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Theorem qnumdenbi 10777
Description: Two numbers are the canonical representation of a rational iff they are coprime and have the right quotient. (Contributed by Stefan O'Rear, 13-Sep-2014.)
Assertion
Ref Expression
qnumdenbi  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( ( B  gcd  C )  =  1  /\  A  =  ( B  /  C ) )  <-> 
( (numer `  A
)  =  B  /\  (denom `  A )  =  C ) ) )

Proof of Theorem qnumdenbi
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 opelxpi 4422 . . . 4  |-  ( ( B  e.  ZZ  /\  C  e.  NN )  -> 
<. B ,  C >.  e.  ( ZZ  X.  NN ) )
213adant1 957 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  <. B ,  C >.  e.  ( ZZ 
X.  NN ) )
3 qredeu 10686 . . . 4  |-  ( A  e.  QQ  ->  E! a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) )
433ad2ant1 960 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  E! a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) )
5 fveq2 5229 . . . . . . 7  |-  ( a  =  <. B ,  C >.  ->  ( 1st `  a
)  =  ( 1st `  <. B ,  C >. ) )
6 fveq2 5229 . . . . . . 7  |-  ( a  =  <. B ,  C >.  ->  ( 2nd `  a
)  =  ( 2nd `  <. B ,  C >. ) )
75, 6oveq12d 5581 . . . . . 6  |-  ( a  =  <. B ,  C >.  ->  ( ( 1st `  a )  gcd  ( 2nd `  a ) )  =  ( ( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. ) ) )
87eqeq1d 2091 . . . . 5  |-  ( a  =  <. B ,  C >.  ->  ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  <-> 
( ( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. )
)  =  1 ) )
95, 6oveq12d 5581 . . . . . 6  |-  ( a  =  <. B ,  C >.  ->  ( ( 1st `  a )  /  ( 2nd `  a ) )  =  ( ( 1st `  <. B ,  C >. )  /  ( 2nd `  <. B ,  C >. ) ) )
109eqeq2d 2094 . . . . 5  |-  ( a  =  <. B ,  C >.  ->  ( A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) )  <->  A  =  ( ( 1st `  <. B ,  C >. )  /  ( 2nd `  <. B ,  C >. )
) ) )
118, 10anbi12d 457 . . . 4  |-  ( a  =  <. B ,  C >.  ->  ( ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) )  <-> 
( ( ( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. ) )  =  1  /\  A  =  ( ( 1st `  <. B ,  C >. )  /  ( 2nd `  <. B ,  C >. )
) ) ) )
1211riota2 5541 . . 3  |-  ( (
<. B ,  C >.  e.  ( ZZ  X.  NN )  /\  E! a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) )  ->  ( (
( ( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. )
)  =  1  /\  A  =  ( ( 1st `  <. B ,  C >. )  /  ( 2nd `  <. B ,  C >. ) ) )  <->  ( iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) )  =  <. B ,  C >. ) )
132, 4, 12syl2anc 403 . 2  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( ( ( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. ) )  =  1  /\  A  =  ( ( 1st `  <. B ,  C >. )  /  ( 2nd `  <. B ,  C >. )
) )  <->  ( iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) )  =  <. B ,  C >. ) )
14 op1stg 5828 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  NN )  ->  ( 1st `  <. B ,  C >. )  =  B )
15 op2ndg 5829 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  NN )  ->  ( 2nd `  <. B ,  C >. )  =  C )
1614, 15oveq12d 5581 . . . . 5  |-  ( ( B  e.  ZZ  /\  C  e.  NN )  ->  ( ( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. )
)  =  ( B  gcd  C ) )
17163adant1 957 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. )
)  =  ( B  gcd  C ) )
1817eqeq1d 2091 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( ( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. )
)  =  1  <->  ( B  gcd  C )  =  1 ) )
19143adant1 957 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( 1st `  <. B ,  C >. )  =  B )
20153adant1 957 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( 2nd `  <. B ,  C >. )  =  C )
2119, 20oveq12d 5581 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( 1st `  <. B ,  C >. )  /  ( 2nd `  <. B ,  C >. )
)  =  ( B  /  C ) )
2221eqeq2d 2094 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( A  =  ( ( 1st `  <. B ,  C >. )  /  ( 2nd `  <. B ,  C >. ) )  <->  A  =  ( B  /  C
