ofStabilizer — Mathlib

English

The apply-map of equality preserves basepoints under the fixing-subgroup equivalence.

Русский

Отображение применения по равенству сохраняет базовые точки при эквивалентности фиксационных подгрупп.

LaTeX

$$$\text{eq\_apply}: (\mathrm{ofFixingSubgroup\_of\_eq}\ M hst)\ x = x$$$

Lean4

/-- Action of the stabilizer of a point on the complement. -/
@[to_additive /-- Action of the stabilizer of a point on the complement. -/
    ]
def ofStabilizer (a : α) : SubMulAction (stabilizer G a) α
    where
  carrier := { a }ᶜ
  smul_mem' g
    x := by
    simp only [Set.mem_compl_iff, Set.mem_singleton_iff]
    rw [not_imp_not, smul_eq_iff_eq_inv_smul]
    intro hgx
    apply symm
    rw [hgx, ← smul_eq_iff_eq_inv_smul]
    exact g.prop

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