Let $W$ be a subspace of the vector space $\mathbb{K}^n$, where $\mathbb{K}$ is a field of characteristic $0$. The support of a vector $v = (v_1,\ldots, v_n) \in \mathbb{K}^n$ is given by $\text{supp}(v) = \left\{i : v_i\neq 0\right\}$. Let $L$ denote the set of supports of all vectors in $W$, ordered by reverse inclusion.
I want to show that $L$ is a semimodular lattice. By Proposition 3.3.2 in Stanley's Enumerative Combinatorics, vol. 1, we can either show that $L$ is graded (every maximal chain has the same length) and that its rank functions $\rho:L\rightarrow \left\{0,1,\ldots, n \right\}$ satisfies $$ \rho(s)+\rho(t) \geq \rho(s\wedge t)+\rho(s\vee t)$$
for any $s,t \in L$ or we can show that if $s$ and $t$ both cover $s\wedge t$, then $s\vee t$ covers both $s$ and $t$.
Here's the problem...I know that the meet of any $s,t \in L$ is $s\cup t$, but I don't know what the join is. Ultimately what I'm trying to show is the $L$ is a geometric lattice, which requires showing that $L$ is semimodular. Perhaps there is a better way of proving this.