Definition df-mod 10545

Description: Define the modulo (remainder) operation. See modqval 10546 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10490 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 10544	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 7998	. . 3 class ℝ
5		crp 9849	. . 3 class ℝ+
6	2	cv 1394	. . . 4 class 𝑥
7	3	cv 1394	. . . . 5 class 𝑦
8		cdiv 8819	. . . . . . 7 class /
9	6, 7, 8	co 6001	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10488	. . . . . 6 class ⌊
11	9, 10	cfv 5318	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 8004	. . . . 5 class ·
13	7, 11, 12	co 6001	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8317	. . . 4 class −
15	6, 13, 14	co 6001	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 6003	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1395	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  10546