`BookProof.ChapterWignerSymmetry.key_complex` (a b c d : ℂ) (h1 : ‖a‖ = ‖c‖) (h2 : ‖b‖ = ‖d‖) (h3 : ‖a + b‖ = ‖c + d‖) (h4 : ‖conj a + Complex.I * conj b‖ = ‖conj...

Proved

BookProof.ChapterWignerSymmetry.key_complex

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterWignerSymmetry.

BookProof.ChapterWignerSymmetry.key_complex (a b c d : ℂ) (h1 : ‖a‖ = ‖c‖) (h2 : ‖b‖ = ‖d‖) (h3 : ‖a + b‖ = ‖c + d‖) (h4 : ‖conj a + Complex.I * conj b‖ = ‖conj c + Complex.I * conj d‖) : conj a * b = conj c * d

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

Preamble

-- Generated from ChapterWignerSymmetry.lean — theorem BookProof.ChapterWignerSymmetry.key_complex
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}

Formal statement

theorem BookProof.ChapterWignerSymmetry.key_complex (a b c d : ℂ) (h1 : ‖a‖ = ‖c‖) (h2 : ‖b‖ = ‖d‖) (h3 : ‖a + b‖ = ‖c + d‖)
    (h4 : ‖conj a + Complex.I * conj b‖ = ‖conj c + Complex.I * conj d‖) :
    conj a * b = conj c * d := by sorry

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