valuation_of_unit_mod_eq — Mathlib

English

For any natural n and unit x of R, the valuation of its image in K modulo n equals 1 in the Selmer context.

Русский

Для любого n и единицы x в R, оценка образа в K по модулю n равна 1 в контексте Selmer.

LaTeX

$$valuation_of_unit_mod_eq (n) (x) : S(valuation)(image_of_x) = 1$$

Lean4

@[simp]
theorem valuation_of_unit_mod_eq (n : ℕ) (x : Rˣ) :
    v.valuationOfNeZeroMod n (Units.map (algebraMap R K : R →* K) x : K / n) = 1 := by
  -- This used to be `rw`, but we need `erw` after https://github.com/leanprover/lean4/pull/2644
  erw [valuationOfNeZeroMod, MonoidHom.comp_apply, ← QuotientGroup.coe_mk',
    QuotientGroup.map_mk' (G := Kˣ) (N := MonoidHom.range (powMonoidHom n)), valuation_of_unit_eq, QuotientGroup.mk_one,
    map_one]

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