Real Addition Identity is Zero/Corollary

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Corollary to Real Addition Identity is Zero

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


Proof

\(\displaystyle x + y\) \(=\) \(\displaystyle x\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {-x} + \paren {x + y}\) \(=\) \(\displaystyle \paren {-x} + x\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {\paren {-x} + x} + y\) \(=\) \(\displaystyle \paren {-x} + x\) Real Number Axioms: $\R A 1$
\(\displaystyle \leadsto \ \ \) \(\displaystyle 0 + y\) \(=\) \(\displaystyle 0\) Real Number Axioms: $\R A 4$
\(\displaystyle \leadsto \ \ \) \(\displaystyle y\) \(=\) \(\displaystyle 0\) Real Addition Identity is Zero

$\blacksquare$


Sources