Iterated multiplication in \( VTC ^0\)

We show that \( VTC ^0\), the basic theory of bounded arithmetic corresponding to the complexity class \(\mathrm {TC}^0\), proves the \( IMUL \) axiom expressing the totality of iterated multiplication satisfying its recursive definition, by formalizing a suitable version of the \(\mathrm {TC}^0\) iterated multiplication algorithm by Hesse, Allender, and Barrington. As a consequence, \( VTC ^0\) can also prove the integer division axiom, and (by our previous results) the \( RSUV \)-translation of induction and minimization for sharply bounded formulas. Similar consequences hold for the related theories \(\Delta ^b_1\text{- } CR \) and \(C^0_2\). As a side result, we also prove that there is a well-behaved \(\Delta _0\) definition of modular powering in \(I\Delta _0+ WPHP (\Delta _0)\).