addEquiv_eq_refl_or_neg — Mathlib

English

Every commutative group H is naturally isomorphic to its abelianization: there exists an isomorphism H ≅ Abelianization(H).

Русский

Каждая коммутативная группа H естественно изоморфна своей абелизации: существует изоморфизм H ≅ Abelianization(H).

LaTeX

$$$ H \cong Abelianization(H) $$$

Lean4

theorem addEquiv_eq_refl_or_neg (e : ℤ ≃+ ℤ) : e = .refl _ ∨ e = .neg _ :=
  by
  suffices e 1 = 1 ∨ -e 1 = 1 by simpa [AddEquiv.ext_int_iff, neg_eq_iff_eq_neg]
  have he : ¬IsOfFinAddOrder (e 1) :=
    not_isOfFinAddOrder_of_isAddTorsionFree ((AddEquiv.map_ne_zero_iff e).mpr Int.one_ne_zero)
  rw [← AddSubgroup.zmultiples_eq_zmultiples_iff he]
  simpa [e.surjective, eq_comm] using (e : ℤ →+ ℤ).map_zmultiples 1

Похожие утверждения