`BookProof.UnboundedSpectralModel.model_ae_real_multiplier_inc` [IsFiniteMeasure mu] : ∀ᵐ z ∂mu, 1 + Complex.I * z.1 = ((z.1.re / ‖z.1‖ ^ 2 : ℝ) : ℂ) * z.1

Proved

BookProof.UnboundedSpectralModel.model_ae_real_multiplier_inc

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterUnboundedSpectralModel.

BookProof.UnboundedSpectralModel.model_ae_real_multiplier_inc [IsFiniteMeasure mu] : ∀ᵐ z ∂mu, 1 + Complex.I * z.1 = ((z.1.re / ‖z.1‖ ^ 2 : ℝ) : ℂ) * z.1

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

Replaces the unuploadable node BookProof_UnboundedSpectralModel_model_ae_real_multiplier, 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_ae_real_multiplier_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_ChapterStoneResolvent
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_ae_real_multiplier_inc [IsFiniteMeasure mu] :
    ∀ᵐ z ∂mu, 1 + Complex.I * z.1 = ((z.1.re / ‖z.1‖ ^ 2 : ℝ) : ℂ) * z.1 := by sorry

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