`BookProof.FriedrichsSquare.IsFriedrichsSqExtension.symmetric` {A : D →ₗ[ℂ] F} {hstab : ∀ v : D, (A v : F) ∈ D} {R : Submodule ℂ (F × F)} (h : IsFriedrichsSqExtension A hstab R) :

Proved

BookProof.FriedrichsSquare.IsFriedrichsSqExtension.symmetric

by leonardopedro · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFriedrichsSquareFactorization.

BookProof.FriedrichsSquare.IsFriedrichsSqExtension.symmetric {A : D →ₗ[ℂ] F} {hstab : ∀ v : D, (A v : F) ∈ D} {R : Submodule ℂ (F × F)} (h : IsFriedrichsSqExtension A hstab R) : ∀ p ∈ R, ∀ q ∈ R, (inner ℂ p.2 q.1 : ℂ) = inner ℂ p.1 q.2

Formalization note: Lean 4 identifier BookProof.FriedrichsSquare.IsFriedrichsSqExtension.symmetric.

Preamble

-- Generated from ChapterFriedrichsSquareFactorization.lean — theorem BookProof.FriedrichsSquare.IsFriedrichsSqExtension.symmetric
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterEsaClosureCore
import Mathlib
import Definitions.Def_ChapterFriedrichsSquareFactorization
import Definitions.Def_ChapterClosureUniqueness
import Definitions.Def_ChapterStoneConverse
open BookProof.ClosureUniqueness
open BookProof.ChapterStoneMeasurable
open BookProof.ChapterStoneMeasurable.WeakMeasurableUnitaryGroup
open BookProof.FriedrichsSquare



open BookProof.FarisLavine BookProof.EsaClosure BookProof.ClosureUniqueness

variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℂ F]
variable {D : Submodule ℂ F}

variable [CompleteSpace F]

Formal statement

theorem BookProof.FriedrichsSquare.IsFriedrichsSqExtension.symmetric {A : D →ₗ[ℂ] F} {hstab : ∀ v : D, (A v : F) ∈ D}
    {R : Submodule ℂ (F × F)} (h : IsFriedrichsSqExtension A hstab R) :
    ∀ p ∈ R, ∀ q ∈ R, (inner ℂ p.2 q.1 : ℂ) = inner ℂ p.1 q.2 := by sorry

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