Theorem: Finset.preimage_image_quotientMk_stabilizer_eq_mul_mulStab (LeanCamCombi/Kneser/MulStab.lean)

State before:
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
ht : t.Nonempty
s : Finset α
⊢ QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s) = ↑s * ↑t.mulStab
Tactic:
rw [QuotientGroup.preimage_image_mk_eq_iUnion_smul, mul_comm, Set.iUnion_subtype]
State after:
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
ht : t.Nonempty
s : Finset α
⊢ ⋃ x, ⋃ (hx : x ∈ stabilizer α t), ⟨x, hx⟩ • ↑s = ↑t.mulStab * ↑s

State before:
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
ht : t.Nonempty
s : Finset α
⊢ ⋃ x, ⋃ (hx : x ∈ stabilizer α t), ⟨x, hx⟩ • ↑s = ↑t.mulStab * ↑s
Tactic:
simp_rw [Subgroup.mk_smul, ← SetLike.mem_coe, ← coe_mulStab ht, ← coe_smul_finset, ← coe_biUnion,
  biUnion_smul_finset, smul_eq_mul, coe_mul]
State after:
no goals