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Proof step 9 from proof equality-01 (previous step)

i / gcd(i, j) = i0 / gcd(i0, j0), j / gcd(i, j) = j0 / gcd(i0, j0), 0 < j, 0 < j0 ⊦ i * j0 = i0 * j, j = 0, i = 0, j0 = 0, i0 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ i ≠ 0 → (i / gcd(i, j) = i0 ↔ i = i0 * gcd(i, j))
Proof step 11:
i ≠ 0, i = i0 / gcd(i0, j0) * gcd(i, j), j / gcd(i, j) = j0 / gcd(i0, j0), 0 < j, 0 < j0 ⊦ i * j0 = i0 * j, j = 0, i = 0, j0 = 0, i0 = 0

Rule argument

seq: ⊦ i ≠ 0 → (i / gcd(i, j) = i0 ↔ i = i0 * gcd(i, j)) subst: [i0, j, i] → [i0 / gcd(i0, j0), j, i] rotate: F spec: int-gcd inst: name: gcd-div-eq

Additional information

Used terms: i0 / gcd(i0, j0), j, i