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Proof step 26 from proof congmod-inverse (previous step)

gcd(i, j) = 1, (i * i0) % abs(j) = 1, 0 < i0, 0 ≤ i ⊦ j | i * i0 - 1, ∃ i0. ≡mod(i * i0, 1, j)

User applied rule insert elim lemma

And obtained the following new goals

Rule argument

spec: int-div inst: name: div-elim-pospos seq: ⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0) subst: [i, j, i0, j0] → [i * i0, abs(j), i1, j0] flags:

Additional information

Used simplifier rules: cong-zero: ⊦ ≡mod(i, j, 0) ↔ i = j;%abs-06: ⊦ 0 < abs(i) ↔ i ≠ 0; from specification int-abs%zero-ge: ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); from specification int-mult%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%gcd-zero-04: ⊦ gcd(i, 0) = abs(i); from specification int-gcd%absle: ⊦ 0 ≤ i → abs(i) = i; from specification int-abs%one-01: ⊦ i = 1 → i * j = j; from specification int-mult%div-zero-zero: ⊦ 0 | i ↔ i = 0; from specification int-gcd%sub-eq: ⊦ 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%⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0); (elim) Used terms: i * i0, abs(j), i1, j0