Arguments sumsmx_sup i0 [P m n A B_].

(* Specifications: *)
Lemma sumsmx_subP P m n (A_ : I -> 'M_n) (B : 'M_(m, n)) :
reflect (forall i, P i -> A_ i <= B)%MS (\sum_(i | P i) A_ i <= B)%MS.
Lemma summx_sub_sums P m n (A : I -> 'M[F]_(m, n)) B :
(forall i, P i -> A i <= B i)%MS ->
((\sum_(i | P i) A i)%R <= \sum_(i | P i) B i)%MS.
Lemma sumsmxS P n (A B : I -> 'M[F]_n) :
(forall i, P i -> A i <= B i)%MS ->
(\sum_(i | P i) A i <= \sum_(i | P i) B i)%MS.
Lemma eqmx_sums P n (A B : I -> 'M[F]_n) :
(forall i, P i -> A i :=: B i)%MS ->
(\sum_(i | P i) A i :=: \sum_(i | P i) B i)%MS.
Lemma sumsmxMr_gen P m n A (B : 'M[F]_(m, n)) :
((\sum_(i | P i) A i)%MS *m B :=: \sum_(i | P i) <<A i *m B>>)%MS.
Lemma sumsmxMr P n (A_ : I -> 'M[F]_n) (B : 'M_n) :
((\sum_(i | P i) A_ i)%MS *m B :=: \sum_(i | P i) (A_ i *m B))%MS.
Lemma rank_pid_mx m n r : r <= m -> r <= n -> \rank (pid_mx r : 'M_(m, n)) = r.
Lemma rank_copid_mx n r : r <= n -> \rank (copid_mx r : 'M_n) = (n - r)%N.
Lemma mxrank_compl m n (A : 'M_(m, n)) : \rank A^C = (n - \rank A)%N.
Lemma mulmxKV_ker m n p (A : 'M_(n, p)) (B : 'M_(m, n)) :
B *m A = 0 -> B *m col_ebase A *m kermx A = B.
Lemma sub_kermxP p m n (A : 'M_(m, n)) (B : 'M_(p, m)) :
reflect (B *m A = 0) (B <= kermx A)%MS.
Lemma mulmx0_rank_max m n p (A : 'M_(m, n)) (B : 'M_(n, p)) :
A *m B = 0 -> \rank A + \rank B <= n.
Lemma mxrank_Frobenius m n p q (A : 'M_(m, n)) B (C : 'M_(p, q)) :
\rank (A *m B) + \rank (B *m C) <= \rank B + \rank (A *m B *m C).
Lemma mxrank_mul_min m n p (A : 'M_(m, n)) (B : 'M_(n, p)) :
\rank A + \rank B - n <= \rank (A *m B).
Lemma addsmx_compl_full m n (A : 'M_(m, n)) : row_full (A + A^C)%MS.
Lemma sub_capmx_gen m1 m2 m3 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) (C : 'M_(m3, n)) :
(A <= capmx_gen B C)%MS = (A <= B)%MS && (A <= C)%MS.