Section IrrClassDef.
Variables (gT : finGroupType) (G : {group gT}).
Let sG := DecSocleType (regular_repr algCF G).
Let sG := DecSocleType (regular_repr algCF G).
Let C'G := algC'G G.
Let closG := @groupC _ G.
Local Notation W i := (@socle_of_Iirr _ G i).
Local Notation "''n_' i" := 'n_(W i).
Local Notation "''R_' i" := 'R_(W i).
Local Notation "''e_' i" := 'e_(W i).
Let aG := regular_repr algCF G.
Let R_G := group_ring algCF G.

Lemma xcfun_mul_id i A :
(A \in R_G)%MS -> ('chi_i).['e_i *m A]%CF = ('chi_i).[A]%CF.