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Proof step 23 from proof divides-prime (previous step)

i < 0, i0 < 0, 2 ≤ i0 * i, ∀ i1. 1 < i1 ∧ i1 < i0 * i → ¬ i1 | i0 * i ⊦ ~1 = i0, ~1 = i, i = 0, i0 = 0, 1 < ~ i0 ∧ ~ i0 < i0 * i

User applied rule case distinction

And closed the goal!

Rule argument

pos: right 5

Additional information

Used simplifier rules: ⊦ ~ i < j ↔ 0 < j + i; (s)% ⊦ i * j + i = i * (j + 1); (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ i < 0 → (0 < i * j ↔ j < 0); (s)% ⊦ i + (* j) < (* j0) ↔ i < j0 - j; (s)% ⊦ i < j ↔ ¬ j ≤ i; (ss)% ⊦ i < 0 → (~1 ≤ i ↔ i = ~1); (s)% ⊦ i < ~ j ↔ i + j < 0; (s)