mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : SMulCommClass ℝ 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace ℝ F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : SMulCommClass ℝ 𝕜 F'
n : ℕ∞
K : Compacts E
inst✝ : LinearMap.CompatibleSMul F F' ℝ 𝕜
i : ℕ
T : F →L[𝕜] F'
f : 𝓓^{n}_{K}(E, F)
T' : F →L[ℝ] F' := ContinuousLinearMap.restrictScalars ℝ T
hi : ↑i ≤ n
x : E
hx : x ∈ K
⊢ ‖T'‖ * (ContDiffMapSupportedIn.seminorm ℝ E F n K i) f ≤ ‖T'‖ * (ContDiffMapSupportedIn.seminorm ℝ E F n K i) f

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
⊢ (∫⁻ (a : α), ENNReal.ofReal (‖↑↑f a‖ ^ 2) ∂μ).toReal = (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ).toReal

pp: case hf
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
⊢ 0 ≤ᵐ[μ] fun x ↦ ‖↑↑f x‖ ^ 2

pp: case hfm
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
⊢ AEStronglyMeasurable (fun x ↦ ‖↑↑f x‖ ^ 2) μ

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
⊢ (∫⁻ (a : α), ENNReal.ofReal (‖↑↑f a‖ ^ 2) ∂μ).toReal = (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ).toReal

case hf
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
⊢ 0 ≤ᵐ[μ] fun x ↦ ‖↑↑f x‖ ^ 2

case hfm
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
⊢ AEStronglyMeasurable (fun x ↦ ‖↑↑f x‖ ^ 2) μ
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: case inr
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁶ : RCLike 𝕜
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : Module 𝕜 A
inst✝¹² : StarRing A
inst✝¹¹ : PartialOrder A
inst✝¹⁰ : StarOrderedRing A
inst✝⁹ : IsScalarTower 𝕜 A A
inst✝⁸ : SMulCommClass 𝕜 A A
inst✝⁷ : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝⁶ : Module ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsTopologicalRing A
inst✝¹ : T2Space A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
f : 𝕜 → 𝕜
a : A
ha : p a
hf : ContinuousOn f ((fun z ↦ ↑‖z‖) '' quasispectrum 𝕜 a)
hf0 : ¬f 0 = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁶ : RCLike 𝕜
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : Module 𝕜 A
inst✝¹² : StarRing A
inst✝¹¹ : PartialOrder A
inst✝¹⁰ : StarOrderedRing A
inst✝⁹ : IsScalarTower 𝕜 A A
inst✝⁸ : SMulCommClass 𝕜 A A
inst✝⁷ : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝⁶ : Module ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsTopologicalRing A
inst✝¹ : T2Space A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : p a
⊢ (fun z ↦ ↑‖z‖) '' quasispectrum 𝕜 a = (fun z ↦ ↑‖z‖) '' quasispectrum 𝕜 a

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: A : Type u_2
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : Algebra ℝ A
inst✝⁵ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsTopologicalRing A
inst✝ : T2Space A
a : A
ha : IsSelfAdjoint a
⊢ cfc (fun x ↦ ‖x‖) a = cfc (fun x ↦ ‖x‖) a

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: case e_f
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
x : α
h_two : 2 = ↑2
⊢ ‖↑↑f x‖ₑ ^ 2 = ↑(‖↑↑f x‖₊ ^ 2)

Tactic:
]

Goal state after:
case e_f
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
x : α
h_two : 2 = ↑2
⊢ ‖↑↑f x‖ₑ ^ 2 = ↑(‖↑↑f x‖₊ ^ 2)
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: case negSucc
A : Type u_2
inst✝¹⁰ : Ring A
inst✝⁹ : StarRing A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Algebra ℝ A
inst✝⁶ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsTopologicalRing A
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
n : ℕ
⊢ ↑↑(n + 1) = ↑|Int.negSucc n|

Tactic:
]

