`BookProof.ChapterWignerSymmetry.exists_zeta` (hS : WignerCoord S o) {i : ι} (hi : i ≠ o) : ∃ ζ : ℂ, (ζ = Complex.I ∨ ζ = -Complex.I) ∧ ∀ v, ‖conj (S v o) + ζ * conj (S v i)‖...

Proved

BookProof.ChapterWignerSymmetry.exists_zeta

by leonardopedro · Oct 11, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterWignerSymmetry.

BookProof.ChapterWignerSymmetry.exists_zeta (hS : WignerCoord S o) {i : ι} (hi : i ≠ o) : ∃ ζ : ℂ, (ζ = Complex.I ∨ ζ = -Complex.I) ∧ ∀ v, ‖conj (S v o) + ζ * conj (S v i)‖ = ‖conj (v o) + Complex.I * conj (v i)‖

Formalization note: Lean 4 identifier BookProof.ChapterWignerSymmetry.exists_zeta.

Preamble

-- Generated from ChapterWignerSymmetry.lean — theorem BookProof.ChapterWignerSymmetry.exists_zeta
import Mathlib
import Definitions.Def_ChapterWignerSymmetry
open BookProof.ChapterWignerSymmetry


open scoped InnerProductSpace ComplexConjugate
open Finset


variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]


variable {T : E → E}

variable {ι : Type*} [Fintype ι] [DecidableEq ι]
variable {S : (ι → ℂ) → (ι → ℂ)} {o : ι}

Formal statement

theorem BookProof.ChapterWignerSymmetry.exists_zeta (hS : WignerCoord S o) {i : ι} (hi : i ≠ o) :
    ∃ ζ : ℂ, (ζ = Complex.I ∨ ζ = -Complex.I) ∧
      ∀ v, ‖conj (S v o) + ζ * conj (S v i)‖ = ‖conj (v o) + Complex.I * conj (v i)‖ := by sorry

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