Theorem bezoutr1 11721

Description: Converse of bezout 11699 for when the greater common divisor is one (sufficient condition for relative primality). (Contributed by Stefan O'Rear, 23-Sep-2014.)

Assertion

Ref	Expression
bezoutr1	|- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) -> ( ( ( A x. X ) + ( B x. Y ) ) = 1 -> ( A gcd B ) = 1 ) )

Proof of Theorem bezoutr1

Step	Hyp	Ref	Expression
1		bezoutr 11720	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) -> ( A gcd B ) || ( ( A x. X ) + ( B x. Y ) ) )
2	1	adantr 274	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) /\ ( ( A x. X ) + ( B x. Y ) ) = 1 ) -> ( A gcd B ) || ( ( A x. X ) + ( B x. Y ) ) )
3		simpr 109	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) /\ ( ( A x. X ) + ( B x. Y ) ) = 1 ) -> ( ( A x. X ) + ( B x. Y ) ) = 1 )
4	2, 3	breqtrd 3954	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) /\ ( ( A x. X ) + ( B x. Y ) ) = 1 ) -> ( A gcd B ) || 1 )
5		gcdcl 11655	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) e. NN0 )
6	5	nn0zd 9171	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) e. ZZ )
7	6	ad2antrr 479	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) /\ ( ( A x. X ) + ( B x. Y ) ) = 1 ) -> ( A gcd B ) e. ZZ )
8		1nn 8731	. . . . . 6 |- 1 e. NN
9	8	a1i 9	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) /\ ( ( A x. X ) + ( B x. Y ) ) = 1 ) -> 1 e. NN )
10		dvdsle 11542	. . . . 5 |- ( ( ( A gcd B ) e. ZZ /\ 1 e. NN ) -> ( ( A gcd B ) || 1 -> ( A gcd B ) <_ 1 ) )
11	7, 9, 10	syl2anc 408	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) /\ ( ( A x. X ) + ( B x. Y ) ) = 1 ) -> ( ( A gcd B ) || 1 -> ( A gcd B ) <_ 1 ) )
12	4, 11	mpd 13	. . 3 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) /\ ( ( A x. X ) + ( B x. Y ) ) = 1 ) -> ( A gcd B ) <_ 1 )
13		simpll 518	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) /\ ( ( A x. X ) + ( B x. Y ) ) = 1 ) -> ( A e. ZZ /\ B e. ZZ ) )
14		oveq1 5781	. . . . . . . . . . . . 13 |- ( A = 0 -> ( A x. X ) = ( 0 x. X ) )
15		oveq1 5781	. . . . . . . . . . . . 13 |- ( B = 0 -> ( B x. Y ) = ( 0 x. Y ) )
16	14, 15	oveqan12d 5793	. . . . . . . . . . . 12 |- ( ( A = 0 /\ B = 0 ) -> ( ( A x. X ) + ( B x. Y ) ) = ( ( 0 x. X ) + ( 0 x. Y ) ) )
17		zcn 9059	. . . . . . . . . . . . . 14 |- ( X e. ZZ -> X e. CC )
18	17	mul02d 8154	. . . . . . . . . . . . 13 |- ( X e. ZZ -> ( 0 x. X ) = 0 )
19		zcn 9059	. . . . . . . . . . . . . 14 |- ( Y e. ZZ -> Y e. CC )
20	19	mul02d 8154	. . . . . . . . . . . . 13 |- ( Y e. ZZ -> ( 0 x. Y ) = 0 )
21	18, 20	oveqan12d 5793	. . . . . . . . . . . 12 |- ( ( X e. ZZ /\ Y e. ZZ ) -> ( ( 0 x. X ) + ( 0 x. Y ) ) = ( 0 + 0 ) )
22	16, 21	sylan9eqr 2194	. . . . . . . . . . 11 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ ( A = 0 /\ B = 0 ) ) -> ( ( A x. X ) + ( B x. Y ) ) = ( 0 + 0 ) )
23		00id 7903	. . . . . . . . . . 11 |- ( 0 + 0 ) = 0
24	22, 23	syl6eq 2188	. . . . . . . . . 10 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ ( A = 0 /\ B = 0 ) ) -> ( ( A x. X ) + ( B x. Y ) ) = 0 )
25	24	adantll 467	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) /\ ( A = 0 /\ B = 0 ) ) -> ( ( A x. X ) + ( B x. Y ) ) = 0 )
26		0ne1 8787	. . . . . . . . . 10 |- 0 =/= 1
27	26	a1i 9	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) /\ ( A = 0 /\ B = 0 ) ) -> 0 =/= 1 )
28	25, 27	eqnetrd 2332	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) /\ ( A = 0 /\ B = 0 ) ) -> ( ( A x. X ) + ( B x. Y ) ) =/= 1 )
29	28	ex 114	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) -> ( ( A = 0 /\ B = 0 ) -> ( ( A x. X ) + ( B x. Y ) ) =/= 1 ) )
30	29	necon2bd 2366	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) -> ( ( ( A x. X ) + ( B x. Y ) ) = 1 -> -. ( A = 0 /\ B = 0 ) ) )
31	30	imp 123	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) /\ ( ( A x. X ) + ( B x. Y ) ) = 1 ) -> -. ( A = 0 /\ B = 0 ) )
32		gcdn0cl 11651	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. NN )
33	13, 31, 32	syl2anc 408	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) /\ ( ( A x. X ) + ( B x. Y ) ) = 1 ) -> ( A gcd B ) e. NN )
34		nnle1eq1 8744	. . . 4 |- ( ( A gcd B ) e. NN -> ( ( A gcd B ) <_ 1 <-> ( A gcd B ) = 1 ) )
35	33, 34	syl 14	. . 3 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) /\ ( ( A x. X ) + ( B x. Y ) ) = 1 ) -> ( ( A gcd B ) <_ 1 <-> ( A gcd B ) = 1 ) )
36	12, 35	mpbid 146	. 2 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) /\ ( ( A x. X ) + ( B x. Y ) ) = 1 ) -> ( A gcd B ) = 1 )
37	36	ex 114	1 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) -> ( ( ( A x. X ) + ( B x. Y ) ) = 1 -> ( A gcd B ) = 1 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   =/= wne 2308   class class class wbr 3929  (class class class)co 5774   0cc0 7620   1c1 7621   + caddc 7623   x. cmul 7625   <_ cle 7801   NNcn 8720   ZZcz 9054   || cdvds 11493   gcd cgcd 11635

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fz 9791  df-fzo 9920  df-fl 10043  df-mod 10096  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-dvds 11494  df-gcd 11636

This theorem is referenced by:  divgcdcoprm0  11782