two_nsmul_lie_lmul_lmul_add_add_eq_zero

English

IsPrimePow n is equivalent to the existence of k ≤ log_2 n with n = n.minFac^k and k > 0 and p prime.

Русский

IsPrimePow n эквивалентно существованию k ≤ log_2 n такое, что n = n.minFac^k и k > 0 и p простое.

LaTeX

$$IsPrimePow(n) \iff ∃ k ≤ log_2 n, 0 < k ∧ ∃ n.minFac, n = (n.minFac)^k ∧ Nat.Prime(n.minFac)$$

Lean4

theorem two_nsmul_lie_lmul_lmul_add_add_eq_zero (a b c : A) :
    2 • (⁅L a, L (b * c)⁆ + ⁅L b, L (c * a)⁆ + ⁅L c, L (a * b)⁆) = 0 :=
  by
  symm
  calc
    0 = ⁅L (a + b + c), L ((a + b + c) * (a + b + c))⁆ := by rw [(commute_lmul_lmul_sq (a + b + c)).lie_eq]
    _ = _ := by rw [aux0, aux1, aux2, aux3, nsmul_add, nsmul_add]

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