Definition df-mod 10432

Description: Define the modulo (remainder) operation. See modqval 10433 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10377 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 10431	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 7895	. . 3 class ℝ
5		crp 9745	. . 3 class ℝ+
6	2	cv 1363	. . . 4 class 𝑥
7	3	cv 1363	. . . . 5 class 𝑦
8		cdiv 8716	. . . . . . 7 class /
9	6, 7, 8	co 5925	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10375	. . . . . 6 class ⌊
11	9, 10	cfv 5259	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 7901	. . . . 5 class ·
13	7, 11, 12	co 5925	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8214	. . . 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  10433