The Lean 4 theorem `qg3D_symmetricOn` in the `ChapterQuantumGravity3DGauge` chapter of the timepiece formalization

-- Generated from ChapterQuantumGravity3DGauge.lean — theorem BookProof.QuantumGravity3DGauge.qg3D_symmetricOn
import Definitions.Def_ChapterYangMillsFriedrichs
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterFriedrichsExtension
import Definitions.Def_ChapterHermiteGalerkinFriedrichs
import Definitions.Def_ChapterComplexShiftCore
import Definitions.Def_ChapterQuantumGravityDensitized
import Mathlib
import Definitions.Def_ChapterQuantumGravity3DGauge
import Definitions.Def_ChapterFarisLavineCore
import Definitions.Def_ChapterHermiteProductCore
import Definitions.Def_ChapterYangMillsHermite
open BookProof.HermiteProductCore
open BookProof.YangMillsHermite
open BookProof.QuantumGravity3DGauge

variable {d : ℕ}
variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℂ F] {D : Submodule ℂ F}
variable {D : Submodule ℂ (L2d 84)}



open MeasureTheory Complex MvPolynomial Filter Topology
open BookProof.HermiteProductCore BookProof.YangMillsHermite BookProof.YangMillsFriedrichs
open BookProof.FarisLavine BookProof.FriedrichsExtension BookProof.HermiteGalerkin
open BookProof.HashimotoShiftInvert BookProof.QuantumGravityDensitized

noncomputable section

theorem BookProof.QuantumGravity3DGauge.qg3D_symmetricOn (Φ : CoreRep 84 D) (x : D) :
    quadForm (qg3DHamiltonian Φ) x
      = 1 / 2 * (∑ j, qgKappa j * ‖((qgMom Φ j x : D) : L2d 84)‖ ^ 2)
        + 1 / 2 * ∑ m, ‖((torsionOps Φ m x : D) : L2d 84)‖ ^ 2 := by sorry