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Proof step 19 from proof mult-def-02 (previous step)

rational(i1, j1) * rational(i0, i3) = rational(i1 * i0, j1 * i3), gcd(i1, j1) = 1, 0 < j1, 0 < i, 1 < i, 0 < i3, ∀ j2, j1, i4, i3. i→n(i3) + i→n(i4) < i→n(i1) + i→n(i * i0) → 0 < j1 ∧ 0 < j2 → rational(i3, j1) * rational(i4, j2) = rational(i3 * i4, j1 * j2) ⊦ rational(i1 * i0, j1 * i3) = rational(i1 * i * i0, j1 * i * i3)

User applied rule cut formula

And obtained the following new goals

Proof step 20:
i1 * i * i0 = i * i1 * i0 ∧ j1 * i * i3 = i * j1 * i3, rational(i1, j1) * rational(i0, i3) = rational(i1 * i0, j1 * i3), gcd(i1, j1) = 1, 0 < j1, 0 < i, 1 < i, 0 < i3, ∀ j2, j1, i4, i3. i→n(i3) + i→n(i4) < i→n(i1) + i→n(i * i0) → 0 < j1 ∧ 0 < j2 → rational(i3, j1) * rational(i4, j2) = rational(i3 * i4, j1 * j2) ⊦ rational(i1 * i0, j1 * i3) = rational(i1 * i * i0, j1 * i * i3)

Rule argument

fma: i1 * i * i0 = i * i1 * i0 ∧ j1 * i * i3 = i * j1 * i3

Additional information

Used simplifier rules: ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i; Used formulas: i1 * i * i0 = i * i1 * i0 ∧ j1 * i * i3 = i * j1 * i3