MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-numer Unicode version

Definition df-numer 12822
Description: The canonical numerator of a rational is the numerator of the rational's reduced fraction representation (no common factors, denominator positive). (Contributed by Stefan O'Rear, 13-Sep-2014.)
Assertion
Ref Expression
df-numer  |- numer  =  ( y  e.  QQ  |->  ( 1st `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1  /\  y  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) ) ) )
Distinct variable group:    x, y

Detailed syntax breakdown of Definition df-numer
StepHypRef Expression
1 cnumer 12820 . 2  class numer
2 vy . . 3  set  y
3 cq 10332 . . 3  class  QQ
4 vx . . . . . . . . . 10  set  x
54cv 1631 . . . . . . . . 9  class  x
6 c1st 6136 . . . . . . . . 9  class  1st
75, 6cfv 5271 . . . . . . . 8  class  ( 1st `  x )
8 c2nd 6137 . . . . . . . . 9  class  2nd
95, 8cfv 5271 . . . . . . . 8  class  ( 2nd `  x )
10 cgcd 12701 . . . . . . . 8  class  gcd
117, 9, 10co 5874 . . . . . . 7  class  ( ( 1st `  x )  gcd  ( 2nd `  x
) )
12 c1 8754 . . . . . . 7  class  1
1311, 12wceq 1632 . . . . . 6  wff  ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1
142cv 1631 . . . . . . 7  class  y
15 cdiv 9439 . . . . . . . 8  class  /
167, 9, 15co 5874 . . . . . . 7  class  ( ( 1st `  x )  /  ( 2nd `  x
) )
1714, 16wceq 1632 . . . . . 6  wff  y  =  ( ( 1st `  x
)  /  ( 2nd `  x ) )
1813, 17wa 358 . . . . 5  wff  ( ( ( 1st `  x
)  gcd  ( 2nd `  x ) )  =  1  /\  y  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) )
19 cz 10040 . . . . . 6  class  ZZ
20 cn 9762 . . . . . 6  class  NN
2119, 20cxp 4703 . . . . 5  class  ( ZZ 
X.  NN )
2218, 4, 21crio 6313 . . . 4  class  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1  /\  y  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) )
2322, 6cfv 5271 . . 3  class  ( 1st `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  y  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) ) )
242, 3, 23cmpt 4093 . 2  class  ( y  e.  QQ  |->  ( 1st `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  y  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) ) ) )
251, 24wceq 1632 1  wff numer  =  ( y  e.  QQ  |->  ( 1st `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1  /\  y  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) ) ) )
Colors of variables: wff set class
This definition is referenced by:  qnumval  12824  fnum  12829
  Copyright terms: Public domain W3C validator