Locally connected implies connected component of identity is open subgroup

Statement
In a uses property satisfaction of::locally connected topological group, the  connected component of identity is an  proves property satisfaction of::open subgroup.

Applications

 * Locally connected and no proper open subgroup implies connected

Proof
Given: A locally connected topological group $$G$$.

To prove: The connected component of identity in $$G$$ is an open subgroup.

Proof: