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

Definition df-numer 13930
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 13928 . 2  class numer
2 vy . . 3  setvar  y
3 cq 11063 . . 3  class  QQ
4 vx . . . . . . . . . 10  setvar  x
54cv 1369 . . . . . . . . 9  class  x
6 c1st 6684 . . . . . . . . 9  class  1st
75, 6cfv 5525 . . . . . . . 8  class  ( 1st `  x )
8 c2nd 6685 . . . . . . . . 9  class  2nd
95, 8cfv 5525 . . . . . . . 8  class  ( 2nd `  x )
10 cgcd 13807 . . . . . . . 8  class  gcd
117, 9, 10co 6199 . . . . . . 7  class  ( ( 1st `  x )  gcd  ( 2nd `  x
) )
12 c1 9393 . . . . . . 7  class  1
1311, 12wceq 1370 . . . . . 6  wff  ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1
142cv 1369 . . . . . . 7  class  y
15 cdiv 10103 . . . . . . . 8  class  /
167, 9, 15co 6199 . . . . . . 7  class  ( ( 1st `  x )  /  ( 2nd `  x
) )
1714, 16wceq 1370 . . . . . 6  wff  y  =  ( ( 1st `  x
)  /  ( 2nd `  x ) )
1813, 17wa 369 . . . . 5  wff  ( ( ( 1st `  x
)  gcd  ( 2nd `  x ) )  =  1  /\  y  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) )
19 cz 10756 . . . . . 6  class  ZZ
20 cn 10432 . . . . . 6  class  NN
2119, 20cxp 4945 . . . . 5  class  ( ZZ 
X.  NN )
2218, 4, 21crio 6159 . . . 4  class  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1  /\  y  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) )
2322, 6cfv 5525 . . 3  class  ( 1st `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  y  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) ) )
242, 3, 23cmpt 4457 . 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 1370 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 setvar class
This definition is referenced by:  qnumval  13932  fnum  13937
  Copyright terms: Public domain W3C validator