Existence and Uniqueness of Direct Limit of Sequence of Groups/Lemma 3

Lemma
Let $h_\infty: G_\infty \to H$ be the mapping defined as:


 * $\eqclass {\tuple {x_n, n} } {} \mapsto \map {h_n} {x_n}$

Then $h_\infty$ is a well-defined group homomorphism.

Well-Definedness of $h_\infty$
Let $\tuple {x_n, n}, \tuple{x_{n'}, n'} \in \eqclass {\tuple {x_n, n} } {}$.

, let $n' \ge n$.

Then we have:
 * $\map {g_{n, n'} } {x_n} = x_{n'}$

and:

This proves that $h_\infty$ is independent of the representative chosen.

That is, $h_\infty$ is well-defined.

Homomorphism Property
Let $\eqclass {\tuple{x_n, n} } {}, \eqclass {\tuple {y_m, m} } {} \in G_\infty$.

By the definition of the group operation,, suppose $n = m$.

See Lemma 2 for details.

It follows that:

Thus $h_\infty$ is a homomorphism.