Definition df-mod 10760

Description: Define the modulo (remainder) operation. See modqval 10761 for its value. For example, ( 5 mod 3 ) = 2 and ( -u 7 mod 2 ) = 1. As with df-fl 10705 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 10759	. 2 class mod
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cr 8178	. . 3 class RR
5		crp 10054	. . 3 class RR+
6	2	cv 1401	. . . 4 class x
7	3	cv 1401	. . . . 5 class y
8		cdiv 9002	. . . . . . 7 class /
9	6, 7, 8	co 6085	. . . . . 6 class ( x / y )
10		cfl 10703	. . . . . 6 class |_
11	9, 10	cfv 5377	. . . . 5 class ( |_ ` ( x / y ) )
12		cmul 8184	. . . . 5 class x.
13	7, 11, 12	co 6085	. . . 4 class ( y x. ( |_ ` ( x / y ) ) )
14		cmin 8497	. . . 4 class -
15	6, 13, 14	co 6085	. . 3 class ( x - ( y x. ( |_ ` ( x / y ) ) ) )
16	2, 3, 4, 5, 15	cmpo 6087	. 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 used by:  modqval  10761