Mod-two reduction of Chern classes

For a complex line $L$ on a path-connected CW base: the underlying real bundle $L_{\mathbb{R}}$ is oriented by the complex orientation, so $w_1(L_{\mathbb{R}})=0$ by [F5]; and by [F3] applied to the rank-two oriented bundle $L_{\mathbb{R}}$ together with [F6], $w_2(L_{\mathbb{R}})={\rho }_{2}e(L_{\mathbb{R}})={\rho }_2{c}_1(L)$. Hence $w(L_{\mathbb{R}})=1+{\rho }_2{c}_1(L)$.

If $n=0$, then $E_{\mathbb{R}}$ is the zero bundle and both total classes are $1$, so the theorem holds directly. Assume henceforth that $n\ge 1$. Pulling back to the flag bundle gives $q^{\ast }E=L_1\oplus \cdots \oplus L_n$ with all lines complex, by [F1]. By [F7] choose a CW model $h:W\to \mathrm{Fl}(E)$ that is a homotopy equivalence, and put $r=qh$ and $M_i=h^{\ast }{L}_i$. The flag space is path connected: each projective stage has path-connected fiber and local path lifting, so a base path followed by a path in its endpoint fiber joins any two points. Thus $W$ is path connected. By [F8], $r^{\ast }E\cong {⨁}_i{M}_i$, and $r^{\ast }=h^{\ast }{q}^{\ast }$ is injective over ${\mathbb{F}}_2$ by [F1] and [F7]. Both bases for Chern multiplicativity and line normalization are now actual CW complexes.

On $W$, multiplicativity [F4] and step 1.1 give $w((r^{\ast }E{)}_{\mathbb{R}})={\prod }_i(1+{\rho }_2{c}_1(M_i))={\rho }_2{\prod }_i(1+c_1(M_i))={\rho }_2{r}^{\ast }c(E)=r^{\ast }{\rho }_{2}c(E)$. Here reduction preserves products since the formula of [F8] gives ${\rho }_2(uv)={\rho }_2(u){\rho }_2(v)$ on every simplex; it preserves the unit as well. Pullback compatibility is [F8], and the Chern identities on the actual CW base are [F2].

By [F8], $(r^{\ast }E{)}_{\mathbb{R}}\cong r^{\ast }(E_{\mathbb{R}})$, so Stiefel–Whitney naturality [F4] identifies the left side of step 2.1 with $r^{\ast }w(E_{\mathbb{R}})$. Comparing degrees gives $r^{\ast }{w}_{2i}(E_{\mathbb{R}})=r^{\ast }{\rho }_2{c}_i(E)$ and $r^{\ast }{w}_{2i+1}(E_{\mathbb{R}})=0$, because each $c_i$ has degree $2i$.

Injectivity of $r^{\ast }$ over ${\mathbb{F}}_2$ from step 1.2 gives $w_{2i+1}(E_{\mathbb{R}})=0$ and $w_{2i}(E_{\mathbb{R}})={\rho }_2{c}_i(E)$.

Boundary cases. For $n=0$ both sides are $1$; for $n=1$ the assertion is step 1.1 specialized to the bundle itself. The empty base is excluded by the path-connected hypothesis, the coefficient field ${\mathbb{F}}_2$ is nonzero, and degrees $i$ above the rank give $c_i=0$ on the right and $w_{2i}=0$ on the left because $2i>2n$ exceeds the real rank. AC is used only through [A1] in the splitting and characteristic-class suppliers.