The multiplicative group of positive rational numbers

Content created by malarbol, Fredrik Bakke, Garrett Figueroa and Louis Wasserman.

Created on 2024-04-25.
Last modified on 2025-08-11.

{-# OPTIONS --lossy-unification #-}

module elementary-number-theory.multiplicative-group-of-positive-rational-numbers where
Imports 
open import elementary-number-theory.inequality-integers
open import elementary-number-theory.inequality-rational-numbers
open import elementary-number-theory.multiplication-integers
open import elementary-number-theory.multiplication-rational-numbers
open import elementary-number-theory.multiplicative-monoid-of-rational-numbers
open import elementary-number-theory.positive-rational-numbers
open import elementary-number-theory.rational-numbers
open import elementary-number-theory.strict-inequality-integers
open import elementary-number-theory.strict-inequality-rational-numbers

open import foundation.binary-transport
open import foundation.cartesian-product-types
open import foundation.dependent-pair-types
open import foundation.identity-types
open import foundation.universe-levels

open import group-theory.abelian-groups
open import group-theory.groups
open import group-theory.monoids
open import group-theory.submonoids

Idea

The submonoid of positive rational numbers in the multiplicative monoid of rational numbers is a commutative group.

Definitions

The positive inverse of a positive rational number

opaque
  unfolding inv-is-positive-ℚ

  inv-ℚ⁺ : ℚ⁺ → ℚ⁺
  pr1 (inv-ℚ⁺ (x , P)) = inv-is-positive-ℚ x P
  pr2 (inv-ℚ⁺ (x , P)) = is-positive-denominator-ℚ x

Inverse laws in the multiplicative group of positive rational numbers

opaque
  unfolding inv-ℚ⁺

  left-inverse-law-mul-ℚ⁺ : (x : ℚ⁺) → (inv-ℚ⁺ x) *ℚ⁺ x ＝ one-ℚ⁺
  left-inverse-law-mul-ℚ⁺ x =
    eq-ℚ⁺
      ( left-inverse-law-mul-is-positive-ℚ
        ( rational-ℚ⁺ x)
        ( is-positive-rational-ℚ⁺ x))

  right-inverse-law-mul-ℚ⁺ : (x : ℚ⁺) → x *ℚ⁺ (inv-ℚ⁺ x) ＝ one-ℚ⁺
  right-inverse-law-mul-ℚ⁺ x =
    eq-ℚ⁺
      ( right-inverse-law-mul-is-positive-ℚ
        ( rational-ℚ⁺ x)
        ( is-positive-rational-ℚ⁺ x))

The multiplicative group of positive rational numbers

group-mul-ℚ⁺ : Group lzero
pr1 group-mul-ℚ⁺ = semigroup-mul-ℚ⁺
pr1 (pr2 group-mul-ℚ⁺) = is-unital-Monoid monoid-mul-ℚ⁺
pr1 (pr2 (pr2 group-mul-ℚ⁺)) = inv-ℚ⁺
pr1 (pr2 (pr2 (pr2 group-mul-ℚ⁺))) = left-inverse-law-mul-ℚ⁺
pr2 (pr2 (pr2 (pr2 group-mul-ℚ⁺))) = right-inverse-law-mul-ℚ⁺

Properties

The multiplicative group of positive rational numbers is commutative

abelian-group-mul-ℚ⁺ : Ab lzero
pr1 abelian-group-mul-ℚ⁺ = group-mul-ℚ⁺
pr2 abelian-group-mul-ℚ⁺ = commutative-mul-ℚ⁺

Inversion reverses inequality on the positive rational numbers

opaque
  unfolding inv-ℚ⁺
  unfolding leq-ℚ-Prop

  inv-leq-ℚ⁺ : (x y : ℚ⁺) → leq-ℚ⁺ (inv-ℚ⁺ x) (inv-ℚ⁺ y) → leq-ℚ⁺ y x
  inv-leq-ℚ⁺ x y =
    binary-tr
      ( leq-ℤ)
      ( commutative-mul-ℤ
        ( denominator-ℚ⁺ x)
        ( numerator-ℚ⁺ y))
      ( commutative-mul-ℤ
        ( denominator-ℚ⁺ y)
        ( numerator-ℚ⁺ x))

Inversion reverses strict inequality on the positive rational numbers

opaque
  unfolding inv-ℚ⁺
  unfolding le-ℚ-Prop

  inv-le-ℚ⁺ : (x y : ℚ⁺) → le-ℚ⁺ (inv-ℚ⁺ x) (inv-ℚ⁺ y) → le-ℚ⁺ y x
  inv-le-ℚ⁺ x y =
    binary-tr
      ( le-ℤ)
      ( commutative-mul-ℤ
        ( denominator-ℚ⁺ x)
        ( numerator-ℚ⁺ y))
      ( commutative-mul-ℤ
        ( denominator-ℚ⁺ y)
        ( numerator-ℚ⁺ x))

  inv-le-ℚ⁺' : (x y : ℚ⁺) → le-ℚ⁺ x y → le-ℚ⁺ (inv-ℚ⁺ y) (inv-ℚ⁺ x)
  inv-le-ℚ⁺' x y =
    binary-tr
      ( le-ℤ)
      ( inv
        ( commutative-mul-ℤ
          ( denominator-ℚ⁺ y)
          ( numerator-ℚ⁺ x)))
      ( inv
        ( commutative-mul-ℤ
          ( denominator-ℚ⁺ x)
          ( numerator-ℚ⁺ y)))

Inversion of positive rational numbers distributes over multiplication

abstract
  distributive-inv-mul-ℚ⁺ :
    (x y : ℚ⁺) → inv-ℚ⁺ (x *ℚ⁺ y) ＝ inv-ℚ⁺ x *ℚ⁺ inv-ℚ⁺ y
  distributive-inv-mul-ℚ⁺ x y =
    distributive-inv-mul-Group'
      ( group-mul-ℚ⁺)
      ( x)
      ( y)
      ( commutative-mul-ℚ⁺ x y)

Inversion on the positive rational numbers interchanges numerator and denominator

module _
  (x : ℚ⁺)
  where

  opaque
    unfolding inv-ℚ⁺

    eq-numerator-inv-denominator-ℚ⁺ :
      numerator-ℚ⁺ (inv-ℚ⁺ x) ＝ denominator-ℚ⁺ x
    eq-numerator-inv-denominator-ℚ⁺ =
      ind-Σ eq-numerator-inv-denominator-is-positive-ℚ x

    eq-denominator-inv-numerator-ℚ⁺ :
      denominator-ℚ⁺ (inv-ℚ⁺ x) ＝ numerator-ℚ⁺ x
    eq-denominator-inv-numerator-ℚ⁺ =
      ind-Σ eq-denominator-inv-numerator-is-positive-ℚ x

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