Definition df-mod 10354

Description: Define the modulo (remainder) operation. See modqval 10355 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10301 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 10353	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 7840	. . 3 class ℝ
5		crp 9683	. . 3 class ℝ+
6	2	cv 1363	. . . 4 class 𝑥
7	3	cv 1363	. . . . 5 class 𝑦
8		cdiv 8659	. . . . . . 7 class /
9	6, 7, 8	co 5896	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10299	. . . . . 6 class ⌊
11	9, 10	cfv 5235	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 7846	. . . . 5 class ·
13	7, 11, 12	co 5896	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8158	. . . 4 class −
15	6, 13, 14	co 5896	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 5898	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1364	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  10355