isSMulRegular_submodule_iff_right_eq_zero_of_smul

English

For N ⊆ M, IsSmulRegular N r is equivalent to: for all x ∈ M with x ∈ N and r x = 0 implies x = 0.

Русский

Для N ⊆ M: IsSmulRegular(N,r) эквивалентно: для всех x ∈ M, если x ∈ N и r x = 0, то x = 0.

LaTeX

$$$IsSmulRegular N r \iff \forall x\in N,\; r\cdot x = 0 \Rightarrow x=0$.$$

Lean4

theorem isSMulRegular_submodule_iff_right_eq_zero_of_smul : IsSMulRegular N r ↔ ∀ x ∈ N, r • x = 0 → x = 0 :=
  isSMulRegular_iff_right_eq_zero_of_smul.trans <|
    Subtype.forall.trans <| by simp only [SetLike.mk_smul_mk, Submodule.mk_eq_zero]

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