valuation_of_unit_eq — Mathlib

English

The set of derivations from a ring R to an R-module M forms an additive commutative group under pointwise addition.

Русский

Множество производных от кольца R в módulo M над R образует аддитивную коммутативную группу под покоординатным сложением.

LaTeX

$$$\\mathrm{Derivation}(R,A,M) \\text{ is an }\\operatorname{AddCommGroup}$$$

Lean4

@[simp]
theorem valuation_of_unit_eq (x : Rˣ) : v.valuationOfNeZero (Units.map (algebraMap R K : R →* K) x) = 1 :=
  by
  rw [← WithZero.coe_inj, valuationOfNeZero_eq, Units.coe_map, eq_iff_le_not_lt]
  constructor
  · exact v.valuation_le_one x
  · obtain ⟨x, _, hx, _⟩ := x
    change ¬v.valuation K (algebraMap R K x) < 1
    apply_fun v.intValuation at hx
    rw [map_one, map_mul] at hx
    rw [not_lt, ← hx, ← mul_one <| v.valuation _ _, valuation_of_algebraMap]
    gcongr
    exact v.intValuation_le_one _

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