Definition df-mod 10781

Description: Define the modulo (remainder) operation. See modqval 10782 for its value. For example, (5 mod 3) = 2 and ( -7 mod 2) = 1. As with df-fl 10722 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 10780	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 8185	. . 3 class ℝ
5		crp 10071	. . 3 class ℝ+
6	2	cv 1401	. . . 4 class 𝑥
7	3	cv 1401	. . . . 5 class 𝑦
8		cdiv 9011	. . . . . . 7 class /
9	6, 7, 8	co 6088	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10720	. . . . . 6 class ⌊
11	9, 10	cfv 5380	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 8191	. . . . 5 class ·
13	7, 11, 12	co 6088	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8505	. . . 4 class −
15	6, 13, 14	co 6088	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 6090	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1402	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is used by:  modqval  10782