Theorem qnumdenbi 12146

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 4643	. . . 4 |- ( ( B e. ZZ /\ C e. NN ) -> <. B , C >. e. ( ZZ X. NN ) )
2	1	3adant1 1010	. . 3 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> <. B , C >. e. ( ZZ X. NN ) )
3		qredeu 12051	. . . 4 |- ( A e. QQ -> E! a e. ( ZZ X. NN ) ( ( ( 1st ` a ) gcd ( 2nd ` a ) ) = 1 /\ A = ( ( 1st ` a ) / ( 2nd ` a ) ) ) )
4	3	3ad2ant1 1013	. . 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 5496	. . . . . . 7 |- ( a = <. B , C >. -> ( 1st ` a ) = ( 1st ` <. B , C >. ) )
6		fveq2 5496	. . . . . . 7 |- ( a = <. B , C >. -> ( 2nd ` a ) = ( 2nd ` <. B , C >. ) )
7	5, 6	oveq12d 5871	. . . . . 6 |- ( a = <. B , C >. -> ( ( 1st ` a ) gcd ( 2nd ` a ) ) = ( ( 1st ` <. B , C >. ) gcd ( 2nd ` <. B , C >. ) ) )
8	7	eqeq1d 2179	. . . . 5 |- ( a = <. B , C >. -> ( ( ( 1st ` a ) gcd ( 2nd ` a ) ) = 1 <-> ( ( 1st ` <. B , C >. ) gcd ( 2nd ` <. B , C >. ) ) = 1 ) )
9	5, 6	oveq12d 5871	. . . . . 6 |- ( a = <. B , C >. -> ( ( 1st ` a ) / ( 2nd ` a ) ) = ( ( 1st ` <. B , C >. ) / ( 2nd ` <. B , C >. ) ) )
10	9	eqeq2d 2182	. . . . 5 |- ( a = <. B , C >. -> ( A = ( ( 1st ` a ) / ( 2nd ` a ) ) <-> A = ( ( 1st ` <. B , C >. ) / ( 2nd ` <. B , C >. ) ) ) )
11	8, 10	anbi12d 470	. . . 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 5831	. . 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 6129	. . . . . 6 |- ( ( B e. ZZ /\ C e. NN ) -> ( 1st ` <. B , C >. ) = B )
15		op2ndg 6130	. . . . . 6 |- ( ( B e. ZZ /\ C e. NN ) -> ( 2nd ` <. B , C >. ) = C )
16	14, 15	oveq12d 5871	. . . . 5 |- ( ( B e. ZZ /\ C e. NN ) -> ( ( 1st ` <. B , C >. ) gcd ( 2nd ` <. B , C >. ) ) = ( B gcd C ) )
17	16	3adant1 1010	. . . 4 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( ( 1st ` <. B , C >. ) gcd ( 2nd ` <. B , C >. ) ) = ( B gcd C ) )
18	17	eqeq1d 2179	. . 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 1010	. . . . 5 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( 1st ` <. B , C >. ) = B )
20	15	3adant1 1010	. . . . 5 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( 2nd ` <. B , C >. ) = C )
21	19, 20	oveq12d 5871	. . . 4 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( ( 1st ` <. B , C >. ) / ( 2nd ` <. B , C >. ) ) = ( B / C ) )
22	21	eqeq2d 2182	. . 3 |- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( A = ( ( 1st ` <. B , C >. ) / ( 2nd ` <. B , C >. ) ) <-> A = ( B / C ) ) )
23	18, 22	anbi12d 470	. 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 5823	. . . . . . 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 6154	. . . . . . 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 12139	. . . . . . 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 12140	. . . . . . 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 3773	. . . . . 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 2206	. . . . 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 2179	. . . 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 1013	. . 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 12142	. . . . 5 |- ( A e. QQ -> (numer ` A ) e. ZZ )
34		qdencl 12143	. . . . 5 |- ( A e. QQ -> (denom ` A ) e. NN )
35		opthg 4223	. . . . 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 1013	. . 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 973   = wceq 1348   e. wcel 2141   E!wreu 2450   <.cop 3586   X. cxp 4609   ` cfv 5198   iota_crio 5808  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   1c1 7775   / cdiv 8589   NNcn 8878   ZZcz 9212   QQcq 9578   gcd cgcd 11897  numercnumer 12135  denomcdenom 12136

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-gcd 11898  df-numer 12137  df-denom 12138

This theorem is referenced by:  qnumdencoprm  12147  qeqnumdivden  12148  divnumden  12150  numdensq  12156