Definition df-mod 10434

Description: Define the modulo (remainder) operation. See modqval 10435 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10379 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 10433	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 7897	. . 3 class ℝ
5		crp 9747	. . 3 class ℝ+
6	2	cv 1363	. . . 4 class 𝑥
7	3	cv 1363	. . . . 5 class 𝑦
8		cdiv 8718	. . . . . . 7 class /
9	6, 7, 8	co 5925	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10377	. . . . . 6 class ⌊
11	9, 10	cfv 5259	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 7903	. . . . 5 class ·
13	7, 11, 12	co 5925	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8216	. . . 4 class −
15	6, 13, 14	co 5925	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 5927	. 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  10435