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Proof step 48 from proof RSA-decrypt-ij-vars (previous step)

gcd(i2, (i - 1) * (i0 - 1)) = 1, i2 < i * i0, 2 < i0, 2 < i, prime(i0), prime(i), 0 < i0, 0 < i3, 0 ≤ i2, 1 < i * i0, 0 < i2, 0 < i, j < i * i0, 0 < i1, 0 ≤ (i - 1) * (i0 - 1), ~1 ≤ ((i - 1) * (i0 - 1)) * i1, 0 < j, 0 ≤ j ^ (((i - 1) * (i0 - 1)) * i1 + 1) ⊦ i1 = 0, j = 0, i3 = 0, i = i0, j = j ^ (((i - 1) * (i0 - 1)) * i1 + 1) |%| (i * i0), i0 = 0, j ^ (((i - 1) * (i0 - 1)) * i1 + 1) |%| i0 = j |%| i0 ∧ ≡mod(j ^ (((i - 1) * (i0 - 1)) * i1 + 1), j, i)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 49:
⊦ 0 < j → (≡mod(i, i0, j) ↔ i |%| j = i0 |%| j)
Proof step 50:
0 < i, gcd(i2, (i - 1) * (i0 - 1)) = 1, i2 < i * i0, 2 < i0, 2 < i, prime(i0), prime(i), 0 < i0, 0 < i3, 0 ≤ i2, 1 < i * i0, 0 < i2, 0 < i, j < i * i0, 0 < i1, 0 ≤ (i - 1) * (i0 - 1), ~1 ≤ ((i - 1) * (i0 - 1)) * i1, 0 < j, 0 ≤ j ^ (((i - 1) * (i0 - 1)) * i1 + 1) ⊦ i1 = 0, j = 0, i3 = 0, i = i0, j = j ^ (((i - 1) * (i0 - 1)) * i1 + 1) |%| (i * i0), i0 = 0, j ^ (((i - 1) * (i0 - 1)) * i1 + 1) |%| i0 = j |%| i0 ∧ j ^ (((i - 1) * (i0 - 1)) * i1 + 1) |%| i = j |%| i

Rule argument

spec: inst: name: congmod-def-mod seq: ⊦ 0 < j → (≡mod(i, i0, j) ↔ i |%| j = i0 |%| j) subst: [j, i0, i] → [i, j, j ^ (((i - 1) * (i0 - 1)) * i1 + 1)] flags: rewrite, current

Additional information

Used terms: i, j, j ^ (((i - 1) * (i0 - 1)) * i1 + 1)