Question about proving the sequences of assertion

Question

1. P OR (Q AND R)
2. P → R

These two are given. And goal to prove is R. By those two given formulas, is R OR (Q AND R) valid?

This is my approach:

1. P OR (Q AND R)   - Given
2. P → R            - Given
3. R OR (Q AND R)   - If it's valid, what's the reason for this?
4. R OR R           - Simplification, 3
5. R                - Simplification, 4

Please correct me if my approach is wrong.

Answer 1

It's invalid, because $P \lor (Q \land R)$ if $Q \land R$, regardless if $P$ or $\lnot P$.

One way to prove it would be the following:

 1 │ P ∨ (Q ∧ R).   (given)
 2 │ P ⇒ R.         (given)
 3 │ 
 4 │ If P:          (assumption)
 5 │     R.         (implication elimination, [2, 4])
 6 │ P ⇒ R.         (implication introduction, [4, 5])
 7 │ 
 8 │ If Q ∧ R:      (assumption)
 9 │     R.         (conjunction elimination, [1])
10 │ (Q ∧ R) ⇒ R.   (implication introduction, [8, 9])
11 │ 
12 │ R.             (disjunction elimination, [1, 6, 10])