Theorem bezout 12707

Description: Bézout's identity: For any integers A and B, there are integers x , y such that ( A gcd B ) = A x. x + B x. y. This is Metamath 100 proof #60.

The proof is constructive, in the sense that it applies the Extended Euclidian Algorithm to constuct a number which can be shown to be ( A gcd B ) and which satisfies the rest of the theorem. In the presence of excluded middle, it is common to prove Bézout's identity by taking the smallest number which satisfies the Bézout condition, and showing it is the greatest common divisor. But we do not have the ability to show that number exists other than by providing a way to determine it. (Contributed by Mario Carneiro, 22-Feb-2014.)

Assertion

Ref	Expression
bezout	|- ( ( A e. ZZ /\ B e. ZZ ) -> E. x e. ZZ E. y e. ZZ ( A gcd B ) = ( ( A x. x ) + ( B x. y ) ) )

Distinct variable groups:   x, A, y   x, B, y

Proof of Theorem bezout

Dummy variables d w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bezoutlembi 12701	. 2 |- ( ( A e. ZZ /\ B e. ZZ ) -> E. d e. NN0 ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) )
2		simprrr 542	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( d e. NN0 /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) ) -> E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) )
3		nfv 1577	. . . . 5 |- F/ x ( A e. ZZ /\ B e. ZZ )
4		nfv 1577	. . . . . 6 |- F/ x d e. NN0
5		nfv 1577	. . . . . . 7 |- F/ x A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) )
6		nfre1 2585	. . . . . . 7 |- F/ x E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) )
7	5, 6	nfan 1614	. . . . . 6 |- F/ x ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) )
8	4, 7	nfan 1614	. . . . 5 |- F/ x ( d e. NN0 /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) )
9	3, 8	nfan 1614	. . . 4 |- F/ x ( ( A e. ZZ /\ B e. ZZ ) /\ ( d e. NN0 /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) )
10		nfv 1577	. . . . . 6 |- F/ y ( A e. ZZ /\ B e. ZZ )
11		nfv 1577	. . . . . . 7 |- F/ y d e. NN0
12		nfv 1577	. . . . . . . 8 |- F/ y A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) )
13		nfcv 2384	. . . . . . . . 9 |- F/_ y ZZ
14		nfre1 2585	. . . . . . . . 9 |- F/ y E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) )
15	13, 14	nfrexya 2583	. . . . . . . 8 |- F/ y E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) )
16	12, 15	nfan 1614	. . . . . . 7 |- F/ y ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) )
17	11, 16	nfan 1614	. . . . . 6 |- F/ y ( d e. NN0 /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) )
18	10, 17	nfan 1614	. . . . 5 |- F/ y ( ( A e. ZZ /\ B e. ZZ ) /\ ( d e. NN0 /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) )
19		dfgcd3 12706	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) = ( iota_ w e. NN0 A. z e. ZZ ( z || w <-> ( z || A /\ z || B ) ) ) )
20	19	adantr 276	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( d e. NN0 /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) ) -> ( A gcd B ) = ( iota_ w e. NN0 A. z e. ZZ ( z || w <-> ( z || A /\ z || B ) ) ) )
21		simprrl 541	. . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( d e. NN0 /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) ) -> A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) )
22		simprl 531	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( d e. NN0 /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) ) -> d e. NN0 )
23		simpll 527	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( d e. NN0 /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) ) -> A e. ZZ )
24		simplr 529	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( d e. NN0 /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) ) -> B e. ZZ )
25	23, 24, 22, 21	bezoutlemeu 12703	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( d e. NN0 /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) ) -> E! w e. NN0 A. z e. ZZ ( z || w <-> ( z || A /\ z || B ) ) )
26		breq2 4113	. . . . . . . . . . . 12 |- ( w = d -> ( z || w <-> z || d ) )
27	26	bibi1d 233	. . . . . . . . . . 11 |- ( w = d -> ( ( z || w <-> ( z || A /\ z || B ) ) <-> ( z || d <-> ( z || A /\ z || B ) ) ) )
28	27	ralbidv 2542	. . . . . . . . . 10 |- ( w = d -> ( A. z e. ZZ ( z || w <-> ( z || A /\ z || B ) ) <-> A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) ) )
29	28	riota2 6027	. . . . . . . . 9 |- ( ( d e. NN0 /\ E! w e. NN0 A. z e. ZZ ( z || w <-> ( z || A /\ z || B ) ) ) -> ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) <-> ( iota_ w e. NN0 A. z e. ZZ ( z || w <-> ( z || A /\ z || B ) ) ) = d ) )
30	22, 25, 29	syl2anc 411	. . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( d e. NN0 /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) ) -> ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) <-> ( iota_ w e. NN0 A. z e. ZZ ( z || w <-> ( z || A /\ z || B ) ) ) = d ) )
31	21, 30	mpbid 147	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( d e. NN0 /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) ) -> ( iota_ w e. NN0 A. z e. ZZ ( z || w <-> ( z || A /\ z || B ) ) ) = d )
32	20, 31	eqtrd 2265	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( d e. NN0 /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) ) -> ( A gcd B ) = d )
33	32	eqeq1d 2241	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( d e. NN0 /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) ) -> ( ( A gcd B ) = ( ( A x. x ) + ( B x. y ) ) <-> d = ( ( A x. x ) + ( B x. y ) ) ) )
34	18, 33	rexbid 2541	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( d e. NN0 /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) ) -> ( E. y e. ZZ ( A gcd B ) = ( ( A x. x ) + ( B x. y ) ) <-> E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) )
35	9, 34	rexbid 2541	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( d e. NN0 /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) ) -> ( E. x e. ZZ E. y e. ZZ ( A gcd B ) = ( ( A x. x ) + ( B x. y ) ) <-> E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) )
36	2, 35	mpbird 167	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( d e. NN0 /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) ) -> E. x e. ZZ E. y e. ZZ ( A gcd B ) = ( ( A x. x ) + ( B x. y ) ) )
37	1, 36	rexlimddv 2665	1 |- ( ( A e. ZZ /\ B e. ZZ ) -> E. x e. ZZ E. y e. ZZ ( A gcd B ) = ( ( A x. x ) + ( B x. y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   A.wral 2520   E.wrex 2521   E!wreu 2522   class class class wbr 4109   iota_crio 6002  (class class class)co 6050   + caddc 8130   x. cmul 8132   NN0cn0 9496   ZZcz 9577   || cdvds 12473   gcd cgcd 12649

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-sup 7275  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-dvds 12474  df-gcd 12650

This theorem is referenced by:  dvdsgcd  12708  dvdsmulgcd  12721  lcmgcdlem  12774  divgcdcoprm0  12798  znunit  14807