instIsMulTorsionFree — Mathlib

English

The product of torsion-free monoids is torsion-free: if (a,b)^n = e for n > 0, then (a,b) = e.

Русский

Произведение безtorsion моноидов тождественно без torsion: если (a,b)^n = e при n>0, то (a,b) = e.

LaTeX

$$$\forall (a,b)\in M\times N, \forall n \in \mathbb{N}_{>0}, (a,b)^n = (1,1) \Rightarrow (a,b) = (1,1)$$$

Lean4

@[to_additive]
instance instIsMulTorsionFree [Monoid M] [Monoid N] [IsMulTorsionFree M] [IsMulTorsionFree N] : IsMulTorsionFree (M × N)
    where
  pow_left_injective n hn a b hab := by ext <;> apply pow_left_injective hn; exacts [congr(($hab).1), congr(($hab).2)]

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