Goal state after:
case negSucc
A : Type u_2
inst✝¹⁰ : Ring A
inst✝⁹ : StarRing A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Algebra ℝ A
inst✝⁶ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsTopologicalRing A
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
n : ℕ
⊢ ↑↑(n + 1) = ↑|Int.negSucc n|
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: case negSucc
A : Type u_2
inst✝¹⁰ : Ring A
inst✝⁹ : StarRing A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Algebra ℝ A
inst✝⁶ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsTopologicalRing A
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
n : ℕ
⊢ abs ↑(Int.negSucc n) = ↑|Int.negSucc n|

Tactic:
rw [Int.cast_negSucc, abs_neg, abs_natCast, ← Int.cast_natCast]

Goal state after:
case negSucc
A : Type u_2
inst✝¹⁰ : Ring A
inst✝⁹ : StarRing A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Algebra ℝ A
inst✝⁶ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsTopologicalRing A
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
n : ℕ
⊢ ↑↑(n + 1) = ↑|Int.negSucc n|
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
h_two : ENNReal.toReal 2 = 2
⊢ eLpNorm (↑↑f) 2 μ ^ 2 = ∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ

pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
h_two : ENNReal.toReal 2 = 2
⊢ ∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ ≠ ∞

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
h_two : ENNReal.toReal 2 = 2
⊢ eLpNorm (↑↑f) 2 μ ^ 2 = ∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ

α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
h_two : ENNReal.toReal 2 = 2
⊢ ∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ ≠ ∞
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : RCLike 𝕜
inst✝¹¹ : Ring A
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : StarRing A
inst✝⁸ : PartialOrder A
inst✝⁷ : StarOrderedRing A
inst✝⁶ : Algebra 𝕜 A
inst✝⁵ : ContinuousFunctionalCalculus 𝕜 A p
inst✝⁴ : Algebra ℝ A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsTopologicalRing A
inst✝¹ : T2Space A
inst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
f : 𝕜 → 𝕜
a : A
ha : p a
hf : ContinuousOn f ((fun z ↦ ↑‖z‖) '' spectrum 𝕜 a)
⊢ cfc (fun x ↦ f ↑‖x‖) a = cfc (fun x ↦ f ↑‖x‖) a

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
h_two : ENNReal.toReal 2 = 2
⊢ (∫⁻ (a : α), ‖↑↑f a‖ₑ ^ 2 ∂μ) ^ (2⁻¹ * ↑2) = ∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
h_two : ENNReal.toReal 2 = 2
⊢ (∫⁻ (a : α), ‖↑↑f a‖ₑ ^ 2 ∂μ) ^ (2⁻¹ * ↑2) = ∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: case pos
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
n k : ℕ∞
K : Compacts E
i : ℕ
f : 𝓓^{n}_{K}(E, F)
hk : k + 1 ≤ n
⊢ ↑i ≤ k →
    ∀ x ∈ K, ‖iteratedFDeriv ℝ i (⇑((fderivLM 𝕜 n k) f)) x‖ ≤ (ContDiffMapSupportedIn.seminorm 𝕜 E F n K (i + 1)) f

Tactic:
]

Goal state after:
case pos
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
n k : ℕ∞
K : Compacts E
i : ℕ
f : 𝓓^{n}_{K}(E, F)
hk : k + 1 ≤ n
⊢ ↑i ≤ k →
    ∀ x ∈ K, ‖iteratedFDeriv ℝ i (⇑((fderivLM 𝕜 n k) f)) x‖ ≤ (ContDiffMapSupportedIn.seminorm 𝕜 E F n K (i + 1)) f
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: A : Type u_2
inst✝⁹ : Ring A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : StarRing A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsTopologicalRing A
inst✝¹ : T2Space A
inst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
⊢ star a * a = star a * a

