Definition df-mod 10773

Description: Define the modulo (remainder) operation. See modqval 10774 for its value. For example, ( 5 mod 3 ) = 2 and ( -u 7 mod 2 ) = 1. As with df-fl 10715 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 10772	. 2 class mod
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cr 8178	. . 3 class RR
5		crp 10064	. . 3 class RR+
6	2	cv 1401	. . . 4 class x
7	3	cv 1401	. . . . 5 class y
8		cdiv 9004	. . . . . . 7 class /
9	6, 7, 8	co 6085	. . . . . 6 class ( x / y )
10		cfl 10713	. . . . . 6 class |_
11	9, 10	cfv 5377	. . . . 5 class ( |_ ` ( x / y ) )
12		cmul 8184	. . . . 5 class x.
13	7, 11, 12	co 6085	. . . 4 class ( y x. ( |_ ` ( x / y ) ) )
14		cmin 8498	. . . 4 class -
15	6, 13, 14	co 6085	. . 3 class ( x - ( y x. ( |_ ` ( x / y ) ) ) )
16	2, 3, 4, 5, 15	cmpo 6087	. 2 class ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )
17	1, 16	wceq 1402	1 wff mod = ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )

This definition is used by:  modqval  10774