Definition df-mod 10337

Description: Define the modulo (remainder) operation. See modqval 10338 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10284 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 10336	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 7824	. . 3 class ℝ
5		crp 9667	. . 3 class ℝ+
6	2	cv 1362	. . . 4 class 𝑥
7	3	cv 1362	. . . . 5 class 𝑦
8		cdiv 8643	. . . . . . 7 class /
9	6, 7, 8	co 5888	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10282	. . . . . 6 class ⌊
11	9, 10	cfv 5228	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 7830	. . . . 5 class ·
13	7, 11, 12	co 5888	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8142	. . . 4 class −
15	6, 13, 14	co 5888	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 5890	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1363	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  10338