Definition df-mod 10397

Description: Define the modulo (remainder) operation. See modqval 10398 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10342 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 10396	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 7873	. . 3 class ℝ
5		crp 9722	. . 3 class ℝ+
6	2	cv 1363	. . . 4 class 𝑥
7	3	cv 1363	. . . . 5 class 𝑦
8		cdiv 8693	. . . . . . 7 class /
9	6, 7, 8	co 5919	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10340	. . . . . 6 class ⌊
11	9, 10	cfv 5255	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 7879	. . . . 5 class ·
13	7, 11, 12	co 5919	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8192	. . . 4 class −
15	6, 13, 14	co 5919	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 5921	. 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  10398