instNoZeroSMulDivisorsOfIsDomain — Mathlib

English

If e: M ≃_R N, then N is reflexive (repeated formulation of reflexivity under isomorphism).

Русский

Если e: M ≃_R N, тогда N рефлексивен (повторная формулировка сохранения рефлексивности сквозь изоморфизм).

LaTeX

$$∀ e: M ≃_R N, IsReflexive(R,N).$$

Lean4

instance (priority := 100) [IsDomain R] : NoZeroSMulDivisors R M :=
  by
  refine (noZeroSMulDivisors_iff R M).mpr ?_
  intro r m hrm
  rw [or_iff_not_imp_left]
  intro hr
  suffices Dual.eval R M m = Dual.eval R M 0 from (bijective_dual_eval R M).injective this
  ext n
  simp only [Dual.eval_apply, map_zero, LinearMap.zero_apply]
  suffices r • n m = 0 from eq_zero_of_ne_zero_of_mul_left_eq_zero hr this
  rw [← LinearMap.map_smul_of_tower, hrm, LinearMap.map_zero]

Похожие утверждения