basis_le — Mathlib

English

For rings with nontrivial top, the inclusion of bases is equivalent to the conjunction J ≤ K and e ≤ d.

Русский

Для колец с ненулевой топологией включение баз эквивалентно сочетанию J ≤ K и e ≤ d.

LaTeX

$$theorem basis_le_iff {J K : TwoSidedIdeal R} {d e : σ →₀ ℕ} (hK : K ≠ ⊤) : basis σ R ⟨J, d⟩ ≤ basis σ R ⟨K, e⟩ ↔ J ≤ K ∧ e ≤ d$$

Lean4

/-- If `J ≤ K` and `e ≤ d`, then we have the inclusion of two-sided ideals
`basis σ R ⟨J, d⟩ ≤ basis σ R ⟨K, e,>`. -/
theorem basis_le {Jd Ke : TwoSidedIdeal R × (σ →₀ ℕ)} (hJK : Jd.1 ≤ Ke.1) (hed : Ke.2 ≤ Jd.2) :
    basis σ R Jd ≤ basis σ R Ke := fun _ ↦ forall_imp (fun _ h hue ↦ hJK (h (le_trans hue hed)))

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