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Theorem bezout 13043
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. (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, y, A    x, B, y

Proof of Theorem bezout
Dummy variables  t  u  v  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2443 . . . . . . . 8  |-  ( z  =  t  ->  (
z  =  ( ( A  x.  x )  +  ( B  x.  y ) )  <->  t  =  ( ( A  x.  x )  +  ( B  x.  y ) ) ) )
212rexbidv 2749 . . . . . . 7  |-  ( z  =  t  ->  ( E. x  e.  ZZ  E. y  e.  ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y
) )  <->  E. x  e.  ZZ  E. y  e.  ZZ  t  =  ( ( A  x.  x
)  +  ( B  x.  y ) ) ) )
3 oveq2 6090 . . . . . . . . . 10  |-  ( x  =  u  ->  ( A  x.  x )  =  ( A  x.  u ) )
43oveq1d 6097 . . . . . . . . 9  |-  ( x  =  u  ->  (
( A  x.  x
)  +  ( B  x.  y ) )  =  ( ( A  x.  u )  +  ( B  x.  y
) ) )
54eqeq2d 2448 . . . . . . . 8  |-  ( x  =  u  ->  (
t  =  ( ( A  x.  x )  +  ( B  x.  y ) )  <->  t  =  ( ( A  x.  u )  +  ( B  x.  y ) ) ) )
6 oveq2 6090 . . . . . . . . . 10  |-  ( y  =  v  ->  ( B  x.  y )  =  ( B  x.  v ) )
76oveq2d 6098 . . . . . . . . 9  |-  ( y  =  v  ->  (
( A  x.  u
)  +  ( B  x.  y ) )  =  ( ( A  x.  u )  +  ( B  x.  v
) ) )
87eqeq2d 2448 . . . . . . . 8  |-  ( y  =  v  ->  (
t  =  ( ( A  x.  u )  +  ( B  x.  y ) )  <->  t  =  ( ( A  x.  u )  +  ( B  x.  v ) ) ) )
95, 8cbvrex2v 2942 . . . . . . 7  |-  ( E. x  e.  ZZ  E. y  e.  ZZ  t  =  ( ( A  x.  x )  +  ( B  x.  y
) )  <->  E. u  e.  ZZ  E. v  e.  ZZ  t  =  ( ( A  x.  u
)  +  ( B  x.  v ) ) )
102, 9syl6bb 254 . . . . . 6  |-  ( z  =  t  ->  ( E. x  e.  ZZ  E. y  e.  ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y
) )  <->  E. u  e.  ZZ  E. v  e.  ZZ  t  =  ( ( A  x.  u
)  +  ( B  x.  v ) ) ) )
1110cbvrabv 2956 . . . . 5  |-  { z  e.  NN  |  E. x  e.  ZZ  E. y  e.  ZZ  z  =  ( ( A  x.  x
)  +  ( B  x.  y ) ) }  =  { t  e.  NN  |  E. u  e.  ZZ  E. v  e.  ZZ  t  =  ( ( A  x.  u
)  +  ( B  x.  v ) ) }
12 simpll 732 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  A  e.  ZZ )
13 simplr 733 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  B  e.  ZZ )
14 eqid 2437 . . . . 5  |-  sup ( { z  e.  NN  |  E. x  e.  ZZ  E. y  e.  ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y
) ) } ,  RR ,  `'  <  )  =  sup ( { z  e.  NN  |  E. x  e.  ZZ  E. y  e.  ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y
) ) } ,  RR ,  `'  <  )
15 simpr 449 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  -.  ( A  =  0  /\  B  =  0 ) )
1611, 12, 13, 14, 15bezoutlem4 13042 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  { z  e.  NN  |  E. x  e.  ZZ  E. y  e.  ZZ  z  =  ( ( A  x.  x
)  +  ( B  x.  y ) ) } )
17 eqeq1 2443 . . . . . . 7  |-  ( z  =  ( A  gcd  B )  ->  ( z  =  ( ( A  x.  x )  +  ( B  x.  y
) )  <->  ( A  gcd  B )  =  ( ( A  x.  x
)  +  ( B  x.  y ) ) ) )
18172rexbidv 2749 . . . . . 6  |-  ( z  =  ( A  gcd  B )  ->  ( E. x  e.  ZZ  E. y  e.  ZZ  z  =  ( ( A  x.  x
)  +  ( B  x.  y ) )  <->  E. x  e.  ZZ  E. y  e.  ZZ  ( A  gcd  B )  =  ( ( A  x.  x )  +  ( B  x.  y ) ) ) )
1918elrab 3093 . . . . 5  |-  ( ( A  gcd  B )  e.  { z  e.  NN  |  E. x  e.  ZZ  E. y  e.  ZZ  z  =  ( ( A  x.  x
)  +  ( B  x.  y ) ) }  <->  ( ( A  gcd  B )  e.  NN  /\  E. x  e.  ZZ  E. y  e.  ZZ  ( A  gcd  B )  =  ( ( A  x.  x )  +  ( B  x.  y ) ) ) )
2019simprbi 452 . . . 4  |-  ( ( A  gcd  B )  e.  { z  e.  NN  |  E. x  e.  ZZ  E. y  e.  ZZ  z  =  ( ( A  x.  x
)  +  ( B  x.  y ) ) }  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( A  gcd  B )  =  ( ( A  x.  x )  +  ( B  x.  y ) ) )
