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Proof step 23 from proof fermat-pos (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 24:
⊦ (∀ j. j ∈ ints → i | j) → i | Σ ints
Proof step 25:
¬(∀ j0. j0 ∈ countFun(1, j - 1, λ j0. i1 ^ j0 * binom_coef(j, j0)) → j | j0), 0 < j, 0 ≤ j, 0 ≤ i1, ≡mod(i1 ^ j, i1, j), prime(j), ∀ i0, j. i→n(i0) < i→n(i1 + 1) → prime(j) ∧ 0 < i0 → ≡mod(i0 ^ j, i0, j) ⊦ j | Σ countFun(1, j - 1, λ j0. i1 ^ j0 * binom_coef(j, j0)), ≡mod(Σ countFun(1, j - 1, λ j0. i1 ^ j0 * binom_coef(j, j0)) + i1 ^ j, i1, j)

Rule argument

spec: int-fac inst: name: divides-sum seq: ⊦ (∀ j. j ∈ ints → i | j) → i | Σ ints subst: [ints, i] → [countFun(1, j - 1, λ j0. i1 ^ j0 * binom_coef(j, j0)), j] flags: rewrite

Additional information

Used terms: countFun(1, j - 1, λ j0. i1 ^ j0 * binom_coef(j, j0)), j