isLeftRegular_iff_right_eq_zero_of_mul

English

IsLeftRegular r iff ∀ x, r x = 0 ⇒ x = 0.

Русский

Левое регулярное: IsLeftRegular r еквивалентно ∀ x, r x = 0 ⇒ x = 0.

LaTeX

$$$\text{IsLeftRegular}(r) \iff \forall x,\ r\cdot x = 0 \Rightarrow x = 0$$$

Lean4

theorem isLeftRegular_iff_right_eq_zero_of_mul : IsLeftRegular r ↔ ∀ x, r * x = 0 → x = 0
    where
  mp h r' eq := h (by simp_rw [eq, mul_zero])
  mpr h r₁ r₂ eq := sub_eq_zero.mp <| h _ <| by simp_rw [mul_sub, eq, sub_self]

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