Theorem bezoutr1 12066

Description: Converse of bezout 12044 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 12065	. . . . . 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 276	. . . . 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 110	. . . . 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 4044	. . . 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 11999	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) e. NN0 )
6	5	nn0zd 9403	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) e. ZZ )
7	6	ad2antrr 488	. . . . 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 8960	. . . . . 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 11882	. . . . 5 |- ( ( ( A gcd B ) e. ZZ /\ 1 e. NN ) -> ( ( A gcd B ) || 1 -> ( A gcd B ) <_ 1 ) )
11	7, 9, 10	syl2anc 411	. . . 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 527	. . . . 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 5903	. . . . . . . . . . . . 13 |- ( A = 0 -> ( A x. X ) = ( 0 x. X ) )
15		oveq1 5903	. . . . . . . . . . . . 13 |- ( B = 0 -> ( B x. Y ) = ( 0 x. Y ) )
16	14, 15	oveqan12d 5915	. . . . . . . . . . . 12 |- ( ( A = 0 /\ B = 0 ) -> ( ( A x. X ) + ( B x. Y ) ) = ( ( 0 x. X ) + ( 0 x. Y ) ) )
17		zcn 9288	. . . . . . . . . . . . . 14 |- ( X e. ZZ -> X e. CC )
18	17	mul02d 8379	. . . . . . . . . . . . 13 |- ( X e. ZZ -> ( 0 x. X ) = 0 )
19		zcn 9288	. . . . . . . . . . . . . 14 |- ( Y e. ZZ -> Y e. CC )
20	19	mul02d 8379	. . . . . . . . . . . . 13 |- ( Y e. ZZ -> ( 0 x. Y ) = 0 )
21	18, 20	oveqan12d 5915	. . . . . . . . . . . 12 |- ( ( X e. ZZ /\ Y e. ZZ ) -> ( ( 0 x. X ) + ( 0 x. Y ) ) = ( 0 + 0 ) )
22	16, 21	sylan9eqr 2244	. . . . . . . . . . 11 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ ( A = 0 /\ B = 0 ) ) -> ( ( A x. X ) + ( B x. Y ) ) = ( 0 + 0 ) )
23		00id 8128	. . . . . . . . . . 11 |- ( 0 + 0 ) = 0
24	22, 23	eqtrdi 2238	. . . . . . . . . 10 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ ( A = 0 /\ B = 0 ) ) -> ( ( A x. X ) + ( B x. Y ) ) = 0 )
25	24	adantll 476	. . . . . . . . 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 9016	. . . . . . . . . 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 2384	. . . . . . . 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 115	. . . . . . 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 2418	. . . . . 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 124	. . . . 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 11995	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. NN )
33	13, 31, 32	syl2anc 411	. . . 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 8973	. . . 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 147	. 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 115	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 104   <-> wb 105   = wceq 1364   e. wcel 2160   =/= wne 2360   class class class wbr 4018  (class class class)co 5896   0cc0 7841   1c1 7842   + caddc 7844   x. cmul 7846   <_ cle 8023   NNcn 8949   ZZcz 9283   || cdvds 11826   gcd cgcd 11975

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959  ax-arch 7960  ax-caucvg 7961

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-frec 6416  df-sup 7013  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-2 9008  df-3 9009  df-4 9010  df-n0 9207  df-z 9284  df-uz 9559  df-q 9650  df-rp 9684  df-fz 10039  df-fzo 10173  df-fl 10301  df-mod 10354  df-seqfrec 10477  df-exp 10551  df-cj 10883  df-re 10884  df-im 10885  df-rsqrt 11039  df-abs 11040  df-dvds 11827  df-gcd 11976

This theorem is referenced by:  divgcdcoprm0  12133