Theorem qnumdenbi 12709

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.

Step	Hyp	Ref	Expression
1		opelxpi 4750	. . . 4 |- ( ( B e. ZZ /\ C e. NN ) -> <. B , C >. e. ( ZZ X. NN ) )
2	1	3adant1 1039	. . 3 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> <. B , C >. e. ( ZZ X. NN ) )
3		qredeu 12614	. . . 4 |- ( A e. QQ -> E! a e. ( ZZ X. NN ) ( ( ( 1st ` a ) gcd ( 2nd ` a ) ) = 1 /\ A = ( ( 1st ` a ) / ( 2nd ` a ) ) ) )
4	3	3ad2ant1 1042	. . 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 5626	. . . . . . 7 |- ( a = <. B , C >. -> ( 1st ` a ) = ( 1st ` <. B , C >. ) )
6		fveq2 5626	. . . . . . 7 |- ( a = <. B , C >. -> ( 2nd ` a ) = ( 2nd ` <. B , C >. ) )
7	5, 6	oveq12d 6018	. . . . . 6 |- ( a = <. B , C >. -> ( ( 1st ` a ) gcd ( 2nd ` a ) ) = ( ( 1st ` <. B , C >. ) gcd ( 2nd ` <. B , C >. ) ) )
8	7	eqeq1d 2238	. . . . 5 |- ( a = <. B , C >. -> ( ( ( 1st ` a ) gcd ( 2nd ` a ) ) = 1 <-> ( ( 1st ` <. B , C >. ) gcd ( 2nd ` <. B , C >. ) ) = 1 ) )
9	5, 6	oveq12d 6018	. . . . . 6 |- ( a = <. B , C >. -> ( ( 1st ` a ) / ( 2nd ` a ) ) = ( ( 1st ` <. B , C >. ) / ( 2nd ` <. B , C >. ) ) )
10	9	eqeq2d 2241	. . . . 5 |- ( a = <. B , C >. -> ( A = ( ( 1st ` a ) / ( 2nd ` a ) ) <-> A = ( ( 1st ` <. B , C >. ) / ( 2nd ` <. B , C >. ) ) ) )
11	8, 10	anbi12d 473	. . . 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 >. ) ) ) ) )
12	11	riota2 5977	. . 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 >. ) )
13	2, 4, 12	syl2anc 411	. 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 6294	. . . . . 6 |- ( ( B e. ZZ /\ C e. NN ) -> ( 1st ` <. B , C >. ) = B )
15		op2ndg 6295	. . . . . 6 |- ( ( B e. ZZ /\ C e. NN ) -> ( 2nd ` <. B , C >. ) = C )
16	14, 15	oveq12d 6018	. . . . 5 |- ( ( B e. ZZ /\ C e. NN ) -> ( ( 1st ` <. B , C >. ) gcd ( 2nd ` <. B , C >. ) ) = ( B gcd C ) )
17	16	3adant1 1039	. . . 4 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( ( 1st ` <. B , C >. ) gcd ( 2nd ` <. B , C >. ) ) = ( B gcd C ) )
18	17	eqeq1d 2238	. . 3 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( ( ( 1st ` <. B , C >. ) gcd ( 2nd ` <. B , C >. ) ) = 1 <-> ( B gcd C ) = 1 ) )
19	14	3adant1 1039	. . . . 5 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( 1st ` <. B , C >. ) = B )
20	15	3adant1 1039	. . . . 5 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( 2nd ` <. B , C >. ) = C )
21	19, 20	oveq12d 6018	. . . 4 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( ( 1st ` <. B , C >. ) / ( 2nd ` <. B , C >. ) ) = ( B / C ) )
22	21	eqeq2d 2241	. . 3 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( A = ( ( 1st ` <. B , C >. ) / ( 2nd ` <. B , C >. ) ) <-> A = ( B / C ) ) )
23	18, 22	anbi12d 473	. 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 5969	. . . . . . 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 6319	. . . . . . 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 ) ) ) ) ) >. )
26	3, 24, 25	3syl 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 12702	. . . . . . 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 12703	. . . . . . 7 |- ( A e. QQ -> (denom ` A ) = ( 2nd ` ( iota_ a e. ( ZZ X. NN ) ( ( ( 1st ` a ) gcd ( 2nd ` a ) ) = 1 /\ A = ( ( 1st ` a ) / ( 2nd ` a ) ) ) ) ) )
29	27, 28	opeq12d 3864	. . . . . 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 ) ) ) ) ) >. )
30	26, 29	eqtr4d 2265	. . . . 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 ) >. )
31	30	eqeq1d 2238	. . . 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 >. ) )
32	31	3ad2ant1 1042	. . 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 12705	. . . . 5 |- ( A e. QQ -> (numer ` A ) e. ZZ )
34		qdencl 12706	. . . . 5 |- ( A e. QQ -> (denom ` A ) e. NN )
35		opthg 4323	. . . . 5 |- ( ( (numer ` A ) e. ZZ /\ (denom ` A ) e. NN ) -> ( <. (numer ` A ) , (denom ` A ) >. = <. B , C >. <-> ( (numer ` A ) = B /\ (denom ` A ) = C ) ) )
36	33, 34, 35	syl2anc 411	. . . 4 |- ( A e. QQ -> ( <. (numer ` A ) , (denom ` A ) >. = <. B , C >. <-> ( (numer ` A ) = B /\ (denom ` A ) = C ) ) )
37	36	3ad2ant1 1042	. . 3 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( <. (numer ` A ) , (denom ` A ) >. = <. B , C >. <-> ( (numer ` A ) = B /\ (denom ` A ) = C ) ) )
38	32, 37	bitrd 188	. 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 ) ) )
39	13, 23, 38	3bitr3d 218	1 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( ( ( B gcd C ) = 1 /\ A = ( B / C ) ) <-> ( (numer ` A ) = B /\ (denom ` A ) = C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   E!wreu 2510   <.cop 3669   X. cxp 4716   ` cfv 5317   iota_crio 5952  (class class class)co 6000   1stc1st 6282   2ndc2nd 6283   1c1 7996   / cdiv 8815   NNcn 9106   ZZcz 9442   QQcq 9810   gcd cgcd 12469  numercnumer 12698  denomcdenom 12699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-sup 7147  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-n0 9366  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-fz 10201  df-fzo 10335  df-fl 10485  df-mod 10540  df-seqfrec 10665  df-exp 10756  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-dvds 12294  df-gcd 12470  df-numer 12700  df-denom 12701

This theorem is referenced by:  qnumdencoprm  12710  qeqnumdivden  12711  divnumden  12713  numdensq  12719