Definition df-mod 10343

Description: Define the modulo (remainder) operation. See modqval 10344 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10290 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 10342	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 7830	. . 3 class ℝ
5		crp 9673	. . 3 class ℝ+
6	2	cv 1363	. . . 4 class 𝑥
7	3	cv 1363	. . . . 5 class 𝑦
8		cdiv 8649	. . . . . . 7 class /
9	6, 7, 8	co 5892	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10288	. . . . . 6 class ⌊
11	9, 10	cfv 5232	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 7836	. . . . 5 class ·
13	7, 11, 12	co 5892	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8148	. . . 4 class −
15	6, 13, 14	co 5892	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 5894	. 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  10344