nonunits — Mathlib

English

The nonunits of a valuation subring A form a subsemigroup of K, namely the set {x ∈ K : v(x) < 1}, closed under multiplication.

Русский

Неединицы valuation-подкольца A образуют подпол semigroup K: {x ∈ K : v(x) < 1}, замкнутое относительно умножения.

LaTeX

$$$A_{\mathrm{nonunits}} = \{x \in K : v_A(x) < 1\}$ and this is a Subsemigroup of K$$

Lean4

/-- The nonunits of a valuation subring of `K`, as a subsemigroup of `K` -/
def nonunits : Subsemigroup K where
  carrier := {x | A.valuation x < 1}
  mul_mem' ha hb := (mul_lt_mul'' (Set.mem_setOf.mp ha) (Set.mem_setOf.mp hb) zero_le' zero_le').trans_eq <| mul_one _

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