Definition df-mod 10685

Description: Define the modulo (remainder) operation. See modqval 10686 for its value. For example, ( 5 mod 3 ) = 2 and ( -u 7 mod 2 ) = 1. As with df-fl 10630 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 10684	. 2 class mod
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cr 8126	. . 3 class RR
5		crp 9986	. . 3 class RR+
6	2	cv 1397	. . . 4 class x
7	3	cv 1397	. . . . 5 class y
8		cdiv 8946	. . . . . . 7 class /
9	6, 7, 8	co 6050	. . . . . 6 class ( x / y )
10		cfl 10628	. . . . . 6 class |_
11	9, 10	cfv 5352	. . . . 5 class ( |_ ` ( x / y ) )
12		cmul 8132	. . . . 5 class x.
13	7, 11, 12	co 6050	. . . 4 class ( y x. ( |_ ` ( x / y ) ) )
14		cmin 8444	. . . 4 class -
15	6, 13, 14	co 6050	. . 3 class ( x - ( y x. ( |_ ` ( x / y ) ) ) )
16	2, 3, 4, 5, 15	cmpo 6052	. 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  10686