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Theorem qnumdenbi 12831
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 qredeu 12802 . . . . . . 7  |-  ( A  e.  QQ  ->  E! a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) )
2 riotacl 6335 . . . . . . 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 ) )
3 1st2nd2 6175 . . . . . . 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 ) ) ) ) ) >. )
41, 2, 33syl 18 . . . . . 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 ) ) ) ) ) >. )
5 qnumval 12824 . . . . . . 7  |-  ( A  e.  QQ  ->  (numer `  A )  =  ( 1st `  ( iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) ) ) )
6 qdenval 12825 . . . . . . 7  |-  ( A  e.  QQ  ->  (denom `  A )  =  ( 2nd `  ( iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) ) ) )
75, 6opeq12d 3820 . . . . . 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 ) ) ) ) ) >. )
84, 7eqtr4d 2331 . . . . 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 ) >. )
98eqeq1d 2304 . . . 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 >. )
)
1093ad2ant1 976 . . 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 >. )
)
11 fvex 5555 . . . 4  |-  (numer `  A )  e.  _V
12 fvex 5555 . . . 4  |-  (denom `  A )  e.  _V
1311, 12opth 4261 . . 3  |-  ( <.
(numer `  A ) ,  (denom `  A ) >.  =  <. B ,  C >.  <-> 
( (numer `  A
)  =  B  /\  (denom `  A )  =  C ) )
1410, 13syl6rbb 253 . 2  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( (numer `  A
)  =  B  /\  (denom `  A )  =  C )  <->  ( iota_ a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) )  =  <. B ,  C >. ) )
15 opelxpi 4737 . . . 4  |-  ( ( B  e.  ZZ  /\  C  e.  NN )  -> 
<. B ,  C >.  e.  ( ZZ  X.  NN ) )
16153adant1 973 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  <. B ,  C >.  e.  ( ZZ 
X.  NN ) )
1713ad2ant1 976 . . 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 ) ) ) )
18 fveq2 5541 . . . . . . 7  |-  ( a  =  <. B ,  C >.  ->  ( 1st `  a
)  =  ( 1st `  <. B ,  C >. ) )
19 fveq2 5541 . . . . . . 7  |-  ( a  =  <. B ,  C >.  ->  ( 2nd `  a
)  =  ( 2nd `  <. B ,  C >. ) )
2018, 19oveq12d 5892 . . . . . 6  |-  ( a  =  <. B ,  C >.  ->  ( ( 1st `  a )  gcd  ( 2nd `  a ) )  =  ( ( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. ) ) )
2120eqeq1d 2304 . . . . 5  |-  ( a  =  <. B ,  C >.  ->  ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  <-> 
( ( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. )
)  =  1 ) )
2218, 19oveq12d 5892 . . . . . 6  |-  ( a  =  <. B ,  C >.  ->  ( ( 1st `  a )  /  ( 2nd `  a ) )  =  ( ( 1st `  <. B ,  C >. )  /  ( 2nd `  <. B ,  C >. ) ) )
2322eqeq2d 2307 . . . . 5  |-  ( a  =  <. B ,  C >.  ->  ( A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) )  <->  A  =  ( ( 1st `  <. B ,  C >. )  /  ( 2nd `  <. B ,  C >. )
) ) )
2421, 23anbi12d 691 . . . 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 >. )
) ) ) )
2524riota2 6343 . . 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 >. ) )
2616, 17, 25syl2anc 642 . 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 >. ) )
27 op1stg 6148 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  NN )  ->  ( 1st `  <. B ,  C >. )  =  B )
28 op2ndg 6149 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  NN )  ->  ( 2nd `  <. B ,  C >. )  =  C )
2927, 28oveq12d 5892 . . . . 5  |-  ( ( B  e.  ZZ  /\  C  e.  NN )  ->  ( ( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. )
)  =  ( B  gcd  C ) )
30293adant1 973 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. )
)  =  ( B  gcd  C ) )
3130eqeq1d 2304 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( ( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. )
)  =  1  <->  ( B  gcd  C )  =  1 ) )
32273adant1 973 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( 1st `  <. B ,  C >. )  =  B )
33283adant1 973 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( 2nd `  <. B ,  C >. )  =  C )
3432, 33oveq12d 5892 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( 1st `  <. B ,  C >. )  /  ( 2nd `  <. B ,  C >. )
)  =  ( B  /  C ) )
3534eqeq2d 2307 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( A  =  ( ( 1st `  <. B ,  C >. )  /  ( 2nd `  <. B ,  C >. ) )  <->  A  =  ( B  /  C
) ) )
3631, 35anbi12d 691 . 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
) ) ) )
3714, 26, 363bitr2rd 273 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E!wreu 2558   <.cop 3656    X. cxp 4703   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   iota_crio 6313   1c1 8754    / cdiv 9439   NNcn 9762   ZZcz 10040   QQcq 10332    gcd cgcd 12701  numercnumer 12820  denomcdenom 12821
This theorem is referenced by:  qnumdencoprm  12832  qeqnumdivden  12833  divnumden  12835  numdensq  12841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-numer 12822  df-denom 12823
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