I am trying to understand proposition 23.2. of Bott Tu proof. It wants to prove that $H^{*}(G_k(V))=\frac{\mathbb{R}[c(S),c(Q)]}{(c(S)c(Q)=1)}$. It claims that:
1.$H^{*}(F(V))=\frac{\mathbb{R}[x_1,\dots,x_{n-k},y_1,\dots,y_k]}{(\prod(1+x_i)\prod(1+y_j)=1)}$ (which is true using previous proposition)
2.$H^{*}(F(V))=\frac{H^{*}(G_k(V))[x_1,\dots,x_{n-k},y_1,\dots,y_k]}{(\prod(1+x_i)=c(S),\prod(1+y_j)=c(Q))}$ (also true with previous proposition)
3.$c(S)c(Q)=1$ in $H^{*}(Gl_k(V))$ (True using Whitney's formula and $S\oplus Q$ trivial bundle of $Gl_k(V)$)
Then it asserts that it follows an injection of algebras:
$$\frac{\mathbb{R}[c(S),c(Q)]}{(c(S)c(Q)=1)}\to H^{*}(G_k(V))$$
This is the point I do not understand. How can it be followed by the 3 previous hypothesis that you have this injection?
Any help will be appreciated. Thanks in advance!