Proposition: For all $m, n, p, q$ contained in Natural Numbers, $(m-n)p = mp - np$

Question

I have to prove the following proposition using the axioms for integers, and the definition of subtraction to be $m - n = m + (-n).$

Proof:

$(m-n)p = (m+(-n))p$ by definition of subtraction

$(m+(-n))p = mp + (-n)p$ by Distributivity

$mp + (-n)p = mp - np$ by definition of subtraction

My last line was marked wrong; is this not a valid step to finish the proof?

Answer 1

What you did on the last line was to multiply $(-n)$ with $m$, which is not using the definition of subtraction

I am guessing you have some other principle that states that $(-x)y=(-xy)$, which you can use to justify that:

$mp+(-n)p=mp+(-np)$

And then you use the definition of subtraction to say that:

$mp+(-np)=mp-np$

so that you end up with:

$mp+(-n)p=mp-np$