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Definition df-mod 13228
Description: Define the modulo (remainder) operation. See modval 13229 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1 (ex-mod 28156). (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 13227 . 2 class mod
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cr 10525 . . 3 class
5 crp 12379 . . 3 class +
62cv 1527 . . . 4 class 𝑥
73cv 1527 . . . . 5 class 𝑦
8 cdiv 11286 . . . . . . 7 class /
96, 7, 8co 7145 . . . . . 6 class (𝑥 / 𝑦)
10 cfl 13150 . . . . . 6 class
119, 10cfv 6349 . . . . 5 class (⌊‘(𝑥 / 𝑦))
12 cmul 10531 . . . . 5 class ·
137, 11, 12co 7145 . . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14 cmin 10859 . . . 4 class
156, 13, 14co 7145 . . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
162, 3, 4, 5, 15cmpo 7147 . 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
171, 16wceq 1528 1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
Colors of variables: wff setvar class
This definition is referenced by:  modval  13229
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