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : RCLike 𝕜
inst✝¹¹ : Ring A
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : StarRing A
inst✝⁸ : PartialOrder A
inst✝⁷ : StarOrderedRing A
inst✝⁶ : Algebra 𝕜 A
inst✝⁵ : ContinuousFunctionalCalculus 𝕜 A p
inst✝⁴ : Algebra ℝ A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsTopologicalRing A
inst✝¹ : T2Space A
inst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : p a
⊢ (fun z ↦ ↑‖z‖) '' spectrum 𝕜 a = (fun z ↦ ↑‖z‖) '' spectrum 𝕜 a

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
h_two : ENNReal.toReal 2 = 2
⊢ eLpNorm (↑↑f) 2 μ < ∞

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
h_two : ENNReal.toReal 2 = 2
⊢ eLpNorm (↑↑f) 2 μ < ∞
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: A : Type u_2
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : CStarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
⊢ a = 0 ↔ a = 0

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f f' g : ↥(Lp E 2 μ)
⊢ ⟪f + f', g⟫ = ⟪f, g⟫ + ⟪f', g⟫

Tactic:
simp_rw [inner_def, ← integral_add (integrable_inner (𝕜 := 𝕜) f g) (integrable_inner f' g),
    ← inner_add_left]

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f f' g : ↥(Lp E 2 μ)
⊢ ∫ (a : α), ⟪↑↑(f + f') a, ↑↑g a⟫ ∂μ = ∫ (a : α), ⟪↑↑f a + ↑↑f' a, ↑↑g a⟫ ∂μ
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: A : Type u_2
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : CStarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
⊢ ‖a‖ * ‖a‖ = ‖a‖ * ‖a‖

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f g : ↥(Lp E 2 μ)
r : 𝕜
⊢ ∫ (a : α), ⟪↑↑(r • f) a, ↑↑g a⟫ ∂μ = ∫ (a : α), (starRingEnd 𝕜) r • ⟪↑↑f a, ↑↑g a⟫ ∂μ

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f g : ↥(Lp E 2 μ)
r : 𝕜
⊢ ∫ (a : α), ⟪↑↑(r • f) a, ↑↑g a⟫ ∂μ = ∫ (a : α), (starRingEnd 𝕜) r • ⟪↑↑f a, ↑↑g a⟫ ∂μ
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f g : ↥(Lp E 2 μ)
r : 𝕜
x : α
hx : ↑↑(r • f) x = (r • ↑↑f) x
⊢ ⟪r • ↑↑f x, ↑↑g x⟫ = ⟪r • ↑↑f x, ↑↑g x⟫

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
n : ℕ∞
K : Compacts E
m : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
μ : Measure E
μ_finite : IsFiniteMeasure (μ.restrict ↑K)
f : 𝓓^{n}_{K}(E, F)
⊢ IntegrableOn (⇑f) (↑K) μ

Tactic:
]

Goal state after:
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
n : ℕ∞
K : Compacts E
m : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
μ : Measure E
μ_finite : IsFiniteMeasure (μ.restrict ↑K)
f : 𝓓^{n}_{K}(E, F)
⊢ IntegrableOn (⇑f) (↑K) μ
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
s : Set α
f : ↥(Lp E 2 μ)
hs : MeasurableSet s
c : E
hμs : μ s ≠ ∞
⊢ ∫ (a : α), ⟪↑↑(indicatorConstLp 2 hs hμs c) a, ↑↑f a⟫ ∂μ = ∫ (x : α), s.indicator (fun x ↦ ⟪c, ↑↑f x⟫) x ∂μ

