Extensionality of multiplication of real numbers as a bilinear form

Content created by Louis Wasserman.

Created on 2025-11-26.
Last modified on 2026-01-09.

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

module real-numbers.extensionality-multiplication-bilinear-form-real-numbers where
Imports 
open import foundation.identity-types
open import foundation.transport-along-identifications
open import foundation.universe-levels

open import real-numbers.absolute-value-real-numbers
open import real-numbers.dedekind-real-numbers
open import real-numbers.nonnegative-real-numbers
open import real-numbers.raising-universe-levels-real-numbers
open import real-numbers.rational-real-numbers
open import real-numbers.similarity-real-numbers
open import real-numbers.square-roots-nonnegative-real-numbers
open import real-numbers.squares-real-numbers
open import real-numbers.zero-real-numbers

Idea

If the square of a real number is equal to 0, the real number is 0. This property is extensionality of multiplication as a bilinear form over the real vector space of real numbers.

Proof

abstract
  eq-zero-eq-zero-square-ℝ :
    {l : Level} (x : ℝ l) → square-ℝ x ＝ raise-zero-ℝ l →
    x ＝ raise-zero-ℝ l
  eq-zero-eq-zero-square-ℝ {l} x x²=0 =
    eq-raise-zero-is-zero-ℝ
      ( is-zero-is-zero-square-ℝ
        ( is-zero-eq-raise-zero-ℝ x²=0))

Recent changes