2116, 20syl 16 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( A  gcd  B )  =  ( ( A  x.  x )  +  ( B  x.  y ) ) )
2221ex 425 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( -.  ( A  =  0  /\  B  =  0 )  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( A  gcd  B )  =  ( ( A  x.  x )  +  ( B  x.  y ) ) ) )
23 0z 10294 . . . 4  |-  0  e.  ZZ
24 00id 9242 . . . . 5  |-  ( 0  +  0 )  =  0
25 0cn 9085 . . . . . . 7  |-  0  e.  CC
2625mul01i 9257 . . . . . 6  |-  ( 0  x.  0 )  =  0
2726, 26oveq12i 6094 . . . . 5  |-  ( ( 0  x.  0 )  +  ( 0  x.  0 ) )  =  ( 0  +  0 )
28 gcd0val 13010 . . . . 5  |-  ( 0  gcd  0 )  =  0
2924, 27, 283eqtr4ri 2468 . . . 4  |-  ( 0  gcd  0 )  =  ( ( 0  x.  0 )  +  ( 0  x.  0 ) )
30 oveq2 6090 . . . . . . 7  |-  ( x  =  0  ->  (
0  x.  x )  =  ( 0  x.  0 ) )
3130oveq1d 6097 . . . . . 6  |-  ( x  =  0  ->  (
( 0  x.  x
)  +  ( 0  x.  y ) )  =  ( ( 0  x.  0 )  +  ( 0  x.  y
) ) )
3231eqeq2d 2448 . . . . 5  |-  ( x  =  0  ->  (
( 0  gcd  0
)  =  ( ( 0  x.  x )  +  ( 0  x.  y ) )  <->  ( 0  gcd  0 )  =  ( ( 0  x.  0 )  +  ( 0  x.  y ) ) ) )
33 oveq2 6090 . . . . . . 7  |-  ( y  =  0  ->  (
0  x.  y )  =  ( 0  x.  0 ) )
3433oveq2d 6098 . . . . . 6  |-  ( y  =  0  ->  (
( 0  x.  0 )  +  ( 0  x.  y ) )  =  ( ( 0  x.  0 )  +  ( 0  x.  0 ) ) )
3534eqeq2d 2448 . . . . 5  |-  ( y  =  0  ->  (
( 0  gcd  0
)  =  ( ( 0  x.  0 )  +  ( 0  x.  y ) )  <->  ( 0  gcd  0 )  =  ( ( 0  x.  0 )  +  ( 0  x.  0 ) ) ) )
3632, 35rspc2ev 3061 . . . 4  |-  ( ( 0  e.  ZZ  /\  0  e.  ZZ  /\  (
0  gcd  0 )  =  ( ( 0  x.  0 )  +  ( 0  x.  0 ) ) )  ->  E. x  e.  ZZ  E. y  e.  ZZ  (
0  gcd  0 )  =  ( ( 0  x.  x )  +  ( 0  x.  y
) ) )
3723, 23, 29, 36mp3an 1280 . . 3  |-  E. x  e.  ZZ  E. y  e.  ZZ  ( 0  gcd  0 )  =  ( ( 0  x.  x
)  +  ( 0  x.  y ) )
38 oveq12 6091 . . . . 5  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  gcd  B )  =  ( 0  gcd  0 ) )
39 oveq1 6089 . . . . . 6  |-  ( A  =  0  ->  ( A  x.  x )  =  ( 0  x.  x ) )
40 oveq1 6089 . . . . . 6  |-  ( B  =  0  ->  ( B  x.  y )  =  ( 0  x.  y ) )
4139, 40oveqan12d 6101 . . . . 5  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( A  x.  x )  +  ( B  x.  y
) )  =  ( ( 0  x.  x
)  +  ( 0  x.  y ) ) )
4238, 41eqeq12d 2451 . . . 4  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( A  gcd  B )  =  ( ( A  x.  x )  +  ( B  x.  y ) )  <->  ( 0  gcd  0 )  =  ( ( 0  x.  x
)  +  ( 0  x.  y ) ) ) )
43422rexbidv 2749 . . 3  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( 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  ( 0  gcd  0 )  =  ( ( 0  x.  x
)  +  ( 0  x.  y ) ) ) )
4437, 43mpbiri 226 . 2  |-  ( ( A  =  0  /\  B  =  0 )  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( A  gcd  B )  =  ( ( A  x.  x )  +  ( B  x.  y ) ) )
4522, 44pm2.61d2 155 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:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2707   {crab 2710   `'ccnv 4878  (class class class)co 6082   supcsup 7446   RRcr 8990   0cc0 8991    + caddc 8994    x. cmul 8996    < clt 9121   NNcn 10001   ZZcz 10283    gcd cgcd 13007
This theorem is referenced by:  dvdsgcd  13044  dvdsmulgcd  13055  odbezout  15195  ablfacrp  15625  pgpfac1lem3  15636  znunit  16845  2sqb  21163  ostth3  21333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-sup 7447  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-n0 10223  df-z 10284  df-uz 10490  df-rp 10614  df-fl 11203  df-mod 11252  df-seq 11325  df-exp 11384  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-dvds 12854  df-gcd 13008
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