Definition df-mod 10322

Description: Define the modulo (remainder) operation. See modqval 10323 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10269 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 10321	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 7809	. . 3 class ℝ
5		crp 9652	. . 3 class ℝ+
6	2	cv 1352	. . . 4 class 𝑥
7	3	cv 1352	. . . . 5 class 𝑦
8		cdiv 8628	. . . . . . 7 class /
9	6, 7, 8	co 5874	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10267	. . . . . 6 class ⌊
11	9, 10	cfv 5216	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 7815	. . . . 5 class ·
13	7, 11, 12	co 5874	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8127	. . . 4 class −
15	6, 13, 14	co 5874	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 5876	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1353	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  10323