awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
X : Type u_6
f : X → ℝ
m0 : MeasurableSpace X
μ : Measure X
g : SimpleFunc X ℝ
hf : Integrable f μ
c : ℝ
s : Set X
hs : MeasurableSet s
this : s.indicator (const X c) * f = s.indicator (c • f)
⊢ Integrable (s.indicator (const X c) * f) μ
have rw : IntegrableOn (c • f) s μ
Tactic: rw [this, integrable_indicator_iff hs]
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
X : Type u_6
f : X → ℝ
m m0 : MeasurableSpace X
μ : Measure X
hm : m ≤ m0
g : SimpleFunc X ℝ
hf : Integrable f μ
⊢ Integrable ((⇑g : (a : X) → ℝ) * f) μ
have rw : Integrable ((⇑(SimpleFunc.toLargerSpace hm g) : (a : X) → ℝ) * f) μ
Tactic: rw [← SimpleFunc.coe_toLargerSpace_eq hm g]
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
Y : Type u_2
E : Type u_3
X : Type u_5
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace ℝ E
inst_6 : FirstCountableTopology X
inst_7 : LocallyCompactSpace X
inst_8 : SecondCountableTopologyEither Y E
inst_9 : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
⊢ Continuous[inst, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦ ∫ (y : Y) in s, f x y ∂μ
have rw : ∀ (x : X), ContinuousAt.{u_5, u_3} (Y := E) (fun (x : X) ↦ ∫ (y : Y) in s, f x y ∂μ) x
Tactic: rw [continuous_iff_continuousAt]
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
Y : Type u_2
E : Type u_3
X : Type u_5
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace ℝ E
inst_6 : FirstCountableTopology X
inst_7 : LocallyCompactSpace X
inst_8 : SecondCountableTopologyEither Y E
inst_9 : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
⊢ ContinuousAt.{u_5, u_3} (Y := E) (fun (x : X) ↦ ∫ (y : Y) in s, f x y ∂μ) x₀
have rcases : ∀ (U : Set X) (U_cpct : IsCompact U) (U_nhds : U ∈ 𝓝 x₀),
  ContinuousAt.{u_5, u_3} (Y := E) (fun (x : X) ↦ ∫ (y : Y) in s, f x y ∂μ) x₀
Tactic: rcases exists_compact_mem_nhds x₀ with ⟨U, U_cpct, U_nhds⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
Y : Type u_2
E : Type u_3
X : Type u_5
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace ℝ E
inst_6 : FirstCountableTopology X
inst_7 : LocallyCompactSpace X
inst_8 : SecondCountableTopologyEither Y E
inst_9 : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
U : Set X
U_cpct : IsCompact U
U_nhds : U ∈ 𝓝 x₀
⊢ ContinuousAt.{u_5, u_3} (Y := E) (fun (x : X) ↦ ∫ (y : Y) in s, f x y ∂μ) x₀
have rcases : ∀ (M : ℝ) (hM : Membership.mem (γ := Set ℝ) (upperBounds ((fun (x : X × Y) ↦ ‖uncurry f x‖) '' U ×ˢ s)) M),
  ContinuousAt.{u_5, u_3} (Y := E) (fun (x : X) ↦ ∫ (y : Y) in s, f x y ∂μ) x₀
Tactic: rcases (U_cpct.prod hs).bddAbove_image hf.norm.continuousOn with ⟨M, hM⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
Y : Type u_2
E : Type u_3
X : Type u_5
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace ℝ E
inst_6 : FirstCountableTopology X
inst_7 : LocallyCompactSpace X
inst_8 : SecondCountableTopologyEither Y E
inst_9 : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
U : Set X
U_cpct : IsCompact U
U_nhds : U ∈ 𝓝 x₀
M : ℝ
