`BookProof.UnboundedSpectralModel.cayley_real_multiplier_inc` {w : ℂ} (h : ‖w‖ ^ 2 = w.im) (hw : w ≠ 0) : 1 + Complex.I * w = ((w.re / ‖w‖ ^ 2 : ℝ) : ℂ) * w

Proved

BookProof.UnboundedSpectralModel.cayley_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.cayley_real_multiplier_inc {w : ℂ} (h : ‖w‖ ^ 2 = w.im) (hw : w ≠ 0) : 1 + Complex.I * w = ((w.re / ‖w‖ ^ 2 : ℝ) : ℂ) * w

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

Replaces the unuploadable node BookProof_UnboundedSpectralModel_cayley_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.cayley_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
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]

Formal statement

theorem BookProof.UnboundedSpectralModel.cayley_real_multiplier_inc {w : ℂ} (h : ‖w‖ ^ 2 = w.im) (hw : w ≠ 0) :
    1 + Complex.I * w = ((w.re / ‖w‖ ^ 2 : ℝ) : ℂ) * w := by sorry

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