Definition df-mod 10258

Description: Define the modulo (remainder) operation. See modqval 10259 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10205 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 10257	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 7752	. . 3 class ℝ
5		crp 9589	. . 3 class ℝ+
6	2	cv 1342	. . . 4 class 𝑥
7	3	cv 1342	. . . . 5 class 𝑦
8		cdiv 8568	. . . . . . 7 class /
9	6, 7, 8	co 5842	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10203	. . . . . 6 class ⌊
11	9, 10	cfv 5188	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 7758	. . . . 5 class ·
13	7, 11, 12	co 5842	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8069	. . . 4 class −
15	6, 13, 14	co 5842	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 5844	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1343	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  10259