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Definition df-mod 12482
Description: Define the modulo (remainder) operation. See modval 12483 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1 (ex-mod 26460). (Contributed by NM, 10-Nov-2008.)
Assertion
Ref Expression
df-mod mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-mod
StepHypRef Expression
1 cmo 12481 . 2 class mod
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cr 9787 . . 3 class
5 crp 11660 . . 3 class +
62cv 1473 . . . 4 class 𝑥
73cv 1473 . . . . 5 class 𝑦
8 cdiv 10529 . . . . . . 7 class /
96, 7, 8co 6523 . . . . . 6 class (𝑥 / 𝑦)
10 cfl 12404 . . . . . 6 class
119, 10cfv 5786 . . . . 5 class (⌊‘(𝑥 / 𝑦))
12 cmul 9793 . . . . 5 class ·
137, 11, 12co 6523 . . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14 cmin 10113 . . . 4 class
156, 13, 14co 6523 . . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
162, 3, 4, 5, 15cmpt2 6525 . 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
171, 16wceq 1474 1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
Colors of variables: wff setvar class
This definition is referenced by:  modval  12483
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