hM : Membership.mem (γ := Set ℝ) (upperBounds ((fun (x : X × Y) ↦ ‖uncurry f x‖) '' U ×ˢ s)) M
⊢ ContinuousAt.{u_5, u_3} (Y := E) (fun (x : X) ↦ ∫ (y : Y) in s, f x y ∂μ) x₀
have hF_meas : ∀ᶠ (x : X) in 𝓝 x₀, AEStronglyMeasurable (f x) (μ.restrict s)
Tactic: apply continuousAt_of_dominated
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
Y : Type u_2
E : Type u_3
X : Type u_5
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace ℝ E
inst_6 : FirstCountableTopology X
inst_7 : LocallyCompactSpace X
inst_8 : SecondCountableTopologyEither Y E
inst_9 : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
U : Set X
U_cpct : IsCompact U
U_nhds : U ∈ 𝓝 x₀
M : ℝ
hM : Membership.mem (γ := Set ℝ) (upperBounds ((fun (x : X × Y) ↦ ‖uncurry f x‖) '' U ×ˢ s)) M
hF_meas : ∀ᶠ (x : X) in 𝓝 x₀, AEStronglyMeasurable (f x) (μ.restrict s)
⊢ ContinuousAt.{u_5, u_3} (Y := E) (fun (x : X) ↦ ∫ (y : Y) in s, f x y ∂μ) x₀
have h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (a : Y) ∂μ.restrict s, ‖f x a‖ ≤ M
Tactic: apply continuousAt_of_dominated
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
Y : Type u_2
E : Type u_3
X : Type u_5
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace ℝ E
inst_6 : FirstCountableTopology X
inst_7 : LocallyCompactSpace X
inst_8 : SecondCountableTopologyEither Y E
inst_9 : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
U : Set X
U_cpct : IsCompact U
U_nhds : U ∈ 𝓝 x₀
M : ℝ
hM : Membership.mem (γ := Set ℝ) (upperBounds ((fun (x : X × Y) ↦ ‖uncurry f x‖) '' U ×ˢ s)) M
hF_meas : ∀ᶠ (x : X) in 𝓝 x₀, AEStronglyMeasurable (f x) (μ.restrict s)
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (a : Y) ∂μ.restrict s, ‖f x a‖ ≤ M
⊢ ContinuousAt.{u_5, u_3} (Y := E) (fun (x : X) ↦ ∫ (y : Y) in s, f x y ∂μ) x₀
have bound_integrable : Integrable (ε := ℝ) (fun (t : Y) ↦ M) (μ.restrict s)
Tactic: apply continuousAt_of_dominated
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
Y : Type u_2
E : Type u_3
X : Type u_5
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace ℝ E
inst_6 : FirstCountableTopology X
inst_7 : LocallyCompactSpace X
inst_8 : SecondCountableTopologyEither Y E
inst_9 : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
U : Set X
U_cpct : IsCompact U
U_nhds : U ∈ 𝓝 x₀
M : ℝ
hM : Membership.mem (γ := Set ℝ) (upperBounds ((fun (x : X × Y) ↦ ‖uncurry f x‖) '' U ×ˢ s)) M
hF_meas : ∀ᶠ (x : X) in 𝓝 x₀, AEStronglyMeasurable (f x) (μ.restrict s)
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (a : Y) ∂μ.restrict s, ‖f x a‖ ≤ M
bound_integrable : Integrable (ε := ℝ) (fun (t : Y) ↦ M) (μ.restrict s)
⊢ ContinuousAt.{u_5, u_3} (Y := E) (fun (x : X) ↦ ∫ (y : Y) in s, f x y ∂μ) x₀
have h_cont : ∀ᵐ (a : Y) ∂μ.restrict s, ContinuousAt.{u_5, u_3} (Y := E) (fun (x : X) ↦ f x a) x₀
Tactic: apply continuousAt_of_dominated
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
Y : Type u_2
E : Type u_3
X : Type u_5
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace ℝ E
inst_6 : FirstCountableTopology X
inst_7 : LocallyCompactSpace X
