Definition df-mod 10578

Description: Define the modulo (remainder) operation. See modqval 10579 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10523 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 10577	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 8024	. . 3 class ℝ
5		crp 9881	. . 3 class ℝ+
6	2	cv 1394	. . . 4 class 𝑥
7	3	cv 1394	. . . . 5 class 𝑦
8		cdiv 8845	. . . . . . 7 class /
9	6, 7, 8	co 6013	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10521	. . . . . 6 class ⌊
11	9, 10	cfv 5324	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 8030	. . . . 5 class ·
13	7, 11, 12	co 6013	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8343	. . . 4 class −
15	6, 13, 14	co 6013	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 6015	. 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  10579