Definition df-mod 10512

Description: Define the modulo (remainder) operation. See modqval 10513 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10457 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 10511	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 7966	. . 3 class ℝ
5		crp 9817	. . 3 class ℝ+
6	2	cv 1374	. . . 4 class 𝑥
7	3	cv 1374	. . . . 5 class 𝑦
8		cdiv 8787	. . . . . . 7 class /
9	6, 7, 8	co 5974	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10455	. . . . . 6 class ⌊
11	9, 10	cfv 5294	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 7972	. . . . 5 class ·
13	7, 11, 12	co 5974	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8285	. . . 4 class −
15	6, 13, 14	co 5974	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 5976	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1375	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  10513