Real Zero is Zero Element

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

$\blacksquare$