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Theorem qnumdencl 13123
Description: Lemma for qnumcl 13124 and qdencl 13125. (Contributed by Stefan O'Rear, 13-Sep-2014.)
Assertion
Ref Expression
qnumdencl  |-  ( A  e.  QQ  ->  (
(numer `  A )  e.  ZZ  /\  (denom `  A )  e.  NN ) )

Proof of Theorem qnumdencl
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 qredeu 13099 . . 3  |-  ( A  e.  QQ  ->  E! a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) )
2 riotacl 6556 . . 3  |-  ( E! a  e.  ( ZZ 
X.  NN ) ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) )  ->  ( iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) )  e.  ( ZZ 
X.  NN ) )
31, 2syl 16 . 2  |-  ( A  e.  QQ  ->  ( iota_ a  e.  ( ZZ 
X.  NN ) ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) )  e.  ( ZZ 
X.  NN ) )
4 elxp6 6370 . . 3  |-  ( (
iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) )  e.  ( ZZ 
X.  NN )  <->  ( ( iota_ a  e.  ( ZZ 
X.  NN ) ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) )  =  <. ( 1st `  ( iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) ) ) ,  ( 2nd `  ( iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) ) ) >.  /\  (
( 1st `  ( iota_ a  e.  ( ZZ 
X.  NN ) ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) ) )  e.  ZZ  /\  ( 2nd `  ( iota_ a  e.  ( ZZ 
X.  NN ) ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) ) )  e.  NN ) ) )
5 qnumval 13121 . . . . . . 7  |-  ( A  e.  QQ  ->  (numer `  A )  =  ( 1st `  ( iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) ) ) )
65eleq1d 2501 . . . . . 6  |-  ( A  e.  QQ  ->  (
(numer `  A )  e.  ZZ  <->  ( 1st `  ( iota_ a  e.  ( ZZ 
X.  NN ) ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) ) )  e.  ZZ ) )
7 qdenval 13122 . . . . . . 7  |-  ( A  e.  QQ  ->  (denom `  A )  =  ( 2nd `  ( iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) ) ) )
87eleq1d 2501 . . . . . 6  |-  ( A  e.  QQ  ->  (
(denom `  A )  e.  NN  <->  ( 2nd `  ( iota_ a  e.  ( ZZ 
X.  NN ) ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) ) )  e.  NN ) )
96, 8anbi12d 692 . . . . 5  |-  ( A  e.  QQ  ->  (
( (numer `  A
)  e.  ZZ  /\  (denom `  A )  e.  NN )  <->  ( ( 1st `  ( iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) ) )  e.  ZZ  /\  ( 2nd `  ( iota_ a  e.  ( ZZ 
X.  NN ) ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) ) )  e.  NN ) ) )
109biimprd 215 . . . 4  |-  ( A  e.  QQ  ->  (
( ( 1st `  ( iota_ a  e.  ( ZZ 
X.  NN ) ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) ) )  e.  ZZ  /\  ( 2nd `  ( iota_ a  e.  ( ZZ 
X.  NN ) ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) ) )  e.  NN )  ->  ( (numer `  A )  e.  ZZ  /\  (denom `  A )  e.  NN ) ) )
1110adantld 454 . . 3  |-  ( A  e.  QQ  ->  (
( ( iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) )  =  <. ( 1st `  ( iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) ) ) ,  ( 2nd `  ( iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) ) ) >.  /\  (
( 1st `  ( iota_ a  e.  ( ZZ 
X.  NN ) ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) ) )  e.  ZZ  /\  ( 2nd `  ( iota_ a  e.  ( ZZ 
X.  NN ) ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) ) )  e.  NN ) )  ->  (
(numer `  A )  e.  ZZ  /\  (denom `  A )  e.  NN ) ) )
124, 11syl5bi 209 . 2  |-  ( A  e.  QQ  ->  (
( iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a )  /  ( 2nd `  a ) ) ) )  e.  ( ZZ  X.  NN )  ->  ( (numer `  A )  e.  ZZ  /\  (denom `  A )  e.  NN ) ) )
133, 12mpd 15 1  |-  ( A  e.  QQ  ->  (
(numer `  A )  e.  ZZ  /\  (denom `  A )  e.  NN ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E!wreu 2699   <.cop 3809    X. cxp 4868   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   iota_crio 6534   1c1 8983    / cdiv 9669   NNcn 9992   ZZcz 10274   QQcq 10566    gcd cgcd 12998  numercnumer 13117  denomcdenom 13118
This theorem is referenced by:  qnumcl  13124  qdencl  13125
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999  df-numer 13119  df-denom 13120
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