State/Homomorphic images of subgroups are subgroups

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Proposition
Proof

Proposition: (Homomorphic images of subgroups are subgroups) Suppose that $G$ and $H$ are groups, and $\phi \colon G \rightarrow H$ is a homomorphism. Let $L$ be a subgroup of $G$. Then $\phi(L)$ is a subgroup of $H$.

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This statement logically relies on the following definitions and statements: Def/Group, Def/Group homomorphism

The following statements and definitions rely on the material of this page: Def/Conjugate

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State/Homomorphic images of subgroups are subgroups

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	Suppose that $G$ and $H$ are groups, and $\phi \colon G \rightarrow H$ is a homomorphism. Let $L$ be a subgroup of $G$.
	Suppose that $h_1 h_2 \in \phi(L)$. Then, there exist $\ell_1, \ell_2 \in L$, such that $\phi(\ell_1) = h_1$ and $\phi(\ell_2) = h_2$. It follows that: $$\phi(\ell_1 \ell_2^{-1} ) = \phi(\ell_1) \phi(\ell_2)^{-1} = h_1 h_2^{-1}.$$ Hence $h_1 h_2^{-1} \in \phi(L)$.
	Therefore, $\phi(L)$ is a subgroup of $H$.