Theorem qnumdenbi 11906

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 4579	. . . 4 |- ( ( B e. ZZ /\ C e. NN ) -> <. B , C >. e. ( ZZ X. NN ) )
2	1	3adant1 1000	. . 3 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> <. B , C >. e. ( ZZ X. NN ) )
3		qredeu 11814	. . . 4 |- ( A e. QQ -> E! a e. ( ZZ X. NN ) ( ( ( 1st ` a ) gcd ( 2nd ` a ) ) = 1 /\ A = ( ( 1st ` a ) / ( 2nd ` a ) ) ) )
4	3	3ad2ant1 1003	. . 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 5429	. . . . . . 7 |- ( a = <. B , C >. -> ( 1st ` a ) = ( 1st ` <. B , C >. ) )
6		fveq2 5429	. . . . . . 7 |- ( a = <. B , C >. -> ( 2nd ` a ) = ( 2nd ` <. B , C >. ) )
7	5, 6	oveq12d 5800	. . . . . 6 |- ( a = <. B , C >. -> ( ( 1st ` a ) gcd ( 2nd ` a ) ) = ( ( 1st ` <. B , C >. ) gcd ( 2nd ` <. B , C >. ) ) )
8	7	eqeq1d 2149	. . . . 5 |- ( a = <. B , C >. -> ( ( ( 1st ` a ) gcd ( 2nd ` a ) ) = 1 <-> ( ( 1st ` <. B , C >. ) gcd ( 2nd ` <. B , C >. ) ) = 1 ) )
9	5, 6	oveq12d 5800	. . . . . 6 |- ( a = <. B , C >. -> ( ( 1st ` a ) / ( 2nd ` a ) ) = ( ( 1st ` <. B , C >. ) / ( 2nd ` <. B , C >. ) ) )
10	9	eqeq2d 2152	. . . . 5 |- ( a = <. B , C >. -> ( A = ( ( 1st ` a ) / ( 2nd ` a ) ) <-> A = ( ( 1st ` <. B , C >. ) / ( 2nd ` <. B , C >. ) ) ) )
11	8, 10	anbi12d 465	. . . 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 5760	. . 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 409	. 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 6056	. . . . . 6 |- ( ( B e. ZZ /\ C e. NN ) -> ( 1st ` <. B , C >. ) = B )
15		op2ndg 6057	. . . . . 6 |- ( ( B e. ZZ /\ C e. NN ) -> ( 2nd ` <. B , C >. ) = C )
16	14, 15	oveq12d 5800	. . . . 5 |- ( ( B e. ZZ /\ C e. NN ) -> ( ( 1st ` <. B , C >. ) gcd ( 2nd ` <. B , C >. ) ) = ( B gcd C ) )
17	16	3adant1 1000	. . . 4 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( ( 1st ` <. B , C >. ) gcd ( 2nd ` <. B , C >. ) ) = ( B gcd C ) )
18	17	eqeq1d 2149	. . 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 1000	. . . . 5 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( 1st ` <. B , C >. ) = B )
20	15	3adant1 1000	. . . . 5 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( 2nd ` <. B , C >. ) = C )
21	19, 20	oveq12d 5800	. . . 4 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( ( 1st ` <. B , C >. ) / ( 2nd ` <. B , C >. ) ) = ( B / C ) )
22	21	eqeq2d 2152	. . 3 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( A = ( ( 1st ` <. B , C >. ) / ( 2nd ` <. B , C >. ) ) <-> A = ( B / C ) ) )
23	18, 22	anbi12d 465	. 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 5752	. . . . . . 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 6081	. . . . . . 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 11899	. . . . . . 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 11900	. . . . . . 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 3721	. . . . . 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 2176	. . . . 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 2149	. . . 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 1003	. . 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 11902	. . . . 5 |- ( A e. QQ -> (numer ` A ) e. ZZ )
34		qdencl 11903	. . . . 5 |- ( A e. QQ -> (denom ` A ) e. NN )
35		opthg 4168	. . . . 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 409	. . . 4 |- ( A e. QQ -> ( <. (numer ` A ) , (denom ` A ) >. = <. B , C >. <-> ( (numer ` A ) = B /\ (denom ` A ) = C ) ) )
37	36	3ad2ant1 1003	. . 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 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 ) ) )
39	13, 23, 38	3bitr3d 217	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 103   <-> wb 104   /\ w3a 963   = wceq 1332   e. wcel 1481   E!wreu 2419   <.cop 3535   X. cxp 4545   ` cfv 5131   iota_crio 5737  (class class class)co 5782   1stc1st 6044   2ndc2nd 6045   1c1 7645   / cdiv 8456   NNcn 8744   ZZcz 9078   QQcq 9438   gcd cgcd 11671  numercnumer 11895  denomcdenom 11896

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-sup 6879  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-fz 9822  df-fzo 9951  df-fl 10074  df-mod 10127  df-seqfrec 10250  df-exp 10324  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-dvds 11530  df-gcd 11672  df-numer 11897  df-denom 11898

This theorem is referenced by:  qnumdencoprm  11907  qeqnumdivden  11908  divnumden  11910  numdensq  11916