Definition df-mod 10089

Description: Define the modulo (remainder) operation. See modqval 10090 for its value. For example, ( 5 mod 3 ) = 2 and ( -u 7 mod 2 ) = 1. As with df-fl 10036 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 10088	. 2 class mod
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cr 7612	. . 3 class RR
5		crp 9434	. . 3 class RR+
6	2	cv 1330	. . . 4 class x
7	3	cv 1330	. . . . 5 class y
8		cdiv 8425	. . . . . . 7 class /
9	6, 7, 8	co 5767	. . . . . 6 class ( x / y )
10		cfl 10034	. . . . . 6 class |_
11	9, 10	cfv 5118	. . . . 5 class ( |_ ` ( x / y ) )
12		cmul 7618	. . . . 5 class x.
13	7, 11, 12	co 5767	. . . 4 class ( y x. ( |_ ` ( x / y ) ) )
14		cmin 7926	. . . 4 class -
15	6, 13, 14	co 5767	. . 3 class ( x - ( y x. ( |_ ` ( x / y ) ) ) )
16	2, 3, 4, 5, 15	cmpo 5769	. 2 class ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )
17	1, 16	wceq 1331	1 wff mod = ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )

This definition is referenced by:  modqval  10090