Definition df-mod 10712

Description: Define the modulo (remainder) operation. See modqval 10713 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10657 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 10711	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 8142	. . 3 class ℝ
5		crp 10007	. . 3 class ℝ+
6	2	cv 1397	. . . 4 class 𝑥
7	3	cv 1397	. . . . 5 class 𝑦
8		cdiv 8966	. . . . . . 7 class /
9	6, 7, 8	co 6058	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10655	. . . . . 6 class ⌊
11	9, 10	cfv 5357	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 8148	. . . . 5 class ·
13	7, 11, 12	co 6058	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8461	. . . 4 class −
15	6, 13, 14	co 6058	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 6060	. 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  10713