Stable cancellation for complex vector bundles

Let $E_0,E_1$ be complex vector bundles of rank $d$ over a finite CW complex of dimension at most $2d$. If $E_0\oplus {\mathbb{C}}^r\cong E_1\oplus {\mathbb{C}}^r$ for some $r$, then $E_0\cong E_1$. This follows because the stabilization map $BU(d)\to BU$ is $2d$-connected, so stable equivalence is injective on bundles over such a complex.

Two rank-$d$ complex vector bundles over ${\mathbb{CP}}^d$ are isomorphic exactly when their Chern classes agree. Equality of Chern classes gives equality of their Chern characters; the Chern character is injective here because $K^0({\mathbb{CP}}^d)$ is torsion-free. The bundles are therefore stably isomorphic, and stable cancellation for complex vector bundles applies because ${\mathbb{CP}}^d$ has real dimension $2d$.