the problem is verify this statement: $$\lnot ( \lnot p \land q)^\land (p \lor q)=p$$
That turns into $p \lor ( \lnot q \land q)$ - which that last part of the statement is a contradiction and is always false
which means now I have $p \lor F =p$
Does that mean that the original statement is true and valid or does it mean it's false? Could someone please explain.
So, the task is to determine whether $$\lnot( \lnot p \land q)\land (p \lor q) \equiv p \tag{Q}$$
We'll start on the left-hand side (LHS) and use equivalencies to prove that $(Q)$ is true:
$$\begin{align} \lnot( \lnot p \land q)\land (p \lor q) &\equiv (p \lor \lnot q)\land (p \lor q)\tag{De Morgan's} \\ \\ &\equiv p \lor (\underbrace{\lnot q \land q}_{\text{False}})\tag{distributive property} \\ \\ &\equiv p \lor \text {False}\\ \\ &\equiv p \end{align}$$
This means that $$\lnot( \lnot p \land q)\land (p \lor q) \equiv p\tag{Q}$$ is true, since we've shown that the left hand side of $(Q)$ is equivalent to the right hand side, hence the equivalence is true.