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Proof step 25 from proof bertrand-middle-coef-smaller (previous step)

all_prime_divisors(binom_coef(2 * i, i)) = ints + ints0 + (i0 ' + ints3) + ints2, Π ints * Π ints0 * (i0 * Π ints3) * Π ints2 = binom_coef(2 * i, i), 0 < Π ints0, 0 < Π ints, 0 < i, i0 ≤ i, (2 * i) / 3 < i0, 4 ^ ((2 * i) / 3) < Π ints0 * i0 * Π ints3, 0 ≤ Π ints, 0 ≤ Π ints2, (2 * i) ^ (sqrt(2 * i) - 1) * 4 ^ ((2 * i) / 3) < Π ints * Π ints0 * i0 * Π ints3, 2 < i, 0 < Π ints2, 0 < i0 * Π ints3, ∀ j. j ∈ ints0 → sqrt(2 * i) < j ∧ j ≤ (2 * i) / 3, ∀ j. j ∈ ints → j ≤ sqrt(2 * i), ∀ j. j ∈ ints2 → i < j ∧ j ≤ 2 * i, ∀ j. j = i0 ∨ j ∈ ints3 → (2 * i) / 3 < j ∧ j ≤ i ⊦

User applied rule cut formula

And obtained the following new goals

Proof step 26:
i0 ∈ all_prime_divisors(binom_coef(2 * i, i)), all_prime_divisors(binom_coef(2 * i, i)) = ints + ints0 + (i0 ' + ints3) + ints2, Π ints * Π ints0 * (i0 * Π ints3) * Π ints2 = binom_coef(2 * i, i), 0 < Π ints0, 0 < Π ints, 0 < i, i0 ≤ i, (2 * i) / 3 < i0, 4 ^ ((2 * i) / 3) < Π ints0 * i0 * Π ints3, 0 ≤ Π ints, 0 ≤ Π ints2, (2 * i) ^ (sqrt(2 * i) - 1) * 4 ^ ((2 * i) / 3) < Π ints * Π ints0 * i0 * Π ints3, 2 < i, 0 < Π ints2, 0 < i0 * Π ints3, ∀ j. j ∈ ints0 → sqrt(2 * i) < j ∧ j ≤ (2 * i) / 3, ∀ j. j ∈ ints → j ≤ sqrt(2 * i), ∀ j. j ∈ ints2 → i < j ∧ j ≤ 2 * i, ∀ j. j = i0 ∨ j ∈ ints3 → (2 * i) / 3 < j ∧ j ≤ i ⊦

Rule argument

fma: i0 ∈ all_prime_divisors(binom_coef(2 * i, i))

Additional information

Used simplifier rules: neq: ⊦ ¬ i ∈ ints → ints ≠ ints0 + i ' + ints1; from specification list[intlista]%a: ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1; from specification list[intlista] Used formulas: i0 ∈ all_prime_divisors(binom_coef(2 * i, i))