Integral submanifolds of an involutive manifold over $M$ posess an unique structure of smooth submanifold of $M$

Question

Consider a smooth manifold $M$, an involutive distribution $D$ over $M$, and $N\subset M$. We need to prove that if $N$ admits a structure of integral submanifold of $D$, then $N$ admits an unique structure of smooth manifold of $M$.

I'm sure that the Global Frobenius Theorem has to be used somewhere, because we can rewrite it as "any involutive distribution has asociated one and only one foliation", which is similar to what I want to achieve. The Local Frobenius Theorem also assures us that $D$ has to be integrable, but I don't know if this is of any help.

Could anyone please help me out? All of these concepts are very new to me, so any lead of what to do would be helpful (wouldn't like to have the full answer, just a hint).

Answer 1

I think I figured it out. It uses a result that says the following:

Theorem: Let $F:M\to N$ a smooth function between two smooth manifolds. If we consider $P$ an integral submanifold of $D$ such that $F(M)\subset P$, then $F_0:M\to P$ is smooth.

The thing is to prove that if we consider two smooth structures, prove that they are the same. If we consider $N_1$ and $N_2$ the set $N$ with two structures of submanifold, we can see that $Id_N:N_1\to N_2$ is a diffeomorphism thanks to the previous theorem.

Considering $i_1:N_1\to M$, we have that $i_1(N_1)\subset N_2$ ,and so $(i_1)_0 = Id_N:N_1\to N_2$ is smooth, and by considering $i_2:N_2\to M$, then $i_2(N_2)\subset N_1$, and then $(i_2)_0=(Id_N)^{-1}:N_2\to N_1$ is smooth.

It was a similar prove to check that embedded submanifolds have an unique structure, since a similar theorem to the one writen above was used. I don't know if there are something specific I need to address.