`BookProof.ChapterWignerSymmetry.coord_of_key_of_ne_zero` (hS : WignerCoord S o) (κ : ℂ →+* ℂ) (hκn : ∀ z, ‖κ z‖ = ‖z‖) (hκc : ∀ z, κ (conj z) = conj (κ z)) (hkey : ∀ i,...

Proved

BookProof.ChapterWignerSymmetry.coord_of_key_of_ne_zero

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterWignerSymmetry.

BookProof.ChapterWignerSymmetry.coord_of_key_of_ne_zero (hS : WignerCoord S o) (κ : ℂ →+* ℂ) (hκn : ∀ z, ‖κ z‖ = ‖z‖) (hκc : ∀ z, κ (conj z) = conj (κ z)) (hkey : ∀ i, i ≠ o → ∀ v, conj (S v o) * S v i = κ (conj (v o) * v i)) {v : ι → ℂ} (hv : v o ≠ 0) : ∃ lam : ℂ, ‖lam‖ = 1 ∧ ∀ k, S v k = lam * κ (v k)

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

Preamble

-- Generated from ChapterWignerSymmetry.lean — theorem BookProof.ChapterWignerSymmetry.coord_of_key_of_ne_zero
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.coord_of_key_of_ne_zero (hS : WignerCoord S o) (κ : ℂ →+* ℂ) (hκn : ∀ z, ‖κ z‖ = ‖z‖)
    (hκc : ∀ z, κ (conj z) = conj (κ z))
    (hkey : ∀ i, i ≠ o → ∀ v, conj (S v o) * S v i = κ (conj (v o) * v i))
    {v : ι → ℂ} (hv : v o ≠ 0) : ∃ lam : ℂ, ‖lam‖ = 1 ∧ ∀ k, S v k = lam * κ (v k) := by sorry

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