ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-mod Unicode version

Definition df-mod 10249
Description: Define the modulo (remainder) operation. See modqval 10250 for its value. For example,  ( 5  mod  3 )  =  2 and  ( -u 7  mod  2 )  =  1. As with df-fl 10196 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
StepHypRef Expression
1 cmo 10248 . 2  class  mod
2 vx . . 3  setvar  x
3 vy . . 3  setvar  y
4 cr 7744 . . 3  class  RR
5 crp 9581 . . 3  class  RR+
62cv 1341 . . . 4  class  x
73cv 1341 . . . . 5  class  y
8 cdiv 8560 . . . . . . 7  class  /
96, 7, 8co 5837 . . . . . 6  class  ( x  /  y )
10 cfl 10194 . . . . . 6  class  |_
119, 10cfv 5183 . . . . 5  class  ( |_
`  ( x  / 
y ) )
12 cmul 7750 . . . . 5  class  x.
137, 11, 12co 5837 . . . 4  class  ( y  x.  ( |_ `  ( x  /  y
) ) )
14 cmin 8061 . . . 4  class  -
156, 13, 14co 5837 . . 3  class  ( x  -  ( y  x.  ( |_ `  (
x  /  y ) ) ) )
162, 3, 4, 5, 15cmpo 5839 . 2  class  ( x  e.  RR ,  y  e.  RR+  |->  ( x  -  ( y  x.  ( |_ `  (
x  /  y ) ) ) ) )
171, 16wceq 1342 1  wff  mod  =  ( x  e.  RR ,  y  e.  RR+  |->  ( x  -  ( y  x.  ( |_ `  (
x  /  y ) ) ) ) )
Colors of variables: wff set class
This definition is referenced by:  modqval  10250
  Copyright terms: Public domain W3C validator