Real Zero is Zero Element

Theorem

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

$\ds 0 \times x$

$=$

$\ds \paren {0 + 0} \times x$

Real Number Axiom $\R \text A3$: Identity for Addition

$\ds $

$=$

$\ds 0 \times x + 0 \times x$

Real Number Axiom $\R \text D$: Distributivity of Multiplication over Addition

$\ds \leadsto \ \ $

$\ds 0 \times x$

$=$

$\ds 0$

$\blacksquare$

1973: G. Stephenson: Mathematical Methods for Science Students (2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.2$ Operations with Real Numbers: $(7)$
2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $1 \ \text{(b)}$