inst_8 : SecondCountableTopologyEither Y E
inst_9 : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
U : Set X
U_cpct : IsCompact U
U_nhds : U ∈ 𝓝 x₀
M : ℝ
hM : Membership.mem (γ := Set ℝ) (upperBounds ((fun (x : X × Y) ↦ ‖uncurry f x‖) '' U ×ˢ s)) M
hF_meas : ∀ᶠ (x : X) in 𝓝 x₀, AEStronglyMeasurable (f x) (μ.restrict s)
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (a : Y) ∂μ.restrict s, ‖f x a‖ ≤ M
bound_integrable : Integrable (ε := ℝ) (fun (t : Y) ↦ M) (μ.restrict s)
h_cont : ∀ᵐ (a : Y) ∂μ.restrict s, ContinuousAt.{u_5, u_3} (Y := E) (fun (x : X) ↦ f x a) x₀
⊢ ContinuousAt.{u_5, u_3} (Y := E) (fun (x : X) ↦ ∫ (y : Y) in s, f x y ∂μ) x₀
have bound : Y → ℝ
Tactic: apply continuousAt_of_dominated
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
Y : Type u_2
E : Type u_3
X : Type u_5
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace ℝ E
inst_6 : FirstCountableTopology X
inst_7 : LocallyCompactSpace X
inst_8 : SecondCountableTopologyEither Y E
inst_9 : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
U : Set X
U_cpct : IsCompact U
U_nhds : U ∈ 𝓝 x₀
M : ℝ
hM : Membership.mem (γ := Set ℝ) (upperBounds ((fun (x : X × Y) ↦ ‖uncurry f x‖) '' U ×ˢ s)) M
⊢ ∀ᶠ (x : X) in 𝓝 x₀, AEStronglyMeasurable (f x) (μ.restrict s)
have hF_meas : ∀ (a : X), Membership.mem (γ := Set X) {x : X | (fun (x : X) ↦ AEStronglyMeasurable (f x) (μ.restrict s)) x} a
Tactic: filter_upwards with x using Continuous.aestronglyMeasurable (by fun_prop)
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
Y : Type u_2
E : Type u_3
X : Type u_5
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace ℝ E
inst_6 : FirstCountableTopology X
inst_7 : LocallyCompactSpace X
inst_8 : SecondCountableTopologyEither Y E
inst_9 : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
U : Set X
U_cpct : IsCompact U
U_nhds : U ∈ 𝓝 x₀
M : ℝ
hM : Membership.mem (γ := Set ℝ) (upperBounds ((fun (x : X × Y) ↦ ‖uncurry f x‖) '' U ×ˢ s)) M
hF_meas : ∀ (a : X), Membership.mem (γ := Set X) {x : X | (fun (x : X) ↦ AEStronglyMeasurable (f x) (μ.restrict s)) x} a
⊢ ∀ᶠ (x : X) in 𝓝 x₀, AEStronglyMeasurable (f x) (μ.restrict s)
have hF_meas₁ : ∀ (a : X), AEStronglyMeasurable (f a) (μ.restrict s)
Tactic: filter_upwards with x using Continuous.aestronglyMeasurable (by fun_prop)
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
Y : Type u_2
E : Type u_3
X : Type u_5
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace ℝ E
inst_6 : FirstCountableTopology X
inst_7 : LocallyCompactSpace X
inst_8 : SecondCountableTopologyEither Y E
inst_9 : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
U : Set X
U_cpct : IsCompact U
U_nhds : U ∈ 𝓝 x₀
M : ℝ
hM : Membership.mem (γ := Set ℝ) (upperBounds ((fun (x : X × Y) ↦ ‖uncurry f x‖) '' U ×ˢ s)) M
⊢ ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (a : Y) ∂μ.restrict s, ‖f x a‖ ≤ M
have h : ∀ (x : X) (x_in : x ∈ U), ∀ᵐ (a : Y) ∂μ.restrict s, ‖f x a‖ ≤ M
Tactic: filter_upwards [U_nhds] with x x_in
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
