Theorem bezoutr1 11988

Description: Converse of bezout 11966 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 11987	. . . . . 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 4015	. . . 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 11921	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) e. NN0 )
6	5	nn0zd 9332	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) e. ZZ )
7	6	ad2antrr 485	. . . . 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 8889	. . . . . 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 11804	. . . . 5 |- ( ( ( A gcd B ) e. ZZ /\ 1 e. NN ) -> ( ( A gcd B ) || 1 -> ( A gcd B ) <_ 1 ) )
11	7, 9, 10	syl2anc 409	. . . 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 524	. . . . 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 5860	. . . . . . . . . . . . 13 |- ( A = 0 -> ( A x. X ) = ( 0 x. X ) )
15		oveq1 5860	. . . . . . . . . . . . 13 |- ( B = 0 -> ( B x. Y ) = ( 0 x. Y ) )
16	14, 15	oveqan12d 5872	. . . . . . . . . . . 12 |- ( ( A = 0 /\ B = 0 ) -> ( ( A x. X ) + ( B x. Y ) ) = ( ( 0 x. X ) + ( 0 x. Y ) ) )
17		zcn 9217	. . . . . . . . . . . . . 14 |- ( X e. ZZ -> X e. CC )
18	17	mul02d 8311	. . . . . . . . . . . . 13 |- ( X e. ZZ -> ( 0 x. X ) = 0 )
19		zcn 9217	. . . . . . . . . . . . . 14 |- ( Y e. ZZ -> Y e. CC )
20	19	mul02d 8311	. . . . . . . . . . . . 13 |- ( Y e. ZZ -> ( 0 x. Y ) = 0 )
21	18, 20	oveqan12d 5872	. . . . . . . . . . . 12 |- ( ( X e. ZZ /\ Y e. ZZ ) -> ( ( 0 x. X ) + ( 0 x. Y ) ) = ( 0 + 0 ) )
22	16, 21	sylan9eqr 2225	. . . . . . . . . . 11 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ ( A = 0 /\ B = 0 ) ) -> ( ( A x. X ) + ( B x. Y ) ) = ( 0 + 0 ) )
23		00id 8060	. . . . . . . . . . 11 |- ( 0 + 0 ) = 0
24	22, 23	eqtrdi 2219	. . . . . . . . . 10 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ ( A = 0 /\ B = 0 ) ) -> ( ( A x. X ) + ( B x. Y ) ) = 0 )
25	24	adantll 473	. . . . . . . . 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 8945	. . . . . . . . . 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 2364	. . . . . . . 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 2398	. . . . . 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 11917	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. NN )
33	13, 31, 32	syl2anc 409	. . . 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 8902	. . . 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 1348   e. wcel 2141   =/= wne 2340   class class class wbr 3989  (class class class)co 5853   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   <_ cle 7955   NNcn 8878   ZZcz 9212   || cdvds 11749   gcd cgcd 11897

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

This theorem is referenced by:  divgcdcoprm0  12055