I don't understand the homology computation of real projective $\mathbb{RP}^n$ associated to the decomposition of $\mathbb{S}^i$ as union of two $i$-cells, i.e $E_+^i,E_-^i$.
I'm going to introduce the notation and give a sketch of the proof given to me to underline the details I'm missing.
Since $H_i(S^i,S^{i-1}) \simeq H_i(E_+^i,S^{i-1}) \oplus H_i(E_-^i,S^{i-1}) \simeq \mathbb{Z}\oplus \mathbb{Z}$ we have that the differential map are $$\mathbb{Z}\oplus \mathbb{Z} \overset{d_n}{\longrightarrow} \mathbb{Z}\oplus \mathbb{Z} \overset{d_{n-1}}{\longrightarrow} \cdots \mathbb{Z} \oplus \mathbb{Z} \overset{d_{n-2}}{\longrightarrow} \cdots \overset{d_1}{\longrightarrow} \mathbb{Z} \oplus \mathbb{Z}$$
Called $\alpha$ the antipodal map (which restricts to the skeletons $\mathbb{S}^i$ with this given decomposition) we know that $\alpha_*$ send a generator $x_i$ of $H_i(E_+^i,S^{i-1}) \simeq \mathbb{Z}$ into a generator $\alpha_*(x_i) \in H_i(E_-^i,S^{i-1}) \simeq \mathbb{Z}$.
From this Lemma I know that $d_n = (-1)^n +1$.
So the algebraic complex associated to $\mathbb{RP}^n$ is
$$0 \longrightarrow \mathbb{Z} \overset{(-1)^n +1 }{\longrightarrow} \mathbb{Z} \overset{(-1)^{n-1}+1}{\longrightarrow} \cdots \mathbb{Z} \overset{0}{\longrightarrow} \mathbb{Z}$$
In other words $d_n$ are the multiplication by $\pm 2$. It follows that $$H_n(\mathbb{RP}^n) = \begin{cases} 0 & 2 \mid n \\ \mathbb{Z} & 2 \nmid n\end{cases}$$
From this it should follow that $$H_i(\mathbb{RP}^n) = \begin{cases} \mathbb{Z}_2 & 0 < i < n & 2 \nmid i \\ 0 & 2 \mid i, i > n \\ \mathbb{Z} & i = 0,n\end{cases}$$
The proof given both in Hatcher and my notes in the same spirit takes $\alpha : \mathbb{S}^n \longrightarrow \mathbb{RP}^n$ which induces a map of algebric complex $C_i^{CW}(\mathbb{S}^i) = H_i(E_+^i,S^{i-1}) \oplus H_i(E_-^i,S^{i-1})$ and says :
"Both terms maps identically on $C_i^{CW}(\mathbb{RP}^n) = H_i(\mathbb{D}_+^i,\mathbb{S}^i)$ so $x_i$ and $\alpha_*(x_i)$ are mapped in the same generator $y_i$, hence $d_i(y_i) = \pm (1+(-1)^i)y_i"$.
I really don't understand this proof or how should conclude. I don't understand neither how it should follow that the homology of projective are those written above. In particular my difficulties involve being able to distinguish all the pieces of this $CW$-complex associated homology and being able to assemble those objects in order to compute the original homology of the space I'm studying, i.e $\mathbb{RP}^n$.
Any help or reference in order to understand or shed light on my doubts and difficulties would be appreciated.