isSMulRegular_on_submodule_iff_disjoint_ker_lsmul_submodule

English

Let N be a submodule of M. The action by r is regular on N if and only if ker(lsmul R M r) and N are disjoint (intersect trivially).

Русский

Пусть N — подмодуль M. Регулярность по r на N эквивалентна тому, что ker(lsmul R M r) и N⋂ имеютTrivial intersection.

LaTeX

$$$IsSMulRegular(N,r) \iff Disjoint(\ker(\operatorname{lsmul} R M r), N)$$$

Lean4

theorem isSMulRegular_on_submodule_iff_disjoint_ker_lsmul_submodule :
    IsSMulRegular N r ↔ Disjoint (LinearMap.ker (LinearMap.lsmul R M r)) N :=
  Iff.trans (isSMulRegular_submodule_iff_right_eq_zero_of_smul N r) <| Iff.symm <| Iff.trans disjoint_comm disjoint_def

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