Definition df-mod 10481

Description: Define the modulo (remainder) operation. See modqval 10482 for its value. For example, ( 5 mod 3 ) = 2 and ( -u 7 mod 2 ) = 1. As with df-fl 10426 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 10480	. 2 class mod
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cr 7937	. . 3 class RR
5		crp 9788	. . 3 class RR+
6	2	cv 1372	. . . 4 class x
7	3	cv 1372	. . . . 5 class y
8		cdiv 8758	. . . . . . 7 class /
9	6, 7, 8	co 5954	. . . . . 6 class ( x / y )
10		cfl 10424	. . . . . 6 class |_
11	9, 10	cfv 5277	. . . . 5 class ( |_ ` ( x / y ) )
12		cmul 7943	. . . . 5 class x.
13	7, 11, 12	co 5954	. . . 4 class ( y x. ( |_ ` ( x / y ) ) )
14		cmin 8256	. . . 4 class -
15	6, 13, 14	co 5954	. . 3 class ( x - ( y x. ( |_ ` ( x / y ) ) ) )
16	2, 3, 4, 5, 15	cmpo 5956	. 2 class ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )
17	1, 16	wceq 1373	1 wff mod = ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )

This definition is referenced by:  modqval  10482