submonoid — Mathlib

English

The stabilizer submonoid of m inside p equals the stabilizer submonoid of m viewed in M.

Русский

ПодмонНой стабилизатор m внутри p равен стабилизатору m, рассматриваемому в M.

LaTeX

$$$ MulAction.stabilizerSubmonoid\\ R\\ m = MulAction.stabilizerSubmonoid\\ R\\ (m : M) $$$

Lean4

/-- Stabilizers in monoid SubMulAction coincide with stabilizers in the ambient space -/
@[to_additive]
theorem submonoid {p : SubMulAction R M} (m : p) :
    MulAction.stabilizerSubmonoid R m = MulAction.stabilizerSubmonoid R (m : M) :=
  by
  ext
  simp only [MulAction.mem_stabilizerSubmonoid_iff, ← SubMulAction.val_smul, SetLike.coe_eq_coe]

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