`BookProof.UnboundedSpectralModel.model_symmetry_relation_inc` (u : Lp ℂ 2 mu) : (inner ℂ u (mulRep mu (coordFn (resOp T)) u) : ℂ) - inner ℂ (mulRep mu (coordFn (resOp T)) u) u = (2 *

Proved

BookProof.UnboundedSpectralModel.model_symmetry_relation_inc

by leonardopedro · Oct 21, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterUnboundedSpectralModel.

BookProof.UnboundedSpectralModel.model_symmetry_relation_inc (u : Lp ℂ 2 mu) : (inner ℂ u (mulRep mu (coordFn (resOp T)) u) : ℂ) - inner ℂ (mulRep mu (coordFn (resOp T)) u) u = (2 * Complex.I) * inner ℂ (mulRep mu (coordFn (resOp T)) u) (mulRep mu (coordFn (resOp T)) u)

Formalization note: Lean 4 identifier BookProof.UnboundedSpectralModel.model_symmetry_relation_inc.

Replaces the unuploadable node BookProof_UnboundedSpectralModel_model_symmetry_relation, whose statement name is already declared inside its published definition bundle; the _inc suffix publishes the same statement as its own node.

Preamble

-- Generated from ChapterUnboundedSpectralModel.lean — theorem BookProof.UnboundedSpectralModel.model_symmetry_relation_inc
import Definitions.Def_ChapterUnitaryTransport
import Definitions.Def_ChapterSirkTrotterKato
import Definitions.Def_ChapterAbelianGelfandModel
import Definitions.Def_ChapterSpectralMultiplication
import Definitions.Def_ChapterSpectralDirectSum
import Mathlib
import Definitions.Def_ChapterUnboundedSpectralModel
import Definitions.Def_ChapterStoneConverse
import Definitions.Def_ChapterStoneResolvent
open BookProof.ChapterStoneMeasurable
open BookProof.ChapterStoneMeasurable.WeakMeasurableUnitaryGroup
open BookProof.UnboundedSpectralModel


noncomputable section

open MeasureTheory Complex
open scoped InnerProductSpace


open BookProof.ChapterUnitaryTransport BookProof.ChapterStoneResolvent
open BookProof.ChapterAbelianGelfandModel BookProof.ChapterSpectralMultiplication
open BookProof.ChapterSpectralDirectSum

variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

variable (T : UnboundedSelfAdjoint H) {mu : Measure (spectrum ℂ (resOp T))}
  (V : Lp ℂ 2 mu →ₗᵢ[ℂ] H)
  (hV : ∀ u : Lp ℂ 2 mu, V (mulRep mu (coordFn (resOp T)) u) = resOp T (V u))

include hV in

Formal statement

theorem BookProof.UnboundedSpectralModel.model_symmetry_relation_inc (u : Lp ℂ 2 mu) :
    (inner ℂ u (mulRep mu (coordFn (resOp T)) u) : ℂ)
        - inner ℂ (mulRep mu (coordFn (resOp T)) u) u
      = (2 * Complex.I)
          * inner ℂ (mulRep mu (coordFn (resOp T)) u) (mulRep mu (coordFn (resOp T)) u) := by sorry

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