Theorem: MeasureTheory.ContinuousMap.inner_toLp
File: Mathlib/MeasureTheory/Function/L2Space.lean

Proof steps:
state_before: 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)
⊢ ⟪(ContinuousMap.toLp 2 μ 𝕜) f, (ContinuousMap.toLp 2 μ 𝕜) g⟫_𝕜 = ∫ (x : α), (starRingEnd 𝕜) (f x) * g x ∂μ
state_after: case h
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)
⊢ (fun a => ⟪↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a, ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a⟫_𝕜) =ᶠ[ae μ] fun a =>
    (starRingEnd 𝕜) (f a) * g a
tactic: apply integral_congr_ae
state_before: case h
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)
⊢ (fun a => ⟪↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a, ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a⟫_𝕜) =ᶠ[ae μ] fun a =>
    (starRingEnd 𝕜) (f a) * g a
state_after: case h
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) =ᶠ[ae μ] ⇑f
⊢ (fun a => ⟪↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a, ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a⟫_𝕜) =ᶠ[ae μ] fun a =>
    (starRingEnd 𝕜) (f a) * g a
tactic: have hf_ae := f.coeFn_toLp (p := 2) (𝕜 := 𝕜) μ
state_before: case h
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) =ᶠ[ae μ] ⇑f
⊢ (fun a => ⟪↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a, ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a⟫_𝕜) =ᶠ[ae μ] fun a =>
    (starRingEnd 𝕜) (f a) * g a
state_after: case h
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) =ᶠ[ae μ] ⇑f
hg_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) =ᶠ[ae μ] ⇑g
⊢ (fun a => ⟪↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a, ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a⟫_𝕜) =ᶠ[ae μ] fun a =>
    (starRingEnd 𝕜) (f a) * g a
tactic: have hg_ae := g.coeFn_toLp (p := 2) (𝕜 := 𝕜) μ
state_before: case h
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) =ᶠ[ae μ] ⇑f
hg_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) =ᶠ[ae μ] ⇑g
⊢ (fun a => ⟪↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a, ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a⟫_𝕜) =ᶠ[ae μ] fun a =>
    (starRingEnd 𝕜) (f a) * g a
state_after: case h
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) =ᶠ[ae μ] ⇑f
hg_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) =ᶠ[ae μ] ⇑g
a✝ : α
hf : ↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a✝ = f a✝
hg : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a✝ = g a✝
⊢ ⟪↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a✝, ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a✝⟫_𝕜 = (starRingEnd 𝕜) (f a✝) * g a✝
tactic: filter_upwards [hf_ae, hg_ae] with _ hf hg
state_before: case h
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) =ᶠ[ae μ] ⇑f
hg_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) =ᶠ[ae μ] ⇑g
a✝ : α
hf : ↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a✝ = f a✝
hg : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a✝ = g a✝
⊢ ⟪↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a✝, ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a✝⟫_𝕜 = (starRingEnd 𝕜) (f a✝) * g a✝
state_after: case h
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) =ᶠ[ae μ] ⇑f
hg_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) =ᶠ[ae 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✝⟫_𝕜 = (starRingEnd 𝕜) (f a✝) * g a✝
tactic: rw [hf, hg]
state_before: case h
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) =ᶠ[ae μ] ⇑f
hg_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) =ᶠ[ae 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✝⟫_𝕜 = (starRingEnd 𝕜) (f a✝) * g a✝
state_after: no goals
tactic: simp