Definition df-mod 9989

Description: Define the modulo (remainder) operation. See modqval 9990 for its value. For example, ( 5 mod 3 ) = 2 and ( -u 7 mod 2 ) = 1. As with df-fl 9936 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 9988	. 2 class mod
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cr 7546	. . 3 class RR
5		crp 9343	. . 3 class RR+
6	2	cv 1313	. . . 4 class x
7	3	cv 1313	. . . . 5 class y
8		cdiv 8345	. . . . . . 7 class /
9	6, 7, 8	co 5728	. . . . . 6 class ( x / y )
10		cfl 9934	. . . . . 6 class |_
11	9, 10	cfv 5081	. . . . 5 class ( |_ ` ( x / y ) )
12		cmul 7552	. . . . 5 class x.
13	7, 11, 12	co 5728	. . . 4 class ( y x. ( |_ ` ( x / y ) ) )
14		cmin 7856	. . . 4 class -
15	6, 13, 14	co 5728	. . 3 class ( x - ( y x. ( |_ ` ( x / y ) ) ) )
16	2, 3, 4, 5, 15	cmpo 5730	. 2 class ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )
17	1, 16	wceq 1314	1 wff mod = ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )

This definition is referenced by:  modqval  9990