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Proof step 14 from proof bertrand-middle-primes (previous step)

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

User applied rule cut formula

And obtained the following new goals

Proof step 15:
i0 ^ 2 ≤ 2 * i, all_prime_divisors(binom_coef(2 * i, i)) = ints + ints4 + (i0 ' + ints3 + i0 ' + ints2) + ints1, 4 ^ ((2 * i) / 3) < Π ints4 * i0 * Π ints3 * i0 * Π ints2, 0 < i, 2 < i, i0 ^ max_exp(i0, binom_coef(2 * i, i)) ≤ 2 * i, 2 ≤ max_exp(i0, binom_coef(2 * i, i)), ∀ j. j ∈ ints4 ∨ j = i0 ∨ j ∈ ints3 ∨ j ∈ ints2 → sqrt(2 * i) < j ∧ j ≤ (2 * i) / 3 ⊦ max_exp(i0, binom_coef(2 * i, i)) = 0
Proof step 16:
all_prime_divisors(binom_coef(2 * i, i)) = ints + ints4 + (i0 ' + ints3 + i0 ' + ints2) + ints1, 4 ^ ((2 * i) / 3) < Π ints4 * i0 * Π ints3 * i0 * Π ints2, 0 < i, 2 < i, i0 ^ max_exp(i0, binom_coef(2 * i, i)) ≤ 2 * i, 2 ≤ max_exp(i0, binom_coef(2 * i, i)), ∀ j. j ∈ ints4 ∨ j = i0 ∨ j ∈ ints3 ∨ j ∈ ints2 → sqrt(2 * i) < j ∧ j ≤ (2 * i) / 3 ⊦ i0 ^ 2 ≤ 2 * i, max_exp(i0, binom_coef(2 * i, i)) = 0

Rule argument

fma: i0 ^ 2 ≤ 2 * i

Additional information

Used formulas: i0 ^ 2 ≤ 2 * i