English
The inverse of quotientEquiv is the quotient map induced by f⁻¹ with hIJ adjusted accordingly.
Русский
Обратное quotientEquiv — это косательное отображение, индуцированное f⁻¹ с корректировкой hIJ.
LaTeX
$$$(\text{quotientEquiv } I J f hIJ)^{-1} = \text{quotientEquiv } J I f^{-1} hIJ^{-1}$$$
Lean4
/-- `H` and `h` are kept as separate hypothesis since H is used in constructing the quotient map. -/
theorem quotientMap_injective' {J : Ideal R} {I : Ideal S} [I.IsTwoSided] [J.IsTwoSided] {f : R →+* S}
{H : J ≤ I.comap f} (h : I.comap f ≤ J) : Function.Injective (quotientMap I f H) :=
by
refine (injective_iff_map_eq_zero (quotientMap I f H)).2 fun a ha => ?_
obtain ⟨r, rfl⟩ := Quotient.mk_surjective a
rw [quotientMap_mk, Quotient.eq_zero_iff_mem] at ha
exact Quotient.eq_zero_iff_mem.mpr (h ha)
