Definition df-mod 9658

Description: Define the modulo (remainder) operation. See modqval 9659 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 9605 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 9657	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 7293	. . 3 class ℝ
5		crp 9066	. . 3 class ℝ+
6	2	cv 1286	. . . 4 class 𝑥
7	3	cv 1286	. . . . 5 class 𝑦
8		cdiv 8078	. . . . . . 7 class /
9	6, 7, 8	co 5613	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 9603	. . . . . 6 class ⌊
11	9, 10	cfv 4981	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 7299	. . . . 5 class ·
13	7, 11, 12	co 5613	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 7597	. . . 4 class −
15	6, 13, 14	co 5613	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpt2 5615	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1287	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  9659