Whenever I have solved circuits (only cells, resistors and wires), it has only been necessary to use Ohm's law (or things derived from Ohm's law, such as how two resistors $R_1$ and $R_2$ have effective resistance $R_1+R_2$ as a whole) and Kirchoff's loop rules. However, I was curious as to whether this always works.
Given any finitely sized circuit, consisting of only wires (with zero resistance), batteries (with known voltage) and (Ohmic) resistors (with known resistance), is it always possible to use Ohm's law and Kirchhoff's loop and junction rules to calculate the current at any given point on the circuit and the voltage between any two points?
I don't see why this isn't the case, but I do not know of any definitive proof.
(This might lean too heavily into logic, specifically the idea of completeness, so if this question isn't appropriate I'll migrate it over to MSE.)