Definition df-mod 10254

Description: Define the modulo (remainder) operation. See modqval 10255 for its value. For example, ( 5 mod 3 ) = 2 and ( -u 7 mod 2 ) = 1. As with df-fl 10201 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 10253	. 2 class mod
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cr 7748	. . 3 class RR
5		crp 9585	. . 3 class RR+
6	2	cv 1342	. . . 4 class x
7	3	cv 1342	. . . . 5 class y
8		cdiv 8564	. . . . . . 7 class /
9	6, 7, 8	co 5841	. . . . . 6 class ( x / y )
10		cfl 10199	. . . . . 6 class |_
11	9, 10	cfv 5187	. . . . 5 class ( |_ ` ( x / y ) )
12		cmul 7754	. . . . 5 class x.
13	7, 11, 12	co 5841	. . . 4 class ( y x. ( |_ ` ( x / y ) ) )
14		cmin 8065	. . . 4 class -
15	6, 13, 14	co 5841	. . 3 class ( x - ( y x. ( |_ ` ( x / y ) ) ) )
16	2, 3, 4, 5, 15	cmpo 5843	. 2 class ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )
17	1, 16	wceq 1343	1 wff mod = ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )

This definition is referenced by:  modqval  10255