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Theorem qdenval 13927
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 2455 . . . . 5  |-  ( a  =  A  ->  (
a  =  ( ( 1st `  x )  /  ( 2nd `  x
) )  <->  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) )
21anbi2d 703 . . . 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 6156 . . 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 5796 . 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 13925 . 2  |- denom  =  ( a  e.  QQ  |->  ( 2nd `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1  /\  a  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) ) ) )
6 fvex 5802 . 2  |-  ( 2nd `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) ) )  e. 
_V
74, 5, 6fvmpt 5876 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    X. cxp 4939   ` cfv 5519   iota_crio 6153  (class class class)co 6193   1stc1st 6678   2ndc2nd 6679   1c1 9387    / cdiv 10097   NNcn 10426   ZZcz 10750   QQcq 11057    gcd cgcd 13801  denomcdenom 13923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-iota 5482  df-fun 5521  df-fv 5527  df-riota 6154  df-denom 13925
This theorem is referenced by:  qnumdencl  13928  fden  13932  qnumdenbi  13933
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