exists_span_set_encard_eq_spanFinrank

English

Under the Snake Lemma hypotheses, both δ' left and δ' right yield exactness: F → δ' → G, and δ' → G → 0 are exact with the appropriate maps.

Русский

При гипотезах змейки точны как левая часть δ', так и правая часть δ', образуя точные цепочки через δ'.

LaTeX

$$$\operatorname{Im}(F) = \ker(\delta')$ и $\operatorname{Im}(\delta') = \ker(G)$$$

Lean4

/-- Constructs a generating set with cardinality equal to the `spanFinrank` of the submodule when
  the submodule is finitely generated. -/
theorem exists_span_set_encard_eq_spanFinrank {p : Submodule R M} (h : p.FG) :
    ∃ s : Set M, s.encard = p.spanFinrank ∧ span R s = p :=
  by
  obtain ⟨s, ⟨hs₁, hs₂⟩⟩ := exists_span_set_card_eq_spanRank p
  refine ⟨s, ⟨?_, hs₂⟩⟩
  have := fg_iff_spanRank_eq_spanFinrank.mpr h
  rw [Set.encard, ENat.card, spanFinrank, hs₁, this]
  simp

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