Theorem bezout 12148

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 12142	. 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 540	. . 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 1539	. . . . 5 |- F/ x ( A e. ZZ /\ B e. ZZ )
4		nfv 1539	. . . . . 6 |- F/ x d e. NN0
5		nfv 1539	. . . . . . 7 |- F/ x A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) )
6		nfre1 2537	. . . . . . 7 |- F/ x E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) )
7	5, 6	nfan 1576	. . . . . 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 1576	. . . . 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 1576	. . . 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 1539	. . . . . 6 |- F/ y ( A e. ZZ /\ B e. ZZ )
11		nfv 1539	. . . . . . 7 |- F/ y d e. NN0
12		nfv 1539	. . . . . . . 8 |- F/ y A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) )
13		nfcv 2336	. . . . . . . . 9 |- F/_ y ZZ
14		nfre1 2537	. . . . . . . . 9 |- F/ y E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) )
15	13, 14	nfrexya 2535	. . . . . . . 8 |- F/ y E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) )
16	12, 15	nfan 1576	. . . . . . 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 1576	. . . . . 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 1576	. . . . 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 12147	. . . . . . . 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 539	. . . . . . . 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 529	. . . . . . . . 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 528	. . . . . . . . . 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 12144	. . . . . . . . 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 4033	. . . . . . . . . . . 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 2494	. . . . . . . . . 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 5896	. . . . . . . . 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 2226	. . . . . 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 2202	. . . . 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 2493	. . . 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 2493	. . 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 2616	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 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   E!wreu 2474   class class class wbr 4029   iota_crio 5872  (class class class)co 5918   + caddc 7875   x. cmul 7877   NN0cn0 9240   ZZcz 9317   || cdvds 11930   gcd cgcd 12079

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-gcd 12080

This theorem is referenced by:  dvdsgcd  12149  dvdsmulgcd  12162  lcmgcdlem  12215  divgcdcoprm0  12239  znunit  14147