Using transformation rules to manipulate expressions

Question

I'm trying out Mathematica as a replacement for LaTeX for typesetting math. It works very well, but I would like to take advantage of Mathematica's computer algebra capabilities as well. Specifically, instead of manually figuring out and typing the result of a transformation to an expression, I would like Mathematica to do it.

For example, suppose I am trying to prove that associativity of multiplication holds in the complex numbers. The first step might be the following:

(a+bi)((c+di)(e+fi)) = (a+bi)(ce + cfi + edi - df)

How can I automate this using transformation rules? I can do the following:

In:  (a+bi)((c+di)(e+fi)) /. x_*(y_*z_)->x+y+z
Out: a + bi + c + di + e + fi

which works as expected, so the pattern is matching properly (I've tried various other simple transformations as well). However, the following does not work as expected:

In:  (a+bi)((c+di)(e+fi)) /. x_*(y_*z_)->x*(Expand[y*z])
Out: (a+bi)(c+di)(e+fi)

What am I missing? Am I even on the right track, or is there a better way to accomplish this?

Answer 1

(a+bi)((c+di)(e+fi)) /. x_*(y_*z_):>x*(Expand[y*z])
(* (a + bi) (c e + di e + c fi + di fi) *)

Check RuleDelayed in docs

x_*(y_*z_) :> x*(Expand[y*z]) 
x_*(y_*z_) -> x*(Expand[y*z])

(* x_ y_ z_ :> x Expand[y z] *)
(* x_ y_ z_ -> x y z *)