Definition df-mod 10394

Description: Define the modulo (remainder) operation. See modqval 10395 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10339 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 10393	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 7871	. . 3 class ℝ
5		crp 9719	. . 3 class ℝ+
6	2	cv 1363	. . . 4 class 𝑥
7	3	cv 1363	. . . . 5 class 𝑦
8		cdiv 8691	. . . . . . 7 class /
9	6, 7, 8	co 5918	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10337	. . . . . 6 class ⌊
11	9, 10	cfv 5254	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 7877	. . . . 5 class ·
13	7, 11, 12	co 5918	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8190	. . . 4 class −
15	6, 13, 14	co 5918	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 5920	. 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  10395