of_subsingleton — Mathlib

English

If the action is not trivial, the trivial blocks condition implies preprimitivity (pretransitivity is automatic).

Русский

Если действие не тривиально, условие тривиальных блоков имплицирует предпримитивность (предпретTransitivity автоматично).

LaTeX

$$IsPretransitive G X → (∀ {B}, a ∈ B → IsBlock G B → IsTrivialBlock B) → IsPreprimitive G X$$

Lean4

@[to_additive (attr := nontriviality)]
theorem of_subsingleton [SMul G X] [Nonempty G] [Subsingleton X] : IsPreprimitive G X
    where
  exists_smul_eq
    (x y) := by
    use Classical.arbitrary G
    rw [eq_iff_true_of_subsingleton]
    trivial
  isTrivialBlock_of_isBlock
    B := by
    left
    exact Set.subsingleton_of_subsingleton

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