Virtual algebraic calculation within proofs It seems to me that the undergraduates I teach have particular difficulty with proofs that involve reasoning about algebraic calculations that arise only theoretically.  Since I have in mind doing some pedagogical/expository writing around this issue, I want to assemble for myself as many characteristic examples as I can.
I'll give two examples now, and more on demand if it turns out I haven't made myself clear.
1) The proof of the transcendentality of $\pi$ such as one finds in Baker's Transcendental Number Theory, involves a product of many factors, each corresponding to hypothetical conjugate of $\pi$, and leading to a sum having size exponential in the hypothetical degree of $\pi$, all this to set up an application of the fundamental theorem on symmetric polynomials. 
2) The proof of the Bruck-Ryser theorem on finite projective planes involves using geometry to create a hypothetical identity and then doing repeated substitutions to reduce the number of variables.
Challenges to clearly illustrating the mechanism within these two proofs include counterfactual hypotheses (for contradiction) and exponential complexity.  Many of my students simply have very little feeling for what the distributive law says beyond the multiplication of two binomials, so I have a particular interest on proofs that hinge on analyzing the result of products with many factors each of which has many summands.
Thanks in advance for all contributions.  Please give citations for paradigmatic proofs rather then mentioning theorems only, by name or statement.  Please, if in doubt, post, but I'm particular interested in important results within the reach of advanced undergraduates and/or beginning graduate students.
 A: A very elementary example (simpler than the ones you've given) is the generating function for the number of partitions of $n$, denoted $p_n$:
$$\sum_{n\geq 0} p_n q^n = \prod_{i\geq 0} \frac{1}{1-q^i}.$$
A: I'm going to define the characteristic polynomial to my students next monday, and I'm going to tell them about its degree and three special coefficients. The constant coefficient is trivial by evaluation at zero. For both others and the degree, the reasoning is an example of what you ask, if I got your question right.
I will write the determinant on the board (written with parenthesis because I don't remember how to make the straight lines...):
$$\begin{pmatrix}a_{1,1}-X & a_{1,2} & \dots \\\
a_{2,1} & a_{2,2}-X & \dots \\\
\dots & \dots & \dots
\end{pmatrix}$$
and consider the development along the first column : the first term is $a_{1,1}-X$ times a cofactor which looks the same as the original determinant, and all others will have lost two terms of the form $\text{something}-X$ so will at least two lower in degree. That means if I were to really develop the determinant, it would end up looking like $(a_{1,1}-X)\dots(a_{n,n}-X)+\text{at most degree}(n-2)$, so the characteristic polynomial is degree $n$, its dominant coefficient is $(-1)^n$, and the coefficient just behind will be $(-1)^{n-1}\rm{Tr}(A)$.
