support_subset_of_surjective — Mathlib

English

If f: M → N is injective, then the support of M is contained in the support of N.

Русский

Если отображение f: M → N инъективно, то опора M ⊆ опора N.

LaTeX

$$Supp_R(M) ⊆ Supp_R(N) if f is injective$$

Lean4

@[stacks 00L3 "(3)"]
theorem support_subset_of_surjective (hf : Function.Surjective f) : Module.support R N ⊆ Module.support R M :=
  by
  simp_rw [Set.subset_def, mem_support_iff']
  rintro x ⟨m, hm⟩
  obtain ⟨m, rfl⟩ := hf m
  exact ⟨m, fun r hr e ↦ hm r hr (by simpa using congr(f $e))⟩

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