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Proof step 7 from proof mod-crt (previous step)

j * j0 < i, j * j0 ≤ 0, 0 ≤ i0, 0 < i, prime(j0), prime(j), ≡mod(i ^ i0, i, j0), ≡mod(i ^ i0, i, j), ≡mod(i ^ i0, i, j * j0) ⊦ j = j0, i ^ i0 |%| (j * j0) = i |%| (j * j0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ 0 < i → (i * j ≤ 0 ↔ j ≤ 0)

Rule argument

spec: int-mult inst: name: zero-le seq: ⊦ 0 < i → (i * j ≤ 0 ↔ j ≤ 0) subst: [j, i] → [j0, j] flags: rewrite

Additional information

Used simplifier rules: prime-not-neg-01: ⊦ j ≤ 0 → ¬ prime(j); from specification int-gcd%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%zero-le-03: ⊦ i ≤ 0 → (i * j ≤ 0 ↔ ¬ ¬(i = 0 ∨ 0 ≤ j)); from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: j0, j