mem_subgroup_orbit_iff — Mathlib

English

For a subgroup H ≤ G and a,b in orbit G x, membership equivalence transfers to α-level orbits.

Русский

Для подгруппы H ≤ G и элементов a,b в orbit_G x переход membership к орбитам в α.

LaTeX

$$$$ a \\in \\mathrm{orbit}(H, b) \\iff (a \\text{ в } \\alpha) \\in \\mathrm{orbit}(H, b) $$$$

Lean4

@[to_additive]
theorem mem_subgroup_orbit_iff {H : Subgroup G} {x : α} {a b : orbit G x} :
    a ∈ MulAction.orbit H b ↔ (a : α) ∈ MulAction.orbit H (b : α) :=
  by
  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
  · rcases h with ⟨g, rfl⟩
    exact MulAction.mem_orbit _ g
  · rcases h with ⟨g, h⟩
    dsimp at h
    rw [subgroup_smul_def, ← orbit.coe_smul, ← Subtype.ext_iff] at h
    subst h
    exact MulAction.mem_orbit _ g

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