Real Zero is Zero Element

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Theorem

$\forall x \in \R: 0 \times x = 0$


Proof

\(\displaystyle 0 \times x\) \(=\) \(\displaystyle \paren {0 + 0} \times x\) Real Number Axioms: $\R A 3$
\(\displaystyle \) \(=\) \(\displaystyle 0 \times x + 0 \times x\) Real Number Axioms: $\R D$
\(\displaystyle \leadsto \ \ \) \(\displaystyle 0 \times x\) \(=\) \(\displaystyle 0\) Real Addition Identity is Zero: Corollary

$\blacksquare$


Sources