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
s : Set α
f : ↥(Lp E 2 μ)
hs : MeasurableSet s
c : E
hμs : μ s ≠ ∞
⊢ ∫ (a : α), ⟪↑↑(indicatorConstLp 2 hs hμs c) a, ↑↑f a⟫ ∂μ = ∫ (x : α), s.indicator (fun x ↦ ⟪c, ↑↑f x⟫) x ∂μ
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
n : ℕ∞
K : Compacts E
m : MeasurableSpace E
inst✝⁷ : OpensMeasurableSpace E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝⁶ : NormedAddCommGroup F₁
inst✝⁵ : NormedSpace 𝕜 F₁
inst✝⁴ : NormedSpace ℝ F₁
inst✝³ : NormedAddCommGroup F₂
inst✝² : NormedSpace 𝕜 F₂
inst✝¹ : NormedAddCommGroup F₃
inst✝ : NormedSpace 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
hφ : IntegrableOn φ (↑K) μ
f : 𝓓^{n}_{K}(E, F₁)
this : Integrable (fun x ↦ (B (f x)) (φ x)) μ
⊢ Integrable (fun x ↦ (B (f x)) (φ x)) μ

pp: 𝕜 : Type u_1
E : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
n : ℕ∞
K : Compacts E
m : MeasurableSpace E
inst✝⁷ : OpensMeasurableSpace E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝⁶ : NormedAddCommGroup F₁
inst✝⁵ : NormedSpace 𝕜 F₁
inst✝⁴ : NormedSpace ℝ F₁
inst✝³ : NormedAddCommGroup F₂
inst✝² : NormedSpace 𝕜 F₂
inst✝¹ : NormedAddCommGroup F₃
inst✝ : NormedSpace 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
hφ : IntegrableOn φ (↑K) μ
f : 𝓓^{n}_{K}(E, F₁)
this : IntegrableOn (fun x ↦ (B (f x)) (φ x)) (↑K) μ
⊢ (support fun x ↦ (B (f x)) (φ x)) ⊆ ↑K

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
n : ℕ∞
K : Compacts E
m : MeasurableSpace E
inst✝⁷ : OpensMeasurableSpace E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝⁶ : NormedAddCommGroup F₁
inst✝⁵ : NormedSpace 𝕜 F₁
inst✝⁴ : NormedSpace ℝ F₁
inst✝³ : NormedAddCommGroup F₂
inst✝² : NormedSpace 𝕜 F₂
inst✝¹ : NormedAddCommGroup F₃
inst✝ : NormedSpace 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
hφ : IntegrableOn φ (↑K) μ
f : 𝓓^{n}_{K}(E, F₁)
this : Integrable (fun x ↦ (B (f x)) (φ x)) μ
⊢ Integrable (fun x ↦ (B (f x)) (φ x)) μ

𝕜 : Type u_1
E : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
n : ℕ∞
K : Compacts E
m : MeasurableSpace E
inst✝⁷ : OpensMeasurableSpace E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝⁶ : NormedAddCommGroup F₁
inst✝⁵ : NormedSpace 𝕜 F₁
inst✝⁴ : NormedSpace ℝ F₁
inst✝³ : NormedAddCommGroup F₂
inst✝² : NormedSpace 𝕜 F₂
inst✝¹ : NormedAddCommGroup F₃
inst✝ : NormedSpace 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
hφ : IntegrableOn φ (↑K) μ
f : 𝓓^{n}_{K}(E, F₁)
this : IntegrableOn (fun x ↦ (B (f x)) (φ x)) (↑K) μ
⊢ (support fun x ↦ (B (f x)) (φ x)) ⊆ ↑K
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
s : Set α
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E
f : ↥(Lp E 2 μ)
⊢ ∫ (x : α) in s, ⟪c, ↑↑f x⟫ ∂μ = ∫ (x : α) in s, ⟪c, ↑↑f x⟫ ∂μ

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
𝕜 : Type u_4
inst✝ : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
f : ↥(Lp 𝕜 2 μ)
⊢ ⟪1, ∫ (x : α) in s, ↑↑f x ∂μ⟫ = ∫ (x : α) in s, ↑↑f x ∂μ

Tactic:
]

