Definition df-mod 10631

Description: Define the modulo (remainder) operation. See modqval 10632 for its value. For example, ( 5 mod 3 ) = 2 and ( -u 7 mod 2 ) = 1. As with df-fl 10576 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 e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )

Distinct variable group:   x, y

Detailed syntax breakdown of Definition df-mod

Step	Hyp	Ref	Expression
1		cmo 10630	. 2 class mod
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cr 8074	. . 3 class RR
5		crp 9932	. . 3 class RR+
6	2	cv 1397	. . . 4 class x
7	3	cv 1397	. . . . 5 class y
8		cdiv 8894	. . . . . . 7 class /
9	6, 7, 8	co 6028	. . . . . 6 class ( x / y )
10		cfl 10574	. . . . . 6 class |_
11	9, 10	cfv 5333	. . . . 5 class ( |_ ` ( x / y ) )
12		cmul 8080	. . . . 5 class x.
13	7, 11, 12	co 6028	. . . 4 class ( y x. ( |_ ` ( x / y ) ) )
14		cmin 8392	. . . 4 class -
15	6, 13, 14	co 6028	. . 3 class ( x - ( y x. ( |_ ` ( x / y ) ) ) )
16	2, 3, 4, 5, 15	cmpo 6030	. 2 class ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )
17	1, 16	wceq 1398	1 wff mod = ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )

This definition is referenced by:  modqval  10632