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Theorem qdenval 10944
Description: Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
Assertion
Ref Expression
qdenval  |-  ( A  e.  QQ  ->  (denom `  A )  =  ( 2nd `  ( 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 qdenval
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2089 . . . . 5  |-  ( a  =  A  ->  (
a  =  ( ( 1st `  x )  /  ( 2nd `  x
) )  <->  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) )
21anbi2d 452 . . . 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 5549 . . 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 5257 . 2  |-  ( a  =  A  ->  ( 2nd `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1  /\  a  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) ) )  =  ( 2nd `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1  /\  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) ) ) )
5 df-denom 10942 . 2  |- denom  =  ( a  e.  QQ  |->  ( 2nd `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1  /\  a  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) ) ) )
6 zex 8655 . . . 4  |-  ZZ  e.  _V
7 nnex 8322 . . . 4  |-  NN  e.  _V
86, 7xpex 4511 . . 3  |-  ( ZZ 
X.  NN )  e. 
_V
9 riotaexg 5551 . . 3  |-  ( ( ZZ  X.  NN )  e.  _V  ->  ( iota_ x  e.  ( ZZ 
X.  NN ) ( ( ( 1st `  x
)  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) )  e.  _V )
10 2ndexg 5874 . . 3  |-  ( (
iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x
)  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) )  e.  _V  ->  ( 2nd `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1  /\  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) ) )  e.  _V )
118, 9, 10mp2b 8 . 2  |-  ( 2nd `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) ) )  e. 
_V
124, 5, 11fvmpt 5326 1  |-  ( A  e.  QQ  ->  (denom `  A )  =  ( 2nd `  ( 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 102    = wceq 1285    e. wcel 1434   _Vcvv 2612    X. cxp 4399   ` cfv 4969   iota_crio 5546  (class class class)co 5591   1stc1st 5844   2ndc2nd 5845   1c1 7254    / cdiv 8037   NNcn 8316   ZZcz 8646   QQcq 8999    gcd cgcd 10718  denomcdenom 10940
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-cnex 7339  ax-resscn 7340  ax-1re 7342  ax-addrcl 7345
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-fo 4975  df-fv 4977  df-riota 5547  df-ov 5594  df-2nd 5847  df-neg 7559  df-inn 8317  df-z 8647  df-denom 10942
This theorem is referenced by:  qnumdencl  10945  fden  10949  qnumdenbi  10950
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