stabilizer_subgroup_op — Mathlib

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

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