# Preimage of Zero of Homomorphism is Submagma

## Theorem

Let $\struct {S, *}$ be a magma.

Let $\struct {T, \circ}$ be a magma with a zero element $0$.

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

Then $\struct {\phi^{-1} \sqbrk 0, *}$ is a submagma of $\struct {S, *}$.

## Proof

Let $x, y \in \phi^{-1} \sqbrk 0$.

It is to be shown that:

$x * y \in \phi^{-1} \sqbrk 0$

Thus:

 $\displaystyle x, y \in \phi^{-1} \sqbrk 0$ $\leadstoandfrom$ $\displaystyle \paren {\map \phi x = 0} \land \paren {\map \phi y = 0}$ Definition of Preimage of Element under Mapping $\displaystyle$ $\leadstoandfrom$ $\displaystyle \map \phi x \circ \map \phi y = 0$ Definition of Zero Element $\displaystyle$ $\leadstoandfrom$ $\displaystyle \map \phi {x * y} = 0$ Definition of Homomorphism (Abstract Algebra) $\displaystyle$ $\leadstoandfrom$ $\displaystyle x * y \in \phi^{-1} \sqbrk 0$ Definition of Preimage of Element under Mapping

Hence the result.

$\blacksquare$