Real Addition Identity is Zero/Corollary

Corollary to Real Addition Identity is Zero

$\forall x, y \in \R: x + y = x \implies y = 0$

Proof

\(\ds x + y\)

\(=\)

\(\ds x\)

\(\ds \leadsto \ \ \)

\(\ds \paren {-x} + \paren {x + y}\)

\(=\)

\(\ds \paren {-x} + x\)

\(\ds \leadsto \ \ \)

\(\ds \paren {\paren {-x} + x} + y\)

\(=\)

\(\ds \paren {-x} + x\)

Real Number Axiom $\R \text A1$: Associativity of Addition

\(\ds \leadsto \ \ \)

\(\ds 0 + y\)

\(=\)

\(\ds 0\)

Real Number Axiom $\R \text A4$: Inverses for Addition

\(\ds \leadsto \ \ \)

\(\ds y\)

\(=\)

\(\ds 0\)

Real Addition Identity is Zero

$\blacksquare$

Sources

• 1967: Michael Spivak: Calculus ... (previous) ... (next): Part $\text I$: Prologue: Chapter $1$: Basic Properties of Numbers
• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $1 \ \text{(a)}$