Goal state after:
α : Type u_1
𝕜 : Type u_4
inst✝ : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
f : ↥(Lp 𝕜 2 μ)
⊢ ⟪1, ∫ (x : α) in s, ↑↑f x ∂μ⟫ = ∫ (x : α) in s, ↑↑f x ∂μ
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
n : ℕ∞
K : Compacts E
m : MeasurableSpace E
inst✝⁷ : OpensMeasurableSpace E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝⁶ : NormedAddCommGroup F₁
inst✝⁵ : NormedSpace 𝕜 F₁
inst✝⁴ : NormedSpace ℝ F₁
inst✝³ : NormedAddCommGroup F₂
inst✝² : NormedSpace 𝕜 F₂
inst✝¹ : NormedAddCommGroup F₃
inst✝ : NormedSpace 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
hφ : MemLp φ 1 (μ.restrict ↑K)
f : 𝓓^{n}_{K}(E, F₁)
⊢ MemLp (fun x ↦ (B (f x)) (φ x)) 1 (μ.restrict ↑K)

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
n : ℕ∞
K : Compacts E
m : MeasurableSpace E
inst✝⁷ : OpensMeasurableSpace E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝⁶ : NormedAddCommGroup F₁
inst✝⁵ : NormedSpace 𝕜 F₁
inst✝⁴ : NormedSpace ℝ F₁
inst✝³ : NormedAddCommGroup F₂
inst✝² : NormedSpace 𝕜 F₂
inst✝¹ : NormedAddCommGroup F₃
inst✝ : NormedSpace 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
hφ : MemLp φ 1 (μ.restrict ↑K)
f : 𝓓^{n}_{K}(E, F₁)
⊢ MemLp (fun x ↦ (B (f x)) (φ x)) 1 (μ.restrict ↑K)
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝³ : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
s t : Set α
inst✝ : CompleteSpace E
hs : MeasurableSet s
ht : MeasurableSet t
hμs : μ s ≠ ∞
hμt : μ t ≠ ∞
a b : E
this : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E
⊢ μ.real (s ∩ t) • ⟪a, b⟫ = μ.real (s ∩ t) • ⟪a, b⟫

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
inst✝¹ : RCLike 𝕜
μ : Measure α
inst✝ : IsFiniteMeasure μ
f g : α →ᵇ 𝕜
hf_ae : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) f) =ᵐ[μ] ⇑f
hg_ae : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) g) =ᵐ[μ] ⇑g
a✝ : α
hf : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) f) a✝ = f a✝
hg : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) g) a✝ = g a✝
⊢ ⟪f a✝, g a✝⟫ = g a✝ * (starRingEnd 𝕜) (f a✝)

Tactic:
]

Goal state after:
α : Type u_1
𝕜 : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
inst✝¹ : RCLike 𝕜
μ : Measure α
inst✝ : IsFiniteMeasure μ
f g : α →ᵇ 𝕜
hf_ae : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) f) =ᵐ[μ] ⇑f
hg_ae : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) g) =ᵐ[μ] ⇑g
a✝ : α
hf : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) f) a✝ = f a✝
hg : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) g) a✝ = g a✝
⊢ ⟪f a✝, g a✝⟫ = g a✝ * (starRingEnd 𝕜) (f a✝)
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
inst✝² : RCLike 𝕜
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : CompactSpace α
f g : C(α, 𝕜)
hf_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) f) =ᵐ[μ] ⇑f
hg_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) =ᵐ[μ] ⇑g
a✝ : α
hf : ↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a✝ = f a✝
hg : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a✝ = g a✝
⊢ ⟪f a✝, g a✝⟫ = g a✝ * (starRingEnd 𝕜) (f a✝)

Tactic:
]

Goal state after:
α : Type u_1
𝕜 : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
inst✝² : RCLike 𝕜
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : CompactSpace α
f g : C(α, 𝕜)
hf_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) f) =ᵐ[μ] ⇑f
hg_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) =ᵐ[μ] ⇑g
a✝ : α
hf : ↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a✝ = f a✝
hg : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a✝ = g a✝
⊢ ⟪f a✝, g a✝⟫ = g a✝ * (starRingEnd 𝕜) (f a✝)