if $p = charD \ \ $ then $ px = 0 \ \ \ \forall x \in D $.

Question

The characteristic of an integral domain D is defined as:

$charD := min\{ n \in \mathbb{N} : na = 0 , \text{for some} \ 0 \neq a \in D \} $

I want to prove that if $p = charD \ \ $ then $ px = 0 \ \ \ \forall x \in D $.

Here's my attempt.
Let $ 0 \neq a\in D\ \ $ such that $pa = 0 $ this implies that $$(pa)x = (a+a+\cdots +a)x = (ax+ax+\cdots +ax) = \\ a(x+x+\cdots+ x) = a(px) = 0$$ $\forall x \in D $ then $px = 0 \ \ $ since $a\neq 0$

what do you think, is it correct?...if it's not, how would you do it?. Thank you.