Definition df-mod 10323

Description: Define the modulo (remainder) operation. See modqval 10324 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10270 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 10322	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 7810	. . 3 class ℝ
5		crp 9653	. . 3 class ℝ+
6	2	cv 1352	. . . 4 class 𝑥
7	3	cv 1352	. . . . 5 class 𝑦
8		cdiv 8629	. . . . . . 7 class /
9	6, 7, 8	co 5875	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10268	. . . . . 6 class ⌊
11	9, 10	cfv 5217	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 7816	. . . . 5 class ·
13	7, 11, 12	co 5875	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8128	. . . 4 class −
15	6, 13, 14	co 5875	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 5877	. 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  10324