surjective_rootsOfUnityCircleEquiv_comp_rootsOfUnityAddChar

English

The composition of the circle equivalence with the addChar from ZMod is surjective onto the circle roots of unity.

Русский

Сложение эквивалентности круга и AddChar на ZMod образует всю окружность корней unity.

LaTeX

$$Surjective (rootsOfUnityCircleEquiv n ∘ ZMod.rootsOfUnityAddChar n)$$

Lean4

theorem surjective_rootsOfUnityCircleEquiv_comp_rootsOfUnityAddChar (n : ℕ) [NeZero n] :
    Surjective (rootsOfUnityCircleEquiv n ∘ ZMod.rootsOfUnityAddChar n) := fun ⟨w, hw⟩ ↦
  by
  obtain ⟨j, hj1, hj2⟩ := (Complex.mem_rootsOfUnity n w).mp hw
  exact ⟨j, by simp [Units.ext_iff, Subtype.ext_iff, ← hj2, ZMod.toCircle_natCast, mul_div_assoc]⟩

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