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Proof step 12 from proof congmod-inverse-smallest (previous step)

gcd(i, j) = 1, 0 < j, j ≤ i0, 0 < i0, ≡mod(i * i0, 1, j), 0 ≤ i0, 0 < i, i0 |%| j < j, i0 |%| j < i0, 0 ≤ i0 |%| j ⊦ i = 0, j = 0, ≡mod(i * i0 |%| j, 1, j)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 13:
⊦ 0 < j → (≡mod(i, i0, j) ↔ i |%| j = i0 |%| j)
Proof step 14:
0 < j, gcd(i, j) = 1, 0 < j, j ≤ i0, 0 < i0, ≡mod(i * i0, 1, j), 0 ≤ i0, 0 < i, i0 |%| j < j, i0 |%| j < i0, 0 ≤ i0 |%| j ⊦ i = 0, j = 0, (i * i0 |%| j) |%| j = 1 |%| j

Rule argument

spec: inst: name: congmod-def-mod seq: ⊦ 0 < j → (≡mod(i, i0, j) ↔ i |%| j = i0 |%| j) subst: [j, i0, i] → [j, 1, i * i0 |%| j] flags: rewrite, current

Additional information

Used terms: j, 1, i * i0 |%| j