The sumcheck protocol, introduced in 1992, is an interactive proof which is a key component of many probabilistic proof systems in computational complexity theory and cryptography, some of which have been deployed. However, none of these proof systems based on the sumcheck protocol enjoy a formally-verified security analysis. In this paper, we make progress in this direction by providing a formally verified security analysis of the sumcheck protocol using the interactive theorem prover Isabelle/HOL. We follow a general and modular approach. First, we give a general formalization of public-coin interactive proofs. We then define a generalized sumcheck protocol for which we axiomatize the underlying mathematical structure and we establish its soundness and completeness. Finally, we prove that these axioms hold for multivariate polynomials, the original setting of the sumcheck protocol. Our modular analysis facilitates formal verification of sumcheck instances based on different mathematical structures with little effort, by simply proving that these structures satisfy the axioms. Moreover, the analysis supports the development and formal verification of future cryptographic protocols using the sumcheck protocol as a building block.