English
Analyzes a coordinatewise endomorphism compatibility with the isotypic decomposition; coordinates map to componentwise endomorphisms.
Русский
Анализируется совместимость эндоморфизмов по координатам относительно изотипного разложения; координаты переходят к поперечным эндоморфизмам компонент.
LaTeX
$$$\\text{EndAlgEquiv}_{\\_\\text{proof3}}(R,M, c)\\;: \\text{coordinatewise compatibility}$$$
Lean4
/-- The **Wedderburn–Artin Theorem**, algebra form: an Artinian simple algebra is isomorphic
to a matrix algebra over the opposite of the endomorphism algebra of its simple module. -/
theorem exists_algEquiv_matrix_end_mulOpposite :
∃ (n : ℕ) (_ : NeZero n) (I : Ideal R) (_ : IsSimpleModule R I),
Nonempty (R ≃ₐ[R₀] Matrix (Fin n) (Fin n) (Module.End R I)ᵐᵒᵖ) :=
by
have ⟨n, hn, S, hS, ⟨e⟩⟩ := (isIsotypic R).linearEquiv_fun
refine ⟨n, hn, S, hS, ⟨.trans (.opOp R₀ R) <| .trans (.op ?_) (.symm .mopMatrix)⟩⟩
exact .trans (.moduleEndSelf R₀) <| .trans (e.algConj R₀) (endVecAlgEquivMatrixEnd ..)
