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Theorem bezoutlemzz 10598
Description: Lemma for Bézout's identity. Like bezoutlemex 10597 but where ' z ' is any integer, not just a nonnegative one. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.)
Assertion
Ref Expression
bezoutlemzz  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  ->  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 ) ) ) )
Distinct variable groups:    A, d, x, y    B, d, x, y   
z, A, d    z, B

Proof of Theorem bezoutlemzz
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 bezoutlemex 10597 . 2  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  ->  E. d  e.  NN0  ( A. z  e.  NN0  ( z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) )  /\  E. x  e.  ZZ  E. y  e.  ZZ  d  =  ( ( A  x.  x
)  +  ( B  x.  y ) ) ) )
2 nfv 1462 . . . . . . 7  |-  F/ z ( ( A  e. 
NN0  /\  B  e.  NN0 )  /\  d  e. 
NN0 )
3 nfra1 2402 . . . . . . 7  |-  F/ z A. z  e.  NN0  ( z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) )
42, 3nfan 1498 . . . . . 6  |-  F/ z ( ( ( A  e.  NN0  /\  B  e. 
NN0 )  /\  d  e.  NN0 )  /\  A. z  e.  NN0  ( z 
||  d  ->  (
z  ||  A  /\  z  ||  B ) ) )
5 simpr 108 . . . . . . . . . 10  |-  ( ( ( A. z  e. 
NN0  ( z  ||  d  ->  ( z  ||  A  /\  z  ||  B
) )  /\  z  e.  ZZ )  /\  z  e.  NN0 )  ->  z  e.  NN0 )
6 rsp 2416 . . . . . . . . . . 11  |-  ( A. z  e.  NN0  ( z 
||  d  ->  (
z  ||  A  /\  z  ||  B ) )  ->  ( z  e. 
NN0  ->  ( z  ||  d  ->  ( z  ||  A  /\  z  ||  B
) ) ) )
76ad2antrr 472 . . . . . . . . . 10  |-  ( ( ( A. z  e. 
NN0  ( z  ||  d  ->  ( z  ||  A  /\  z  ||  B
) )  /\  z  e.  ZZ )  /\  z  e.  NN0 )  ->  (
z  e.  NN0  ->  ( z  ||  d  -> 
( z  ||  A  /\  z  ||  B ) ) ) )
85, 7mpd 13 . . . . . . . . 9  |-  ( ( ( A. z  e. 
NN0  ( z  ||  d  ->  ( z  ||  A  /\  z  ||  B
) )  /\  z  e.  ZZ )  /\  z  e.  NN0 )  ->  (
z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) ) )
98adantlll 464 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  d  e.  NN0 )  /\  A. z  e.  NN0  (
z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) ) )  /\  z  e.  ZZ )  /\  z  e.  NN0 )  ->  (
z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) ) )
10 breq1 3808 . . . . . . . . . . . 12  |-  ( w  =  -u z  ->  (
w  ||  d  <->  -u z  ||  d ) )
11 breq1 3808 . . . . . . . . . . . . 13  |-  ( w  =  -u z  ->  (
w  ||  A  <->  -u z  ||  A ) )
12 breq1 3808 . . . . . . . . . . . . 13  |-  ( w  =  -u z  ->  (
w  ||  B  <->  -u z  ||  B ) )
1311, 12anbi12d 457 . . . . . . . . . . . 12  |-  ( w  =  -u z  ->  (
( w  ||  A  /\  w  ||  B )  <-> 
( -u z  ||  A  /\  -u z  ||  B
) ) )
1410, 13imbi12d 232 . . . . . . . . . . 11  |-  ( w  =  -u z  ->  (
( w  ||  d  ->  ( w  ||  A  /\  w  ||  B ) )  <->  ( -u z  ||  d  ->  ( -u z  ||  A  /\  -u z  ||  B ) ) ) )
15 breq1 3808 . . . . . . . . . . . . . . 15  |-  ( z  =  w  ->  (
z  ||  d  <->  w  ||  d
) )
16 breq1 3808 . . . . . . . . . . . . . . . 16  |-  ( z  =  w  ->  (
z  ||  A  <->  w  ||  A
) )
17 breq1 3808 . . . . . . . . . . . . . . . 16  |-  ( z  =  w  ->  (
z  ||  B  <->  w  ||  B
) )
1816, 17anbi12d 457 . . . . . . . . . . . . . . 15  |-  ( z  =  w  ->  (
( z  ||  A  /\  z  ||  B )  <-> 
( w  ||  A  /\  w  ||  B ) ) )
1915, 18imbi12d 232 . . . . . . . . . . . . . 14  |-  ( z  =  w  ->  (
( z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) )  <->  ( w  ||  d  ->  ( w  ||  A  /\  w  ||  B
) ) ) )
2019cbvralv 2582 . . . . . . . . . . . . 13  |-  ( A. z  e.  NN0  ( z 
||  d  ->  (
z  ||  A  /\  z  ||  B ) )  <->  A. w  e.  NN0  ( w  ||  d  -> 
( w  ||  A  /\  w  ||  B ) ) )
2120biimpi 118 . . . . . . . . . . . 12  |-  ( A. z  e.  NN0  ( z 
||  d  ->  (
z  ||  A  /\  z  ||  B ) )  ->  A. w  e.  NN0  ( w  ||  d  -> 
( w  ||  A  /\  w  ||  B ) ) )
2221ad2antrr 472 . . . . . . . . . . 11  |-  ( ( ( A. z  e. 
