Definition df-mod 10475

Description: Define the modulo (remainder) operation. See modqval 10476 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10420 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 10474	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 7931	. . 3 class ℝ
5		crp 9782	. . 3 class ℝ+
6	2	cv 1372	. . . 4 class 𝑥
7	3	cv 1372	. . . . 5 class 𝑦
8		cdiv 8752	. . . . . . 7 class /
9	6, 7, 8	co 5951	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10418	. . . . . 6 class ⌊
11	9, 10	cfv 5276	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 7937	. . . . 5 class ·
13	7, 11, 12	co 5951	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8250	. . . 4 class −
15	6, 13, 14	co 5951	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 5953	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1373	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  10476