-- Module: Mathlib.GroupTheory.Transfer
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
A : Type u_2
inst_1 : CommGroup A
ϕ : MonoidHom.{u_1, u_2} (↥H) A
inst_2 : H.FiniteIndex
g : G
inst_3 : Fintype (Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H)))
this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)
simp :
  Eq.{u_2 + 1} (α := A)
    (∏ x : ZMod (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q)),
        (ϕ : (a : ↥H) → A)
          (Subtype.mk (p := fun (x : G) ↦ x ∈ H)
            ((HPow.hPow (β := ℤ) g x.cast * Quotient.out (Quotient.out q))⁻¹ *
              if x = 0 then
                g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q) * Quotient.out (Quotient.out q)
              else HPow.hPow (β := ℤ) g x.cast * Quotient.out (Quotient.out q))
            ⋯) :
      A)
    ((ϕ : (a : ↥H) → A)
        (Subtype.mk (p := fun (x : G) ↦ x ∈ H)
          ((Quotient.out (Quotient.out q))⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q) *
            Quotient.out (Quotient.out q))
          ⋯) :
      A)
⊢ Eq.{u_2 + 1} (α := A)
    (∏ s : ZMod (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q)),
        (ϕ : (a : ↥H) → A)
          (Subtype.mk (p := fun (x : G) ↦ x ∈ H)
            (HMul.hMul (α := G)
              (↑((⋯.leftQuotientEquiv (S := (↑(H.transferTransversal g) : Set G)) :
                        (a : G ⧸ H) → (↑(↑(H.transferTransversal g) : Set G) : Type u_1))
                      (((H.quotientEquivSigmaZMod g).symm :
                          (a :
                              (q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)) ×
                                ZMod (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q))) →
                            G ⧸ H)
                        (Sigma.mk (β := fun (q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)) ↦
                          ZMod (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q))) q s))) :
                  G)⁻¹
              (↑((⋯.leftQuotientEquiv (S := (↑(g • H.transferTransversal g) : Set G)) :
                      (a : G ⧸ H) → (↑(↑(g • H.transferTransversal g) : Set G) : Type u_1))
                    (((H.quotientEquivSigmaZMod g).symm :
                        (a :
                            (q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)) ×
                              ZMod (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q))) →
                          G ⧸ H)
                      (Sigma.mk (β := fun (q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)) ↦
                        ZMod (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q))) q s))) :
                G))
            ⋯) :
      A)
    ((ϕ : (a : ↥H) → A)
        (Subtype.mk (p := fun (x : G) ↦ x ∈ H)
          ((Quotient.out (Quotient.out q))⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q) *
            Quotient.out (Quotient.out q))
          ⋯) :
      A)

Tactic:
simp only [quotientEquivSigmaZMod_symm_apply, transferTransversal_apply',
          transferTransversal_apply'']

Premises used: Subgroup.transferTransversal_apply', Subgroup.transferTransversal_apply'', Subgroup.quotientEquivSigmaZMod_symm_apply