Definition df-mod 10738

Description: Define the modulo (remainder) operation. See modqval 10739 for its value. For example, ( 5 mod 3 ) = 2 and ( -u 7 mod 2 ) = 1. As with df-fl 10683 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 10737	. 2 class mod
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cr 8168	. . 3 class RR
5		crp 10033	. . 3 class RR+
6	2	cv 1401	. . . 4 class x
7	3	cv 1401	. . . . 5 class y
8		cdiv 8992	. . . . . . 7 class /
9	6, 7, 8	co 6075	. . . . . 6 class ( x / y )
10		cfl 10681	. . . . . 6 class |_
11	9, 10	cfv 5372	. . . . 5 class ( |_ ` ( x / y ) )
12		cmul 8174	. . . . 5 class x.
13	7, 11, 12	co 6075	. . . 4 class ( y x. ( |_ ` ( x / y ) ) )
14		cmin 8487	. . . 4 class -
15	6, 13, 14	co 6075	. . 3 class ( x - ( y x. ( |_ ` ( x / y ) ) ) )
16	2, 3, 4, 5, 15	cmpo 6077	. 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 referenced by:  modqval  10739