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Proof step 4 from proof fermat-pos (previous step)

0 < i, prime(j), ∀ i0, j. i→n(i0) < i→n(i) → prime(j) ∧ 0 < i0 → ≡mod(i0 ^ j, i0, j) ⊦ ≡mod(i ^ j, i, j)

User applied rule apply induction

And obtained the following new goals

Proof step 5:
≡mod((i - 1) ^ j, i - 1, j), 0 < i, prime(j), ∀ i0, j. i→n(i0) < i→n(i) → prime(j) ∧ 0 < i0 → ≡mod(i0 ^ j, i0, j) ⊦ ≡mod(i ^ j, i, j)

Rule argument

subst: [i0, j] → [i - 1, j]

Additional information

Used simplifier rules: cong-same: ⊦ ≡mod(i, i, j);%sub-ls-01: ⊦ i < j - i0 ↔ i + i0 < j; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%bet: ⊦ 0 < i → (i ≤ 1 ↔ i = 1); from specification int-abs%one-01: ⊦ 1 ^ j = 1; from specification int-pot%abs-pred: ⊦ i→n(i - 1) < i→n(i) ↔ 0 < i; from specification intnat%lels-03: ⊦ i < j → (¬ ¬ j ≤ i ↔ false); from specification int-abs Used terms: i - 1, j