Section Vspace.
Variables (K : fieldType) (vT : vectType K).
Local Coercion dim : vectType >-> nat.
Definition b2mx n (X : n.-tuple vT) := \matrix_i v2r (tnth X i).
Let lin_b2mx n (X : n.-tuple vT) k :
\sum_(i < n) k i *: X`_i = r2v (\row_i k i *m b2mx X).
Fact memv_submod_closed U : submod_closed U.
Canonical memv_opprPred U := OpprPred (memv_submod_closed U).
Canonical memv_addrPred U := AddrPred (memv_submod_closed U).
Canonical memv_zmodPred U := ZmodPred (memv_submod_closed U).
Canonical memv_submodPred U := SubmodPred (memv_submod_closed U).

Lemma mem0v U : 0 \in U. Proof. exact: rpred0. Qed.
Lemma memvN U v : (- v \in U) = (v \in U). Proof. exact: rpredN. Qed.
Lemma memvD U : {in U &, forall u v, u + v \in U}. Proof. exact: rpredD. Qed.
Lemma memvB U : {in U &, forall u v, u - v \in U}. Proof. exact: rpredB. Qed.
Lemma memvZ U k : {in U, forall v, k *: v \in U}. Proof. exact: rpredZ. Qed.