English
The isLocalizedModule_iff_isLocalization' equivalence states that IsLocalizedModule S (IsScalarTower.toAlgHom R A A_s).toLinearMap holds exactly when IsLocalization S A holds.
Русский
Эквивалентность isLocalizedModule_iff_isLocalization' утверждает, что IsLocalizedModule S (IsScalarTower.toAlgHom R A A_s).toLinearMap истинно тогда и только тогда, когда IsLocalization S A истинно.
LaTeX
$$$\\IsLocalizedModule S (\\IsScalarTower.toAlgHom R A A_s).toLinearMap \\iff \\IsLocalization S A.$$$
Lean4
theorem localized'_eq_span : localized' S p f M' = span S (f '' M') :=
by
refine le_antisymm ?_ (span_le.mpr <| by rintro _ ⟨m, hm, rfl⟩; exact ⟨m, hm, 1, by simp⟩)
rintro _ ⟨m, hm, s, rfl⟩
rw [← one_smul R m, ← mul_one s, ← IsLocalizedModule.mk'_smul_mk' S]
exact smul_mem _ _ (subset_span ⟨m, hm, by simp⟩)
