A Gaussian binomial coefficient identity

Let $u$ and $i$ be nonnegative integers and let $q$ be a prime power. Write $\binom{n}{r}_{q}$ for the Gaussian binomial coefficient, namely the number of $r$-dimensional subspaces of $\mathbb{F}_{q}^{n}$. Gaussian binomial identity.

$$\binom{u}{i}_{q^2}\sum_{j=0}^i q^{j}\binom{i}{j}_{q^2}=\binom{u}{i}_{q}\prod_{j=0}^{i-1}(1+q^{u-j}).$$

This identity arises in the study of Gaussian binomial coefficients and is stated here without a resolution; its validity is relevant to the paper's enumeration of linearized Reed–Solomon codes.