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

Definition df-denom 11789
Description: The canonical denominator of a rational is the denominator of the rational's reduced fraction representation (no common factors, denominator positive). (Contributed by Stefan O'Rear, 13-Sep-2014.)
Assertion
Ref Expression
df-denom  |- denom  =  ( y  e.  QQ  |->  ( 2nd `  ( 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-denom
StepHypRef Expression
1 cdenom 11787 . 2  class denom
2 vy . . 3  setvar  y
3 cq 9379 . . 3  class  QQ
4 vx . . . . . . . . . 10  setvar  x
54cv 1315 . . . . . . . . 9  class  x
6 c1st 6004 . . . . . . . . 9  class  1st
75, 6cfv 5093 . . . . . . . 8  class  ( 1st `  x )
8 c2nd 6005 . . . . . . . . 9  class  2nd
95, 8cfv 5093 . . . . . . . 8  class  ( 2nd `  x )
10 cgcd 11562 . . . . . . . 8  class  gcd
117, 9, 10co 5742 . . . . . . 7  class  ( ( 1st `  x )  gcd  ( 2nd `  x
) )
12 c1 7589 . . . . . . 7  class  1
1311, 12wceq 1316 . . . . . 6  wff  ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1
142cv 1315 . . . . . . 7  class  y
15 cdiv 8400 . . . . . . . 8  class  /
167, 9, 15co 5742 . . . . . . 7  class  ( ( 1st `  x )  /  ( 2nd `  x
) )
1714, 16wceq 1316 . . . . . 6  wff  y  =  ( ( 1st `  x
)  /  ( 2nd `  x ) )
1813, 17wa 103 . . . . 5  wff  ( ( ( 1st `  x
)  gcd  ( 2nd `  x ) )  =  1  /\  y  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) )
19 cz 9022 . . . . . 6  class  ZZ
20 cn 8688 . . . . . 6  class  NN
2119, 20cxp 4507 . . . . 5  class  ( ZZ 
X.  NN )
2218, 4, 21crio 5697 . . . 4  class  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1  /\  y  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) )
2322, 8cfv 5093 . . 3  class  ( 2nd `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  y  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) ) )
242, 3, 23cmpt 3959 . 2  class  ( y  e.  QQ  |->  ( 2nd `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  y  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) ) ) )
251, 24wceq 1316 1  wff denom  =  ( y  e.  QQ  |->  ( 2nd `  ( 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:  qdenval  11791  fden  11796
  Copyright terms: Public domain W3C validator