Section MapField.
Variables (aF rF : fieldType) (f : {rmorphism aF -> rF}).
Local Notation "A ^f" := (map_mx f A) : ring_scope.
Local Notation fp := (map_poly f).
Variables (n' : nat) (A : 'M[aF]_n'.+1) (p : {poly aF}).

Lemma map_mx_companion (e := congr1 predn (size_map_poly _ _)) :
(companionmx p)^f = castmx (e, e) (companionmx (fp p)).