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_inv_horner u : fp (mx_inv_horner A u) = mx_inv_horner A^f u^f.