Real Multiplication Identity is One/Corollary

Theorem

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

Proof

\(\ds x \times y\)

\(=\)

\(\ds x\)

\(\ds \leadsto \ \ \)

\(\ds \frac 1 x \times \paren {x \times y}\)

\(=\)

\(\ds \frac 1 x \times x\)

as long as $x \ne 0$

\(\ds \leadsto \ \ \)

\(\ds \paren {\frac 1 x \times x} \times y\)

\(=\)

\(\ds \frac 1 x \times x\)

Real Number Axiom $\R \text M1$: Associativity of Multiplication

\(\ds \leadsto \ \ \)

\(\ds 1 \times y\)

\(=\)

\(\ds 1\)

Real Number Axiom $\R \text M4$: Inverses for Multiplication

\(\ds \leadsto \ \ \)

\(\ds y\)

\(=\)

\(\ds 1\)

Real Multiplication Identity is One

$\blacksquare$

Sources

• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $1 \ \text{(i)}$