NN0  ( z  ||  d  ->  ( z  ||  A  /\  z  ||  B
) )  /\  z  e.  ZZ )  /\  -u z  e.  NN0 )  ->  A. w  e.  NN0  ( w  ||  d  ->  ( w  ||  A  /\  w  ||  B
) ) )
23 simpr 108 . . . . . . . . . . 11  |-  ( ( ( A. z  e. 
NN0  ( z  ||  d  ->  ( z  ||  A  /\  z  ||  B
) )  /\  z  e.  ZZ )  /\  -u z  e.  NN0 )  ->  -u z  e.  NN0 )
2414, 22, 23rspcdva 2715 . . . . . . . . . 10  |-  ( ( ( A. z  e. 
NN0  ( z  ||  d  ->  ( z  ||  A  /\  z  ||  B
) )  /\  z  e.  ZZ )  /\  -u z  e.  NN0 )  ->  ( -u z  ||  d  -> 
( -u z  ||  A  /\  -u z  ||  B
) ) )
2524adantlll 464 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  d  e.  NN0 )  /\  A. z  e.  NN0  (
z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) ) )  /\  z  e.  ZZ )  /\  -u z  e.  NN0 )  ->  ( -u z  ||  d  -> 
( -u z  ||  A  /\  -u z  ||  B
) ) )
26 simplr 497 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  d  e.  NN0 )  /\  A. z  e.  NN0  (
z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) ) )  /\  z  e.  ZZ )  /\  -u z  e.  NN0 )  ->  z  e.  ZZ )
27 simpllr 501 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  NN0  /\  B  e. 
NN0 )  /\  d  e.  NN0 )  /\  A. z  e.  NN0  ( z 
||  d  ->  (
z  ||  A  /\  z  ||  B ) ) )  /\  z  e.  ZZ )  ->  d  e.  NN0 )
2827adantr 270 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  d  e.  NN0 )  /\  A. z  e.  NN0  (
z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) ) )  /\  z  e.  ZZ )  /\  -u z  e.  NN0 )  ->  d  e.  NN0 )
2928nn0zd 8600 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  d  e.  NN0 )  /\  A. z  e.  NN0  (
z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) ) )  /\  z  e.  ZZ )  /\  -u z  e.  NN0 )  ->  d  e.  ZZ )
30 negdvdsb 10419 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  d  e.  ZZ )  ->  ( z  ||  d  <->  -u z  ||  d ) )
3126, 29, 30syl2anc 403 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  d  e.  NN0 )  /\  A. z  e.  NN0  (
z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) ) )  /\  z  e.  ZZ )  /\  -u z  e.  NN0 )  ->  (
z  ||  d  <->  -u z  ||  d ) )
32 simplll 500 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
NN0  /\  B  e.  NN0 )  /\  d  e. 
NN0 )  /\  A. z  e.  NN0  ( z 
||  d  ->  (
z  ||  A  /\  z  ||  B ) ) )  ->  A  e.  NN0 )
3332ad2antrr 472 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  d  e.  NN0 )  /\  A. z  e.  NN0  (
z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) ) )  /\  z  e.  ZZ )  /\  -u z  e.  NN0 )  ->  A  e.  NN0 )
3433nn0zd 8600 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  d  e.  NN0 )  /\  A. z  e.  NN0  (
z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) ) )  /\  z  e.  ZZ )  /\  -u z  e.  NN0 )  ->  A  e.  ZZ )
35 negdvdsb 10419 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  A  e.  ZZ )  ->  ( z  ||  A  <->  -u z  ||  A ) )
3626, 34, 35syl2anc 403 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  d  e.  NN0 )  /\  A. z  e.  NN0  (
z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) ) )  /\  z  e.  ZZ )  /\  -u z  e.  NN0 )  ->  (
z  ||  A  <->  -u z  ||  A ) )
37 simpllr 501 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
NN0  /\  B  e.  NN0 )  /\  d  e. 
