Let $A,\ B,\ C \leq V$ such that $A + C = B + C,\ A \cap C = B \cap C,\ A \subset B$. Prove $A = B$

Solution 1:

You have proved that $B+C=A+C$, but that's already among the assumptions.

You want to take $b\in B$ and show that $b\in A$.

Since $B\subset B+C=A+C$, you can write $b=a+c$ for some $a\in A$ and $c\in C$. But then $c=b-a\in B$, due to $A\subset B$.

This implies $c\in B\cap C$. Can you finish?

More generally, the lattice $L(V)$ of subspaces of a vector space is modular, that is, it satisfies

for all $A,B,C\in L(V)$, if $A\subset B$, then $A+(B\cap C)=(A+C)\cap B$

In your case $A+(B\cap C)=A+(A\cap C)=A$ and $(A+C)\cap B=(B+C)\cap B=B$.

You should try and prove the above modular identity.