`BookProof.ChapterMackeyQuasiInvariant.proj_symm` (μ : Measure X) {E : Set X} (hE : MeasurableSet E) (f f' : Lp K 2 μ) : (inner ℂ (proj μ hE f) f' : ℂ) = inner ℂ f (proj μ hE f')

Proved

BookProof.ChapterMackeyQuasiInvariant.proj_symm

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterMackeyQuasiInvariant.

BookProof.ChapterMackeyQuasiInvariant.proj_symm (μ : Measure X) {E : Set X} (hE : MeasurableSet E) (f f' : Lp K 2 μ) : (inner ℂ (proj μ hE f) f' : ℂ) = inner ℂ f (proj μ hE f')

Formalization note: Lean 4 identifier BookProof.ChapterMackeyQuasiInvariant.proj_symm.

Preamble

-- Generated from ChapterMackeyQuasiInvariant.lean — theorem BookProof.ChapterMackeyQuasiInvariant.proj_symm
import Mathlib
import Definitions.Def_ChapterMackeyQuasiInvariant
import Definitions.Def_ChapterElectroweakFieldStrength
open BookProof.ChapterElectroweakFieldStrength
open BookProof.ChapterMackeyQuasiInvariant


open MeasureTheory Measure


variable {G X K : Type*} [Group G] [MeasurableSpace X] [MulAction G X]
variable [NormedAddCommGroup K] [InnerProductSpace ℂ K]

variable (μ : Measure X) (L : G → X → (K ≃ₗᵢ[ℂ] K))
variable {μ L}

Formal statement

theorem BookProof.ChapterMackeyQuasiInvariant.proj_symm (μ : Measure X) {E : Set X} (hE : MeasurableSet E) (f f' : Lp K 2 μ) :
    (inner ℂ (proj μ hE f) f' : ℂ) = inner ℂ f (proj μ hE f') := by sorry

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