NN0 )  /\  A. z  e.  NN0  ( z 
||  d  ->  (
z  ||  A  /\  z  ||  B ) ) )  ->  B  e.  NN0 )
3837ad2antrr 472 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  d  e.  NN0 )  /\  A. z  e.  NN0  (
z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) ) )  /\  z  e.  ZZ )  /\  -u z  e.  NN0 )  ->  B  e.  NN0 )
3938nn0zd 8600 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  d  e.  NN0 )  /\  A. z  e.  NN0  (
z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) ) )  /\  z  e.  ZZ )  /\  -u z  e.  NN0 )  ->  B  e.  ZZ )
40 negdvdsb 10419 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  B  e.  ZZ )  ->  ( z  ||  B  <->  -u z  ||  B ) )
4126, 39, 40syl2anc 403 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  d  e.  NN0 )  /\  A. z  e.  NN0  (
z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) ) )  /\  z  e.  ZZ )  /\  -u z  e.  NN0 )  ->  (
z  ||  B  <->  -u z  ||  B ) )
4236, 41anbi12d 457 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  d  e.  NN0 )  /\  A. z  e.  NN0  (
z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) ) )  /\  z  e.  ZZ )  /\  -u z  e.  NN0 )  ->  (
( z  ||  A  /\  z  ||  B )  <-> 
( -u z  ||  A  /\  -u z  ||  B
) ) )
4325, 31, 423imtr4d 201 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  d  e.  NN0 )  /\  A. z  e.  NN0  (
z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) ) )  /\  z  e.  ZZ )  /\  -u z  e.  NN0 )  ->  (
z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) ) )
44 elznn0 8499 . . . . . . . . . 10  |-  ( z  e.  ZZ  <->  ( z  e.  RR  /\  ( z  e.  NN0  \/  -u z  e.  NN0 ) ) )
4544simprbi 269 . . . . . . . . 9  |-  ( z  e.  ZZ  ->  (
z  e.  NN0  \/  -u z  e.  NN0 )
)
4645adantl 271 . . . . . . . 8  |-  ( ( ( ( ( A  e.  NN0  /\  B  e. 
NN0 )  /\  d  e.  NN0 )  /\  A. z  e.  NN0  ( z 
||  d  ->  (
z  ||  A  /\  z  ||  B ) ) )  /\  z  e.  ZZ )  ->  (
z  e.  NN0  \/  -u z  e.  NN0 )
)
479, 43, 46mpjaodan 745 . . . . . . 7  |-  ( ( ( ( ( A  e.  NN0  /\  B  e. 
NN0 )  /\  d  e.  NN0 )  /\  A. z  e.  NN0  ( z 
||  d  ->  (
z  ||  A  /\  z  ||  B ) ) )  /\  z  e.  ZZ )  ->  (
z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) ) )
4847ex 113 . . . . . 6  |-  ( ( ( ( A  e. 
NN0  /\  B  e.  NN0 )  /\  d  e. 
NN0 )  /\  A. z  e.  NN0  ( z 
||  d  ->  (
z  ||  A  /\  z  ||  B ) ) )  ->  ( z  e.  ZZ  ->  ( z  ||  d  ->  ( z 
||  A  /\  z  ||  B ) ) ) )
494, 48ralrimi 2437 . . . . 5  |-  ( ( ( ( A  e. 
NN0  /\  B  e.  NN0 )  /\  d  e. 
NN0 )  /\  A. z  e.  NN0  ( z 
||  d  ->  (
z  ||  A  /\  z  ||  B ) ) )  ->  A. z  e.  ZZ  ( z  ||  d  ->  ( z  ||  A  /\  z  ||  B
) ) )
5049ex 113 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  d  e.  NN0 )  ->  ( A. z  e.  NN0  ( z  ||  d  ->  ( z  ||  A  /\  z  ||  B
) )  ->  A. z  e.  ZZ  ( z  ||  d  ->  ( z  ||  A  /\  z  ||  B
) ) ) )
5150anim1d 329 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  d  e.  NN0 )  ->  ( ( A. z  e.  NN0  ( 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 ) )  /\  E. x  e.  ZZ  E. y  e.  ZZ  d  =  ( ( A  x.  x
)  +  ( B  x.  y ) ) ) ) )
5251reximdva 2468 . 2  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( E. d  e. 
NN0  ( A. z  e.  NN0  ( z  ||  d  ->  ( z  ||  A  /\  z  ||  B
) )  /\  E. x  e.  ZZ  E. y  e.  ZZ  d  =  ( ( A  x.  x
)  +  ( B  x.  y ) ) )  ->  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 ) ) ) ) )
531, 52mpd 13 1  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  ->  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 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    = wceq 1285    e. wcel 1434   A.wral 2353   E.wrex 2354   class class class wbr 3805  (class class class)co 5563   RRcr 7094    + caddc 7098    x. cmul 7100   -ucneg 7399   NN0cn0 8407   ZZcz 8484    || cdvds 10403
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 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209
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 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-n0 8408  df-z 8485  df-uz 8753  df-q 8838  df-rp 8868  df-fz 9158  df-fl 9404  df-mod 9457  df-iseq 9574  df-iexp 9625  df-cj 9930  df-re 9931  df-im 9932  df-rsqrt 10085  df-abs 10086  df-dvds 10404
This theorem is referenced by:  bezoutlemaz  10599
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