`BookProof.AbelianDiagonal.vonNeumann_abelian_typeI_case` : Function.Injective (diagonalStarAlgHom : (n → ℂ) →⋆ₐ[ℂ] Matrix n n ℂ) ∧ (∀ d e : n → ℂ, Matrix.diagonal d *...

Proved

BookProof.AbelianDiagonal.vonNeumann_abelian_typeI_case

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAbelianDiagonal.

BookProof.AbelianDiagonal.vonNeumann_abelian_typeI_case : Function.Injective (diagonalStarAlgHom : (n → ℂ) →⋆ₐ[ℂ] Matrix n n ℂ) ∧ (∀ d e : n → ℂ, Matrix.diagonal d * Matrix.diagonal e = Matrix.diagonal e * Matrix.diagonal d) ∧ (∀ M : Matrix n n ℂ, (∀ d : n → ℂ, M * Matrix.diagonal d = Matrix.diagonal d * M) ↔ ∃ e : n → ℂ, M = Matrix.diagonal e)

Formalization note: Lean 4 identifier BookProof.AbelianDiagonal.vonNeumann_abelian_typeI_case.

Preamble

-- Generated from ChapterAbelianDiagonal.lean — theorem BookProof.AbelianDiagonal.vonNeumann_abelian_typeI_case
import Mathlib
import Definitions.Def_ChapterAbelianDiagonal
open BookProof.AbelianDiagonal



open Matrix

variable {n : Type*} [Fintype n] [DecidableEq n]

Formal statement

theorem BookProof.AbelianDiagonal.vonNeumann_abelian_typeI_case :
    Function.Injective (diagonalStarAlgHom : (n → ℂ) →⋆ₐ[ℂ] Matrix n n ℂ) ∧
      (∀ d e : n → ℂ,
        Matrix.diagonal d * Matrix.diagonal e = Matrix.diagonal e * Matrix.diagonal d) ∧
      (∀ M : Matrix n n ℂ,
        (∀ d : n → ℂ, M * Matrix.diagonal d = Matrix.diagonal d * M) ↔
          ∃ e : n → ℂ, M = Matrix.diagonal e) := by sorry

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