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Theorem bezout 10632
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.
StepHypRef Expression
1 bezoutlembi 10626 . 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 507 . . 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 1462 . . . . 5  |-  F/ x
( A  e.  ZZ  /\  B  e.  ZZ )
4 nfv 1462 . . . . . 6  |-  F/ x  d  e.  NN0
5 nfv 1462 . . . . . . 7  |-  F/ x A. z  e.  ZZ  ( z  ||  d  <->  ( z  ||  A  /\  z  ||  B ) )
6 nfre1 2413 . . . . . . 7  |-  F/ x E. x  e.  ZZ  E. y  e.  ZZ  d  =  ( ( A  x.  x )  +  ( B  x.  y
) )
75, 6nfan 1498 . . . . . 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 ) ) )
84, 7nfan 1498 . . . . 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 ) ) ) )
93, 8nfan 1498 . . . 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 1462 . . . . . 6  |-  F/ y ( A  e.  ZZ  /\  B  e.  ZZ )
11 nfv 1462 . . . . . . 7  |-  F/ y  d  e.  NN0
12 nfv 1462 . . . . . . . 8  |-  F/ y A. z  e.  ZZ  ( z  ||  d  <->  ( z  ||  A  /\  z  ||  B ) )
13 nfcv 2223 . . . . . . . . 9  |-  F/_ y ZZ
14 nfre1 2413 . . . . . . . . 9  |-  F/ y E. y  e.  ZZ  d  =  ( ( A  x.  x )  +  ( B  x.  y ) )
1513, 14nfrexya 2411 . . . . . . . 8  |-  F/ y E. x  e.  ZZ  E. y  e.  ZZ  d  =  ( ( A  x.  x )  +  ( B  x.  y
) )
1612, 15nfan 1498 . . . . . . 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 ) ) )
1711, 16nfan 1498 . . . . . 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 ) ) ) )
1810, 17nfan 1498 . . . . 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 10631 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  =  ( iota_ w  e.  NN0  A. z  e.  ZZ  ( z  ||  w 
<->  ( z  ||  A  /\  z  ||  B ) ) ) )
2019adantr 270 . . . . . . 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 506 . . . . . . . 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 498 . . . . . . . . 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 496 . . . . . . . . . 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 497 . . . . . . . . . 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 )
2523, 24, 22, 21bezoutlemeu 10628 . . . . . . . . 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 3810 . . . . . . . . . . . 12  |-  ( w  =  d  ->  (
z  ||  w  <->  z  ||  d ) )
2726bibi1d 231 . . . . . . . . . . 11  |-  ( w  =  d  ->  (
( z  ||  w  <->  ( z  ||  A  /\  z  ||  B ) )  <-> 
( z  ||  d  <->  ( z  ||  A  /\  z  ||  B ) ) ) )
2827ralbidv 2374 . . . . . . . . . 10  |-  ( w  =  d  ->  ( A. z  e.  ZZ  ( z  ||  w  <->  ( z  ||  A  /\  z  ||  B ) )  <->  A. z  e.  ZZ  ( z  ||  d  <->  ( z  ||  A  /\  z  ||  B ) ) ) )
2928riota2 5543 . . . . . . . . 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 ) )
3022, 25, 29syl2anc 403 . . . . . . . 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 ) )
3121, 30mpbid 145 . . . . . . 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 )
3220, 31eqtrd 2115 . . . . . 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 )
3332eqeq1d 2091 . . . . 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 ) ) ) )
3418, 33rexbid 2373 . . . 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 ) ) ) )
359, 34rexbid 2373 . . 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 ) ) ) )
362, 35mpbird 165 . 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 ) ) )
371, 36rexlimddv 2487 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( A  gcd  B )  =  ( ( A  x.  x )  +  ( B  x.  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2353   E.wrex 2354   E!wreu 2355   class class class wbr 3806   iota_crio 5520  (class class class)co 5565    + caddc 7123    x. cmul 7125   NN0cn0 8432   ZZcz 8509    || cdvds 10428    gcd cgcd 10570
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358  ax-cnex 7206  ax-resscn 7207  ax-1cn 7208  ax-1re 7209  ax-icn 7210  ax-addcl 7211  ax-addrcl 7212  ax-mulcl 7213  ax-mulrcl 7214  ax-addcom 7215  ax-mulcom 7216  ax-addass 7217  ax-mulass 7218  ax-distr 7219  ax-i2m1 7220  ax-0lt1 7221  ax-1rid 7222  ax-0id 7223  ax-rnegex 7224  ax-precex 7225  ax-cnre 7226  ax-pre-ltirr 7227  ax-pre-ltwlin 7228  ax-pre-lttrn 7229  ax-pre-apti 7230  ax-pre-ltadd 7231  ax-pre-mulgt0 7232  ax-pre-mulext 7233  ax-arch 7234  ax-caucvg 7235
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2613  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-if 3370  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-id 4077  df-po 4080  df-iso 4081  df-iord 4150  df-on 4152  df-ilim 4153  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-riota 5521  df-ov 5568  df-oprab 5569  df-mpt2 5570  df-1st 5820  df-2nd 5821  df-recs 5976  df-frec 6062  df-sup 6493  df-pnf 7294  df-mnf 7295  df-xr 7296  df-ltxr 7297  df-le 7298  df-sub 7425  df-neg 7426  df-reap 7819  df-ap 7826  df-div 7905  df-inn 8184  df-2 8242  df-3 8243  df-4 8244  df-n0 8433  df-z 8510  df-uz 8778  df-q 8863  df-rp 8893  df-fz 9183  df-fzo 9307  df-fl 9429  df-mod 9482  df-iseq 9599  df-iexp 9650  df-cj 9955  df-re 9956  df-im 9957  df-rsqrt 10110  df-abs 10111  df-dvds 10429  df-gcd 10571
This theorem is referenced by:  dvdsgcd  10633  dvdsmulgcd  10646  lcmgcdlem  10691  divgcdcoprm0  10715
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