Theorem bezout 12766

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 12760	. 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 546	. . 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 1581	. . . . 5 |- F/ x ( A e. ZZ /\ B e. ZZ )
4		nfv 1581	. . . . . 6 |- F/ x d e. NN0
5		nfv 1581	. . . . . . 7 |- F/ x A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) )
6		nfre1 2593	. . . . . . 7 |- F/ x E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) )
7	5, 6	nfan 1618	. . . . . 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 1618	. . . . 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 1618	. . . 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 1581	. . . . . 6 |- F/ y ( A e. ZZ /\ B e. ZZ )
11		nfv 1581	. . . . . . 7 |- F/ y d e. NN0
12		nfv 1581	. . . . . . . 8 |- F/ y A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) )
13		nfcv 2392	. . . . . . . . 9 |- F/_ y ZZ
14		nfre1 2593	. . . . . . . . 9 |- F/ y E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) )
15	13, 14	nfrexya 2591	. . . . . . . 8 |- F/ y E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) )
16	12, 15	nfan 1618	. . . . . . 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 1618	. . . . . 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 1618	. . . . 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 12765	. . . . . . . 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 545	. . . . . . . 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 535	. . . . . . . . 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 531	. . . . . . . . . 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 533	. . . . . . . . . 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 12762	. . . . . . . . 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 4129	. . . . . . . . . . . 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 2550	. . . . . . . . . 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 6052	. . . . . . . . 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 415	. . . . . . . 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 2271	. . . . . 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 2247	. . . . 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 2549	. . . 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 2549	. . 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 2673	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 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   E!wreu 2530   class class class wbr 4125   iota_crio 6027  (class class class)co 6075   + caddc 8172   x. cmul 8174   NN0cn0 9542   ZZcz 9623   || cdvds 12532   gcd cgcd 12708

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-sup 7314  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-fl 10683  df-mod 10738  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-dvds 12533  df-gcd 12709

This theorem is referenced by:  dvdsgcd  12767  dvdsmulgcd  12780  lcmgcdlem  12833  divgcdcoprm0  12857  znunit  14966