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Proof step 17 from proof modmod-mult-07 (previous step)

i0 < 0, 0 < j, i < 0, i ≤ 0 ⊦ (~(~ i % j) * i0) % j = (i * i0) % j, j = 0

User applied rule cut formula

And obtained the following new goals

Proof step 18:
~(~ i % j) * i0 = ~ i % j * ~ i0, i < 0, 0 < j, i0 < 0, i ≤ 0 ⊦ (~(~ i % j) * i0) % j = (i * i0) % j, j = 0

Rule argument

fma: ~(~ i % j) * i0 = ~ i % j * ~ i0

Additional information

Used simplifier rules: mod-neg-01: j ≠ 0 ⊦ ~ i % j = ~(i % j);% ⊦ ~ ~ i = i; (s)% ⊦ ~ i * j = ~(i * j); (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ i < 0 → (i * j = i * j0 ↔ j = j0); (s) Used formulas: ~(~ i % j) * i0 = ~ i % j * ~ i0