Definition df-mod 10562

Description: Define the modulo (remainder) operation. See modqval 10563 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10507 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 10561	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 8014	. . 3 class ℝ
5		crp 9866	. . 3 class ℝ+
6	2	cv 1394	. . . 4 class 𝑥
7	3	cv 1394	. . . . 5 class 𝑦
8		cdiv 8835	. . . . . . 7 class /
9	6, 7, 8	co 6010	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10505	. . . . . 6 class ⌊
11	9, 10	cfv 5321	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 8020	. . . . 5 class ·
13	7, 11, 12	co 6010	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8333	. . . 4 class −
15	6, 13, 14	co 6010	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 6012	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1395	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  10563