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Proof step 39 from proof make (previous step)

0 < i, i0 < 0 ⊦ i0 = 0, i = 0, rational(~(i / gcd(i, i0)), ~(i0 / gcd(i, i0))).num + i / gcd(i, i0) = 0 ∧ i0 / gcd(i, i0) + rational(~(i / gcd(i, i0)), ~(i0 / gcd(i, i0))).denom = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 40:
⊦ i ≠ 0 ∧ 0 < j → rational(i, j).num = i / gcd(i, j)
Proof step 41:
¬(~(i / gcd(i, i0)) ≠ 0 ∧ 0 < ~(i0 / gcd(i, i0))), 0 < i, i0 < 0 ⊦ i0 = 0, i = 0, rational(~(i / gcd(i, i0)), ~(i0 / gcd(i, i0))).num + i / gcd(i, i0) = 0 ∧ i0 / gcd(i, i0) + rational(~(i / gcd(i, i0)), ~(i0 / gcd(i, i0))).denom = 0
Proof step 42:
~(i / gcd(i, i0)) ≠ 0 ∧ 0 < ~(i0 / gcd(i, i0)), 0 < i, i0 < 0 ⊦ i0 = 0, i = 0, ~(i / gcd(i, i0)) / gcd(~(i / gcd(i, i0)), ~(i0 / gcd(i, i0))) + i / gcd(i, i0) = 0 ∧ i0 / gcd(i, i0) + rational(~(i / gcd(i, i0)), ~(i0 / gcd(i, i0))).denom = 0

Rule argument

name: numerator-pos rotate: F subst: [j, i] → [~(i0 / gcd(i, i0)), ~(i / gcd(i, i0))]

Additional information

Used terms: ~(i0 / gcd(i, i0)), ~(i / gcd(i, i0))