English
Let a ∈ R be left-regular. Then IsLeftRegular (a * b) is equivalent to IsLeftRegular b for all b ∈ R.
Русский
Пусть a ∈ R — левая регулярность. Тогда для всех b ∈ R выполняется IsLeftRegular(a * b) ⇔ IsLeftRegular b.
LaTeX
$$$ IsLeftRegular a \rightarrow (IsLeftRegular (a * b) \leftrightarrow IsLeftRegular b) $$$
Lean4
/-- An element is right-regular if and only if multiplying it on the right with a right-regular
element is right-regular. -/
@[to_additive (attr := simp) /-- An element is add-right-regular if and only if adding it on the right to
an add-right-regular element is add-right-regular. -/
]
theorem mul_isRightRegular_iff (b : R) (ha : IsRightRegular a) : IsRightRegular (b * a) ↔ IsRightRegular b :=
⟨fun ab => IsRightRegular.of_mul ab, fun ab => IsRightRegular.mul ab ha⟩
