Theorem qnumdenbi 12194

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 4660	. . . 4 |- ( ( B e. ZZ /\ C e. NN ) -> <. B , C >. e. ( ZZ X. NN ) )
2	1	3adant1 1015	. . 3 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> <. B , C >. e. ( ZZ X. NN ) )
3		qredeu 12099	. . . 4 |- ( A e. QQ -> E! a e. ( ZZ X. NN ) ( ( ( 1st ` a ) gcd ( 2nd ` a ) ) = 1 /\ A = ( ( 1st ` a ) / ( 2nd ` a ) ) ) )
4	3	3ad2ant1 1018	. . 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 5517	. . . . . . 7 |- ( a = <. B , C >. -> ( 1st ` a ) = ( 1st ` <. B , C >. ) )
6		fveq2 5517	. . . . . . 7 |- ( a = <. B , C >. -> ( 2nd ` a ) = ( 2nd ` <. B , C >. ) )
7	5, 6	oveq12d 5895	. . . . . 6 |- ( a = <. B , C >. -> ( ( 1st ` a ) gcd ( 2nd ` a ) ) = ( ( 1st ` <. B , C >. ) gcd ( 2nd ` <. B , C >. ) ) )
8	7	eqeq1d 2186	. . . . 5 |- ( a = <. B , C >. -> ( ( ( 1st ` a ) gcd ( 2nd ` a ) ) = 1 <-> ( ( 1st ` <. B , C >. ) gcd ( 2nd ` <. B , C >. ) ) = 1 ) )
9	5, 6	oveq12d 5895	. . . . . 6 |- ( a = <. B , C >. -> ( ( 1st ` a ) / ( 2nd ` a ) ) = ( ( 1st ` <. B , C >. ) / ( 2nd ` <. B , C >. ) ) )
10	9	eqeq2d 2189	. . . . 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 5855	. . 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 6153	. . . . . 6 |- ( ( B e. ZZ /\ C e. NN ) -> ( 1st ` <. B , C >. ) = B )
15		op2ndg 6154	. . . . . 6 |- ( ( B e. ZZ /\ C e. NN ) -> ( 2nd ` <. B , C >. ) = C )
16	14, 15	oveq12d 5895	. . . . 5 |- ( ( B e. ZZ /\ C e. NN ) -> ( ( 1st ` <. B , C >. ) gcd ( 2nd ` <. B , C >. ) ) = ( B gcd C ) )
17	16	3adant1 1015	. . . 4 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( ( 1st ` <. B , C >. ) gcd ( 2nd ` <. B , C >. ) ) = ( B gcd C ) )
18	17	eqeq1d 2186	. . 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 1015	. . . . 5 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( 1st ` <. B , C >. ) = B )
20	15	3adant1 1015	. . . . 5 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( 2nd ` <. B , C >. ) = C )
21	19, 20	oveq12d 5895	. . . 4 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( ( 1st ` <. B , C >. ) / ( 2nd ` <. B , C >. ) ) = ( B / C ) )
22	21	eqeq2d 2189	. . 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 5847	. . . . . . 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 6178	. . . . . . 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 12187	. . . . . . 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 12188	. . . . . . 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 3788	. . . . . 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 2213	. . . . 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 2186	. . . 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 1018	. . 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 12190	. . . . 5 |- ( A e. QQ -> (numer ` A ) e. ZZ )
34		qdencl 12191	. . . . 5 |- ( A e. QQ -> (denom ` A ) e. NN )
35		opthg 4240	. . . . 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 1018	. . 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 978   = wceq 1353   e. wcel 2148   E!wreu 2457   <.cop 3597   X. cxp 4626   ` cfv 5218   iota_crio 5832  (class class class)co 5877   1stc1st 6141   2ndc2nd 6142   1c1 7814   / cdiv 8631   NNcn 8921   ZZcz 9255   QQcq 9621   gcd cgcd 11945  numercnumer 12183  denomcdenom 12184

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-sup 6985  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-fzo 10145  df-fl 10272  df-mod 10325  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-dvds 11797  df-gcd 11946  df-numer 12185  df-denom 12186

This theorem is referenced by:  qnumdencoprm  12195  qeqnumdivden  12196  divnumden  12198  numdensq  12204