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Proof step 11 from proof primes-divide (previous step)

i0 * i1 < i0 * Π ints0, 0 < i0 * Π ints0, 1 < i0 * Π ints0, prime(i0), 0 < i0 * i1, j ∈ ints0 → prime(j) ∧ j | i0 * i1, ∀ j. j ∈ ints0 → prime(j) ∧ j | i0 * i1, (j ∈ ints0 → prime(j) ∧ j | i1) ∧ 0 < i1 → Π ints0 ≤ i1 ⊦ i0 = 0, dups(ints0), i0 ∈ ints0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult inst: name: ls-pos seq: ⊦ 0 < i → (i * j < i * j0 ↔ j < j0) subst: [j0, j, i] → [Π ints0, i1, i0] flags: rewrite

Additional information

Used simplifier rules: lst-01: ⊦ ¬ i < j → (i ≠ j ↔ j < i); from specification int-basic1%ls-neg: ⊦ i < 0 → (i * j < i * j0 ↔ j0 < 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%zero-gr-01: ⊦ i < 0 → (0 < i * j ↔ j < 0); from specification int-mult%prime-not-neg: ⊦ j < 0 → ¬ prime(j); from specification int-gcd Used terms: Π ints0, i1, i0