Y : Type u_2
E : Type u_3
X : Type u_5
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace ℝ E
inst_6 : FirstCountableTopology X
inst_7 : LocallyCompactSpace X
inst_8 : SecondCountableTopologyEither Y E
inst_9 : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
U : Set X
U_cpct : IsCompact U
U_nhds : U ∈ 𝓝 x₀
M : ℝ
hM : Membership.mem (γ := Set ℝ) (upperBounds ((fun (x : X × Y) ↦ ‖uncurry f x‖) '' U ×ˢ s)) M
x : X
x_in : x ∈ U
⊢ ∀ᵐ (a : Y) ∂μ.restrict s, ‖f x a‖ ≤ M
have h : ∀ᵐ (x_1 : Y) ∂μ, x_1 ∈ s → ‖f x x_1‖ ≤ M
Tactic: rw [ae_restrict_iff]
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
Y : Type u_2
E : Type u_3
X : Type u_5
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace ℝ E
inst_6 : FirstCountableTopology X
inst_7 : LocallyCompactSpace X
inst_8 : SecondCountableTopologyEither Y E
inst_9 : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
U : Set X
U_cpct : IsCompact U
U_nhds : U ∈ 𝓝 x₀
M : ℝ
hM : Membership.mem (γ := Set ℝ) (upperBounds ((fun (x : X × Y) ↦ ‖uncurry f x‖) '' U ×ˢ s)) M
x : X
x_in : x ∈ U
h : ∀ᵐ (x_1 : Y) ∂μ, x_1 ∈ s → ‖f x x_1‖ ≤ M
⊢ ∀ᵐ (a : Y) ∂μ.restrict s, ‖f x a‖ ≤ M
have h₁ : MeasurableSet.{u_2} (α := Y) {x_1 : Y | ‖f x x_1‖ ≤ M}
Tactic: rw [ae_restrict_iff]
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
Y : Type u_2
E : Type u_3
X : Type u_5
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace ℝ E
inst_6 : FirstCountableTopology X
inst_7 : LocallyCompactSpace X
inst_8 : SecondCountableTopologyEither Y E
inst_9 : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
U : Set X
U_cpct : IsCompact U
U_nhds : U ∈ 𝓝 x₀
M : ℝ
hM : Membership.mem (γ := Set ℝ) (upperBounds ((fun (x : X × Y) ↦ ‖uncurry f x‖) '' U ×ˢ s)) M
x : X
x_in : x ∈ U
⊢ ∀ᵐ (x_1 : Y) ∂μ, x_1 ∈ s → ‖f x x_1‖ ≤ M
have h : ∀ (a : Y), Membership.mem (γ := Set Y) {x_1 : Y | (fun (x_2 : Y) ↦ x_2 ∈ s → ‖f x x_2‖ ≤ M) x_1} a
Tactic: filter_upwards with t t_in using hM (mem_image_of_mem _ <| mk_mem_prod x_in t_in)
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
Y : Type u_2
E : Type u_3
X : Type u_5
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace ℝ E
inst_6 : FirstCountableTopology X
inst_7 : LocallyCompactSpace X
inst_8 : SecondCountableTopologyEither Y E
inst_9 : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
U : Set X
U_cpct : IsCompact U
U_nhds : U ∈ 𝓝 x₀
M : ℝ
hM : Membership.mem (γ := Set ℝ) (upperBounds ((fun (x : X × Y) ↦ ‖uncurry f x‖) '' U ×ˢ s)) M
x : X
x_in : x ∈ U
h : ∀ (a : Y), Membership.mem (γ := Set Y) {x_1 : Y | (fun (x_2 : Y) ↦ x_2 ∈ s → ‖f x x_2‖ ≤ M) x_1} a
⊢ ∀ᵐ (x_1 : Y) ∂μ, x_1 ∈ s → ‖f x x_1‖ ≤ M
have h₁ : ∀ a ∈ s, ‖f x a‖ ≤ M
Tactic: filter_upwards with t t_in using hM (mem_image_of_mem _ <| mk_mem_prod x_in t_in)
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
Y : Type u_2
E : Type u_3
X : Type u_5
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace ℝ E
inst_6 : FirstCountableTopology X
inst_7 : LocallyCompactSpace X
inst_8 : SecondCountableTopologyEither Y E
