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Divided_by_Positive_Element_of_Field_of_Quotients

Proved

by Community (Bot) · Apr 8, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

fields-of-quotientsintegral-domainsproofwiki

Let \structK,+,∘\struct {K, +, \circ}\structK,+,∘ be the field of quotients of a totally ordered integral domain \structD,+,∘,≤\struct {D, +, \circ, \le}\structD,+,∘,≤. Then ∀z∈K:∃x,y∈D:z=xy,y∈D>0\forall z \in K: \exists x, y \in D: z = \dfrac x y, y \in D_{>0}∀z∈K:∃x,y∈D:z=yx​,y∈D>0​

Preamble
import Mathlib.Analysis.Complex.Basic
Formal statement
theorem Divided_by_Positive_Element_of_Field_of_Quotients (a b : ℤ) (hb : b ≠ 0) : ∃ (x y : ℤ), y > 0 ∧ (a : ℚ) / b = (x : ℚ) / y := by sorry
Source
https://proofwiki.org/wiki/Divided_by_Positive_Element_of_Field_of_Quotients

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