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Definition df-denom 13935
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 13933 . 2  class denom
2 vy . . 3  setvar  y
3 cq 11067 . . 3  class  QQ
4 vx . . . . . . . . . 10  setvar  x
54cv 1369 . . . . . . . . 9  class  x
6 c1st 6688 . . . . . . . . 9  class  1st
75, 6cfv 5529 . . . . . . . 8  class  ( 1st `  x )
8 c2nd 6689 . . . . . . . . 9  class  2nd
95, 8cfv 5529 . . . . . . . 8  class  ( 2nd `  x )
10 cgcd 13811 . . . . . . . 8  class  gcd
117, 9, 10co 6203 . . . . . . 7  class  ( ( 1st `  x )  gcd  ( 2nd `  x
) )
12 c1 9397 . . . . . . 7  class  1
1311, 12wceq 1370 . . . . . 6  wff  ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1
142cv 1369 . . . . . . 7  class  y
15 cdiv 10107 . . . . . . . 8  class  /
167, 9, 15co 6203 . . . . . . 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 10760 . . . . . 6  class  ZZ
20 cn 10436 . . . . . 6  class  NN
2119, 20cxp 4949 . . . . 5  class  ( ZZ 
X.  NN )
2218, 4, 21crio 6163 . . . 4  class  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1  /\  y  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) )
2322, 8cfv 5529 . . 3  class  ( 2nd `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  y  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) ) )
242, 3, 23cmpt 4461 . 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 1370 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 setvar class
This definition is referenced by:  qdenval  13937  fden  13942
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