Definition df-mod 10553

Description: Define the modulo (remainder) operation. See modqval 10554 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10498 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 10552	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 8006	. . 3 class ℝ
5		crp 9857	. . 3 class ℝ+
6	2	cv 1394	. . . 4 class 𝑥
7	3	cv 1394	. . . . 5 class 𝑦
8		cdiv 8827	. . . . . . 7 class /
9	6, 7, 8	co 6007	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10496	. . . . . 6 class ⌊
11	9, 10	cfv 5318	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 8012	. . . . 5 class ·
13	7, 11, 12	co 6007	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8325	. . . 4 class −
15	6, 13, 14	co 6007	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 6009	. 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  10554