Definition df-mod 10591

Description: Define the modulo (remainder) operation. See modqval 10592 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10536 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 10590	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 8036	. . 3 class ℝ
5		crp 9893	. . 3 class ℝ+
6	2	cv 1396	. . . 4 class 𝑥
7	3	cv 1396	. . . . 5 class 𝑦
8		cdiv 8857	. . . . . . 7 class /
9	6, 7, 8	co 6023	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10534	. . . . . 6 class ⌊
11	9, 10	cfv 5328	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 8042	. . . . 5 class ·
13	7, 11, 12	co 6023	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8355	. . . 4 class −
15	6, 13, 14	co 6023	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 6025	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1397	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  10592