MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bezout Unicode version

Theorem bezout 12997
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 2410 . . . . . . . 8  |-  ( z  =  t  ->  (
z  =  ( ( A  x.  x )  +  ( B  x.  y ) )  <->  t  =  ( ( A  x.  x )  +  ( B  x.  y ) ) ) )
212rexbidv 2709 . . . . . . 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 6048 . . . . . . . . . 10  |-  ( x  =  u  ->  ( A  x.  x )  =  ( A  x.  u ) )
43oveq1d 6055 . . . . . . . . 9  |-  ( x  =  u  ->  (
( A  x.  x
)  +  ( B  x.  y ) )  =  ( ( A  x.  u )  +  ( B  x.  y
) ) )
54eqeq2d 2415 . . . . . . . 8  |-  ( x  =  u  ->  (
t  =  ( ( A  x.  x )  +  ( B  x.  y ) )  <->  t  =  ( ( A  x.  u )  +  ( B  x.  y ) ) ) )
6 oveq2 6048 . . . . . . . . . 10  |-  ( y  =  v  ->  ( B  x.  y )  =  ( B  x.  v ) )
76oveq2d 6056 . . . . . . . . 9  |-  ( y  =  v  ->  (
( A  x.  u
)  +  ( B  x.  y ) )  =  ( ( A  x.  u )  +  ( B  x.  v
) ) )
87eqeq2d 2415 . . . . . . . 8  |-  ( y  =  v  ->  (
t  =  ( ( A  x.  u )  +  ( B  x.  y ) )  <->  t  =  ( ( A  x.  u )  +  ( B  x.  v ) ) ) )
95, 8cbvrex2v 2901 . . . . . . 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 253 . . . . . 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 2915 . . . . 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 731 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  A  e.  ZZ )
13 simplr 732 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  B  e.  ZZ )
14 eqid 2404 . . . . 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 448 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  -.  ( A  =  0  /\  B  =  0 ) )
1611, 12, 13, 14, 15bezoutlem4 12996 . . . 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 2410 . . . . . . 7  |-  ( z  =  ( A  gcd  B )  ->  ( z  =  ( ( A  x.  x )  +  ( B  x.  y
) )  <->  ( A  gcd  B )  =  ( ( A  x.  x
)  +  ( B  x.  y ) ) ) )
18172rexbidv 2709 . . . . . 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 3052 . . . . 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 451 . . . 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 424 . 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 10249 . . . 4  |-  0  e.  ZZ
24 00id 9197 . . . . 5  |-  ( 0  +  0 )  =  0
25 0cn 9040 . . . . . . 7  |-  0  e.  CC
2625mul01i 9212 . . . . . 6  |-  ( 0  x.  0 )  =  0
2726, 26oveq12i 6052 . . . . 5  |-  ( ( 0  x.  0 )  +  ( 0  x.  0 ) )  =  ( 0  +  0 )
28 gcd0val 12964 . . . . 5  |-  ( 0  gcd  0 )  =  0
2924, 27, 283eqtr4ri 2435 . . . 4  |-  ( 0  gcd  0 )  =  ( ( 0  x.  0 )  +  ( 0  x.  0 ) )
30 oveq2 6048 . . . . . . 7  |-  ( x  =  0  ->  (
0  x.  x )  =  ( 0  x.  0 ) )
3130oveq1d 6055 . . . . . 6  |-  ( x  =  0  ->  (
( 0  x.  x
)  +  ( 0  x.  y ) )  =  ( ( 0  x.  0 )  +  ( 0  x.  y
) ) )
3231eqeq2d 2415 . . . . 5  |-  ( x  =  0  ->  (
( 0  gcd  0
)  =  ( ( 0  x.  x )  +  ( 0  x.  y ) )  <->  ( 0  gcd  0 )  =  ( ( 0  x.  0 )  +  ( 0  x.  y ) ) ) )
33 oveq2 6048 . . . . . . 7  |-  ( y  =  0  ->  (
0  x.  y )  =  ( 0  x.  0 ) )
3433oveq2d 6056 . . . . . 6  |-  ( y  =  0  ->  (
( 0  x.  0 )  +  ( 0  x.  y ) )  =  ( ( 0  x.  0 )  +  ( 0  x.  0 ) ) )
3534eqeq2d 2415 . . . . 5  |-  ( y  =  0  ->  (
( 0  gcd  0
)  =  ( ( 0  x.  0 )  +  ( 0  x.  y ) )  <->  ( 0  gcd  0 )  =  ( ( 0  x.  0 )  +  ( 0  x.  0 ) ) ) )
3632, 35rspc2ev 3020 . . . 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 1279 . . 3  |-  E. x  e.  ZZ  E. y  e.  ZZ  ( 0  gcd  0 )  =  ( ( 0  x.  x
)  +  ( 0  x.  y ) )
38 oveq12 6049 . . . . 5  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  gcd  B )  =  ( 0  gcd  0 ) )
39 oveq1 6047 . . . . . 6  |-  ( A  =  0  ->  ( A  x.  x )  =  ( 0  x.  x ) )
40 oveq1 6047 . . . . . 6  |-  ( B  =  0  ->  ( B  x.  y )  =  ( 0  x.  y ) )
4139, 40oveqan12d 6059 . . . . 5  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( A  x.  x )  +  ( B  x.  y
) )  =  ( ( 0  x.  x
)  +  ( 0  x.  y ) ) )
4238, 41eqeq12d 2418 . . . 4  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( A  gcd  B )  =  ( ( A  x.  x )  +  ( B  x.  y ) )  <->  ( 0  gcd  0 )  =  ( ( 0  x.  x
)  +  ( 0  x.  y ) ) ) )
43422rexbidv 2709 . . 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 225 . 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 154 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 359    = wceq 1649    e. wcel 1721   E.wrex 2667   {crab 2670   `'ccnv 4836  (class class class)co 6040   supcsup 7403   RRcr 8945   0cc0 8946    + caddc 8949    x. cmul 8951    < clt 9076   NNcn 9956   ZZcz 10238    gcd cgcd 12961
This theorem is referenced by:  dvdsgcd  12998  dvdsmulgcd  13009  odbezout  15149  ablfacrp  15579  pgpfac1lem3  15590  znunit  16799  2sqb  21115  ostth3  21285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962
  Copyright terms: Public domain W3C validator