Let $ G $ be a Lie group and $ H\subset G $ be a Lie subgroup, which is itself a Lie group and an immersed submanifold of $ G $. Is the following correct:
for any smooth map $ f:M\to G $ where $ M $ is a smooth manifold, if the image of $ f $ lies in $ H $, then $ \hat{f}:M\to H $ is continuous(and consequently smooth) when $ H $ is regarded as the immersed submanifold of $ G $(this is just the definition of being weakly embedded)?