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Theorem bezoutlemzz 10857
Description: Lemma for Bézout's identity. Like bezoutlemex 10856 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 10856 . 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 1464 . . . . . . 7  |-  F/ z ( ( A  e. 
NN0  /\  B  e.  NN0 )  /\  d  e. 
NN0 )
3 nfra1 2405 . . . . . . 7  |-  F/ z A. z  e.  NN0  ( z  ||  d  ->  ( z  ||  A  /\  z  ||  B ) )
42, 3nfan 1500 . . . . . 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 2419 . . . . . . . . . . 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 3822 . . . . . . . . . . . 12  |-  ( w  =  -u z  ->  (
w  ||  d  <->  -u z  ||  d ) )
11 breq1 3822 . . . . . . . . . . . . 13  |-  ( w  =  -u z  ->  (
w  ||  A  <->  -u z  ||  A ) )
12 breq1 3822 . . . . . . . . . . . . 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 3822 . . . . . . . . . . . . . . 15  |-  ( z  =  w  ->  (
z  ||  d  <->  w  ||  d
) )
16 breq1 3822 . . . . . . . . . . . . . . . 16  |-  ( z  =  w  ->  (
z  ||  A  <->  w  ||  A
) )
17 breq1 3822 . . . . . . . . . . . . . . . 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 2586 . . . . . . . . . . . . 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 2720 . . . . . . . . . 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 8791 . . . . . . . . . 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 10678 . . . . . . . . . 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 8791 . . . . . . . . . . 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 10678 . . . . . . . . . . 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 8791 . . . . . . . . . . 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 10678 . . . . . . . . . . 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 8690 . . . . . . . . . 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 2440 . . . . 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 2471 . 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 1287    e. wcel 1436   A.wral 2355   E.wrex 2356   class class class wbr 3819  (class class class)co 5606   RRcr 7285    + caddc 7289    x. cmul 7291   -ucneg 7590   NN0cn0 8598   ZZcz 8675    || cdvds 10662
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3927  ax-sep 3930  ax-nul 3938  ax-pow 3982  ax-pr 4008  ax-un 4232  ax-setind 4324  ax-iinf 4374  ax-cnex 7372  ax-resscn 7373  ax-1cn 7374  ax-1re 7375  ax-icn 7376  ax-addcl 7377  ax-addrcl 7378  ax-mulcl 7379  ax-mulrcl 7380  ax-addcom 7381  ax-mulcom 7382  ax-addass 7383  ax-mulass 7384  ax-distr 7385  ax-i2m1 7386  ax-0lt1 7387  ax-1rid 7388  ax-0id 7389  ax-rnegex 7390  ax-precex 7391  ax-cnre 7392  ax-pre-ltirr 7393  ax-pre-ltwlin 7394  ax-pre-lttrn 7395  ax-pre-apti 7396  ax-pre-ltadd 7397  ax-pre-mulgt0 7398  ax-pre-mulext 7399  ax-arch 7400
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-if 3380  df-pw 3416  df-sn 3436  df-pr 3437  df-op 3439  df-uni 3636  df-int 3671  df-iun 3714  df-br 3820  df-opab 3874  df-mpt 3875  df-tr 3910  df-id 4092  df-po 4095  df-iso 4096  df-iord 4165  df-on 4167  df-ilim 4168  df-suc 4170  df-iom 4377  df-xp 4415  df-rel 4416  df-cnv 4417  df-co 4418  df-dm 4419  df-rn 4420  df-res 4421  df-ima 4422  df-iota 4942  df-fun 4979  df-fn 4980  df-f 4981  df-f1 4982  df-fo 4983  df-f1o 4984  df-fv 4985  df-riota 5562  df-ov 5609  df-oprab 5610  df-mpt2 5611  df-1st 5861  df-2nd 5862  df-recs 6017  df-frec 6103  df-pnf 7460  df-mnf 7461  df-xr 7462  df-ltxr 7463  df-le 7464  df-sub 7591  df-neg 7592  df-reap 7985  df-ap 7992  df-div 8071  df-inn 8350  df-2 8408  df-n0 8599  df-z 8676  df-uz 8944  df-q 9029  df-rp 9059  df-fz 9349  df-fl 9597  df-mod 9650  df-iseq 9772  df-iexp 9845  df-cj 10163  df-re 10164  df-im 10165  df-rsqrt 10318  df-abs 10319  df-dvds 10663
This theorem is referenced by:  bezoutlemaz  10858
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