inst_9 : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
U : Set X
U_cpct : IsCompact U
U_nhds : U ∈ 𝓝 x₀
M : ℝ
hM : Membership.mem (γ := Set ℝ) (upperBounds ((fun (x : X × Y) ↦ ‖uncurry f x‖) '' U ×ˢ s)) M
⊢ ∀ᵐ (a : Y) ∂μ.restrict s, ContinuousAt.{u_5, u_3} (Y := E) (fun (x : X) ↦ f x a) x₀
have h_cont : ∀ (a : Y),
  Membership.mem (γ := Set Y) {x : Y | (fun (a : Y) ↦ ContinuousAt.{u_5, u_3} (Y := E) (fun (x : X) ↦ f x a) x₀) x} a
Tactic: filter_upwards using (by fun_prop)
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
Y : Type u_2
E : Type u_3
X : Type u_5
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace ℝ E
inst_6 : FirstCountableTopology X
inst_7 : LocallyCompactSpace X
inst_8 : SecondCountableTopologyEither Y E
inst_9 : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
U : Set X
U_cpct : IsCompact U
U_nhds : U ∈ 𝓝 x₀
M : ℝ
hM : Membership.mem (γ := Set ℝ) (upperBounds ((fun (x : X × Y) ↦ ‖uncurry f x‖) '' U ×ˢ s)) M
h_cont :
  ∀ (a : Y),
    Membership.mem (γ := Set Y) {x : Y | (fun (a : Y) ↦ ContinuousAt.{u_5, u_3} (Y := E) (fun (x : X) ↦ f x a) x₀) x} a
⊢ ∀ᵐ (a : Y) ∂μ.restrict s, ContinuousAt.{u_5, u_3} (Y := E) (fun (x : X) ↦ f x a) x₀
have h_cont₁ : ∀ (a : Y), ContinuousAt.{u_5, u_3} (Y := E) (fun (x : X) ↦ f x a) x₀
Tactic: filter_upwards using (by fun_prop)
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
Y : Type u_2
E : Type u_3
F : Type u_4
X : Type u_5
G : Type u_6
𝕜 : Type u_7
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NontriviallyNormedField 𝕜
inst_5 : NormedAddCommGroup E
inst_6 : NormedSpace ℝ E
inst_7 : NormedAddCommGroup F
inst_8 : NormedSpace 𝕜 F
inst_9 : NormedAddCommGroup G
inst_10 : NormedSpace 𝕜 G
inst_11 : NormedSpace 𝕜 E
L : F →L[𝕜] G →L[𝕜] E
f : X → Y → G
s : Set X
k : Set Y
g : Y → F
hk : IsCompact k
hf : ContinuousOn (uncurry f) (SProd.sprod (β := Set Y) s univ)
hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0
hg : IntegrableOn g k μ
p : X
hp : p ∈ s
⊢ Continuous[inst_1, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f p)
have refine : ∀ (y : Y), (p, y) ∈ SProd.sprod (β := Set Y) s univ
Tactic: refine hf.comp_continuous (.prodMk_right _) fun y => ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (hypothesis)
Goal:
Y : Type u_2
E : Type u_3
F : Type u_4
X : Type u_5
G : Type u_6
𝕜 : Type u_7
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NontriviallyNormedField 𝕜
inst_5 : NormedAddCommGroup E
inst_6 : NormedSpace ℝ E
inst_7 : NormedAddCommGroup F
inst_8 : NormedSpace 𝕜 F
inst_9 : NormedAddCommGroup G
inst_10 : NormedSpace 𝕜 G
inst_11 : NormedSpace 𝕜 E
L : F →L[𝕜] G →L[𝕜] E
f : X → Y → G
s : Set X
k : Set Y
g : Y → F
hk : IsCompact k
hf : ContinuousOn (uncurry f) (SProd.sprod (β := Set Y) s univ)
hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0
hg : IntegrableOn g k μ
⊢ ContinuousOn.{u_5, u_3} (Y := E)
    (fun (x : X) ↦ ∫ (y : Y), (((L : (a : F) → G →L[𝕜] E) (g y) : G →L[𝕜] E) : (a : G) → E) (f x y) ∂μ) s
have A : ∀ p ∈ s, Continuous[inst_1, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f p)
