not_isMulTorsionFree_of_isTorsion

English

A nontrivial torsion abelian group is not torsion-free.

Русский

Непустая торовая абелевация не торовоподобна.

LaTeX

$$Not (IsMulTorsionFree(G)) given (IsTorsion G) and Nontrivial G$$

Lean4

/-- A nontrivial torsion abelian group is not torsion-free. -/
@[to_additive /-- A nontrivial additive torsion abelian group is not torsion-free. -/
    ]
theorem not_isMulTorsionFree_of_isTorsion [Nontrivial G] (hG : IsTorsion G) : ¬IsMulTorsionFree G :=
  not_isMulTorsionFree_iff_isOfFinOrder.2 <|
    let ⟨x, hx⟩ := exists_ne (1 : G);
    ⟨x, hx, hG x⟩

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