Definition df-mod 10681

Description: Define the modulo (remainder) operation. See modqval 10682 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10626 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 10680	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 8122	. . 3 class ℝ
5		crp 9982	. . 3 class ℝ+
6	2	cv 1397	. . . 4 class 𝑥
7	3	cv 1397	. . . . 5 class 𝑦
8		cdiv 8942	. . . . . . 7 class /
9	6, 7, 8	co 6049	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10624	. . . . . 6 class ⌊
11	9, 10	cfv 5351	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 8128	. . . . 5 class ·
13	7, 11, 12	co 6049	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8440	. . . 4 class −
15	6, 13, 14	co 6049	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 6051	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1398	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  10682