Definition df-mod 9165

Description: Define the modulo (remainder) operation. See modqval 9166 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 9114 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 9164	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 6888	. . 3 class ℝ
5		crp 8583	. . 3 class ℝ+
6	2	cv 1242	. . . 4 class 𝑥
7	3	cv 1242	. . . . 5 class 𝑦
8		cdiv 7651	. . . . . . 7 class /
9	6, 7, 8	co 5512	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 9112	. . . . . 6 class ⌊
11	9, 10	cfv 4902	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 6894	. . . . 5 class ·
13	7, 11, 12	co 5512	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 7182	. . . 4 class −
15	6, 13, 14	co 5512	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpt2 5514	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1243	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  9166