Theorem bezoutr1 12603

Description: Converse of bezout 12581 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 12602	. . . . . 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 4114	. . . 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 12536	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) e. NN0 )
6	5	nn0zd 9599	. . . . . 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 9153	. . . . . 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 12404	. . . . 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 6024	. . . . . . . . . . . . 13 |- ( A = 0 -> ( A x. X ) = ( 0 x. X ) )
15		oveq1 6024	. . . . . . . . . . . . 13 |- ( B = 0 -> ( B x. Y ) = ( 0 x. Y ) )
16	14, 15	oveqan12d 6036	. . . . . . . . . . . 12 |- ( ( A = 0 /\ B = 0 ) -> ( ( A x. X ) + ( B x. Y ) ) = ( ( 0 x. X ) + ( 0 x. Y ) ) )
17		zcn 9483	. . . . . . . . . . . . . 14 |- ( X e. ZZ -> X e. CC )
18	17	mul02d 8570	. . . . . . . . . . . . 13 |- ( X e. ZZ -> ( 0 x. X ) = 0 )
19		zcn 9483	. . . . . . . . . . . . . 14 |- ( Y e. ZZ -> Y e. CC )
20	19	mul02d 8570	. . . . . . . . . . . . 13 |- ( Y e. ZZ -> ( 0 x. Y ) = 0 )
21	18, 20	oveqan12d 6036	. . . . . . . . . . . 12 |- ( ( X e. ZZ /\ Y e. ZZ ) -> ( ( 0 x. X ) + ( 0 x. Y ) ) = ( 0 + 0 ) )
22	16, 21	sylan9eqr 2286	. . . . . . . . . . 11 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ ( A = 0 /\ B = 0 ) ) -> ( ( A x. X ) + ( B x. Y ) ) = ( 0 + 0 ) )
23		00id 8319	. . . . . . . . . . 11 |- ( 0 + 0 ) = 0
24	22, 23	eqtrdi 2280	. . . . . . . . . 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 9209	. . . . . . . . . 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 2426	. . . . . . . 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 2460	. . . . . 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 12532	. . . . 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 9166	. . . 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 1397   e. wcel 2202   =/= wne 2402   class class class wbr 4088  (class class class)co 6017   0cc0 8031   1c1 8032   + caddc 8034   x. cmul 8036   <_ cle 8214   NNcn 9142   ZZcz 9478   || cdvds 12347   gcd cgcd 12523

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-sup 7182  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-dvds 12348  df-gcd 12524

This theorem is referenced by:  divgcdcoprm0  12672