Definition df-mod 10279

Description: Define the modulo (remainder) operation. See modqval 10280 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10226 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 10278	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 7773	. . 3 class ℝ
5		crp 9610	. . 3 class ℝ+
6	2	cv 1347	. . . 4 class 𝑥
7	3	cv 1347	. . . . 5 class 𝑦
8		cdiv 8589	. . . . . . 7 class /
9	6, 7, 8	co 5853	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 10224	. . . . . 6 class ⌊
11	9, 10	cfv 5198	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 7779	. . . . 5 class ·
13	7, 11, 12	co 5853	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 8090	. . . 4 class −
15	6, 13, 14	co 5853	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpo 5855	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1348	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modqval  10280