Definition df-mod 10774

Description: Define the modulo (remainder) operation. See modqval 10775 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10716 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 10773	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 8179	. . 3 class ℝ
5		crp 10065	. . 3 class ℝ+
6	2	cv 1401	. . . 4 class 𝑥
7	3	cv 1401	. . . . 5 class 𝑦
8		cdiv 9005	. . . . . . 7 class /
9	6, 7, 8	co 6085	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10714	. . . . . 6 class ⌊
11	9, 10	cfv 5377	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 8185	. . . . 5 class ·
13	7, 11, 12	co 6085	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8499	. . . 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  10775