Definition df-mod 10762

Description: Define the modulo (remainder) operation. See modqval 10763 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10707 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 10761	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 8178	. . 3 class ℝ
5		crp 10056	. . 3 class ℝ+
6	2	cv 1401	. . . 4 class 𝑥
7	3	cv 1401	. . . . 5 class 𝑦
8		cdiv 9003	. . . . . . 7 class /
9	6, 7, 8	co 6085	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10705	. . . . . 6 class ⌊
11	9, 10	cfv 5377	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 8184	. . . . 5 class ·
13	7, 11, 12	co 6085	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8497	. . . 4 class −
15	6, 13, 14	co 6085	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 6087	. 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  10763