Definition df-mod 10586

Description: Define the modulo (remainder) operation. See modqval 10587 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10531 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 10585	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 8031	. . 3 class ℝ
5		crp 9888	. . 3 class ℝ+
6	2	cv 1396	. . . 4 class 𝑥
7	3	cv 1396	. . . . 5 class 𝑦
8		cdiv 8852	. . . . . . 7 class /
9	6, 7, 8	co 6018	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10529	. . . . . 6 class ⌊
11	9, 10	cfv 5326	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 8037	. . . . 5 class ·
13	7, 11, 12	co 6018	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8350	. . . 4 class −
15	6, 13, 14	co 6018	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 6020	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1397	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  10587