mul_comm_of_exponent_two — Mathlib

English

In a cancellative monoid of exponent two, all elements commute.

Русский

В(cancel) cancellative моноиде с экспонентой 2 все элементы коммутируют.

LaTeX

$$$[IsCancelMul G] \land (\operatorname{exponent} G = 2) \Rightarrow \forall a,b \in G,\ ab=ba$$$

Lean4

/-- In a cancellative monoid of exponent two, all elements commute. -/
@[to_additive]
theorem mul_comm_of_exponent_two [IsCancelMul G] (hG : Monoid.exponent G = 2) (a b : G) : a * b = b * a :=
  Commute.of_orderOf_dvd_two (fun g => hG ▸ Monoid.order_dvd_exponent g) a b

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