) ) )
2318, 22anbi12d 457 . 2  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( ( ( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. ) )  =  1  /\  A  =  ( ( 1st `  <. B ,  C >. )  /  ( 2nd `  <. B ,  C >. )
) )  <->  ( ( B  gcd  C )  =  1  /\  A  =  ( B  /  C
) ) ) )
24 riotacl 5533 . . . . . . 7  |-  ( 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 ) )
25 1st2nd2 5852 . . . . . . 7  |-  ( (
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 ) ) ) ) ) >. )
263, 24, 253syl 17 . . . . . 6  |-  ( 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 ) ) ) ) ) >. )
27 qnumval 10770 . . . . . . 7  |-  ( A  e.  QQ  ->  (numer `  A )  =  ( 1st `  ( iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) ) ) )
28 qdenval 10771 . . . . . . 7  |-  ( A  e.  QQ  ->  (denom `  A )  =  ( 2nd `  ( iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) ) ) )
2927, 28opeq12d 3598 . . . . . 6  |-  ( A  e.  QQ  ->  <. (numer `  A ) ,  (denom `  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 ) ) ) ) ) >. )
3026, 29eqtr4d 2118 . . . . 5  |-  ( A  e.  QQ  ->  ( iota_ a  e.  ( ZZ 
X.  NN ) ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) )  =  <. (numer `  A ) ,  (denom `  A ) >. )
3130eqeq1d 2091 . . . 4  |-  ( A  e.  QQ  ->  (
( iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a )  /  ( 2nd `  a ) ) ) )  =  <. B ,  C >.  <->  <. (numer `  A ) ,  (denom `  A ) >.  =  <. B ,  C >. )
)
32313ad2ant1 960 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a )  /  ( 2nd `  a ) ) ) )  =  <. B ,  C >.  <->  <. (numer `  A ) ,  (denom `  A ) >.  =  <. B ,  C >. )
)
33 qnumcl 10773 . . . . 5  |-  ( A  e.  QQ  ->  (numer `  A )  e.  ZZ )
34 qdencl 10774 . . . . 5  |-  ( A  e.  QQ  ->  (denom `  A )  e.  NN )
35 opthg 4021 . . . . 5  |-  ( ( (numer `  A )  e.  ZZ  /\  (denom `  A )  e.  NN )  ->  ( <. (numer `  A ) ,  (denom `  A ) >.  =  <. B ,  C >.  <->  ( (numer `  A )  =  B  /\  (denom `  A
)  =  C ) ) )
3633, 34, 35syl2anc 403 . . . 4  |-  ( A  e.  QQ  ->  ( <. (numer `  A ) ,  (denom `  A ) >.  =  <. B ,  C >.  <-> 
( (numer `  A
)  =  B  /\  (denom `  A )  =  C ) ) )
37363ad2ant1 960 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( <. (numer `  A ) ,  (denom `  A ) >.  =  <. B ,  C >.  <-> 
( (numer `  A
)  =  B  /\  (denom `  A )  =  C ) ) )
3832, 37bitrd 186 . 2  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a )  /  ( 2nd `  a ) ) ) )  =  <. B ,  C >.  <->  ( (numer `  A )  =  B  /\  (denom `  A
)  =  C ) ) )
3913, 23, 383bitr3d 216 1  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( ( B  gcd  C )  =  1  /\  A  =  ( B  /  C ) )  <-> 
( (numer `  A
)  =  B  /\  (denom `  A )  =  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   E!wreu 2355   <.cop 3419    X. cxp 4389   ` cfv 4952   iota_crio 5518  (class class class)co 5563   1stc1st 5816   2ndc2nd 5817   1c1 7096    / cdiv 7879   NNcn 8158   ZZcz 8484   QQcq 8837    gcd cgcd 10545  numercnumer 10766  denomcdenom 10767
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-in1 577  ax-in2 578  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-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209  ax-caucvg 7210
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  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-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-sup 6491  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-3 8218  df-4 8219  df-n0 8408  df-z 8485  df-uz 8753  df-q 8838  df-rp 8868  df-fz 9158  df-fzo 9282  df-fl 9404  df-mod 9457  df-iseq 9574  df-iexp 9625  df-cj 9930  df-re 9931  df-im 9932  df-rsqrt 10085  df-abs 10086  df-dvds 10404  df-gcd 10546  df-numer 10768  df-denom 10769
This theorem is referenced by:  qnumdencoprm  10778  qeqnumdivden  10779  divnumden  10781  numdensq  10787
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