Tactic: have A : ∀ p ∈ s, Continuous (f p) := fun p hp ↦ by
    refine hf.comp_continuous (.prodMk_right _) fun y => ?_
    simpa only [prodMk_mem_set_prod_eq, mem_univ, and_true] using hp
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
Y : Type u_2
E : Type u_3
F : Type u_4
X : Type u_5
G : Type u_6
𝕜 : Type u_7
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NontriviallyNormedField 𝕜
inst_5 : NormedAddCommGroup E
inst_6 : NormedSpace ℝ E
inst_7 : NormedAddCommGroup F
inst_8 : NormedSpace 𝕜 F
inst_9 : NormedAddCommGroup G
inst_10 : NormedSpace 𝕜 G
inst_11 : NormedSpace 𝕜 E
L : F →L[𝕜] G →L[𝕜] E
f : X → Y → G
s : Set X
k : Set Y
g : Y → F
hk : IsCompact k
hf : ContinuousOn (uncurry f) (SProd.sprod (β := Set Y) s univ)
hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0
hg : IntegrableOn g k μ
A : ∀ p ∈ s, Continuous[inst_1, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f p)
q : X
hq : q ∈ s
⊢ ContinuousWithinAt.{u_5, u_3} (Y := E)
    (fun (x : X) ↦ ∫ (y : Y), (((L : (a : F) → G →L[𝕜] E) (g y) : G →L[𝕜] E) : (a : G) → E) (f x y) ∂μ) s q
have apply : ∀ (ε : ℝ) (εpos : ε > 0),
  ∀ᶠ (x : X) in 𝓝[s] q,
    dist.{u_3} (α := E) (∫ (y : Y), (((L : (a : F) → G →L[𝕜] E) (g y) : G →L[𝕜] E) : (a : G) → E) (f x y) ∂μ)
        (∫ (y : Y), (((L : (a : F) → G →L[𝕜] E) (g y) : G →L[𝕜] E) : (a : G) → E) (f q y) ∂μ) <
      ε
Tactic: apply Metric.continuousWithinAt_iff'.2 (fun ε εpos ↦ ?_)
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Bochner.Set (goal)
Goal:
Y : Type u_2
E : Type u_3
F : Type u_4
X : Type u_5
G : Type u_6
𝕜 : Type u_7
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : MeasurableSpace Y
inst_3 : OpensMeasurableSpace Y
μ : Measure Y
inst_4 : NontriviallyNormedField 𝕜
inst_5 : NormedAddCommGroup E
inst_6 : NormedSpace ℝ E
inst_7 : NormedAddCommGroup F
inst_8 : NormedSpace 𝕜 F
inst_9 : NormedAddCommGroup G
inst_10 : NormedSpace 𝕜 G
inst_11 : NormedSpace 𝕜 E
L : F →L[𝕜] G →L[𝕜] E
f : X → Y → G
s : Set X
k : Set Y
g : Y → F
hk : IsCompact k
hf : ContinuousOn (uncurry f) (SProd.sprod (β := Set Y) s univ)
hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0
hg : IntegrableOn g k μ
A : ∀ p ∈ s, Continuous[inst_1, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f p)
q : X
hq : q ∈ s
ε : ℝ
εpos : ε > 0
⊢ ∀ᶠ (x : X) in 𝓝[s] q,
    dist.{u_3} (α := E) (∫ (y : Y), (((L : (a : F) → G →L[𝕜] E) (g y) : G →L[𝕜] E) : (a : G) → E) (f x y) ∂μ)
        (∫ (y : Y), (((L : (a : F) → G →L[𝕜] E) (g y) : G →L[𝕜] E) : (a : G) → E) (f q y) ∂μ) <
      ε
have obtain : ∀ (δ : ℝ) (δpos : 0 < δ) (hδ : ∫ (x : Y) in k, ‖L‖ * ‖g x‖ * δ ∂μ < ε),
  ∀ᶠ (x : X) in 𝓝[s] q,
    dist.{u_3} (α := E) (∫ (y : Y), (((L : (a : F) → G →L[𝕜] E) (g y) : G →L[𝕜] E) : (a : G) → E) (f x y) ∂μ)
        (∫ (y : Y), (((L : (a : F) → G →L[𝕜] E) (g y) : G →L[𝕜] E) : (a : G) → E) (f q y) ∂μ) <
      ε
Tactic: obtain ⟨δ, δpos, hδ⟩ : ∃ (δ : ℝ), 0 < δ ∧ ∫ x in k, ‖L‖ * ‖g x‖ * δ ∂μ < ε := by
    simpa [integral_mul_const] using exists_pos_mul_lt εpos _