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Proof step 9 from proof bertrand-estimate (previous step)

i / 3 ≤ i - (2 * i) / 3, (2 * i) / 3 ≤ i, 0 ≤ 4 ^ i, 4 ^ (i - (2 * i) / 3) ≤ (2 * i) ^ sqrt(2 * i), 0 ≤ i, 0 < i, 1024 ≤ i, 4 ^ i ≤ 4 ^ ((2 * i) / 3) * (2 * i) ^ sqrt(2 * i), 0 < 4 ^ ((2 * i) / 3), 0 ≤ (2 * i) ^ sqrt(2 * i), 0 ≤ (2 * i) / 3 ⊦

User applied rule cut formula

And obtained the following new goals

Proof step 10:
4 ^ (i / 3) ≤ 4 ^ (i - (2 * i) / 3), i / 3 ≤ i - (2 * i) / 3, (2 * i) / 3 ≤ i, 0 ≤ 4 ^ i, 4 ^ (i - (2 * i) / 3) ≤ (2 * i) ^ sqrt(2 * i), 0 ≤ i, 0 < i, 1024 ≤ i, 4 ^ i ≤ 4 ^ ((2 * i) / 3) * (2 * i) ^ sqrt(2 * i), 0 < 4 ^ ((2 * i) / 3), 0 ≤ (2 * i) ^ sqrt(2 * i), 0 ≤ (2 * i) / 3 ⊦
Proof step 13:
i / 3 ≤ i - (2 * i) / 3, (2 * i) / 3 ≤ i, 0 ≤ 4 ^ i, 4 ^ (i - (2 * i) / 3) ≤ (2 * i) ^ sqrt(2 * i), 0 ≤ i, 0 < i, 1024 ≤ i, 4 ^ i ≤ 4 ^ ((2 * i) / 3) * (2 * i) ^ sqrt(2 * i), 0 < 4 ^ ((2 * i) / 3), 0 ≤ (2 * i) ^ sqrt(2 * i), 0 ≤ (2 * i) / 3 ⊦ 4 ^ (i / 3) ≤ 4 ^ (i - (2 * i) / 3)

Rule argument

fma: 4 ^ (i / 3) ≤ 4 ^ (i - (2 * i) / 3)

Additional information

Used formulas: 4 ^ (i / 3) ≤ 4 ^ (i - (2 * i) / 3)