rw [range_eq_Ico, ← Ico_insert_succ_left (succ_pos _), sum_insert, ArithmeticFunction.map_zero,
  mul_zero, zero_add]
N : ℕ
hN : 1 ≤ N
⊢ ↑(∑ x ∈ Icc 1 N, ↑(N / x) * μ x) = ↑(∑ x ∈ range (N + 1), ↑N / ↑x * μ x)
N : ℕ
hN : 1 ≤ N
⊢ ↑(∑ x ∈ Icc 1 N, ↑(N / x) * μ x) = ↑(∑ x ∈ Ico (succ 0) N.succ, ↑N / ↑x * μ x)

N : ℕ
hN : 1 ≤ N
⊢ 0 ∉ Ico (succ 0) N.succ