OP is using the notation C(A) to mean "the column space of matrix A", whereas here that notation is being interpreted as "multiplication of matrix C by matrix A

OP is using the notation C(A) to mean "the column space of matrix A", whereas here that notation is being interpreted as "multiplication of matrix C by matrix A, which happens to be inside of parentheses". These are totally different questions. It's a reasonable misunderstanding as OP's notation is nonstandard. But also the following math/logic is nonsense.