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Proof step 36 from proof congmod-inverse (previous step)

gcd(i, j) = 1, ~1 = (i * i0) % abs(j), 0 < i0, i < 0, i < i0 ⊦ j = 0, j | i * i0 - 1, ≡mod(~(i0 * i), 1, j), ∃ i0. ≡mod(i * i0, 1, j)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 37:
⊦ ≡mod(i, i0, j) ↔ j | abs(i - i0)
Proof step 38:
gcd(i, j) = 1, ~1 = (i * i0) % abs(j), 0 < i0, i < 0, i < i0 ⊦ j = 0, j | i * i0 - 1, j | abs(~(i0 * i) - 1), ∃ i0. ≡mod(i * i0, 1, j)

Rule argument

spec: int-mod inst: name: congmod-def seq: ⊦ ≡mod(i, i0, j) ↔ j | abs(i - i0) subst: [j, i0, i] → [j, 1, ~(i0 * i)] flags: rewrite, current

Additional information

Used terms: j, 1, ~(i0 * i)