Zero of Field is Unique/Proof 2
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Theorem
Let $\struct {F, +, \times}$ be a field.
The zero of $F$ is unique.
Proof
Let $0_1$ and $0_2$ both be elements of $F$ such that:
- $\forall a \in F: a + 0_1 = a$
- $\forall a \in F: a + 0_2 = a$
Then:
- $0_1 + 0_2 = 0_2$
because $0_1$ is a zero element
- $0_1 + 0_2 = 0_1$
because $0_2$ is a zero element
Hence:
- $0_1 = 0_2$
and the two zero elements are the same.
$\blacksquare$
Sources
- 1973: C.R.J. Clapham: Introduction to Mathematical Analysis ... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields: Theorem $1 \ \text{(i)}$