`BookProof.ChapterLpRestrictSplit.inner_restrictEmbed_eq_zero` {A B : Set α} (hA : MeasurableSet A) (hB : MeasurableSet B) (hAB : Disjoint A B) (u : Lp ℂ 2 (mu.restrict A)) (v : Lp

Proved

BookProof.ChapterLpRestrictSplit.inner_restrictEmbed_eq_zero

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterLpRestrictSplit.

BookProof.ChapterLpRestrictSplit.inner_restrictEmbed_eq_zero {A B : Set α} (hA : MeasurableSet A) (hB : MeasurableSet B) (hAB : Disjoint A B) (u : Lp ℂ 2 (mu.restrict A)) (v : Lp ℂ 2 (mu.restrict B)) : inner ℂ (restrictEmbed hA u) (restrictEmbed hB v) = 0

Formalization note: Lean 4 identifier BookProof.ChapterLpRestrictSplit.inner_restrictEmbed_eq_zero.

Preamble

-- Generated from ChapterLpRestrictSplit.lean — theorem BookProof.ChapterLpRestrictSplit.inner_restrictEmbed_eq_zero
import Definitions.Def_ChapterLinftyMultiplication
import Mathlib
import Definitions.Def_ChapterLpRestrictSplit
open BookProof.ChapterLpRestrictSplit


noncomputable section

open MeasureTheory


open BookProof.ChapterLinftyMultiplication

variable {α : Type*} [MeasurableSpace α] {mu : Measure α}

Formal statement

theorem BookProof.ChapterLpRestrictSplit.inner_restrictEmbed_eq_zero {A B : Set α} (hA : MeasurableSet A) (hB : MeasurableSet B)
    (hAB : Disjoint A B) (u : Lp ℂ 2 (mu.restrict A)) (v : Lp ℂ 2 (mu.restrict B)) :
    inner ℂ (restrictEmbed hA u) (restrictEmbed hB v) = 0 := by sorry

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