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Theorem qnumval 12808
Description: Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
Assertion
Ref Expression
qnumval  |-  ( A  e.  QQ  ->  (numer `  A )  =  ( 1st `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1  /\  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) ) ) )
Distinct variable group:    x, A

Proof of Theorem qnumval
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2289 . . . . 5  |-  ( a  =  A  ->  (
a  =  ( ( 1st `  x )  /  ( 2nd `  x
) )  <->  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) )
21anbi2d 684 . . . 4  |-  ( a  =  A  ->  (
( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  a  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) )  <->  ( ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1  /\  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) ) )
32riotabidv 6306 . . 3  |-  ( a  =  A  ->  ( iota_ x  e.  ( ZZ 
X.  NN ) ( ( ( 1st `  x
)  gcd  ( 2nd `  x ) )  =  1  /\  a  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) )  =  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1  /\  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) ) )
43fveq2d 5529 . 2  |-  ( a  =  A  ->  ( 1st `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1  /\  a  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) ) )  =  ( 1st `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1  /\  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) ) ) )
5 df-numer 12806 . 2  |- numer  =  ( a  e.  QQ  |->  ( 1st `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1  /\  a  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) ) ) )
6 fvex 5539 . 2  |-  ( 1st `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) ) )  e. 
_V
74, 5, 6fvmpt 5602 1  |-  ( A  e.  QQ  ->  (numer `  A )  =  ( 1st `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1  /\  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    X. cxp 4687   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   iota_crio 6297   1c1 8738    / cdiv 9423   NNcn 9746   ZZcz 10024   QQcq 10316    gcd cgcd 12685  numercnumer 12804
This theorem is referenced by:  qnumdencl  12810  fnum  12813  qnumdenbi  12815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-riota 6304  df-numer 12806
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