Definition df-mod 10096

Description: Define the modulo (remainder) operation. See modqval 10097 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10043 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 10095	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 7619	. . 3 class ℝ
5		crp 9441	. . 3 class ℝ+
6	2	cv 1330	. . . 4 class 𝑥
7	3	cv 1330	. . . . 5 class 𝑦
8		cdiv 8432	. . . . . . 7 class /
9	6, 7, 8	co 5774	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10041	. . . . . 6 class ⌊
11	9, 10	cfv 5123	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 7625	. . . . 5 class ·
13	7, 11, 12	co 5774	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 7933	. . . 4 class −
15	6, 13, 14	co 5774	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 5776	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1331	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  10097