCalculus R B pb
inst✝¹ : ContinuousMap.UniqueHom R A
inst✝ : ContinuousMap.UniqueHom R B
f : R → R
a : A
b : B
hf : ContinuousOn f (spectrum R a ∪ spectrum R b)
hab : pab (a, b)
ha : pa a
hb : pb b
⊢ (cfc f (a, b)).2 = (cfc f a, cfc f b).2
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
b : A
a : A := star b * b
a_def : a = star b * b
ha : IsSelfAdjoint a
c : A := b * a⁻
h_eq_negPart_a : -(star c * c) = a⁻ ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x
⊢ SpectrumRestricts (2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)) ⇑ContinuousMap.realToNNReal

Tactic:
refine SpectrumRestricts.nnreal_add ?_ ?_ ?_ h_c_spec₀

Goal state after:
case refine_1
A : Type u_1
inst✝ : CStarAlgebra A
b : A
a : A := star b * b
a_def : a = star b * b
ha : IsSelfAdjoint a
c : A := b * a⁻
h_eq_negPart_a : -(star c * c) = a⁻ ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x
⊢ IsSelfAdjoint (2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2))

case refine_2
A : Type u_1
inst✝ : CStarAlgebra A
b : A
a : A := star b * b
a_def : a = star b * b
ha : IsSelfAdjoint a
c : A := b * a⁻
h_eq_negPart_a : -(star c * c) = a⁻ ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x
⊢ IsSelfAdjoint (-(star c * c))

case refine_3
A : Type u_1
inst✝ : CStarAlgebra A
b : A
a : A := star b * b
a_def : a = star b * b
ha : IsSelfAdjoint a
c : A := b * a⁻
h_eq_negPart_a : -(star c * c) = a⁻ ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x
⊢ SpectrumRestricts (2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2)) ⇑ContinuousMap.realToNNReal
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝ : PartialOrder A := spectralOrder A
x y : A
⊢ x ≤ y ↔ ∃ p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s), y = x + p

Tactic:
constructor

Goal state after:
case mp
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝ : PartialOrder A := ⋯
x y : A
⊢ x ≤ y → ∃ p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s), y = x + p

case mpr
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝ : PartialOrder A := ⋯
x y : A
⊢ (∃ p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s), y = x + p) → x ≤ y
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝¹³ : NonUnitalCStarAlgebra A
R : Type u_2
inst✝¹² : Semifield R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : Algebra R ℂ
inst✝³ : IsScalarTower R ℂ A
p : A → Prop
p' : Unitization ℂ A → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R A p
inst✝¹ : ContinuousFunctionalCalculus R (Unitization ℂ A) p'
inst✝ : ContinuousMapZero.UniqueHom R (Unitization ℂ A)
hp : ∀ {a : A}, p' ↑a ↔ p a
a : A
f : R → R
hf₀ : f 0 = 0
⊢ ↑(cfcₙ f a) = cfc f ↑a

Tactic:
by_cases h : ContinuousOn f (σₙ R a) ∧ p a
  · obtain ⟨hf, ha⟩ := h
    rw [← cfcₙ_eq_cfc (quasispectrum_inr_eq R ℂ a ▸ hf)]
    exact (inrNonUnitalStarAlgHom ℂ A).map_cfcₙ f a
  · obtain (hf | ha) := not_and_or.mp h
    · rw [cfcₙ_apply_of_not_continuousOn a hf, inr_zero,
        cfc_apply_of_not_continuousOn _ (quasispectrum_eq_spectrum_inr' R ℂ a ▸ hf)]
    · rw [cfcₙ_apply_of_not_predicate a ha, inr_zero,
        cfc_apply_of_not_predicate _ (not_iff_not.mpr hp |>.mpr ha)]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: case inr.inr
A : Type u_1
inst✝¹¹ : PartialOrder A
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : StarRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
x y : ℝ≥0
ha : 0 ≤ a
hx : 0 < x
hy : 0 < y
u : ℝ≥0
⊢ ((fun x ↦ x.nnrpow y) ∘ fun x_1 ↦ x_1.nnrpow x) u = u.nnrpow (x * y)

