Definition df-mod 10532

Description: Define the modulo (remainder) operation. See modqval 10533 for its value. For example, ( 5 mod 3 ) = 2 and ( -u 7 mod 2 ) = 1. As with df-fl 10477 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 10531	. 2 class mod
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cr 7986	. . 3 class RR
5		crp 9837	. . 3 class RR+
6	2	cv 1394	. . . 4 class x
7	3	cv 1394	. . . . 5 class y
8		cdiv 8807	. . . . . . 7 class /
9	6, 7, 8	co 5994	. . . . . 6 class ( x / y )
10		cfl 10475	. . . . . 6 class |_
11	9, 10	cfv 5314	. . . . 5 class ( |_ ` ( x / y ) )
12		cmul 7992	. . . . 5 class x.
13	7, 11, 12	co 5994	. . . 4 class ( y x. ( |_ ` ( x / y ) ) )
14		cmin 8305	. . . 4 class -
15	6, 13, 14	co 5994	. . 3 class ( x - ( y x. ( |_ ` ( x / y ) ) ) )
16	2, 3, 4, 5, 15	cmpo 5996	. 2 class ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )
17	1, 16	wceq 1395	1 wff mod = ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )

This definition is referenced by:  modqval  10533