Real Zero is Zero Element
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Theorem
- $\forall x \in \R: 0 \times x = 0$
Proof
| \(\ds 0 \times x\) | \(=\) | \(\ds \paren {0 + 0} \times x\) | 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 | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 0 \times x\) | \(=\) | \(\ds 0\) | Real Addition Identity is Zero: Corollary |
$\blacksquare$
Sources
- 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 $1 \ \text{(b)}$