no goals
p : ℕ
hp : Prime p
x✝ : ℕ
⊢ multiplicity p x✝ ≤ multiplicity p (choose x✝ 0) + multiplicity p 0
simp
no goals
p : ℕ
hp : Prime p
n✝ : ℕ
⊢ multiplicity p 0 ≤ multiplicity p (choose 0 (n✝ + 1)) + multiplicity p (n✝ + 1)
simp
p : ℕ
hp : Prime p
n k : ℕ
⊢ multiplicity p (n + 1) ≤ multiplicity p (choose (n + 1) (k + 1) * (k + 1))
p : ℕ
hp : Prime p
n k : ℕ
⊢ multiplicity p (n + 1) ≤ multiplicity p (choose (n + 1) (k + 1)) + multiplicity p (k + 1)
rw [← hp.multiplicity_mul]
p : ℕ
hp : Prime p
n k : ℕ
⊢ n + 1 ∣ choose (n + 1) (k + 1) * (k + 1)
p : ℕ
hp : Prime p
n k : ℕ
⊢ multiplicity p (n + 1) ≤ multiplicity p (choose (n + 1) (k + 1) * (k + 1))
refine' multiplicity_le_multiplicity_of_dvd_right _
p : ℕ
hp : Prime p
n k : ℕ
⊢ n + 1 ∣ succ n * choose n k
p : ℕ
hp : Prime p
n k : ℕ
⊢ n + 1 ∣ choose (n + 1) (k + 1) * (k + 1)
rw [← succ_mul_choose_eq]
no goals
p : ℕ
hp : Prime p
n k : ℕ
⊢ n + 1 ∣ succ n * choose n k
exact dvd_mul_right _ _