idealFactorsEquivOfQuotEquiv_is_dvd_iso

English

There is a canonical bijection between the sets of normalized factors of I and J, induced by a quotient equivalence f.

Русский

Существует каноническая биекция между множествами нормализованных факторов I и J, индуцированная тquotient эквивалентом f.

LaTeX

$$normalizedFactorsEquivOfQuotEquiv$$

Lean4

theorem idealFactorsEquivOfQuotEquiv_is_dvd_iso {L M : Ideal R} (hL : L ∣ I) (hM : M ∣ I) :
    (idealFactorsEquivOfQuotEquiv f ⟨L, hL⟩ : Ideal A) ∣ idealFactorsEquivOfQuotEquiv f ⟨M, hM⟩ ↔ L ∣ M :=
  by
  suffices
    idealFactorsEquivOfQuotEquiv f ⟨M, hM⟩ ≤ idealFactorsEquivOfQuotEquiv f ⟨L, hL⟩ ↔
      (⟨M, hM⟩ : {p : Ideal R | p ∣ I}) ≤ ⟨L, hL⟩
    by rw [dvd_iff_le, dvd_iff_le, Subtype.coe_le_coe, this, Subtype.mk_le_mk]
  exact (idealFactorsEquivOfQuotEquiv f).le_iff_le

Похожие утверждения