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Proof step 8 from proof mod-mult-add (previous step)

(i + (i0 - 1) * j) |%| j = i |%| j, 0 < j, 0 < j, 0 < i0, ∀ i1, j, i. i→n(i1) < i→n(i0) → 0 < i1 ∧ 0 < j → (i + i1 * j) |%| j = i |%| j ⊦ (i + i0 * j - j) |%| j = i |%| j

User applied rule cut formula

And obtained the following new goals

Proof step 9:
i + (i0 - 1) * j = i + i0 * j - j, (i + (i0 - 1) * j) |%| j = i |%| j, 0 < j, 0 < j, 0 < i0, ∀ i1, j, i. i→n(i1) < i→n(i0) → 0 < i1 ∧ 0 < j → (i + i1 * j) |%| j = i |%| j ⊦ (i + i0 * j - j) |%| j = i |%| j
Proof step 10:
(i + (i0 - 1) * j) |%| j = i |%| j, 0 < j, 0 < j, 0 < i0, ∀ i1, j, i. i→n(i1) < i→n(i0) → 0 < i1 ∧ 0 < j → (i + i1 * j) |%| j = i |%| j ⊦ i + (i0 - 1) * j = i + i0 * j - j, (i + i0 * j - j) |%| j = i |%| j

Rule argument

fma: i + (i0 - 1) * j = i + i0 * j - j

Additional information

Used formulas: i + (i0 - 1) * j = i + i0 * j - j