English
Restricting a multiplicative equivalence and then applying its symmetry equals restricting the inverse equivalence.
Русский
Ограничение мультиэквививалента и применение его симметрии равно ограничению обратного эквививалента.
LaTeX
$$$\bigl(\operatorname{restrictRootsOfUnity} \sigma k\bigr)^{\mathrm{symm}} = \sigma^{\mathrm{symm}} \!\restriction_k$$$
Lean4
theorem card_rootsOfUnity : Fintype.card (rootsOfUnity k R) ≤ k := by
classical
calc
Fintype.card (rootsOfUnity k R) = Fintype.card { x // x ∈ nthRoots k (1 : R) } :=
Fintype.card_congr (rootsOfUnityEquivNthRoots R k)
_ ≤ Multiset.card (nthRoots k (1 : R)).attach := (Multiset.card_le_card (Multiset.dedup_le _))
_ = Multiset.card (nthRoots k (1 : R)) := Multiset.card_attach
_ ≤ k := card_nthRoots k 1
