English
A simp variant asserting union-right behaviour of stabilizers.
Русский
Упрощённый вариант, задающий поведение стабилизаторов при объединении справа.
LaTeX
$$$ stabilizer_G(s \cup t) \le stabilizer_G(s) $$$
Lean4
@[to_additive (attr := simp)]
theorem stabilizer_subgroup_op (s : Subgroup Gᵐᵒᵖ) : stabilizer G (s : Set Gᵐᵒᵖ) = s.unop :=
by
simp_rw [SetLike.ext_iff, mem_stabilizer_set]
refine fun a ↦ ⟨fun h ↦ ?_, fun ha b ↦ s.mul_mem_cancel_right ha⟩
have : 1 * MulOpposite.op a ∈ s := (h 1).2 s.one_mem
simpa only [op_smul_eq_mul, SetLike.mem_coe, one_mul] using this