smul_le_self_of_tower — Mathlib

English

A slightly different formulation: if p contains s • N and p ≤ q, then s • N ≤ q.

Русский

Другой вариант формулировки: если p содержит s • N и p ≤ q, тогда s • N ≤ q.

LaTeX

$$$s \\cdot N \\le q \\quad\\text{implies}\\quad s \\cdot N = p$$

Lean4

@[simp]
theorem smul_le_self_of_tower {α : Type*} [Monoid α] [SMul α R] [DistribMulAction α M] [SMulCommClass α R M]
    [IsScalarTower α R M] (a : α) (S : Submodule R M) : a • S ≤ S :=
  by
  rintro y ⟨x, hx, rfl⟩
  exact smul_of_tower_mem _ a hx

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