File: Mathlib/Analysis/Calculus/BumpFunction/Normed.lean

full_name: ContDiffBump.normed
code: /-- A bump function normed so that `∫ x, f.normed μ x ∂μ = 1`. -/
protected def normed (μ : Measure E) : E → ℝ 
goal: E → ℝ

full_name: ContDiffBump.normed_def
code: theorem normed_def {μ : Measure E} (x : E) : f.normed μ x = f x / ∫ x, f x ∂μ 
goal: f.normed μ x = ↑f x / ∫ (x : E), ↑f x ∂μ

full_name: ContDiffBump.nonneg_normed
code: theorem nonneg_normed (x : E) : 0 ≤ f.normed μ x 
goal: 0 ≤ f.normed μ x

full_name: ContDiffBump.contDiff_normed
code: theorem contDiff_normed {n : ℕ∞} : ContDiff ℝ n (f.normed μ) 
goal: ContDiff ℝ n (f.normed μ)

full_name: ContDiffBump.continuous_normed
code: theorem continuous_normed : Continuous (f.normed μ) 
goal: Continuous (f.normed μ)

full_name: ContDiffBump.normed_sub
code: theorem normed_sub (x : E) : f.normed μ (c - x) = f.normed μ (c + x) 
goal: f.normed μ (c - x) = f.normed μ (c + x)

full_name: ContDiffBump.normed_neg
code: theorem normed_neg (f : ContDiffBump (0 : E)) (x : E) : f.normed μ (-x) = f.normed μ x 
goal: f.normed μ (-x) = f.normed μ x

full_name: ContDiffBump.integrable
code: protected theorem integrable : Integrable f μ 
goal: MeasureTheory.Integrable (↑f) μ

full_name: ContDiffBump.integrable_normed
code: protected theorem integrable_normed : Integrable (f.normed μ) μ 
goal: MeasureTheory.Integrable (f.normed μ) μ

full_name: ContDiffBump.integral_pos
code: theorem integral_pos : 0 < ∫ x, f x ∂μ 
goal: 0 < ∫ (x : E), ↑f x ∂μ

full_name: ContDiffBump.integral_normed
code: theorem integral_normed : ∫ x, f.normed μ x ∂μ = 1 
goal: ∫ (x : E), f.normed μ x ∂μ = 1

full_name: ContDiffBump.support_normed_eq
code: theorem support_normed_eq : Function.support (f.normed μ) = Metric.ball c f.rOut 
goal: Function.support (f.normed μ) = Metric.ball c f.rOut

full_name: ContDiffBump.tsupport_normed_eq
code: theorem tsupport_normed_eq : tsupport (f.normed μ) = Metric.closedBall c f.rOut 
goal: tsupport (f.normed μ) = Metric.closedBall c f.rOut

full_name: ContDiffBump.hasCompactSupport_normed
code: theorem hasCompactSupport_normed : HasCompactSupport (f.normed μ) 
goal: HasCompactSupport (f.normed μ)

full_name: ContDiffBump.tendsto_support_normed_smallSets
code: theorem tendsto_support_normed_smallSets {ι} {φ : ι → ContDiffBump c} {l : Filter ι}
    (hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)) :
    Tendsto (fun i => Function.support fun x => (φ i).normed μ x) l (𝓝 c).smallSets 
goal: Filter.Tendsto (fun i => Function.support fun x => (φ i).normed μ x) l (𝓝 c).smallSets

full_name: ContDiffBump.integral_normed_smul
code: theorem integral_normed_smul {X} [NormedAddCommGroup X] [NormedSpace ℝ X]
    [CompleteSpace X] (z : X) : ∫ x, f.normed μ x • z ∂μ = z 
goal: ∫ (x : E), f.normed μ x • z ∂μ = z

full_name: ContDiffBump.measure_closedBall_le_integral
code: theorem measure_closedBall_le_integral : (μ (closedBall c f.rIn)).toReal ≤ ∫ x, f x ∂μ 
goal: (μ (Metric.closedBall c f.rIn)).toReal ≤ ∫ (x : E), ↑f x ∂μ

full_name: ContDiffBump.normed_le_div_measure_closedBall_rIn
code: theorem normed_le_div_measure_closedBall_rIn (x : E) :
    f.normed μ x ≤ 1 / (μ (closedBall c f.rIn)).toReal 
goal: f.normed μ x ≤ 1 / (μ (Metric.closedBall c f.rIn)).toReal

full_name: ContDiffBump.integral_le_measure_closedBall
code: theorem integral_le_measure_closedBall : ∫ x, f x ∂μ ≤ (μ (closedBall c f.rOut)).toReal 
goal: ∫ (x : E), ↑f x ∂μ ≤ (μ (Metric.closedBall c f.rOut)).toReal

full_name: ContDiffBump.measure_closedBall_div_le_integral
code: theorem measure_closedBall_div_le_integral [IsAddHaarMeasure μ] (K : ℝ) (h : f.rOut ≤ K * f.rIn) :
    (μ (closedBall c f.rOut)).toReal / K ^ finrank ℝ E ≤ ∫ x, f x ∂μ 
goal: (μ (Metric.closedBall c f.rOut)).toReal / K ^ FiniteDimensional.finrank ℝ E ≤ ∫ (x : E), ↑f x ∂μ

full_name: ContDiffBump.normed_le_div_measure_closedBall_rOut
code: theorem normed_le_div_measure_closedBall_rOut [IsAddHaarMeasure μ] (K : ℝ) (h : f.rOut ≤ K * f.rIn)
    (x : E) :
    f.normed μ x ≤ K ^ finrank ℝ E / (μ (closedBall c f.rOut)).toReal 
goal: f.normed μ x ≤ K ^ FiniteDimensional.finrank ℝ E / (μ (Metric.closedBall c f.rOut)).toReal