Tactic:
ext

Goal state after:
case inr.inr
A : Type u_1
inst✝¹¹ : PartialOrder A
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : StarRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
x y : ℝ≥0
ha : 0 ≤ a
hx : 0 < x
hy : 0 < y
u : ℝ≥0
⊢ ↑(((fun x ↦ x.nnrpow y) ∘ fun x_1 ↦ x_1.nnrpow x) u) = ↑(u.nnrpow (x * y))
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝²⁵ : PartialOrder A
inst✝²⁴ : NonUnitalRing A
inst✝²³ : TopologicalSpace A
inst✝²² : StarRing A
inst✝²¹ : Module ℝ A
inst✝²⁰ : SMulCommClass ℝ A A
inst✝¹⁹ : IsScalarTower ℝ A A
inst✝¹⁸ : StarOrderedRing A
inst✝¹⁷ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹⁶ : NonnegSpectrumClass ℝ A
inst✝¹⁵ : IsSemitopologicalRing A
inst✝¹⁴ : T2Space A
B : Type u_2
inst✝¹³ : PartialOrder B
inst✝¹² : NonUnitalRing B
inst✝¹¹ : TopologicalSpace B
inst✝¹⁰ : StarRing B
inst✝⁹ : Module ℝ B
inst✝⁸ : SMulCommClass ℝ B B
inst✝⁷ : IsScalarTower ℝ B B
inst✝⁶ : NonUnitalContinuousFunctionalCalculus ℝ B IsSelfAdjoint
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ (A × B) IsSelfAdjoint
inst✝⁴ : IsSemitopologicalRing B
inst✝³ : T2Space B
inst✝² : NonnegSpectrumClass ℝ B
inst✝¹ : NonnegSpectrumClass ℝ (A × B)
inst✝ : StarOrderedRing B
a : A
b : B
x : ℝ≥0
ha : 0 ≤ a
hb : 0 ≤ b
⊢ 0.1 ≤ (a, b).1 ∧ 0.2 ≤ (a, b).2

Tactic:
constructor

Goal state after:
case left
A : Type u_1
inst✝²⁵ : PartialOrder A
inst✝²⁴ : NonUnitalRing A
inst✝²³ : TopologicalSpace A
inst✝²² : StarRing A
inst✝²¹ : Module ℝ A
inst✝²⁰ : SMulCommClass ℝ A A
inst✝¹⁹ : IsScalarTower ℝ A A
inst✝¹⁸ : StarOrderedRing A
inst✝¹⁷ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹⁶ : NonnegSpectrumClass ℝ A
inst✝¹⁵ : IsSemitopologicalRing A
inst✝¹⁴ : T2Space A
B : Type u_2
inst✝¹³ : PartialOrder B
inst✝¹² : NonUnitalRing B
inst✝¹¹ : TopologicalSpace B
inst✝¹⁰ : StarRing B
inst✝⁹ : Module ℝ B
inst✝⁸ : SMulCommClass ℝ B B
inst✝⁷ : IsScalarTower ℝ B B
inst✝⁶ : NonUnitalContinuousFunctionalCalculus ℝ B IsSelfAdjoint
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ (A × B) IsSelfAdjoint
inst✝⁴ : IsSemitopologicalRing B
inst✝³ : T2Space B
inst✝² : NonnegSpectrumClass ℝ B
inst✝¹ : NonnegSpectrumClass ℝ (A × B)
inst✝ : StarOrderedRing B
a : A
b : B
x : ℝ≥0
ha : 0 ≤ a
hb : 0 ≤ b
⊢ 0.1 ≤ (a, b).1

case right
A : Type u_1
inst✝²⁵ : PartialOrder A
inst✝²⁴ : NonUnitalRing A
inst✝²³ : TopologicalSpace A
inst✝²² : StarRing A
inst✝²¹ : Module ℝ A
inst✝²⁰ : SMulCommClass ℝ A A
inst✝¹⁹ : IsScalarTower ℝ A A
inst✝¹⁸ : StarOrderedRing A
inst✝¹⁷ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹⁶ : NonnegSpectrumClass ℝ A
inst✝¹⁵ : IsSemitopologicalRing A
inst✝¹⁴ : T2Space A
B : Type u_2
inst✝¹³ : PartialOrder B
inst✝¹² : NonUnitalRing B
inst✝¹¹ : TopologicalSpace B
inst✝¹⁰ : StarRing B
inst✝⁹ : Module ℝ B
inst✝⁸ : SMulCommClass ℝ B B
inst✝⁷ : IsScalarTower ℝ B B
inst✝⁶ : NonUnitalContinuousFunctionalCalculus ℝ B IsSelfAdjoint
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ (A × B) IsSelfAdjoint
inst✝⁴ : IsSemitopologicalRing B
inst✝³ : T2Space B
inst✝² : NonnegSpectrumClass ℝ B
inst✝¹ : NonnegSpectrumClass ℝ (A × B)
inst✝ : StarOrderedRing B
a : A
b : B
x : ℝ≥0
ha : 0 ≤ a
hb : 0 ≤ b
⊢ 0.2 ≤ (a, b).2