supIndep_torsionBy — Mathlib

English

If the p_i are pairwise coprime, the family {torsionBySet R M (p_i)} is SupIndep.

Русский

Если p_i попарноCOPRIME, семейство {torsionBySet R M (p_i)} является SupIndep.

LaTeX

$$$\text{SupIndep}\; i \mapsto \mathrm{torsionBySet}\,R\,M\,(p_i)$ under pairwise condition$$

Lean4

theorem supIndep_torsionBy (hq : (S : Set ι).Pairwise <| (IsCoprime on q)) : S.SupIndep fun i => torsionBy R M <| q i :=
  by
  convert
    supIndep_torsionBySet_ideal (M := M) fun i hi j hj ij => (Ideal.sup_eq_top_iff_isCoprime (q i) _).mpr <| hq hi hj ij
  exact (torsionBySet_span_singleton_eq (R := R) (M := M) _).symm

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