Definition df-mod 10325

Description: Define the modulo (remainder) operation. See modqval 10326 for its value. For example, ( 5 mod 3 ) = 2 and ( -u 7 mod 2 ) = 1. As with df-fl 10272 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 10324	. 2 class mod
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cr 7812	. . 3 class RR
5		crp 9655	. . 3 class RR+
6	2	cv 1352	. . . 4 class x
7	3	cv 1352	. . . . 5 class y
8		cdiv 8631	. . . . . . 7 class /
9	6, 7, 8	co 5877	. . . . . 6 class ( x / y )
10		cfl 10270	. . . . . 6 class |_
11	9, 10	cfv 5218	. . . . 5 class ( |_ ` ( x / y ) )
12		cmul 7818	. . . . 5 class x.
13	7, 11, 12	co 5877	. . . 4 class ( y x. ( |_ ` ( x / y ) ) )
14		cmin 8130	. . . 4 class -
15	6, 13, 14	co 5877	. . 3 class ( x - ( y x. ( |_ ` ( x / y ) ) ) )
16	2, 3, 4, 5, 15	cmpo 5879	. 2 class ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )
17	1, 16	wceq 1353	1 wff mod = ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )

This definition is referenced by:  modqval  10326