English
In Ring.DirectLimit, every element is represented by some component's element under the quotient map.
Русский
В Ring.DirectLimit каждый элемент представлен элементом компоненты через притомление по идентефикации.
LaTeX
$$exists_of holds in Ring.DirectLimit: exists i x with of i x = z$$
Lean4
/-- Every element of the direct limit corresponds to some element in
some component of the directed system. -/
theorem exists_of [Nonempty ι] [IsDirected ι (· ≤ ·)] (z : DirectLimit G f) : ∃ i x, of G f i x = z :=
by
obtain ⟨z, rfl⟩ := Ideal.Quotient.mk_surjective z
refine z.induction_on ⟨Classical.arbitrary ι, -1, by simp⟩ (fun ⟨i, x⟩ ↦ ⟨i, x, rfl⟩) ?_ ?_ <;>
rintro x' y' ⟨i, x, hx⟩ ⟨j, y, hy⟩ <;>
have ⟨k, hik, hjk⟩ := exists_ge_ge i j
· exact ⟨k, f i k hik x + f j k hjk y, by rw [map_add, of_f, of_f, hx, hy]; rfl⟩
· exact ⟨k, f i k hik x * f j k hjk y, by rw [map_mul, of_f, of_f, hx, hy]; rfl⟩
