Definition df-mod 12669

Description: Define the modulo (remainder) operation. See modval 12670 for its value. For example, ( 5 mod 3 ) = 2 and ( -u 7 mod 2 ) = 1 (ex-mod 27306). (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 12668	. 2 class mod
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cr 9935	. . 3 class RR
5		crp 11832	. . 3 class RR+
6	2	cv 1482	. . . 4 class x
7	3	cv 1482	. . . . 5 class y
8		cdiv 10684	. . . . . . 7 class /
9	6, 7, 8	co 6650	. . . . . 6 class ( x / y )
10		cfl 12591	. . . . . 6 class |_
11	9, 10	cfv 5888	. . . . 5 class ( |_ ` ( x / y ) )
12		cmul 9941	. . . . 5 class x.
13	7, 11, 12	co 6650	. . . 4 class ( y x. ( |_ ` ( x / y ) ) )
14		cmin 10266	. . . 4 class -
15	6, 13, 14	co 6650	. . 3 class ( x - ( y x. ( |_ ` ( x / y ) ) ) )
16	2, 3, 4, 5, 15	cmpt2 6652	. 2 class ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )
17	1, 16	wceq 1483	1 wff mod = ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )

This definition is referenced by:  modval  12670