Math Logic

I JUST started with this, and I can't do the exercises, I simply don't know where to start... I have to verify:

Code:

[p^(p=>q)]=>q

What I've got so far is

Code:

~[p^(p=>q)] v q
~p v ~(p=>q) v q
~p v (p ^ ~q) v q

After that step I just don't know what to do, also, just in case my symbols aren't used correctly, ^ conjunction, v as "or" and ~ as "no".
It should be universal language but I hope I didn't do anything stupid just cause I'm asking in english.

1/ Verification: confirmation: additional proof that something that was believed (some fact or hypothesis or theory) is correct. (Just so I don't get off track, I haven't really done any of these exercises)
2/ Symbology:
- ∧ = conjunction: A∧B is satisfied if and only if both A and B are true
- ⇒ = implication: A⇒B means if A is true, then B is true
- ¬ = not: The famous P ∧ ¬P that can derive anything under this or the next sky, including your mom's phone number

[p ∧ (p⇒q)] ⇒ q translates into:
Layer 1: If [p ∧ (p⇒q)] is true, then q is true.
Layer 2: If [p ∧ (p⇒q)] is true, p and (p⇒q) must simultaneously be true.
Layer 3: Well, it's obvious at this point.

Basically, this reads as: "If P is true and if p's trueness leads to q's trueness, then q is true." which is pretty damned annoying if you ask me.
So we flip things around.

If the claim is not true, then:
- p is not true, or
- p implies q is not true
while q is still true.
p implies q being untrue means p and ¬q have to be true at the same time.

q and ¬q can't be true at the same time, so yeah.