finite_quotient — Mathlib

English

If φ is arith Frobenius at Q, then the cardinality of the quotient R/(Q.under R) is positive.

Русский

Если φ — арифметический Фробениус по Q, то кардинал R/(Q.under R) положителен.

LaTeX

$$$$ 0 < \operatorname{card}\left(R \;/\; Q.under R\right). $$$$

Lean4

theorem finite_quotient : _root_.Finite (R ⧸ Q.under R) :=
  by
  by_contra! h
  obtain rfl : Q = ⊤ := by simpa [Nat.card_eq_zero_of_infinite, ← Ideal.eq_top_iff_one] using H 0
  simp only [Ideal.comap_top] at h
  exact not_finite (R ⧸ (⊤ : Ideal R))

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