Given gcd(a,b)=1 and a|bc prove a|c?

This is what I did, I just want someone to confirm it as I don't have any solutions.

as+bt=1 (coprime) (1) and bc=ak (2) for some integer s,t and k.

Multiply (1) by c --> acs+bct=c and use (2) to replace bc with ak

Then we get acs+akt=c --> factorize a and then we get a|c.

Is this simple proof enough, or have I missed out on something?

Thanks