isRegular_iff_eq_zero_of_mul — Mathlib

English

IsRegular r iff both left and right regularity hold: ∀ x, r x = 0 ⇒ x = 0 and ∀ x, x r = 0 ⇒ x = 0.

Русский

IsRegular r эквивалентно тому, что и слева и справа регулярность выполняются: ∀ x, r x = 0 ⇒ x = 0 и ∀ x, x r = 0 ⇒ x = 0.

LaTeX

$$$\text{IsRegular}(r) \iff \Big(\forall x, r x = 0 \Rightarrow x = 0\Big) \land \Big(\forall x, x r = 0 \Rightarrow x = 0\Big)$$$

Lean4

theorem isRegular_iff_eq_zero_of_mul : IsRegular r ↔ (∀ x, r * x = 0 → x = 0) ∧ (∀ x, x * r = 0 → x = 0) := by
  rw [isRegular_iff, isLeftRegular_iff_right_eq_zero_of_mul, isRightRegular_iff_left_eq_zero_of_mul]

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