English
The Apply₂-stability lemma remains valid under simp-marked variants.
Русский
Лемма Apply₂―устойчивость остаётся верной под варианты, помеченные как simp.
LaTeX
$$$ \text{(stabilizer_inf_stabilizer_le_stabilizer_apply₂)} \text{ (simp) }$$$
Lean4
@[to_additive (attr := simp)]
theorem stabilizer_subgroup (s : Subgroup G) : stabilizer G (s : Set G) = s :=
by
simp_rw [SetLike.ext_iff, mem_stabilizer_set]
refine fun a ↦ ⟨fun h ↦ ?_, fun ha b ↦ s.mul_mem_cancel_left ha⟩
simpa only [smul_eq_mul, SetLike.mem_coe, mul_one] using (h 1).2 s.one_mem