Definition df-mod 10248

Description: Define the modulo (remainder) operation. See modqval 10249 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10195 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 10247	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 7743	. . 3 class ℝ
5		crp 9580	. . 3 class ℝ+
6	2	cv 1341	. . . 4 class 𝑥
7	3	cv 1341	. . . . 5 class 𝑦
8		cdiv 8559	. . . . . . 7 class /
9	6, 7, 8	co 5836	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10193	. . . . . 6 class ⌊
11	9, 10	cfv 5182	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 7749	. . . . 5 class ·
13	7, 11, 12	co 5836	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8060	. . . 4 class −
15	6, 13, 14	co 5836	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 5838	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1342	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  10249