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Theorem qnumval 14272
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 2386 . . . . 5  |-  ( a  =  A  ->  (
a  =  ( ( 1st `  x )  /  ( 2nd `  x
) )  <->  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) )
21anbi2d 701 . . . 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 6160 . . 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 5778 . 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 14270 . 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 5784 . 2  |-  ( 1st `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) ) )  e. 
_V
74, 5, 6fvmpt 5857 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826    X. cxp 4911   ` cfv 5496   iota_crio 6157  (class class class)co 6196   1stc1st 6697   2ndc2nd 6698   1c1 9404    / cdiv 10123   NNcn 10452   ZZcz 10781   QQcq 11101    gcd cgcd 14146  numercnumer 14268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-riota 6158  df-numer 14270
This theorem is referenced by:  qnumdencl  14274  fnum  14277  qnumdenbi  14279
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