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Proof step 7 from proof congmod-inverse-pos (previous step)

gcd(i, j) = 1, i0 < 0, ≡mod(i * i0, 1, j), 0 ≤ j, 0 ≤ i ⊦ ∃ i0. ≡mod(i * i0, 1, j) ∧ 0 ≤ i0

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 8:
1 < j, 1 < i, gcd(i, j) = 1 ⊦ ∃ i0. (i * i0) % j = 1 ∧ 0 < i0
Proof step 9:
gcd(i, j) = 1, i0 < 0, ≡mod(i * i0, 1, j), 0 ≤ j, 0 ≤ i ⊦ 1 < j ∧ 1 < i, ∃ i0. ≡mod(i * i0, 1, j) ∧ 0 ≤ i0
Proof step 21:
∃ i0. (i * i0) % j = 1 ∧ 0 < i0, gcd(i, j) = 1, i0 < 0, ≡mod(i * i0, 1, j), 0 ≤ j, 0 ≤ i ⊦ ∃ i0. ≡mod(i * i0, 1, j) ∧ 0 ≤ i0

Rule argument

spec: int-gcd inst: name: gcd-mod-one-01 seq: 1 < j, 1 < i, gcd(i, j) = 1 ⊦ ∃ i0. (i * i0) % j = 1 ∧ 0 < i0 subst: [j, i] → [j, i] flags:

Additional information

Used terms: j, i