English
If two reduced words have equal φ-products, then their leading coordinate lists are identical, and the head-coset relation holds pointwise across fixed entries.
Русский
Если два редуцированных слова имеют равные произведения по φ, то их списки ведущих координат совпадают, а отношение головы по косетам выполняется покомпонентно.
LaTeX
$$Same as 122934 with φ-product equality implies first-coordinate equality and head-coset condition.$$
Lean4
@[to_additive]
theorem toSubmonoid_zpowers {G : Type*} [Group G] (g : G) :
(Subgroup.zpowers g).toSubmonoid = Submonoid.powers g ⊔ Submonoid.powers g⁻¹ := by
rw [zpowers_eq_closure, closure_toSubmonoid, Submonoid.closure_union, Submonoid.powers_eq_closure,
Submonoid.powers_eq_closure, Set.inv_singleton]
