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Theorem qnumdenbi 14132
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 14103 . . . . . . 7  |-  ( A  e.  QQ  ->  E! a  e.  ( ZZ  X.  NN ) ( ( ( 1st `  a
)  gcd  ( 2nd `  a ) )  =  1  /\  A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) ) ) )
2 riotacl 6258 . . . . . . 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 6818 . . . . . . 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 20 . . . . . 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 14125 . . . . . . 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 14126 . . . . . . 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 4221 . . . . . 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 2511 . . . . 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 2469 . . . 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 1017 . . 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 5874 . . . 4  |-  (numer `  A )  e.  _V
12 fvex 5874 . . . 4  |-  (denom `  A )  e.  _V
1311, 12opth 4721 . . 3  |-  ( <.
(numer `  A ) ,  (denom `  A ) >.  =  <. B ,  C >.  <-> 
( (numer `  A
)  =  B  /\  (denom `  A )  =  C ) )
1410, 13syl6rbb 262 . 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 5030 . . . 4  |-  ( ( B  e.  ZZ  /\  C  e.  NN )  -> 
<. B ,  C >.  e.  ( ZZ  X.  NN ) )
16153adant1 1014 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  <. B ,  C >.  e.  ( ZZ 
X.  NN ) )
1713ad2ant1 1017 . . 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 5864 . . . . . . 7  |-  ( a  =  <. B ,  C >.  ->  ( 1st `  a
)  =  ( 1st `  <. B ,  C >. ) )
19 fveq2 5864 . . . . . . 7  |-  ( a  =  <. B ,  C >.  ->  ( 2nd `  a
)  =  ( 2nd `  <. B ,  C >. ) )
2018, 19oveq12d 6300 . . . . . 6  |-  ( a  =  <. B ,  C >.  ->  ( ( 1st `  a )  gcd  ( 2nd `  a ) )  =  ( ( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. ) ) )
2120eqeq1d 2469 . . . . 5  |-  ( a  =  <. B ,  C >.  ->  ( ( ( 1st `  a )  gcd  ( 2nd `  a
) )  =  1  <-> 
( ( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. )
)  =  1 ) )
2218, 19oveq12d 6300 . . . . . 6  |-  ( a  =  <. B ,  C >.  ->  ( ( 1st `  a )  /  ( 2nd `  a ) )  =  ( ( 1st `  <. B ,  C >. )  /  ( 2nd `  <. B ,  C >. ) ) )
2322eqeq2d 2481 . . . . 5  |-  ( a  =  <. B ,  C >.  ->  ( A  =  ( ( 1st `  a
)  /  ( 2nd `  a ) )  <->  A  =  ( ( 1st `  <. B ,  C >. )  /  ( 2nd `  <. B ,  C >. )
) ) )
2421, 23anbi12d 710 . . . 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 6266 . . 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 661 . 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 6793 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  NN )  ->  ( 1st `  <. B ,  C >. )  =  B )
28 op2ndg 6794 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  NN )  ->  ( 2nd `  <. B ,  C >. )  =  C )
2927, 28oveq12d 6300 . . . . 5  |-  ( ( B  e.  ZZ  /\  C  e.  NN )  ->  ( ( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. )
)  =  ( B  gcd  C ) )
30293adant1 1014 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. )
)  =  ( B  gcd  C ) )
3130eqeq1d 2469 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( ( 1st `  <. B ,  C >. )  gcd  ( 2nd `  <. B ,  C >. )
)  =  1  <->  ( B  gcd  C )  =  1 ) )
32273adant1 1014 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( 1st `  <. B ,  C >. )  =  B )
33283adant1 1014 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( 2nd `  <. B ,  C >. )  =  C )
3432, 33oveq12d 6300 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( 1st `  <. B ,  C >. )  /  ( 2nd `  <. B ,  C >. )
)  =  ( B  /  C ) )
3534eqeq2d 2481 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( A  =  ( ( 1st `  <. B ,  C >. )  /  ( 2nd `  <. B ,  C >. ) )  <->  A  =  ( B  /  C
) ) )
3631, 35anbi12d 710 . 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 282 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E!wreu 2816   <.cop 4033    X. cxp 4997   ` cfv 5586   iota_crio 6242  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   1c1 9489    / cdiv 10202   NNcn 10532   ZZcz 10860   QQcq 11178    gcd cgcd 13999  numercnumer 14121  denomcdenom 14122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-rp 11217  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-dvds 13844  df-gcd 14000  df-numer 14123  df-denom 14124
This theorem is referenced by:  qnumdencoprm  14133  qeqnumdivden  14134  divnumden  14136  numdensq  14142  numdenneg  27275  qqh0  27601  qqh1  27602
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