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Proof step 9 from proof prime-def-mod (previous step)

j % i = 0, i * i ≤ j, 2 ≤ j, 0 < j, 0 < i, 1 < i, 0 ≤ j, ∀ i. 1 < i ∧ i < j → ¬ i | j, i < j → ¬ i | j ⊦ i = 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
j % i = 0, 2 < j, i < j, i * i ≤ j, 2 ≤ j, 0 < j, 0 < i, 1 < i, 0 ≤ j, ∀ i. 1 < i ∧ i < j → ¬ i | j ⊦ i = 1, i | j

Rule argument

Additional information

Used simplifier rules: numlslsdel-01: i0 ≠ 0, i0 < i ⊦ (* i) < j → (* i0) < j; from specification int-basic1%lf-01: ⊦ i < j ∧ j < i0 → i +1 < i0; (forward) from specification int-basic1%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%bet: ⊦ 0 < i → (i ≤ 1 ↔ i = 1); from specification int-abs%le-same: ⊦ 0 < i → (i * j ≤ i ↔ j ≤ 1); from specification int-mult%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs