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Definition df-mod 9658
Description: Define the modulo (remainder) operation. See modqval 9659 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 9605 we define this for first and second arguments which are real and positive real, respectively, even though many theorems will need to be more restricted (for example, specify rational arguments). (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 9657 . 2 class mod
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cr 7293 . . 3 class
5 crp 9066 . . 3 class +
62cv 1286 . . . 4 class 𝑥
73cv 1286 . . . . 5 class 𝑦
8 cdiv 8078 . . . . . . 7 class /
96, 7, 8co 5613 . . . . . 6 class (𝑥 / 𝑦)
10 cfl 9603 . . . . . 6 class
119, 10cfv 4981 . . . . 5 class (⌊‘(𝑥 / 𝑦))
12 cmul 7299 . . . . 5 class ·
137, 11, 12co 5613 . . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14 cmin 7597 . . . 4 class
156, 13, 14co 5613 . . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
162, 3, 4, 5, 15cmpt2 5615 . 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
171, 16wceq 1287 1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
Colors of variables: wff set class
This definition is referenced by:  modqval  9659
  Copyright terms: Public domain W3C validator