Definition df-mod 10743

Description: Define the modulo (remainder) operation. See modqval 10744 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10688 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 10742	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 8172	. . 3 class ℝ
5		crp 10037	. . 3 class ℝ+
6	2	cv 1401	. . . 4 class 𝑥
7	3	cv 1401	. . . . 5 class 𝑦
8		cdiv 8996	. . . . . . 7 class /
9	6, 7, 8	co 6079	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10686	. . . . . 6 class ⌊
11	9, 10	cfv 5375	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 8178	. . . . 5 class ·
13	7, 11, 12	co 6079	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8491	. . . 4 class −
15	6, 13, 14	co 6079	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 6081	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1402	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  10744