Positivity of the alternating sum associated to at most five subspaces

Here is a purely combinatorial proof for $n=5$. We first generalize the problem as follows:
Let $B_5$ be the boolean lattice of rank $5$, i.e. the subsets lattice of $\{1,2,3,4,5\}$.

Lemma: Let $\phi: B_5 \to \mathbb{R}_{\ge 0}$ be a map satisfying that $\forall a, b \in B_5$:

$(1)$ $ \ $ $a \le b \Rightarrow \phi(a) \le \phi(b)$ [poset morphism]
$(2)$ $ \ $ $\phi(a \vee b) + \phi(a \wedge b) \ge \phi(a ) + \phi(b)$

and let $a_i= \{i \}^\complement$ the complement of $\{i \}$ in $\{1,2,3,4,5\}$, then $$\sum_{r=1}^5 (-1)^{r+1}\sum_{i_1 < i_2 < \cdots < i_r} \phi( a_{i_1} \wedge \cdots \wedge a_{i_r}) \ge 0$$ proof: we reorganize the alternative sum into the sum of the following components:

• $\phi(\{1,2,3,4\}) - \phi(\{1,2,3\}) - \phi(\{1,2,4\}) + \phi(\{1,2\})$
• $\phi(\{1,3,4,5 \}) - \phi(\{1,3,4\}) - \phi(\{1,3,5\}) + \phi(\{1,3\})$
• $\phi(\{2,3,4,5\}) - \phi(\{2,3,4\}) - \phi(\{3,4,5\}) + \phi(\{3,4\})$
• $\phi(\{1,2,4,5 \}) - \phi(\{2,4,5\}) - \phi(\{1,4,5\}) + \phi(\{4,5\})$
• $\phi(\{1,2,3,5\}) - \phi(\{1,2,5\}) - \phi(\{2,3,5\}) + \phi(\{2,5\})$
• $ \phi(\{1,5\}) - \phi(\{1\})$
• $ \phi(\{2,4\}) - \phi(\{2\})$
• $ \phi(\{2,3\}) - \phi(\{3\})$
• $ \phi(\{1,4\}) - \phi(\{4\})$
• $ \phi(\{3,5\}) - \phi(\{5\})$
• $ \phi(\emptyset) $

but the first five components are positive by $(2)$, the next five components are positive by $(1)$, and the last is positive by definition $\square$.

Now the answer of the question is yes by observing that the map $\phi$ defined by $$\phi( a_{i_1} \wedge \cdots \wedge a_{i_r}) = \dim (V_{i_1} \cap \cdots \cap V_{i_r})$$ checks $(1)$ and $(2)$. For $(1)$ it is immediate. For $(2)$ we use the following equality and inclusion: $\dim(U+V) = \dim(U) + \dim(V) - \dim(U \cap V)$ and $(A\cap B ) + (A\cap C) \subseteq A$.

Here is proof by hand for $n=4$:

Let $X = \bigcap_{i=1}^n V_i$ then by taking $V_i' = V_i \ominus X$ we get that $$\sum_{r=1}^n (-1)^{r+1} \sum_{ \ i_1 < i_2 < \cdots < i_r } \dim(V_{i_1} \cap \cdots \cap V_{i_r}) = \dim(X) + \sum_{r=1}^n (-1)^{r+1} \sum_{ \ i_1 < i_2 < \cdots < i_r } \dim(V'_{i_1} \cap \cdots \cap V'_{i_r})$$ So we can assume $\dim(X) = 0$.

First we apply four times the inequality: $ \dim(U +V + W) \le $ $$ \dim U + \dim V + \dim W - \dim (U \cap V) - \dim (U \cap W) - \dim (V \cap W) + \dim(U \cap V \cap W) $$ and we get $$\alpha \ge \sum_{ \ i_1 < i_2 < i_3 } \dim(V_{i_1} + V_{i_2} + V_{i_3})-2\sum_i\dim(V_i) + \sum_{ \ i_1 < i_2 } \dim(V_{i_1} \cap V_{i_2} )$$ Next we apply six times the equality: $$\dim(U+V) = \dim U + \dim V - \dim (U \cap V)$$ and we get $$\alpha \ge \sum_{ \ i_1 < i_2 < i_3 } \dim(V_{i_1} + V_{i_2} + V_{i_3}) - \sum_{ \ i_1 < i_2 } \dim(V_{i_1} + V_{i_2} ) + \sum_i\dim(V_i)$$ Finally we observe that:
$\dim(V_1+V_2+V_3) \ge \dim(V_1+V_3)$
$\dim(V_1+V_2+V_4) \ge \dim(V_2+V_4)$
$\dim(V_1+V_3+V_4) \ge \dim(V_1+V_4)$
$\dim(V_2+V_3+V_4) \ge \dim(V_2+V_3)$
$\sum_i\dim(V_i) \ge \dim(V_1+V_2) + \dim(V_3+V_4)$

It follows that $\alpha \ge 0$ $\square$