Let R_G := group_ring algCF G.

(* Specifications: *)
Lemma xcfun_annihilate i j A : i != j -> (A \in 'R_j)%MS -> ('chi_i).[A]%CF = 0.
Lemma xcfunG phi x : x \in G -> phi.[aG x]%CF = phi x.
Lemma xcfun_mul_id i A :
(A \in R_G)%MS -> ('chi_i).['e_i *m A]%CF = ('chi_i).[A]%CF.
Lemma xcfun_id i j : ('chi_i).['e_j]%CF = 'chi_i 1%g *+ (i == j).
Lemma irr_free : free (irr G).
Lemma irr_inj : injective (tnth (irr G)).
Lemma eq_sum_nth_irr a : \sum_i a i *: 'chi[G]_i = \sum_i a i *: (irr G)`_i.
Lemma cfRepr_standard n (rG : mx_representation algCF G n) :
cfRepr (standard_grepr rG)
= \sum_i (standard_irr_coef rG (W i))%:R *: 'chi_i.
Lemma cfRepr_inj n1 n2 rG1 rG2 :
@cfRepr _ G n1 rG1 = @cfRepr _ G n2 rG2 -> mx_rsim rG1 rG2.
Lemma cfRepr_rsimP n1 n2 rG1 rG2 :
reflect (mx_rsim rG1 rG2) (@cfRepr _ G n1 rG1 == @cfRepr _ G n2 rG2).
(* This is Isaacs, Theorem (2.12). *)
Lemma Wedderburn_id_expansion i :
'e_i = #|G|%:R^-1 *: \sum_(x in G) 'chi_i 1%g * 'chi_i x^-1%g *: aG x.