Definition df-mod 10415

Description: Define the modulo (remainder) operation. See modqval 10416 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10360 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

Step	Hyp	Ref	Expression
1		cmo 10414	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 7878	. . 3 class ℝ
5		crp 9728	. . 3 class ℝ+
6	2	cv 1363	. . . 4 class 𝑥
7	3	cv 1363	. . . . 5 class 𝑦
8		cdiv 8699	. . . . . . 7 class /
9	6, 7, 8	co 5922	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10358	. . . . . 6 class ⌊
11	9, 10	cfv 5258	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 7884	. . . . 5 class ·
13	7, 11, 12	co 5922	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8197	. . . 4 class −
15	6, 13, 14	co 5922	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 5924	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1364	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  10416