First Power of Element in B-Algebra

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\left({X, \circ}\right)$ be a $B$-algebra.


Then:

$\forall x \in X: x^1 = x$

where $x^k$ for $k \in \N$ denotes the $k$th power of the element $x$.


Proof

\(\ds x^1\) \(=\) \(\ds x^0 \circ \left({0 \circ x}\right)\) Definition of B-Algebra Power of Element
\(\ds \) \(=\) \(\ds 0 \circ \left({0 \circ x}\right)\) Definition of B-Algebra Power of Element
\(\ds \) \(=\) \(\ds 0 \circ x\) $0$ in $B$-Algebra is Left Cancellable Element
\(\ds \) \(=\) \(\ds x\) $0$ in $B$-Algebra is Left Cancellable Element

Hence the result.

$\blacksquare$