I'm studying the theorems that show the fashion in which translating or scaling a random variable by a constant changes the expected value. In these proofs, I've seen constants used within probability functions, e.g. $$E[X+c] = \sum_{x} (x+c)P(X+c=x+c)$$ $$E[X+c] = \sum_{x} xP(X=x) + \sum_{x} cP(X=x)$$ $$E[X+c] = E[X] +c$$
Or similarly, $E[cX] = \sum_{x} cxP(cX=cx)=c\sum_{x} xP(X=x)=cE[X]$.
My question is, why does the constant, $c$, disappear in $P(X+c=x+c)$ and $P(cX=cx)$? Are we simply cancelling the $c$ with a subtraction ($X+c-c=x+c-c$) and division ($\frac{cX}{c}=\frac{cx}{c}$), in each example, respectively? Or is there another explanation?