Definition df-mod 9325

Description: Define the modulo (remainder) operation. See modqval 9326 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 9274 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 9324	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 6980	. . 3 class ℝ
5		crp 8734	. . 3 class ℝ+
6	2	cv 1283	. . . 4 class 𝑥
7	3	cv 1283	. . . . 5 class 𝑦
8		cdiv 7760	. . . . . . 7 class /
9	6, 7, 8	co 5532	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 9272	. . . . . 6 class ⌊
11	9, 10	cfv 4922	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 6986	. . . . 5 class ·
13	7, 11, 12	co 5532	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 7279	. . . 4 class −
15	6, 13, 14	co 5532	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpt2 5534	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1284	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  9326