Definition df-mod 10340

Description: Define the modulo (remainder) operation. See modqval 10341 for its value. For example, ( 5 mod 3 ) = 2 and ( -u 7 mod 2 ) = 1. As with df-fl 10287 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 10339	. 2 class mod
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cr 7827	. . 3 class RR
5		crp 9670	. . 3 class RR+
6	2	cv 1362	. . . 4 class x
7	3	cv 1362	. . . . 5 class y
8		cdiv 8646	. . . . . . 7 class /
9	6, 7, 8	co 5890	. . . . . 6 class ( x / y )
10		cfl 10285	. . . . . 6 class |_
11	9, 10	cfv 5230	. . . . 5 class ( |_ ` ( x / y ) )
12		cmul 7833	. . . . 5 class x.
13	7, 11, 12	co 5890	. . . 4 class ( y x. ( |_ ` ( x / y ) ) )
14		cmin 8145	. . . 4 class -
15	6, 13, 14	co 5890	. . 3 class ( x - ( y x. ( |_ ` ( x / y ) ) ) )
16	2, 3, 4, 5, 15	cmpo 5892	. 2 class ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )
17	1, 16	wceq 1363	1 wff mod = ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )

This definition is referenced by:  modqval  10341