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Theorem qnumdenbi 11797
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 4541 . . . 4  |-  ( ( B  e.  ZZ  /\  C  e.  NN )  -> 
<. B ,  C >.  e.  ( ZZ  X.  NN ) )
213adant1 984 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  <. B ,  C >.  e.  ( ZZ 
X.  NN ) )
3 qredeu 11705 . . . 4  |-  ( A  e.  QQ  ->  E! a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) )
433ad2ant1 987 . . 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 5389 . . . . . . 7  |-  ( a  =  <. B ,  C >.  ->  ( 1st `  a
)  =  ( 1st `  <. B ,  C >. ) )
6 fveq2 5389 . . . . . . 7  |-  ( a  =  <. B ,  C >.  ->  ( 2nd `  a
)  =  ( 2nd `  <. B ,  C >. ) )
75, 6oveq12d 5760 . . . . . 6  |-  ( a  =  <. B ,  C >.  ->  ( ( 1st `  a )  gcd  ( 2nd `  a ) )  =  ( ( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. ) ) )
87eqeq1d 2126 . . . . 5  |-  ( a  =  <. B ,  C >.  ->  ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  <-> 
( ( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. )
)  =  1 ) )
95, 6oveq12d 5760 . . . . . 6  |-  ( a  =  <. B ,  C >.  ->  ( ( 1st `  a )  /  ( 2nd `  a ) )  =  ( ( 1st `  <. B ,  C >. )  /  ( 2nd `  <. B ,  C >. ) ) )
109eqeq2d 2129 . . . . 5  |-  ( a  =  <. B ,  C >.  ->  ( A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) )  <->  A  =  ( ( 1st `  <. B ,  C >. )  /  ( 2nd `  <. B ,  C >. )
) ) )
118, 10anbi12d 464 . . . 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 5720 . . 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 408 . 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 6016 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  NN )  ->  ( 1st `  <. B ,  C >. )  =  B )
15 op2ndg 6017 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  NN )  ->  ( 2nd `  <. B ,  C >. )  =  C )
1614, 15oveq12d 5760 . . . . 5  |-  ( ( B  e.  ZZ  /\  C  e.  NN )  ->  ( ( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. )
)  =  ( B  gcd  C ) )
17163adant1 984 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. )
)  =  ( B  gcd  C ) )
1817eqeq1d 2126 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( ( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. )
)  =  1  <->  ( B  gcd  C )  =  1 ) )
19143adant1 984 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( 1st `  <. B ,  C >. )  =  B )
20153adant1 984 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( 2nd `  <. B ,  C >. )  =  C )
2119, 20oveq12d 5760 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( 1st `  <. B ,  C >. )  /  ( 2nd `  <. B ,  C >. )
)  =  ( B  /  C ) )
2221eqeq2d 2129 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( A  =  ( ( 1st `  <. B ,  C >. )  /  ( 2nd `  <. B ,  C >. ) )  <->  A  =  ( B  /  C
) ) )
2318, 22anbi12d 464 . 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 5712 . . . . . . 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 6041 . . . . . . 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 11790 . . . . . . 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 11791 . . . . . . 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 3683 . . . . . 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 2153 . . . . 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 2126 . . . 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 987 . . 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 11793 . . . . 5  |-  ( A  e.  QQ  ->  (numer `  A )  e.  ZZ )
34 qdencl 11794 . . . . 5  |-  ( A  e.  QQ  ->  (denom `  A )  e.  NN )
35 opthg 4130 . . . . 5  |-  ( ( (numer `  A )  e.  ZZ  /\  (denom `  A )  e.  NN )  ->  ( <. (numer `  A ) ,  (denom `  A ) >.  =  <. B ,  C >.  <->  ( (numer `  A )  =  B  /\  (denom `  A
)  =  C ) ) )
3633, 34, 35syl2anc 408 . . . 4  |-  ( A  e.  QQ  ->  ( <. (numer `  A ) ,  (denom `  A ) >.  =  <. B ,  C >.  <-> 
( (numer `  A
)  =  B  /\  (denom `  A )  =  C ) ) )
37363ad2ant1 987 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( <. (numer `  A ) ,  (denom `  A ) >.  =  <. B ,  C >.  <-> 
( (numer `  A
)  =  B  /\  (denom `  A )  =  C ) ) )
3832, 37bitrd 187 . 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 217 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 103    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465   E!wreu 2395   <.cop 3500    X. cxp 4507   ` cfv 5093   iota_crio 5697  (class class class)co 5742   1stc1st 6004   2ndc2nd 6005   1c1 7589    / cdiv 8400   NNcn 8688   ZZcz 9022   QQcq 9379    gcd cgcd 11562  numercnumer 11786  denomcdenom 11787
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-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-sup 6839  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-n0 8946  df-z 9023  df-uz 9295  df-q 9380  df-rp 9410  df-fz 9759  df-fzo 9888  df-fl 10011  df-mod 10064  df-seqfrec 10187  df-exp 10261  df-cj 10582  df-re 10583  df-im 10584  df-rsqrt 10738  df-abs 10739  df-dvds 11421  df-gcd 11563  df-numer 11788  df-denom 11789
This theorem is referenced by:  qnumdencoprm  11798  qeqnumdivden  11799  divnumden  11801  numdensq  11807
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