Definition df-mod 10692

Description: Define the modulo (remainder) operation. See modqval 10693 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10637 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 10691	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 8131	. . 3 class ℝ
5		crp 9992	. . 3 class ℝ+
6	2	cv 1397	. . . 4 class 𝑥
7	3	cv 1397	. . . . 5 class 𝑦
8		cdiv 8951	. . . . . . 7 class /
9	6, 7, 8	co 6052	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10635	. . . . . 6 class ⌊
11	9, 10	cfv 5354	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 8137	. . . . 5 class ·
13	7, 11, 12	co 6052	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8449	. . . 4 class −
15	6, 13, 14	co 6052	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 6054	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1398	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  10693