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Theorem qdenval 11770
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 2122 . . . . 5  |-  ( a  =  A  ->  (
a  =  ( ( 1st `  x )  /  ( 2nd `  x
) )  <->  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) )
21anbi2d 457 . . . 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 5698 . . 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 5391 . 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 11768 . 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 9017 . . . 4  |-  ZZ  e.  _V
7 nnex 8686 . . . 4  |-  NN  e.  _V
86, 7xpex 4622 . . 3  |-  ( ZZ 
X.  NN )  e. 
_V
9 riotaexg 5700 . . 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 6032 . . 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 5464 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 103    = wceq 1314    e. wcel 1463   _Vcvv 2658    X. cxp 4505   ` cfv 5091   iota_crio 5695  (class class class)co 5740   1stc1st 6002   2ndc2nd 6003   1c1 7585    / cdiv 8395   NNcn 8680   ZZcz 9008   QQcq 9363    gcd cgcd 11542  denomcdenom 11766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-cnex 7675  ax-resscn 7676  ax-1re 7678  ax-addrcl 7681
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fo 5097  df-fv 5099  df-riota 5696  df-ov 5743  df-2nd 6005  df-neg 7900  df-inn 8681  df-z 9009  df-denom 11768
This theorem is referenced by:  qnumdencl  11771  fden  11775  qnumdenbi  11776
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