Definition df-mod 10291

Description: Define the modulo (remainder) operation. See modqval 10292 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10238 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 10290	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 7785	. . 3 class ℝ
5		crp 9622	. . 3 class ℝ+
6	2	cv 1352	. . . 4 class 𝑥
7	3	cv 1352	. . . . 5 class 𝑦
8		cdiv 8601	. . . . . . 7 class /
9	6, 7, 8	co 5865	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10236	. . . . . 6 class ⌊
11	9, 10	cfv 5208	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 7791	. . . . 5 class ·
13	7, 11, 12	co 5865	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8102	. . . 4 class −
15	6, 13, 14	co 5865	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 5867	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1353	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  10292