Theorem bezoutr1 10816

Description: Converse of bezout 10794 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 10815	. . . . . 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 270	. . . . 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 108	. . . . 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 3838	. . . 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 10752	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) e. NN0 )
6	5	nn0zd 8776	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) e. ZZ )
7	6	ad2antrr 472	. . . . 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 8345	. . . . . 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 10639	. . . . 5 |- ( ( ( A gcd B ) e. ZZ /\ 1 e. NN ) -> ( ( A gcd B ) || 1 -> ( A gcd B ) <_ 1 ) )
11	7, 9, 10	syl2anc 403	. . . 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 496	. . . . 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 5601	. . . . . . . . . . . . 13 |- ( A = 0 -> ( A x. X ) = ( 0 x. X ) )
15		oveq1 5601	. . . . . . . . . . . . 13 |- ( B = 0 -> ( B x. Y ) = ( 0 x. Y ) )
16	14, 15	oveqan12d 5613	. . . . . . . . . . . 12 |- ( ( A = 0 /\ B = 0 ) -> ( ( A x. X ) + ( B x. Y ) ) = ( ( 0 x. X ) + ( 0 x. Y ) ) )
17		zcn 8665	. . . . . . . . . . . . . 14 |- ( X e. ZZ -> X e. CC )
18	17	mul02d 7791	. . . . . . . . . . . . 13 |- ( X e. ZZ -> ( 0 x. X ) = 0 )
19		zcn 8665	. . . . . . . . . . . . . 14 |- ( Y e. ZZ -> Y e. CC )
20	19	mul02d 7791	. . . . . . . . . . . . 13 |- ( Y e. ZZ -> ( 0 x. Y ) = 0 )
21	18, 20	oveqan12d 5613	. . . . . . . . . . . 12 |- ( ( X e. ZZ /\ Y e. ZZ ) -> ( ( 0 x. X ) + ( 0 x. Y ) ) = ( 0 + 0 ) )
22	16, 21	sylan9eqr 2139	. . . . . . . . . . 11 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ ( A = 0 /\ B = 0 ) ) -> ( ( A x. X ) + ( B x. Y ) ) = ( 0 + 0 ) )
23		00id 7544	. . . . . . . . . . 11 |- ( 0 + 0 ) = 0
24	22, 23	syl6eq 2133	. . . . . . . . . 10 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ ( A = 0 /\ B = 0 ) ) -> ( ( A x. X ) + ( B x. Y ) ) = 0 )
25	24	adantll 460	. . . . . . . . 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 8401	. . . . . . . . . 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 2275	. . . . . . . 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 113	. . . . . . 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 2309	. . . . . 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 122	. . . . 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 10748	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. NN )
33	13, 31, 32	syl2anc 403	. . . 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 8358	. . . 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 145	. 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 113	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 102   <-> wb 103   = wceq 1287   e. wcel 1436   =/= wne 2251   class class class wbr 3814  (class class class)co 5594   0cc0 7271   1c1 7272   + caddc 7274   x. cmul 7276   <_ cle 7444   NNcn 8334   ZZcz 8660   || cdvds 10590   gcd cgcd 10732

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-mulrcl 7365  ax-addcom 7366  ax-mulcom 7367  ax-addass 7368  ax-mulass 7369  ax-distr 7370  ax-i2m1 7371  ax-0lt1 7372  ax-1rid 7373  ax-0id 7374  ax-rnegex 7375  ax-precex 7376  ax-cnre 7377  ax-pre-ltirr 7378  ax-pre-ltwlin 7379  ax-pre-lttrn 7380  ax-pre-apti 7381  ax-pre-ltadd 7382  ax-pre-mulgt0 7383  ax-pre-mulext 7384  ax-arch 7385  ax-caucvg 7386

This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-if 3377  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-id 4087  df-po 4090  df-iso 4091  df-iord 4160  df-on 4162  df-ilim 4163  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-recs 6005  df-frec 6091  df-sup 6600  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449  df-sub 7576  df-neg 7577  df-reap 7970  df-ap 7977  df-div 8056  df-inn 8335  df-2 8393  df-3 8394  df-4 8395  df-n0 8584  df-z 8661  df-uz 8929  df-q 9014  df-rp 9044  df-fz 9334  df-fzo 9458  df-fl 9580  df-mod 9633  df-iseq 9755  df-iexp 9806  df-cj 10117  df-re 10118  df-im 10119  df-rsqrt 10272  df-abs 10273  df-dvds 10591  df-gcd 10733

This theorem is referenced by:  divgcdcoprm0  10877