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Proof step 9 from proof bertrand-middle-primes (previous step)

all_prime_divisors(binom_coef(2 * i, i)) = ints + ints4 + (i0 ' + ints3 + i0 ' + ints2) + ints1, 2 < i, 0 < i, 4 ^ ((2 * i) / 3) < Π ints4 * i0 * Π ints3 * i0 * Π ints2, ∀ j. j ∈ ints4 ∨ j = i0 ∨ j ∈ ints3 ∨ j ∈ ints2 → sqrt(2 * i) < j ∧ j ≤ (2 * i) / 3 ⊦

User applied rule insert lemma

And obtained the following new goals

Proof step 10:
prime(i), 0 < j ⊦ i ^ max_exp(i, binom_coef(2 * j, j)) ≤ 2 * j
Proof step 11:
all_prime_divisors(binom_coef(2 * i, i)) = ints + ints4 + (i0 ' + ints3 + i0 ' + ints2) + ints1, 2 < i, 0 < i, 4 ^ ((2 * i) / 3) < Π ints4 * i0 * Π ints3 * i0 * Π ints2, ∀ j. j ∈ ints4 ∨ j = i0 ∨ j ∈ ints3 ∨ j ∈ ints2 → sqrt(2 * i) < j ∧ j ≤ (2 * i) / 3 ⊦ prime(i0)
Proof step 12:
i0 ^ max_exp(i0, binom_coef(2 * i, i)) ≤ 2 * i, all_prime_divisors(binom_coef(2 * i, i)) = ints + ints4 + (i0 ' + ints3 + i0 ' + ints2) + ints1, 2 < i, 0 < i, 4 ^ ((2 * i) / 3) < Π ints4 * i0 * Π ints3 * i0 * Π ints2, ∀ j. j ∈ ints4 ∨ j = i0 ∨ j ∈ ints3 ∨ j ∈ ints2 → sqrt(2 * i) < j ∧ j ≤ (2 * i) / 3 ⊦

Rule argument

spec: inst: name: middle-coef-max_exp-stronger seq: prime(i), 0 < j ⊦ i ^ max_exp(i, binom_coef(2 * j, j)) ≤ 2 * j subst: [i, j] → [i0, i] flags: current, []

Additional information

Used terms: i0, i