coe_span_singleton_eq_setOf_le_v_algebraMap

English

The homomorphism algebraMap O F is bijective under the given integers valuation assumption and a subsingleton of units range.

Русский

Гомоморфизм algebraMap O F является биективным при заданном предположении о целостности и подмножестве единиц.

LaTeX

$$IsBijective (algebraMap O F)$$

Lean4

theorem coe_span_singleton_eq_setOf_le_v_algebraMap (hv : Integers v O) (x : O) :
    (Ideal.span { x } : Set O) = {y : O | v (algebraMap O F y) ≤ v (algebraMap O F x)} :=
  by
  rcases eq_or_ne x 0 with rfl | hx
  · simp [Set.singleton_zero, Ideal.span_zero, map_eq_zero_iff _ hv.hom_inj]
  ext
  simp [SetLike.mem_coe, Ideal.mem_span_singleton, hv.dvd_iff_le]

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