surjectiveOnStalks_of_surjective — Mathlib

English

If the ring map f is itself surjective, then f is surjective on stalks.

Русский

Если отображение колец f сюръективно, то оно сюръективно на всех стэках.

LaTeX

$$$\\text{If } f \\text{ is surjective, then } f.SurjectiveOnStalks$$$

Lean4

theorem surjectiveOnStalks_of_surjective (h : Function.Surjective f) : SurjectiveOnStalks f :=
  surjectiveOnStalks_iff_forall_ideal.mpr fun _ _ s ↦
    let ⟨r, hr⟩ := h s
    ⟨r, 1, 1, by simpa [← Ideal.eq_top_iff_one], by simpa [← Ideal.eq_top_iff_one], by simp [hr]⟩

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