Real Multiplication Identity is One/Corollary

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Theorem

$\forall x \in \R_{\ne 0}: x \times y = x \implies y = 1$


Proof

\(\displaystyle x \times y\) \(=\) \(\displaystyle x\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \frac 1 x \times \paren {x \times y}\) \(=\) \(\displaystyle \frac 1 x \times x\) as long as $x \ne 0$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {\frac 1 x \times x} \times y\) \(=\) \(\displaystyle \frac 1 x \times x\) Real Number Axioms: $\R M 1$: Associativity
\(\displaystyle \leadsto \ \ \) \(\displaystyle 1 \times y\) \(=\) \(\displaystyle 1\) Real Number Axioms: $\R M 4$: Inverses
\(\displaystyle \leadsto \ \ \) \(\displaystyle y\) \(=\) \(\displaystyle 1\) Real Multiplication Identity is One

$\blacksquare$


Sources