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Proof step 29 from proof fermat-pos (previous step)

prime(j), ≡mod(i1 ^ j, i1, j), 0 ≤ i1, 0 ≤ j, j1 < j, 0 < j1, 1 < j, 0 < j, ∀ i0, j. i→n(i0) < i→n(i1 + 1) → prime(j) ∧ 0 < i0 → ≡mod(i0 ^ j, i0, j) ⊦ ≡mod(Σ countFun(1, j - 1, λ j0. i1 ^ j0 * binom_coef(j, j0)) + i1 ^ j, i1, j), j | Σ countFun(1, j - 1, λ j0. i1 ^ j0 * binom_coef(j, j0)), j | i1, j | binom_coef(j, j1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 30:
⊦ gcd(i, j) = 1 ∧ j ≤ i ∧ 0 ≤ j → i | binom_coef(i, j)
Proof step 31:
¬(gcd(j, j1) = 1 ∧ j1 ≤ j ∧ 0 ≤ j1), prime(j), ≡mod(i1 ^ j, i1, j), 0 ≤ i1, 0 ≤ j, j1 < j, 0 < j1, 1 < j, 0 < j, ∀ i0, j. i→n(i0) < i→n(i1 + 1) → prime(j) ∧ 0 < i0 → ≡mod(i0 ^ j, i0, j) ⊦ ≡mod(Σ countFun(1, j - 1, λ j0. i1 ^ j0 * binom_coef(j, j0)) + i1 ^ j, i1, j), j | Σ countFun(1, j - 1, λ j0. i1 ^ j0 * binom_coef(j, j0)), j | i1, j | binom_coef(j, j1)

Rule argument

spec: int-fac inst: name: coef-gcd seq: ⊦ gcd(i, j) = 1 ∧ j ≤ i ∧ 0 ≤ j → i | binom_coef(i, j) subst: [j, i] → [j1, j] flags: rewrite

Additional information

Used terms: j1, j