of_orbits — Mathlib

English

Orbits form a partition of the ambient set; equivalently, they are pairwise disjoint or equal covers of each element.

Русский

Орбиты образуют разбиение исходного множества; каждый элемент принадлежит exactly одной орбите.

LaTeX

$$$\\text{IsPartition}(\\{\\operatorname{orbit}(G,a) : a \\in X\\})$$$

Lean4

/-- Orbits of an element form a partition -/
@[to_additive /-- Orbits of an element form a partition -/
    ]
theorem of_orbits : Setoid.IsPartition (Set.range fun a : X => orbit G a) :=
  by
  apply orbit.pairwiseDisjoint.isPartition_of_exists_of_ne_empty
  · intro x
    exact ⟨_, ⟨x, rfl⟩, mem_orbit_self x⟩
  · rintro ⟨a, ha : orbit G a = ∅⟩
    exact (MulAction.nonempty_orbit a).ne_empty ha

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