Definition df-mod 12531

Description: Define the modulo (remainder) operation. See modval 12532 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1 (ex-mod 26698). (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 12530	. 2 class mod
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 9814	. . 3 class ℝ
5		crp 11708	. . 3 class ℝ+
6	2	cv 1474	. . . 4 class 𝑥
7	3	cv 1474	. . . . 5 class 𝑦
8		cdiv 10563	. . . . . . 7 class /
9	6, 7, 8	co 6549	. . . . . 6 class (𝑥 / 𝑦)
10		cfl 12453	. . . . . 6 class ⌊
11	9, 10	cfv 5804	. . . . 5 class (⌊‘(𝑥 / 𝑦))
12		cmul 9820	. . . . 5 class ·
13	7, 11, 12	co 6549	. . . 4 class (𝑦 · (⌊‘(𝑥 / 𝑦)))
14		cmin 10145	. . . 4 class −
15	6, 13, 14	co 6549	. . 3 class (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))
16	2, 3, 4, 5, 15	cmpt2 6551	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
17	1, 16	wceq 1475	1 wff mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

This definition is referenced by:  modval  12532