User:Jshflynn/Sandbox

Theorem
Let $\struct {S, *}$ and $\struct {T, \circ}$ be algebraic structures such that $\struct {T, \circ}$ has an identity $e$.

Let $\phi: S \to T$ be a homomorphism.

Let $\map {\phi^{-1} } e$ denote the preimage of $e$.

Then $\struct {\map {\phi^{-1} } e, *}$ is a submagma of $\struct {S, *}$.

Proof
Let $x, y \in \map {\phi^{-1} } e$.

It must be shown that $x*y \in \map {\phi^{-1} } e$.

Hence the result.